Equivalence classFrom Wikipedia, the free encyclopedia

Contents

1 Antisymmetric relation 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Asymmetric relation 32.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Binary relation 53.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Is a relation more than its graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Special types of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.1 Difunctional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Relations over a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Operations on binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.3 Algebras, categories, and rewriting systems . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Sets versus classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 The number of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Examples of common binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Divisor 154.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Further notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Equality (mathematics) 205.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Types of equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Equalities as predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.5 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Logical formalizations of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Logical formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5 Some basic logical properties of equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.6 Relation with equivalence and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Equivalence class 246.1 Notation and formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 Graphical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 Quotient space in topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Equivalence relation 297.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.3.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.3.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3.3 Relations that are not equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.4 Connections to other relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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7.5 Well-definedness under an equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6 Equivalence class, quotient set, partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.6.1 Equivalence class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6.2 Quotient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6.4 Equivalence kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6.5 Partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.7 Fundamental theorem of equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.8 Comparing equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.9 Generating equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.10 Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.10.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.10.2 Categories and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.10.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.11 Equivalence relations and mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.12 Euclidean relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.14 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367.16 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Homogeneous space 388.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9 Intransitivity 429.1 Intransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.2 Antitransitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.3 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.4 Occurrences in preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.5 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Material conditional 45

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10.1 Definitions of the material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.1.1 As a truth function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.1.2 As a formal connective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3 Philosophical problems with material conditional . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.4.1 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11 Matrilineality 5011.1 Early human kinship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.2 Matrilineal surname . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.3 Cultural patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11.3.1 Clan names vs. surnames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.3.2 Care of children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.3.3 A feminist and patriarchal relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11.4 Matrilineality in specific ethnic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.4.1 In America . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.4.2 In Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.4.3 In Asia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5511.4.4 In Oceania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.5 Matrilineal identification within Judaism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.6 In mythology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5811.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12 Nontransitive dice 6312.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6312.2 Comment regarding the equivalency of intransitive dice . . . . . . . . . . . . . . . . . . . . . . . 6412.3 Variations of nontransitive dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12.3.1 Efron’s dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6412.3.2 Numbered 1 through 24 dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.3.3 Miwin’s dice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6712.3.4 Three-dice set with minimal alterations to standard dice . . . . . . . . . . . . . . . . . . . 68

12.4 Warren Buffett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6812.5 Nontransitive dice set for more than two players . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.5.1 Three players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6912.5.2 Four players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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12.6 Nontransitive dodecahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.6.1 Nontransitive prime-numbers-dodecahedra . . . . . . . . . . . . . . . . . . . . . . . . . 70

12.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7112.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13 Partition of a set 7213.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.4 Refinement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

14 Preorder 8014.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8014.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

15 Quasitransitive relation 8415.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8415.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

16 Quotient category 8616.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8616.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8616.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8716.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

17 Quotient ring 88

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17.1 Formal quotient ring construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8817.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

17.2.1 Alternative complex planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8917.2.2 Quaternions and alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

17.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9017.6 Further references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9117.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

18 Quotient space (linear algebra) 9218.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9218.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9318.4 Quotient of a Banach space by a subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

18.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9318.4.2 Generalization to locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

18.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9418.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

19 Quotient space (topology) 9519.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.2 Quotient map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9619.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9719.5 Compatibility with other topological notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9819.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

19.6.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9819.6.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

19.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

20 Rational choice theory 9920.1 Definition and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9920.2 Actions, assumptions, and individual preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

20.2.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10120.2.2 Additional assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

20.3 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.4 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10220.5 Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10420.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10420.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10520.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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20.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

21 Reflexive relation 10821.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10821.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10921.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11021.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11121.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11121.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

22 Semigroup 11222.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11222.2 Examples of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11322.3 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

22.3.1 Identity and zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11322.3.2 Subsemigroups and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11322.3.3 Homomorphisms and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

22.4 Structure of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.5 Special classes of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11522.6 Structure theorem for commutative semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.7 Group of fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.8 Semigroup methods in partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11622.9 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.10Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11722.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.13Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11822.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

23 Set (mathematics) 12023.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.2 Describing sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12123.3 Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

23.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12323.3.2 Power sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

23.4 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12423.5 Special sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12423.6 Basic operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

23.6.1 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12523.6.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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23.6.3 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12623.6.4 Cartesian product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

23.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12923.8 Axiomatic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12923.9 Principle of inclusion and exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13023.10De Morgan’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13023.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13123.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13123.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13123.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

24 Subset 13224.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13324.2 ⊂ and ⊃ symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13324.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13324.4 Other properties of inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13424.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

25 Symmetric relation 13625.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

25.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13625.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

25.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 13725.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13725.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

26 Total order 13926.1 Strict total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13926.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14026.3 Further concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

26.3.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14026.3.2 Lattice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14026.3.3 Finite total orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.3.4 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.3.5 Order topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.3.6 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14126.3.7 Sums of orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

26.4 Orders on the Cartesian product of totally ordered sets . . . . . . . . . . . . . . . . . . . . . . . . 14226.5 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14226.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

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26.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14226.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

27 Total relation 14427.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14427.2 Properties and related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14427.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14427.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

28 Transitive closure 14628.1 Transitive relations and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14628.2 Existence and description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14628.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14728.4 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14728.5 In logic and computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14728.6 In database query languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14828.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14928.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

29 Transitive reduction 15029.1 In directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15029.2 In graphs with cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15029.3 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15129.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15129.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15229.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

30 Transitive relation 15330.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15330.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15330.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

30.3.1 Closure properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15430.3.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15430.3.3 Properties that require transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

30.4 Counting transitive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15430.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15430.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

30.6.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15530.6.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

30.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

x CONTENTS

31 Weak ordering 15631.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15731.2 Axiomatizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

31.2.1 Strict weak orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15731.2.2 Total preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15831.2.3 Ordered partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15831.2.4 Representation by functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

31.3 Related types of ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15931.4 All weak orders on a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

31.4.1 Combinatorial enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15931.4.2 Adjacency structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

31.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16131.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16131.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 162

31.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16231.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16631.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Chapter 1

Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if there is no pair of distinct elements of X each ofwhich is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X

if R(a,b) and R(b,a), then a = b,

or, equivalently,

if R(a,b) with a ≠ b, then R(b,a) must not hold.

As a simple example, the divisibility order on the natural numbers is an antisymmetric relation. And what antisym-metry means here is that the only way each of two numbers can be divisible by the other is if the two are, in fact, thesame number; equivalently, if n and m are distinct and n is a factor of m, then m cannot be a factor of n.In mathematical notation, this is:

∀a, b ∈ X, R(a, b) ∧R(b, a) ⇒ a = b

or, equivalently,

∀a, b ∈ X, R(a, b) ∧ a ̸= b⇒ ¬R(b, a).

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalitiesx ≤ y and y ≤ x hold then x and y must be equal. Similarly, the subset order ⊆ on the subsets of any given set isantisymmetric: given two sets A and B, if every element in A also is in B and every element in B is also in A, then Aand B must contain all the same elements and therefore be equal:

A ⊆ B ∧B ⊆ A⇒ A = B

Partial and total orders are antisymmetric by definition. A relation can be both symmetric and antisymmetric (e.g.,the equality relation), and there are relations which are neither symmetric nor antisymmetric (e.g., the “preys on”relation on biological species).Antisymmetry is different from asymmetry, which requires both antisymmetry and irreflexivity.

1.1 Examples

The relation "x is even, y is odd” between a pair (x, y) of integers is antisymmetric:

1

2 CHAPTER 1. ANTISYMMETRIC RELATION

Every asymmetric relation is also an antisymmetric relation.

1.2 See also• Symmetric relation

• Asymmetric relation

• Symmetry in mathematics

1.3 References• Weisstein, Eric W., “Antisymmetric Relation”, MathWorld.

• Lipschutz, Seymour; Marc Lars Lipson (1997). Theory and Problems of Discrete Mathematics. McGraw-Hill.p. 33. ISBN 0-07-038045-7.

Chapter 2

Asymmetric relation

In mathematics an asymmetric relation is a binary relation on a set X where:

• For all a and b in X, if a is related to b, then b is not related to a.[1]

In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ ¬(bRa)

2.1 Examples

An example is < (less-than): if x < y, then necessarily y is not less than x. In fact, one of Tarski’s axioms characterizingthe real numbers R is that < over R is asymmetric.An asymmetric relation need not be total. For example, strict subset or ⊊ is asymmetric, and neither of the sets {1,2}and {3,4} is a strict subset of the other. In general, every strict partial order is asymmetric, and conversely, everytransitive asymmetric relation is a strict partial order.Not all asymmetric relations are strict partial orders, however. An example of an asymmetric intransitive relation isthe rock-paper-scissors relation: if X beats Y, then Y does not beat X, but no one choice wins all the time.The ≤ (less than or equal) operator, on the other hand, is not asymmetric, because reversing x ≤ x produces x ≤ xand both are true. In general, any relation in which x R x holds for some x (that is, which is not irreflexive) is also notasymmetric.Asymmetric is not the same thing as “not symmetric": a relation can be neither symmetric nor asymmetric, such as≤, or can be both, only in the case of the empty relation (vacuously).

2.2 Properties• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[2]

• Restrictions and inverses of asymmetric relations are also asymmetric. For example, the restriction of < fromthe reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.

• A transitive relation is asymmetric if and only if it is irreflexive:[3] if a R b and b R a, transitivity gives a R a,contradicting irreflexivity.

2.3 See also• Symmetric relation

3

4 CHAPTER 2. ASYMMETRIC RELATION

• Antisymmetric relation

• Symmetry

• Symmetry in mathematics

2.4 References[1] Gries, David; Schneider, Fred B. (1993), A Logical Approach to Discrete Math, Springer-Verlag, p. 273.

[2] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[3] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

Chapter 3

Binary relation

“Relation (mathematics)" redirects here. For a more general notion of relation, see finitary relation. For a morecombinatorial viewpoint, see theory of relations. For other uses, see Relation § Mathematics.

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is asubset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subsetof A × B. The terms correspondence, dyadic relation and 2-place relation are synonyms for binary relation.An example is the "divides" relation between the set of prime numbers P and the set of integers Z, in which everyprime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). Inthis relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and theprime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and“divides” in arithmetic, "is congruent to" in geometry, “is adjacent to” in graph theory, “is orthogonal to” in linearalgebra and many more. The concept of function is defined as a special kind of binary relation. Binary relations arealso heavily used in computer science.A binary relation is the special case n = 2 of an n-ary relation R ⊆ A1 × … × An, that is, a set of n-tuples where thejth component of each n-tuple is taken from the jth domain Aj of the relation. An example for a ternary relation onZ×Z×Z is “lies between ... and ...”, containing e.g. the triples (5,2,8), (5,8,2), and (−4,9,−7).In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. Thisextension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in settheory, without running into logical inconsistencies such as Russell’s paradox.

3.1 Formal definition

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), andG is a subset of the Cartesian product X × Y. The sets X and Y are called the domain (or the set of departure) andcodomain (or the set of destination), respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y). The latter notation correspondsto viewing R as the characteristic function on X × Y for the set of pairs of G.The order of the elements in each pair ofG is important: if a ≠ b, then aRb and bRa can be true or false, independentlyof each other. Resuming the above example, the prime 3 divides the integer 9, but 9 doesn't divide 3.A relation as defined by the triple (X, Y, G) is sometimes referred to as a correspondence instead.[1] In this case therelation from X to Y is the subset G of X × Y, and “from X to Y" must always be either specified or implied by thecontext when referring to the relation. In practice correspondence and relation tend to be used interchangeably.

5

6 CHAPTER 3. BINARY RELATION

3.1.1 Is a relation more than its graph?

According to the definition above, two relations with identical graphs but different domains or different codomainsare considered different. For example, ifG = {(1, 2), (1, 3), (2, 7)} , then (Z,Z, G) , (R,N, G) , and (N,R, G) arethree distinct relations, where Z is the set of integers and R is the set of real numbers.Especially in set theory, binary relations are often defined as sets of ordered pairs, identifying binary relations withtheir graphs. The domain of a binary relation R is then defined as the set of all x such that there exists at least oney such that (x, y) ∈ R , the range of R is defined as the set of all y such that there exists at least one x such that(x, y) ∈ R , and the field of R is the union of its domain and its range.[2][3][4]

A special case of this difference in points of view applies to the notion of function. Many authors insist on distin-guishing between a function’s codomain and its range. Thus, a single “rule,” like mapping every real number x tox2, can lead to distinct functions f : R → R and f : R → R+ , depending on whether the images under thatrule are understood to be reals or, more restrictively, non-negative reals. But others view functions as simply sets ofordered pairs with unique first components. This difference in perspectives does raise some nontrivial issues. As anexample, the former camp considers surjectivity—or being onto—as a property of functions, while the latter sees itas a relationship that functions may bear to sets.Either approach is adequate for most uses, provided that one attends to the necessary changes in language, notation,and the definitions of concepts like restrictions, composition, inverse relation, and so on. The choice between the twodefinitions usually matters only in very formal contexts, like category theory.

3.1.2 Example

Example: Suppose there are four objects {ball, car, doll, gun} and four persons {John, Mary, Ian, Venus}. Supposethat John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the gun and Ian owns nothing.Then the binary relation “is owned by” is given as

R = ({ball, car, doll, gun}, {John, Mary, Ian, Venus}, {(ball, John), (doll, Mary), (car, Venus)}).

Thus the first element of R is the set of objects, the second is the set of persons, and the last element is a set of orderedpairs of the form (object, owner).The pair (ball, John), denoted by ₐ RJₒ means that the ball is owned by John.Two different relations could have the same graph. For example: the relation

({ball, car, doll, gun}, {John, Mary, Venus}, {(ball, John), (doll, Mary), (car, Venus)})

is different from the previous one as everyone is an owner. But the graphs of the two relations are the same.Nevertheless, R is usually identified or even defined as G(R) and “an ordered pair (x, y) ∈ G(R)" is usually denoted as"(x, y) ∈ R".

3.2 Special types of binary relations

Some important types of binary relations R between two sets X and Y are listed below. To emphasize that X and Ycan be different sets, some authors call such binary relations heterogeneous.[5][6]

Uniqueness properties:

• injective (also called left-unique[7]): for all x and z in X and y in Y it holds that if xRy and zRy then x = z. Forexample, the green relation in the diagram is injective, but the red relation is not, as it relates e.g. both x = −5and z = +5 to y = 25.

• functional (also called univalent[8] or right-unique[7] or right-definite[9]): for all x in X, and y and z in Yit holds that if xRy and xRz then y = z; such a binary relation is called a partial function. Both relations inthe picture are functional. An example for a non-functional relation can be obtained by rotating the red graphclockwise by 90 degrees, i.e. by considering the relation x=y2 which relates e.g. x=25 to both y=−5 and z=+5.

3.2. SPECIAL TYPES OF BINARY RELATIONS 7

Example relations between real numbers. Red: y=x2. Green: y=2x+20.

• one-to-one (also written 1-to-1): injective and functional. The green relation is one-to-one, but the red is not.

Totality properties:

• left-total:[7] for all x in X there exists a y in Y such that xRy. For example R is left-total when it is a functionor a multivalued function. Note that this property, although sometimes also referred to as total, is differentfrom the definition of total in the next section. Both relations in the picture are left-total. The relation x=y2,obtained from the above rotation, is not left-total, as it doesn't relate, e.g., x = −14 to any real number y.

• surjective (also called right-total[7] or onto): for all y in Y there exists an x in X such that xRy. The greenrelation is surjective, but the red relation is not, as it doesn't relate any real number x to e.g. y = −14.

Uniqueness and totality properties:

8 CHAPTER 3. BINARY RELATION

• A function: a relation that is functional and left-total. Both the green and the red relation are functions.

• An injective function: a relation that is injective, functional, and left-total.

• A surjective function or surjection: a relation that is functional, left-total, and right-total.

• A bijection: a surjective one-to-one or surjective injective function is said to be bijective, also known asone-to-one correspondence.[10] The green relation is bijective, but the red is not.

3.2.1 Difunctional

Less commonly encountered is the notion of difunctional (or regular) relation, defined as a relation R such thatR=RR−1R.[11]

To understand this notion better, it helps to consider a relation as mapping every element x∈X to a set xR = { y∈Y| xRy }.[11] This set is sometimes called the successor neighborhood of x in R; one can define the predecessorneighborhood analogously.[12] Synonymous terms for these notions are afterset and respectively foreset.[5]

A difunctional relation can then be equivalently characterized as a relation R such that wherever x1R and x2R have anon-empty intersection, then these two sets coincide; formally x1R ∩ x2R ≠ ∅ implies x1R = x2R.[11]

As examples, any function or any functional (right-unique) relation is difunctional; the converse doesn't hold. If oneconsiders a relation R from set to itself (X = Y), then if R is both transitive and symmetric (i.e. a partial equivalencerelation), then it is also difunctional.[13] The converse of this latter statement also doesn't hold.A characterization of difunctional relations, which also explains their name, is to consider two functions f: A → Cand g: B→ C and then define the following set which generalizes the kernel of a single function as joint kernel: ker(f,g) = { (a, b) ∈ A × B | f(a) = g(b) }. Every difunctional relation R ⊆ A × B arises as the joint kernel of two functionsf: A→ C and g: B → C for some set C.[14]

In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This ter-minology is justified by the fact that when represented as a boolean matrix, the columns and rows of a difunctionalrelation can be arranged in such a way as to present rectangular blocks of true on the (asymmetric) main diagonal.[15]Other authors however use the term “rectangular” to denote any heterogeneous relation whatsoever.[6]

3.3 Relations over a set

If X = Y then we simply say that the binary relation is over X, or that it is an endorelation over X.[16] In computerscience, such a relation is also called a homogeneous (binary) relation.[16][17][6] Some types of endorelations arewidely studied in graph theory, where they are known as simple directed graphs permitting loops.The set of all binary relations Rel(X) on a set X is the set 2X × X which is a Boolean algebra augmented with theinvolution of mapping of a relation to its inverse relation. For the theoretical explanation see Relation algebra.Some important properties of a binary relation R over a set X are:

• reflexive: for all x in X it holds that xRx. For example, “greater than or equal to” (≥) is a reflexive relation but“greater than” (>) is not.

• irreflexive (or strict): for all x in X it holds that not xRx. For example, > is an irreflexive relation, but ≥ is not.

• coreflexive: for all x and y in X it holds that if xRy then x = y. An example of a coreflexive relation is therelation on integers in which each odd number is related to itself and there are no other relations. The equalityrelation is the only example of a both reflexive and coreflexive relation.

The previous 3 alternatives are far from being exhaustive; e.g. the red relation y=x2 from theabove picture is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair(0,0), and (2,4), but not (2,2), respectively.

• symmetric: for all x and y in X it holds that if xRy then yRx. “Is a blood relative of” is a symmetric relation,because x is a blood relative of y if and only if y is a blood relative of x.

3.4. OPERATIONS ON BINARY RELATIONS 9

• antisymmetric: for all x and y in X, if xRy and yRx then x = y. For example, ≥ is anti-symmetric (so is >, butonly because the condition in the definition is always false).[18]

• asymmetric: for all x and y in X, if xRy then not yRx. A relation is asymmetric if and only if it is bothanti-symmetric and irreflexive.[19] For example, > is asymmetric, but ≥ is not.

• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. A transitive relation is irreflexive if andonly if it is asymmetric.[20] For example, “is ancestor of” is transitive, while “is parent of” is not.

• total: for all x and y in X it holds that xRy or yRx (or both). This definition for total is different from left totalin the previous section. For example, ≥ is a total relation.

• trichotomous: for all x and y in X exactly one of xRy, yRx or x = y holds. For example, > is a trichotomousrelation, while the relation “divides” on natural numbers is not.[21]

• Right Euclidean: for all x, y and z in X it holds that if xRy and xRz, then yRz.

• Left Euclidean: for all x, y and z in X it holds that if yRx and zRx, then yRz.

• Euclidean: An Euclidean relation is both left and right Euclidean. Equality is a Euclidean relation because ifx=y and x=z, then y=z.

• serial: for all x in X, there exists y in X such that xRy. "Is greater than" is a serial relation on the integers. Butit is not a serial relation on the positive integers, because there is no y in the positive integers (i.e. the naturalnumbers) such that 1>y.[22] However, "is less than" is a serial relation on the positive integers, the rationalnumbers and the real numbers. Every reflexive relation is serial: for a given x, choose y=x. A serial relation canbe equivalently characterized as every element having a non-empty successor neighborhood (see the previoussection for the definition of this notion). Similarly an inverse serial relation is a relation in which every elementhas non-empty predecessor neighborhood.[12]

• set-like (or local): for every x in X, the class of all y such that yRx is a set. (This makes sense only if relationson proper classes are allowed.) The usual ordering < on the class of ordinal numbers is set-like, while its inverse> is not.

A relation that is reflexive, symmetric, and transitive is called an equivalence relation. A relation that is symmetric,transitive, and serial is also reflexive. A relation that is only symmetric and transitive (without necessarily beingreflexive) is called a partial equivalence relation.A relation that is reflexive, antisymmetric, and transitive is called a partial order. A partial order that is total is calleda total order, simple order, linear order, or a chain.[23] A linear order where every nonempty subset has a least elementis called a well-order.

3.4 Operations on binary relations

If R, S are binary relations over X and Y, then each of the following is a binary relation over X and Y :

• Union: R ∪ S ⊆ X × Y, defined as R ∪ S = { (x, y) | (x, y) ∈ R or (x, y) ∈ S }. For example, ≥ is the union of >and =.

• Intersection: R ∩ S ⊆ X × Y, defined as R ∩ S = { (x, y) | (x, y) ∈ R and (x, y) ∈ S }.

If R is a binary relation over X and Y, and S is a binary relation over Y and Z, then the following is a binary relationover X and Z: (see main article composition of relations)

• Composition: S ∘ R, also denoted R ; S (or more ambiguously R ∘ S), defined as S ∘ R = { (x, z) | there existsy ∈ Y, such that (x, y) ∈ R and (y, z) ∈ S }. The order of R and S in the notation S ∘ R, used here agrees withthe standard notational order for composition of functions. For example, the composition “is mother of” ∘ “isparent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “isgrandmother of”.

10 CHAPTER 3. BINARY RELATION

A relation R on sets X and Y is said to be contained in a relation S on X and Y if R is a subset of S, that is, if x R yalways implies x S y. In this case, if R and S disagree, R is also said to be smaller than S. For example, > is containedin ≥.If R is a binary relation over X and Y, then the following is a binary relation over Y and X:

• Inverse or converse: R −1, defined as R −1 = { (y, x) | (x, y) ∈ R }. A binary relation over a set is equal to itsinverse if and only if it is symmetric. See also duality (order theory). For example, “is less than” (<) is theinverse of “is greater than” (>).

If R is a binary relation over X, then each of the following is a binary relation over X:

• Reflexive closure: R =, defined as R = = { (x, x) | x ∈ X } ∪ R or the smallest reflexive relation over X containingR. This can be proven to be equal to the intersection of all reflexive relations containing R.

• Reflexive reduction: R ≠, defined as R ≠ = R \ { (x, x) | x ∈ X } or the largest irreflexive relation over Xcontained in R.

• Transitive closure: R +, defined as the smallest transitive relation over X containing R. This can be seen to beequal to the intersection of all transitive relations containing R.

• Transitive reduction: R −, defined as a minimal relation having the same transitive closure as R.

• Reflexive transitive closure: R *, defined as R * = (R +) =, the smallest preorder containing R.

• Reflexive transitive symmetric closure: R ≡, defined as the smallest equivalence relation over X containingR.

3.4.1 Complement

If R is a binary relation over X and Y, then the following too:

• The complement S is defined as x S y if not x R y. For example, on real numbers, ≤ is the complement of >.

The complement of the inverse is the inverse of the complement.If X = Y, the complement has the following properties:

• If a relation is symmetric, the complement is too.

• The complement of a reflexive relation is irreflexive and vice versa.

• The complement of a strict weak order is a total preorder and vice versa.

The complement of the inverse has these same properties.

3.4.2 Restriction

The restriction of a binary relation on a set X to a subset S is the set of all pairs (x, y) in the relation for which x andy are in S.If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its restrictions are too.However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in generalnot equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother ofthe woman y"; its transitive closure doesn't relate a woman with her paternal grandmother. On the other hand, thetransitive closure of “is parent of” is “is ancestor of"; its restriction to females does relate a woman with her paternalgrandmother.

3.5. SETS VERSUS CLASSES 11

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions.For example, on the set of real numbers a property of the relation "≤" is that every non-empty subset S of R with anupper bound in R has a least upper bound (also called supremum) in R. However, for a set of rational numbers thissupremum is not necessarily rational, so the same property does not hold on the restriction of the relation "≤" to theset of rational numbers.The left-restriction (right-restriction, respectively) of a binary relation between X and Y to a subset S of its domain(codomain) is the set of all pairs (x, y) in the relation for which x (y) is an element of S.

3.4.3 Algebras, categories, and rewriting systems

Various operations on binary endorelations can be treated as giving rise to an algebraic structure, known as relationalgebra. It should not be confused with relational algebra which deals in finitary relations (and in practice also finiteand many-sorted).For heterogenous binary relations, a category of relations arises.[6]

Despite their simplicity, binary relations are at the core of an abstract computation model known as an abstractrewriting system.

3.5 Sets versus classes

Certain mathematical “relations”, such as “equal to”, “member of”, and “subset of”, cannot be understood to be binaryrelations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems ofaxiomatic set theory. For example, if we try to model the general concept of “equality” as a binary relation =, wemust take the domain and codomain to be the “class of all sets”, which is not a set in the usual set theory.In most mathematical contexts, references to the relations of equality, membership and subset are harmless becausethey can be understood implicitly to be restricted to some set in the context. The usual work-around to this problemis to select a “large enough” set A, that contains all the objects of interest, and work with the restriction =A instead of=. Similarly, the “subset of” relation ⊆ needs to be restricted to have domain and codomain P(A) (the power set ofa specific set A): the resulting set relation can be denoted ⊆A. Also, the “member of” relation needs to be restrictedto have domain A and codomain P(A) to obtain a binary relation ∈A that is a set. Bertrand Russell has shown thatassuming ∈ to be defined on all sets leads to a contradiction in naive set theory.Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory,and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership,and subset are binary relations without special comment. (A minor modification needs to be made to the concept ofthe ordered triple (X, Y, G), as normally a proper class cannot be a member of an ordered tuple; or of course onecan identify the function with its graph in this context.)[24] With this definition one can for instance define a functionrelation between every set and its power set.

3.6 The number of binary relations

The number of distinct binary relations on an n-element set is 2n2 (sequence A002416 in OEIS):Notes:

• The number of irreflexive relations is the same as that of reflexive relations.

• The number of strict partial orders (irreflexive transitive relations) is the same as that of partial orders.

• The number of strict weak orders is the same as that of total preorders.

• The total orders are the partial orders that are also total preorders. The number of preorders that are neithera partial order nor a total preorder is, therefore, the number of preorders, minus the number of partial orders,minus the number of total preorders, plus the number of total orders: 0, 0, 0, 3, and 85, respectively.

• the number of equivalence relations is the number of partitions, which is the Bell number.

12 CHAPTER 3. BINARY RELATION

The binary relations can be grouped into pairs (relation, complement), except that for n = 0 the relation is its owncomplement. The non-symmetric ones can be grouped into quadruples (relation, complement, inverse, inverse com-plement).

3.7 Examples of common binary relations

• order relations, including strict orders:

• greater than• greater than or equal to• less than• less than or equal to• divides (evenly)• is a subset of

• equivalence relations:

• equality• is parallel to (for affine spaces)• is in bijection with• isomorphy

• dependency relation, a finite, symmetric, reflexive relation.

• independency relation, a symmetric, irreflexive relation which is the complement of some dependency relation.

3.8 See also

• Confluence (term rewriting)

• Hasse diagram

• Incidence structure

• Logic of relatives

• Order theory

• Triadic relation

3.9 Notes[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] Suppes, Patrick (1972) [originally published by D. van Nostrand Company in 1960]. Axiomatic Set Theory. Dover. ISBN0-486-61630-4.

[3] Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.

[4] Levy, Azriel (2002) [republication of the work published by Springer-Verlag, Berlin, Heidelberg and New York in 1979].Basic Set Theory. Dover. ISBN 0-486-42079-5.

[5] Christodoulos A. Floudas; PanosM. Pardalos (2008). Encyclopedia of Optimization (2nd ed.). Springer Science&BusinessMedia. pp. 299–300. ISBN 978-0-387-74758-3.

3.10. REFERENCES 13

[6] Michael Winter (2007). Goguen Categories: A Categorical Approach to L-fuzzy Relations. Springer. pp. x–xi. ISBN978-1-4020-6164-6.

[7] Kilp, Knauer and Mikhalev: p. 3. The same four definitions appear in the following:

• Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook.Springer Science & Business Media. p. 506. ISBN 978-3-540-67995-0.

• Eike Best (1996). Semantics of Sequential and Parallel Programs. Prentice Hall. pp. 19–21. ISBN 978-0-13-460643-9.

• Robert-Christoph Riemann (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the HighLevel Petri Net Calculus. Herbert Utz Verlag. pp. 21–22. ISBN 978-3-89675-629-9.

[8] Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7, Chapt. 5

[9] Mäs, Stephan (2007), “Reasoning on Spatial Semantic Integrity Constraints”, Spatial Information Theory: 8th InternationalConference, COSIT 2007, Melbourne, Australiia, September 19–23, 2007, Proceedings, Lecture Notes in Computer Science4736, Springer, pp. 285–302, doi:10.1007/978-3-540-74788-8_18

[10] Note that the use of “correspondence” here is narrower than as general synonym for binary relation.

[11] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer Science &Business Media. p. 200. ISBN 978-3-211-82971-4.

[12] Yao, Y. (2004). “Semantics of Fuzzy Sets in Rough Set Theory”. Transactions on Rough Sets II. Lecture Notes in ComputerScience 3135. p. 297. doi:10.1007/978-3-540-27778-1_15. ISBN 978-3-540-23990-1.

[13] William Craig (2006). Semigroups Underlying First-order Logic. American Mathematical Soc. p. 72. ISBN 978-0-8218-6588-0.

[14] Gumm, H. P.; Zarrad, M. (2014). “Coalgebraic Simulations and Congruences”. Coalgebraic Methods in Computer Science.Lecture Notes in Computer Science 8446. p. 118. doi:10.1007/978-3-662-44124-4_7. ISBN 978-3-662-44123-7.

[15] Julius Richard Büchi (1989). Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions.Springer Science & Business Media. pp. 35–37. ISBN 978-1-4613-8853-1.

[16] M. E. Müller (2012). Relational Knowledge Discovery. Cambridge University Press. p. 22. ISBN 978-0-521-19021-3.

[17] Peter J. Pahl; Rudolf Damrath (2001). Mathematical Foundations of Computational Engineering: A Handbook. SpringerScience & Business Media. p. 496. ISBN 978-3-540-67995-0.

[18] Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006),ATransition to AdvancedMathematics (6th ed.), Brooks/Cole,p. 160, ISBN 0-534-39900-2

[19] Nievergelt, Yves (2002), Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography,Springer-Verlag, p. 158.

[20] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics – Physics Charles University. p. 1. Lemma 1.1 (iv). This source refers to asymmetric relations as “strictlyantisymmetric”.

[21] Since neither 5 divides 3, nor 3 divides 5, nor 3=5.

[22] Yao, Y.Y.; Wong, S.K.M. (1995). “Generalization of rough sets using relationships between attribute values” (PDF).Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33..

[23] Joseph G. Rosenstein, Linear orderings, Academic Press, 1982, ISBN 0-12-597680-1, p. 4

[24] Tarski, Alfred; Givant, Steven (1987). A formalization of set theory without variables. American Mathematical Society. p.3. ISBN 0-8218-1041-3.

3.10 References• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and

Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

14 CHAPTER 3. BINARY RELATION

3.11 External links• Hazewinkel, Michiel, ed. (2001), “Binary relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 4

Divisor

“Divisible” redirects here. For divisibility of groups, see Divisible group.For the second operand of a division, see Division (mathematics). For divisors in algebraic geometry, see Divisor(algebraic geometry). For divisibility in the ring theory, see Divisibility (ring theory).In mathematics a divisor of an integer n , also called a factor of n , is an integer that can be multiplied by some

The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

other integer to produce n .

4.1 Definition

Two versions of the definition of a divisor are commonplace:

• For integers m and n , it is said that m divides n , m is a divisor of n , or n is a multiple of m , and this iswritten asm | n,

if there exists an integer k such thatmk = n .[1] Under this definition, the statement 0 | 0 holds.

• As before, but with the additional constraintm ̸= 0 .[2] Under this definition, the statement 0 | 0 does not hold.

In the remainder of this article, which definition is applied is indicated where this is significant.

15

16 CHAPTER 4. DIVISOR

4.2 General

Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example,there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually bementioned.1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.[3] Every integeris a divisor of 0.[4] Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd.1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as anon-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, whilethe units −1 and 1 and prime numbers have no non-trivial divisors.There are divisibility rules which allow one to recognize certain divisors of a number from the number’s digits.The generalization can be said to be the concept of divisibility in any integral domain.

4.3 Examples

• 7 is a divisor of 42 because 7× 6 = 42 , so we can say 7 | 42 . It can also be said that 42 is divisible by 7, 42is a multiple of 7, 7 divides 42, or 7 is a factor of 42.

• The non-trivial divisors of 6 are 2, −2, 3, −3.

• The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

• 5 | 0 , because 5× 0 = 0 .

• The set of all positive divisors of 60, A = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} , partially ordered by divisi-bility, has the Hasse diagram:

4.4 Further notions and facts

There are some elementary rules:

• If a | b and b | c , then a | c , i.e. divisibility is a transitive relation.

• If a | b and b | a , then a = b or a = −b .

• If a | b and a | c , then a | (b+ c) holds, as does a | (b− c) .[5] However, if a | b and c | b , then (a+ c) | bdoes not always hold (e.g. 2 | 6 and 3 | 6 but 5 does not divide 6).

If a | bc , and gcd (a, b) = 1 , then a | c . This is called Euclid’s lemma.If p is a prime number and p | ab then p | a or p | b .A positive divisor of n which is different from n is called a proper divisor or an aliquot part of n . A number thatdoes not evenly divide n but leaves a remainder is called an aliquant part of n .An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positiveinteger which has exactly two positive factors: 1 and itself.Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of thefundamental theorem of arithmetic.A number n is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisorsis less than n , and abundant if this sum exceeds n .The total number of positive divisors of n is a multiplicative function d(n) , meaning that when two numbersm and nare relatively prime, then d(mn) = d(m)×d(n) . For instance, d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7) ; the eightdivisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicativefunction: if the two numbersm and n share a common divisor, then it might not be true that d(mn) = d(m)× d(n)

4.4. FURTHER NOTIONS AND FACTS 17

0 10 20n

divisors 0 10 20n

30 divisors124610 12

20 2430 3640 4850

606070 72

8080 849090 96

100100108

110 112120120

126130 132

140140 144150150156160160

168170

180180190 192

200210210216

220230

240240250 252

260 264270270280280

288290

300300310 312

320320 324330330 336

340350

360360370380390

396400410

420420430 432

440450450

460468

470480480

490500

504510520

528530

540540550

560560570

576580

588590

600600610 612

620 624630630

640648

650660660

670 672

680 684

690700700

710720720

730740750

756760770

780780790 792

800810820830

840840850860 864

870880890

900900910920 924

930936

940950

960960970980

9909901000

Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers arein bold.

. The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3 × 4 × 8 =σ(2)×σ(3)×σ(7) = 1+2+3+6+7+14+21+42 ). Both of these functions are examples of divisor functions.If the prime factorization of n is given by

n = pν11 pν2

2 · · · pνk

k

18 CHAPTER 4. DIVISOR

then the number of positive divisors of n is

d(n) = (ν1 + 1)(ν2 + 1) · · · (νk + 1),

and each of the divisors has the form

pµ1

1 pµ2

2 · · · pµk

k

where 0 ≤ µi ≤ νi for each 1 ≤ i ≤ k.

For every natural n , d(n) < 2√n .

Also,[6]

d(1) + d(2) + · · ·+ d(n) = n lnn+ (2γ − 1)n+O(√n).

where γ is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer nhas an expected number of divisors of about lnn .

4.5 In abstract algebra

Given the definition for which 0 | 0 holds, the relation of divisibility turns the set N of non-negative integers into apartially ordered set: a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. Themeet operation ∧ is given by the greatest common divisor and the join operation ∨ by the least common multiple.This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z .

4.6 See also• Arithmetic functions

4.7. NOTES 19

• Divisibility rule

• Divisor function

• Euclid’s algorithm

• Fraction (mathematics)

• Table of divisors — A table of prime and non-prime divisors for 1–1000

• Table of prime factors — A table of prime factors for 1–1000

• Unitary divisor

4.7 Notes[1] for instance, Sims 1984, p. 42 or Durbin 1992, p. 61

[2] Herstein 1986, p. 26

[3] This statement either requires 0|0 or needs to be restricted to nonzero integers.

[4] This statement either requires 0|0 or needs to be restricted to nonzero integers.

[5] a | b, a | c ⇒ b = ja, c = ka ⇒ b + c = (j + k)a ⇒ a | (b + c) . Similarly, a | b, a | c ⇒ b = ja, c = ka ⇒b− c = (j − k)a ⇒ a | (b− c)

[6] Hardy, G. H.; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264.ISBN 0-19-853171-0.

4.8 References• Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0-471-51001-7.

• Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7;section B.

• Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1

• Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).

• Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN0-471-09846-9

Chapter 5

Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions,asserting that the quantities have the same value or that the expressions represent the same mathematical object. Theequality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign".

5.1 Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”,“level”, “fair”, “just”).

5.2 Types of equalities

5.2.1 Identities

Main article: Identity (mathematics)

When A and Bmay be viewed as functions of some variables, then A = Bmeans that A and B define the same function.Such an equality of functions is sometimes called an identity. An example is (x + 1)2 = x2 + 2x + 1.

5.2.2 Equalities as predicates

When A and B are not fully specified or depend on some variables, equality is a proposition, which may be truefor some values and false for some other values. Equality is a binary relation, or, in other words, a two-argumentspredicate, which may produce a truth value (false or true) from its arguments. In computer programming, its com-putation from two expressions is known as comparison.

5.2.3 Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties thatare considered. This is, in particular the case in geometry, where two geometric shapes are said equal when one maybe moved to coincide with the other. The word congruence is also used for this kind of equality.

5.2.4 Equations

An equation is the problem of finding values of some variables, called unknowns, for which the specified equalityis true. Equation may also refer to an equality relation that is satisfied only for the values of the variables that oneis interested on. For example x2 + y2 = 1 is the equation of the unit circle. There is no standard notation that

20

5.3. LOGICAL FORMALIZATIONS OF EQUALITY 21

distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriateinterpretation from the semantic of expressions and the context.

5.2.5 Equivalence relations

Main article: Equivalence relation

Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set:those binary relations that are reflexive, symmetric, and transitive. The identity relation is an equivalence relation.Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of allelements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equalityis the smallest equivalence relation on any set S, in the sense that it is the relation that has the smallest equivalenceclasses (every class is reduced to a single element).

5.3 Logical formalizations of equality

There are several formalizations of the notion of equality in mathematical logic, usually by means of axioms, such asthe first few Peano axioms, or the axiom of extensionality in ZF set theory.For example, Azriel Lévy gives as the five axioms for equality, first the three properties of an equivalence relation,and these two:

x = y ∧ x ∈ z⇒ y ∈ z, andx = y ∧ z ∈ x ⇒ z ∈ y.[1]

These extra two conditions allow substitution of equal quantities into complex expressions.There are also some logic systems that do not have any notion of equality. This reflects the undecidability of theequality of two real numbers defined by formulas involving the integers, the basic arithmetic operations, the logarithmand the exponential function. In other words, there cannot exist any algorithm for deciding such an equality.

5.4 Logical formulations

Equality is always defined such that things that are equal have all and only the same properties. Some people defineequality as congruence. Often equality is just defined as identity.A stronger sense of equality is obtained if some form of Leibniz’s law is added as an axiom; the assertion of this axiomrules out “bare particulars”—things that have all and only the same properties but are not equal to each other—whichare possible in some logical formalisms. The axiom states that two things are equal if they have all and only the sameproperties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective “if and only if” can be weakened to “if"; the modified law is equivalent to the original.Instead of considering Leibniz’s law as an axiom, it can also be taken as the definition of equality. The property ofbeing an equivalence relation, as well as the properties given below, can then be proved: they become theorems. Ifa=b, then a can replace b and b can replace a.

5.5 Some basic logical properties of equality

The substitution property states:

• For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e.are well-formed).

22 CHAPTER 5. EQUALITY (MATHEMATICS)

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functionalpredicate).Some specific examples of this are:

• For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);

• For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);

• For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);

• For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.The symmetric property states:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined,is not transitive (it may seem so at first sight, but many small differences can add up to something big). However,equality almost everywhere is transitive.Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitutionand reflexive properties are assumed instead.

5.6 Relation with equivalence and isomorphism

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism.[2] For example, one may distin-guish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 aredistinct as fractions, as different strings of symbols, but they “represent” the same rational number, the same pointon a number line. This distinction gives rise to the notion of a quotient set.Similarly, the sets

{A,B,C} and {1, 2, 3}

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of threeelements, and thus isomorphic, meaning that there is a bijection between them, for example

A 7→ 1,B 7→ 2,C 7→ 3.

However, there are other choices of isomorphism, such as

A 7→ 3,B 7→ 2,C 7→ 1,

and these sets cannot be identified without making such a choice – any statement that identifies them “dependson choice of identification”. This distinction, between equality and isomorphism, is of fundamental importance incategory theory, and is one motivation for the development of category theory.

5.7. SEE ALSO 23

5.7 See also• Equals sign

• Inequality

• Logical equality

• Extensionality

5.8 References[1] Azriel Lévy (1979) Basic Set Theory, page 358, Springer-Verlag

[2] (Mazur 2007)

• Mazur, Barry (12 June 2007),When is one thing equal to some other thing? (PDF)

• Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.

Chapter 6

Equivalence class

This article is about equivalency in mathematics. For equivalency in music, see equivalence class (music).In mathematics, when a set has an equivalence relation defined on its elements, there is a natural grouping of

Congruence is an example of an equivalence relation. The two triangles on the left are congruent, while the third and fourth trianglesare not congruent to any other triangle. Thus, the first two triangles are in the same equivalence class, while the third and fourthtriangles are in their own equivalence class.

elements that are related to one another, forming what are called equivalence classes. Notationally, given a set Xand an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X whichare equivalent to a. It follows from the definition of the equivalence relations that the equivalence classes form apartition of X. The set of equivalence classes is sometimes called the quotient set or the quotient space of X by ~and is denoted by X / ~.When X has some structure, and the equivalence relation is defined with some connection to this structure, thequotient set often inherits some related structure. Examples include quotient spaces in linear algebra, quotient spacesin topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

6.1 Notation and formal definition

An equivalence relation is a binary relation ~ satisfying three properties:[1]

• For every element a in X, a ~ a (reflexivity),

• For every two elements a and b in X, if a ~ b, then b ~ a (symmetry)

• For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

24

6.2. EXAMPLES 25

The equivalence class of an element a is denoted [a] and is defined as the set

[a] = {x ∈ X | a ∼ x}

of elements that are related to a by ~. An alternative notation [a]R can be used to denote the equivalence class of theelement a, specifically with respect to the equivalence relation R. This is said to be the R-equivalence class of a.The set of all equivalence classes in Xwith respect to an equivalence relationR is denoted asX/R and called XmoduloR (or the quotient set of X by R).[2] The surjective map x 7→ [x] from X onto X/R, which maps each element to itsequivalence class, is called the canonical surjection or the canonical projection map.When an element is chosen (often implicitly) in each equivalence class, this defines an injective map called a section. Ifthis section is denoted by s, one has [s(c)] = c for every equivalence class c. The element s(c) is called a representativeof c. Any element of a class may be chosen as a representative of the class, by choosing the section appropriately.Sometimes, there is a section that is more “natural” than the other ones. In this case, the representatives are calledcanonical representatives. For example, in modular arithmetic, consider the equivalence relation on the integersdefined by a ~ b if a − b is a multiple of a given integer n, called the modulus. Each class contains a unique non-negative integer smaller than n, and these integers are the canonical representatives. The class and its representativeare more or less identified, as is witnessed by the fact that the notation a mod n may denote either the class or itscanonical representative (which is the remainder of the division of a by n).

6.2 Examples• If X is the set of all cars, and ~ is the equivalence relation “has the same color as.” then one particular equivalenceclass consists of all green cars. X/~ could be naturally identified with the set of all car colors (cardinality ofX/~ would be the number of all car colors).

• Let X be the set of all rectangles in a plane, and ~ the equivalence relation “has the same area as”. For eachpositive real number A there will be an equivalence class of all the rectangles that have area A.[3]

• Consider the modulo 2 equivalence relation on the set Z of integers: x ~ y if and only if their difference x − yis an even number. This relation gives rise to exactly two equivalence classes: one class consisting of all evennumbers, and the other consisting of all odd numbers. Under this relation [7], [9], and [1] all represent thesame element of Z/~.[4]

• Let X be the set of ordered pairs of integers (a,b) with b not zero, and define an equivalence relation ~ on Xaccording to which (a,b) ~ (c,d) if and only if ad = bc. Then the equivalence class of the pair (a,b) can beidentified with the rational number a/b, and this equivalence relation and its equivalence classes can be used togive a formal definition of the set of rational numbers.[5] The same construction can be generalized to the fieldof fractions of any integral domain.

• If X consists of all the lines in, say the Euclidean plane, and L ~M means that L andM are parallel lines, thenthe set of lines that are parallel to each other form an equivalence class as long as a line is considered parallelto itself. In this situation, each equivalence class determines a point at infinity.

6.3 Properties

Every element x of X is a member of the equivalence class [x]. Every two equivalence classes [x] and [y] are eitherequal or disjoint. Therefore, the set of all equivalence classes of X forms a partition of X: every element of X belongsto one and only one equivalence class.[6] Conversely every partition of X comes from an equivalence relation in thisway, according to which x ~ y if and only if x and y belong to the same set of the partition.[7]

It follows from the properties of an equivalence relation that

x ~ y if and only if [x] = [y].

In other words, if ~ is an equivalence relation on a set X, and x and y are two elements of X, then these statementsare equivalent:

26 CHAPTER 6. EQUIVALENCE CLASS

• x ∼ y

• [x] = [y]

• [x] ∩ [y] ̸= ∅.

6.4 Graphical representation

Any binary relation can be represented by a directed graph and symmetric ones, such as equivalence relations, byundirected graphs. If ~ is an equivalence relation on a set X, let the vertices of the graph be the elements of X andjoin vertices s and t if and only if s ~ t. The equivalence classes are represented in this graph by the maximal cliquesforming the connected components of the graph.[8]

6.5 Invariants

If ~ is an equivalence relation on X, and P(x) is a property of elements of X such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be an invariant of ~, or well-defined under the relation ~.A frequent particular case occurs when f is a function from X to another set Y; if f(x1) = f(x2) whenever x1 ~ x2,then f is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. Some authors use “compatible with ~" or just “respects ~" instead of “invariantunder ~".Any function f : X → Y itself defines an equivalence relation on X according to which x1 ~ x2 if and only if f(x1)= f(x2). The equivalence class of x is the set of all elements in X which get mapped to f(x), i.e. the class [x] is theinverse image of f(x). This equivalence relation is known as the kernel of f.More generally, a function may map equivalent arguments (under an equivalence relation ~X on X) to equivalentvalues (under an equivalence relation ~Y on Y). Such a function is known as a morphism from ~X to ~Y .

6.6 Quotient space in topology

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relationon a topological space using the original space’s topology to create the topology on the set of equivalence classes.In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebraon the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vectorspace formed by taking a quotient group where the quotient homomorphism is a linear map. By extension, in abstractalgebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotientalgebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a groupaction.The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when theorbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroupon the group by left translations, or respectively the left cosets as orbits under right translation.A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the sensesof topology, abstract algebra, and group actions simultaneously.Although the term can be used for any equivalence relation’s set of equivalence classes, possibly with further structure,the intent of using the term is generally to compare that type of equivalence relation on a setX either to an equivalencerelation that induces some structure on the set of equivalence classes from a structure of the same kind on X, or to theorbits of a group action. Both the sense of a structure preserved by an equivalence relation and the study of invariantsunder group actions lead to the definition of invariants of equivalence relations given above.

6.7 See also• Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible

6.8. NOTES 27

program inputs into equivalence classes according to the behavior of the program on those inputs

• Homogeneous space, the quotient space of Lie groups.

• Transversal (combinatorics)

6.8 Notes[1] Devlin 2004, p. 122

[2] Wolf 1998, p. 178

[3] Avelsgaard 1989, p. 127

[4] Devlin 2004, p. 123

[5] Maddox 2002, pp. 77–78

[6] Maddox 2002, p.74, Thm. 2.5.15

[7] Avelsgaard 1989, p.132, Thm. 3.16

[8] Devlin 2004, p. 123

6.9 References• Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8

• Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman& Hall/ CRC Press, ISBN 978-1-58488-449-1

• Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9

• Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X

• Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematician’s Toolbox, Freeman, ISBN 978-0-7167-3050-7

6.10 Further reading

This material is basic and can be found in any text dealing with the fundamentals of proof technique, such as any ofthe following:

• Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall

• Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th Ed.), Thomson (Brooks/Cole)

• Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley,ISBN 0-201-82653-4

• O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall

• Lay (2001), Analysis with an introduction to proof, Prentice Hall

• Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall

• Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent

• Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan

• D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall

28 CHAPTER 6. EQUIVALENCE CLASS

• Cupillari, The Nuts and Bolts of Proofs, Wadsworth

• Bond, Introduction to Abstract Mathematics, Brooks/Cole

• Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall

• Ash, A Primer of Abstract Mathematics, MAA

Chapter 7

Equivalence relation

This article is about the mathematical concept. For the patent doctrine, see Doctrine of equivalents.In mathematics, an equivalence relation is the relation that holds between two elements if and only if they are

members of the same cell within a set that has been partitioned into cells such that every element of the set is amember of one and only one cell of the partition. The intersection of any two different cells is empty; the union ofall the cells equals the original set. These cells are formally called equivalence classes.

7.1 Notation

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalentwith respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is theobvious relation being referenced, and variations of "a ~R b", "a ≡R b", or "aRb" otherwise.

7.2 Definition

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric andtransitive. Equivalently, for all a, b and c in X:

• a ~ a. (Reflexivity)

• if a ~ b then b ~ a. (Symmetry)

• if a ~ b and b ~ c then a ~ c. (Transitivity)

X together with the relation ~ is called a setoid. The equivalence class of a under ~, denoted [a], is defined as[a] = {b ∈ X | a ∼ b} .

7.3 Examples

7.3.1 Simple example

Let the set {a, b, c} have the equivalence relation {(a, a), (b, b), (c, c), (b, c), (c, b)} . The following sets are equivalenceclasses of this relation:[a] = {a}, [b] = [c] = {b, c} .The set of all equivalence classes for this relation is {{a}, {b, c}} .

29

30 CHAPTER 7. EQUIVALENCE RELATION

7.3.2 Equivalence relations

The following are all equivalence relations:

• “Has the same birthday as” on the set of all people.

• “Is similar to” on the set of all triangles.

• “Is congruent to” on the set of all triangles.

• “Is congruent to, modulo n" on the integers.

• “Has the same image under a function" on the elements of the domain of the function.

• “Has the same absolute value” on the set of real numbers

• “Has the same cosine” on the set of all angles.

7.3.3 Relations that are not equivalences

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 doesnot imply that 5 ≥ 7. It is, however, a partial order.

• The relation “has a common factor greater than 1 with” between natural numbers greater than 1, is reflexiveand symmetric, but not transitive. (Example: The natural numbers 2 and 6 have a common factor greater than1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).

• The empty relation R on a non-empty set X (i.e. aRb is never true) is vacuously symmetric and transitive, butnot reflexive. (If X is also empty then R is reflexive.)

• The relation “is approximately equal to” between real numbers, even if more precisely defined, is not an equiv-alence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes canaccumulate to become a big change. However, if the approximation is defined asymptotically, for example bysaying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point,then this defines an equivalence relation.

7.4 Connections to other relations

• A partial order is a relation that is reflexive, antisymmetric, and transitive.

• Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set thatis reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted forone another, a facility that is not available for equivalence related variables. The equivalence classes of anequivalence relation can substitute for one another, but not individuals within a class.

• A strict partial order is irreflexive, transitive, and asymmetric.

• A partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and onlyif for all a ∈ X, there exists a b ∈ X such that a ~ b.

• A reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite.

• A preorder is reflexive and transitive.

• A congruence relation is an equivalence relation whose domain X is also the underlying set for an algebraicstructure, and which respects the additional structure. In general, congruence relations play the role of kernelsof homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many importantcases congruence relations have an alternative representation as substructures of the structure on which theyare defined. E.g. the congruence relations on groups correspond to the normal subgroups.

7.5. WELL-DEFINEDNESS UNDER AN EQUIVALENCE RELATION 31

7.5 Well-definedness under an equivalence relation

If ~ is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x ~ y, P(x) is true ifP(y) is true, then the property P is said to be well-defined or a class invariant under the relation ~.A frequent particular case occurs when f is a function from X to another set Y ; if x1 ~ x2 implies f(x1) = f(x2) thenf is said to be a morphism for ~, a class invariant under ~, or simply invariant under ~. This occurs, e.g. in thecharacter theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. Seealso invariant. Some authors use “compatible with ~" or just “respects ~" instead of “invariant under ~".More generally, a function may map equivalent arguments (under an equivalence relation ~A) to equivalent values(under an equivalence relation ~B). Such a function is known as a morphism from ~A to ~B.

7.6 Equivalence class, quotient set, partition

Let a, b ∈ X . Some definitions:

7.6.1 Equivalence class

Main article: Equivalence class

A subset Y of X such that a ~ b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalenceclass of X by ~. Let [a] := {x ∈ X | a ∼ x} denote the equivalence class to which a belongs. All elements of Xequivalent to each other are also elements of the same equivalence class.

7.6.2 Quotient set

Main article: Quotient set

The set of all possible equivalence classes of X by ~, denoted X/∼ := {[x] | x ∈ X} , is the quotient set of X by~. If X is a topological space, there is a natural way of transforming X/~ into a topological space; see quotient spacefor the details.

7.6.3 Projection

Main article: Projection (relational algebra)

The projection of ~ is the function π : X → X/∼ defined by π(x) = [x] which maps elements of X into theirrespective equivalence classes by ~.

Theorem on projections:[1] Let the function f: X → B be such that a ~ b→ f(a) = f(b). Then there is aunique function g : X/~ → B, such that f = gπ. If f is a surjection and a ~ b ↔ f(a) = f(b), then g is abijection.

7.6.4 Equivalence kernel

The equivalence kernel of a function f is the equivalence relation ~ defined by x ∼ y ⇐⇒ f(x) = f(y) . Theequivalence kernel of an injection is the identity relation.

7.6.5 Partition

Main article: Partition of a set

32 CHAPTER 7. EQUIVALENCE RELATION

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single elementof P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their unionis X.

Counting possible partitions

Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X, andvice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which isthe nth Bell number Bn:

Bn =1

e

∞∑k=0

kn

k!,

where the above is one of the ways to write the nth Bell number.

7.7 Fundamental theorem of equivalence relations

A key result links equivalence relations and partitions:[2][3][4]

• An equivalence relation ~ on a set X partitions X.

• Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongsto a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection from the setof all possible equivalence relations on X and the set of all partitions of X.

7.8 Comparing equivalence relations

If ~ and ≈ are two equivalence relations on the same set S, and a~b implies a≈b for all a,b ∈ S, then ≈ is said to be acoarser relation than ~, and ~ is a finer relation than ≈. Equivalently,

• ~ is finer than ≈ if every equivalence class of ~ is a subset of an equivalence class of ≈, and thus every equivalenceclass of ≈ is a union of equivalence classes of ~.

• ~ is finer than ≈ if the partition created by ~ is a refinement of the partition created by ≈.

The equality equivalence relation is the finest equivalence relation on any set, while the trivial relation that makes allpairs of elements related is the coarsest.The relation "~ is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial orderrelation.

7.9 Generating equivalence relations

• Given any set X, there is an equivalence relation over the set [X→X] of all possible functions X→X. Two suchfunctions are deemed equivalent when their respective sets of fixpoints have the same cardinality, correspondingto cycles of length one in a permutation. Functions equivalent in this manner form an equivalence class on[X→X], and these equivalence classes partition [X→X].

7.10. ALGEBRAIC STRUCTURE 33

• An equivalence relation ~ on X is the equivalence kernel of its surjective projection π : X→ X/~.[5] Conversely,any surjection between sets determines a partition on its domain, the set of preimages of singletons in thecodomain. Thus an equivalence relation over X, a partition of X, and a projection whose domain is X, are threeequivalent ways of specifying the same thing.

• The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of X × X)is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given anybinary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containingR. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x1, x2, ..., xn in Xsuch that a = x1, b = xn, and (xi,xi₊ ₁)∈R or (xi₊₁,xi)∈R, i = 1, ..., n−1.

Note that the equivalence relation generated in this manner can be trivial. For instance, the equivalencerelation ~ generated by:

• • Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y;• Any subset of the identity relation on X has equivalence classes that are the singletons of X.

• Equivalence relations can construct new spaces by “gluing things together.” Let X be the unit Cartesian square[0,1] × [0,1], and let ~ be the equivalence relation on X defined by ∀a, b ∈ [0,1] ((a, 0) ~ (a, 1) ∧ (0, b) ~ (1, b)).Then the quotient space X/~ can be naturally identified (homeomorphism) with a torus: take a square piece ofpaper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder soas to glue together its two open ends, resulting in a torus.

7.10 Algebraic structure

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures themathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics asorder relations, the algebraic structure of equivalences is not as well known as that of orders. The former structuredraws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

7.10.1 Group theory

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalencerelations are grounded in partitioned sets, which are sets closed under bijections and preserve partition structure.Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hencepermutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathe-matical structure of equivalence relations.Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denotethe set of bijective functions over A that preserve the partition structure of A: ∀x ∈ A ∀g ∈ G (g(x) ∈ [x]). Then thefollowing three connected theorems hold:[6]

• ~ partitions A into equivalence classes. (This is the Fundamental Theorem of Equivalence Relations,mentionedabove);

• Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the parti-tion‡;

• Given a transformation groupG overA, there exists an equivalence relation ~ over A, whose equivalence classesare the orbits of G.[7][8]

In sum, given an equivalence relation ~ over A, there exists a transformation group G over A whose orbits are theequivalence classes of A under ~.This transformation group characterisation of equivalence relations differs fundamentally from the way lattices char-acterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe

34 CHAPTER 7. EQUIVALENCE RELATION

A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a setof bijections, A→ A.Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a~ b↔ (ab−1 ∈ H). The equivalence classes of ~—also called the orbits of the action of H on G—are the right cosetsof H in G. Interchanging a and b yields the left cosets.‡Proof.[9] Let function composition interpret group multiplication, and function inverse interpret group inverse. ThenG is a group under composition, meaning that ∀x ∈ A ∀g ∈ G ([g(x)] = [x]), because G satisfies the following fourconditions:

• G is closed under composition. The composition of any two elements of G exists, because the domain andcodomain of any element of G is A. Moreover, the composition of bijections is bijective;[10]

• Existence of identity function. The identity function, I(x)=x, is an obvious element of G;• Existence of inverse function. Every bijective function g has an inverse g−1, such that gg−1 = I;• Composition associates. f(gh) = (fg)h. This holds for all functions over all domains.[11]

Let f and g be any two elements of G. By virtue of the definition of G, [g(f(x))] = [f(x)] and [f(x)] = [x], so that[g(f(x))] = [x]. Hence G is also a transformation group (and an automorphism group) because function compositionpreserves the partitioning of A. □Related thinking can be found in Rosen (2008: chpt. 10).

7.10.2 Categories and groupoids

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing thisequivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there existsa unique morphism from x to y if and only if x~y.The advantages of regarding an equivalence relation as a special case of a groupoid include:

• Whereas the notion of “free equivalence relation” does not exist, that of a free groupoid on a directed graphdoes. Thus it is meaningful to speak of a “presentation of an equivalence relation,” i.e., a presentation of thecorresponding groupoid;

• Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notionof groupoid, a point of view that suggests a number of analogies;

• In many contexts “quotienting,” and hence the appropriate equivalence relations often called congruences, areimportant. This leads to the notion of an internal groupoid in a category.[12]

7.10.3 Lattices

The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called ConX by convention. The canonical map ker: X^X → Con X, relates the monoid X^X of all functions on X and Con X.ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: X→X toits kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

7.11 Equivalence relations and mathematical logic

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation withexactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical forany larger cardinal number.An implication of model theory is that the properties defining a relation can be proved independent of each other(and hence necessary parts of the definition) if and only if, for each property, examples can be found of relationsnot satisfying the given property while satisfying all the other properties. Hence the three defining properties ofequivalence relations can be proved mutually independent by the following three examples:

7.12. EUCLIDEAN RELATIONS 35

• Reflexive and transitive: The relation ≤ on N. Or any preorder;

• Symmetric and transitive: The relation R on N, defined as aRb↔ ab ≠ 0. Or any partial equivalence relation;

• Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3.”Or any dependency relation.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

• The number of equivalence classes is finite or infinite;

• The number of equivalence classes equals the (finite) natural number n;

• All equivalence classes have infinite cardinality;

• The number of elements in each equivalence class is the natural number n.

7.12 Euclidean relations

Euclid's The Elements includes the following “Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing “equal” by “are in relationwith”). By “relation” is meant a binary relation, in which aRb is generally distinct from bRa. An Euclidean relationthus comes in two forms:

(aRc ∧ bRc) → aRb (Left-Euclidean relation)(cRa ∧ cRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive.

Proof for a left-Euclidean relation

(aRc ∧ bRc) → aRb [a/c] = (aRa ∧ bRa) → aRb [reflexive; erase T∧] = bRa→ aRb. Hence R is symmetric.

(aRc ∧ bRc) → aRb [symmetry] = (aRc ∧ cRb) → aRb. Hence R is transitive. □

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclideanand reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed thereflexivity of equality too obvious to warrant explicit mention.

7.13 See also

• Partition of a set

• Equivalence class

• Up to

• Conjugacy class

• Topological conjugacy

36 CHAPTER 7. EQUIVALENCE RELATION

7.14 Notes[1] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 35, Th. 19. Chelsea.

[2] Wallace, D. A. R., 1998. Groups, Rings and Fields. p. 31, Th. 8. Springer-Verlag.

[3] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. p. 3, Prop. 2. John Wiley & Sons.

[4] Karel Hrbacek & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, pages 29–32, Marcel Dekker

[5] Garrett Birkhoff and Saunders Mac Lane, 1999 (1967). Algebra, 3rd ed. p. 33, Th. 18. Chelsea.

[6] Rosen (2008), pp. 243-45. Less clear is §10.3 of Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press.

[7] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 202, Th. 6.

[8] Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed. John Wiley & Sons: 114, Prop. 2.

[9] Bas van Fraassen, 1989. Laws and Symmetry. Oxford Univ. Press: 246.

[10] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 22, Th. 6.

[11] Wallace, D. A. R., 1998. Groups, Rings and Fields. Springer-Verlag: 24, Th. 7.

[12] Borceux, F. and Janelidze, G., 2001. Galois theories, Cambridge University Press, ISBN 0-521-80309-8

7.15 References• Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.

• Castellani, E., 2003, “Symmetry and equivalence” in Brading, Katherine, and E. Castellani, eds., Symmetriesin Physics: Philosophical Reflections. Cambridge Univ. Press: 422-433.

• Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusseshow equivalence relations arise in lattice theory.

• Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.

• John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.

• Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag.Mostly chpts. 9,10.

• Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axiomsdefining equivalence, pp 48–50, John Wiley & Sons.

7.16 External links• Hazewinkel, Michiel, ed. (2001), “Equivalence relation”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009

• Equivalence relation at PlanetMath

• Binary matrices representing equivalence relations at OEIS.

7.16. EXTERNAL LINKS 37

Logical matrices of the 52 equivalence relations on a 5-element set (Colored fields, including those in light gray, stand for ones; whitefields for zeros.)

Chapter 8

Homogeneous space

A torus. The standard torus is homogeneous under its diffeomorphism and homeomorphism groups, and the flat torus is homogeneousunder its diffeomorphism, homeomorphism, and isometry groups.

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneousspace for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements ofG are called the symmetries of X. A special case of this is when the group G in question is the automorphism groupof the space X – here “automorphism group” can mean isometry group, diffeomorphism group, or homeomorphismgroup. In this case X is homogeneous if intuitively X looks locally the same at each point, either in the sense ofisometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authorsinsist that the action of G be faithful (non-identity elements act non-trivially), although the present article does not.Thus there is a group action of G on X which can be thought of as preserving some “geometric structure” on X, andmaking X into a single G-orbit.

8.1 Formal definition

Let X be a non-empty set and G a group. Then X is called a G-space if it is equipped with an action of G on X.[1]Note that automatically G acts by automorphisms (bijections) on the set. If X in addition belongs to some category,

38

8.2. GEOMETRY 39

then the elements of G are assumed to act as automorphisms in the same category. Thus the maps on X effected byG are structure preserving. A homogeneous space is a G-space on which G acts transitively.Succinctly, if X is an object of the category C, then the structure of a G-space is a homomorphism:

ρ : G→ AutC(X)

into the group of automorphisms of the object X in the category C. The pair (X, ρ) defines a homogeneous spaceprovided ρ(G) is a transitive group of symmetries of the underlying set of X.

8.1.1 Examples

For example, if X is a topological space, then group elements are assumed to act as homeomorphisms on X. Thestructure of a G-space is a group homomorphism ρ : G → Homeo(X) into the homeomorphism group of X.Similarly, if X is a differentiable manifold, then the group elements are diffeomorphisms. The structure of a G-spaceis a group homomorphism ρ : G → Diffeo(X) into the diffeomorphism group of X.Riemannian symmetric spaces are an important class of homogeneous spaces, and include many of the exampleslisted below.Concrete examples include:

Isometry groups

• Positive curvature:

1. Sphere (orthogonal group): Sn−1 ∼= O(n)/O(n− 1)

2. Oriented sphere (special orthogonal group): Sn−1 ∼= SO(n)/SO(n− 1)

3. Projective space (projective orthogonal group): Pn−1 ∼= PO(n)/PO(n− 1)

• Flat (zero curvature):

1. Euclidean space (Euclidean group, point stabilizer is orthogonal group): An ≅ E(n)/O(n)

• Negative curvature:

1. Hyperbolic space (orthochronous Lorentz group, point stabilizer orthogonal group, corresponding to hyperboloidmodel): Hn ≅ O+(1, n)/O(n)

2. Oriented hyperbolic space: SO+(1, n)/SO(n)

3. Anti-de Sitter space: AdS ₊₁ = O(2, n)/O(1, n)

Others

• Affine space (for affine group, point stabilizer general linear group): An = Aff(n, K)/GL(n, k).

• Grassmannian: Gr(r, n) = O(n)/(O(r)× O(n− r))

8.2 Geometry

From the point of view of the Erlangen program, onemay understand that “all points are the same”, in the geometry ofX. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenthcentury.

40 CHAPTER 8. HOMOGENEOUS SPACE

Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spacesfor their respective symmetry groups. The same is true of the models found of non-Euclidean geometry of constantcurvature, such as hyperbolic space.A further classical example is the space of lines in projective space of three dimensions (equivalently, the spaceof two-dimensional subspaces of a four-dimensional vector space). It is simple linear algebra to show that GL4 actstransitively on those. We can parameterize them by line co-ordinates: these are the 2×2 minors of the 4×2 matrix withcolumns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometryof Julius Plücker.

8.3 Homogeneous spaces as coset spaces

In general, if X is a homogeneous space, and Ho is the stabilizer of some marked point o in X (a choice of origin),the points of X correspond to the left cosets G/Ho, and the marked point o corresponds to the coset of the identity.Conversely, given a coset space G/H, it is a homogeneous space for G with a distinguished point, namely the coset ofthe identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.In general, a different choice of origin o will lead to a quotient of G by a different subgroup Ho′ which is related toHo by an inner automorphism of G. Specifically,

Ho′ = gHog−1 (1)

where g is any element of G for which go = o′. Note that the inner automorphism (1) does not depend on which suchg is selected; it depends only on g modulo Ho.If the action of G on X is continuous, then H is a closed subgroup of G. In particular, if G is a Lie group, then H isa Lie subgroup by Cartan’s theorem. Hence G/H is a smooth manifold and so X carries a unique smooth structurecompatible with the group action.If H is the identity subgroup {e}, then X is a principal homogeneous space.One can go further to double coset spaces, notably Clifford–Klein forms Γ\G/H, where Γ is a discrete subgroup (ofG) acting properly discontinuously.

8.4 Example

For example in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional generallinear group, GL(4), defined by conditions on the matrix entries

h13 = h14 = h23 = h24 = 0,

by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that X hasdimension 4.Since the homogeneous coordinates given by theminors are 6 in number, this means that the latter are not independentof each other. In fact a single quadratic relation holds between the six minors, as was known to nineteenth-centurygeometers.This example was the first known example of a Grassmannian, other than a projective space. There are many furtherhomogeneous spaces of the classical linear groups in common use in mathematics.

8.5 Prehomogeneous vector spaces

The idea of a prehomogeneous vector space was introduced by Mikio Sato.It is a finite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of Gthat is open for the Zariski topology (and so, dense). An example is GL(1) acting on a one-dimensional space.

8.6. HOMOGENEOUS SPACES IN PHYSICS 41

The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is aclassification of irreducible prehomogeneous vector spaces, up to a transformation known as “castling”.

8.6 Homogeneous spaces in physics

Cosmology using the general theory of relativity makes use of the Bianchi classification system. Homogeneous spacesin relativity represent the space part of background metrics for some cosmological models; for example, the threecases of the Friedmann–Lemaître–Robertson–Walker metric may be represented by subsets of the Bianchi I (flat),V (open), VII (flat or open) and IX (closed) types, while the Mixmaster universe represents an anisotropic exampleof a Bianchi IX cosmology.[2]

A homogeneous space of N dimensions admits a set of 12N(N + 1) Killing vectors.[3] For three dimensions, this

gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one mayuse linear combinations of these to find three everywhere non-vanishing Killing vector fields ξ(a)i ,

ξ(a)[i;k] = Ca

bcξ(b)i ξ

(c)k

where the object Cabc , the “structure constants”, form a constant order-three tensor antisymmetric in its lower two

indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the covariant differentialoperator). In the case of a flat isotropic universe, one possibility is Ca

bc = 0 (type I), but in the case of a closedFLRW universe, Ca

bc = εabc where εabc is the Levi-Civita symbol.

8.7 See also• Erlangen program

• Klein geometry

• Heap (mathematics)

• Homogeneous variety

8.8 References[1] We assume that the action is on the left. The distinction is only important in the description of X as a coset space.

[2] Lev Landau and Evgeny Lifshitz (1980), Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, ISBN 978-0-7506-2768-9

[3] Steven Weinberg (1972), Gravitation and Cosmology, John Wiley and Sons

Chapter 9

Intransitivity

This article is about intransitivity in mathematics. For the linguistics sense, see Intransitive verb.

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are nottransitive relations. This may include any relation that is not transitive, or the stronger property of antitransitiv-ity, which describes a relation that is never transitive.

9.1 Intransitivity

A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C.Some authors call a relation intransitive if it is not transitive, i.e. (if the relation in question is named R )

¬ (∀a, b, c : aRb ∧ bRc⇒ aRc) .

This statement is equivalent to

∃a, b, c : aRb ∧ bRc ∧ ¬(aRc)

For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass.[1] Thus,the feed on relation among life forms is intransitive, in this sense.Another example that does not involve preference loops arises in freemasonry: it may be the case that lodge Arecognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognitionrelation among Masonic lodges is intransitive.

9.2 Antitransitivity

Often the term intransitive is used to refer to the stronger property of antitransitivity.We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feedon rabbits, rabbits feed on carrots, and humans also feed on carrots.A relation is antitransitive if this never occurs at all, i.e.,

∀a, b, c : aRb ∧ bRc⇒ ¬aRc

Many authors use the term intransitivity to mean antitransitivity.[2][3]

An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated playerB and player B defeated player C, A can have never played C, and therefore, A has not defeated C.

42

9.3. CYCLES 43

9.3 Cycles

The term intransitivity is often used when speaking of scenarios in which a relation describes the relative preferencesbetween pairs of options, and weighing several options produces a “loop” of preference:

• A is preferred to B

• B is preferred to C

• C is preferred to A

Rock, paper, scissors is an example.Assuming no option is preferred to itself i.e. the relation is irreflexive, a preference relation with a loop is not transitive.For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this exampleof a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred toC, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.Therefore such a preference loop (or "cycle") is known as an intransitivity.Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, anequivalence relation possesses cycles but is transitive. Now, consider the relation “is an enemy of” and suppose thatthe relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country isnot itself an enemy of the country. This is an example of an antitransitive relation that does not have any cycles. Inparticular, by virtue of being antitransitive the relation is not transitive.Finally, let us work with the example of rock, paper, scissors, calling the three options A, B, and C. Now, the relationover A, B, and C is “defeats” and the standard rules of the game are such that A defeats B, B defeats C, and C defeatsA. Furthermore, it is also true that B does not defeat A, C does not defeat B, and A does not defeat C. Finally, it isalso true that no option defeats itself. This information can be depicted in a table:The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zeroindicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn(with replacement) from the set {A, B, C}: If x defeats y, and y defeats z, then x does not defeat z. Hence the relationis antitransitive.Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.

9.4 Occurrences in preferences• Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcetvoting method in which ranking several candidates can produce a loop of preference when the weights arecompared (see voting paradox). Intransitive dice demonstrate that probabilities are not necessarily transitive.

• In psychology, intransitivity often occurs in a person’s system of values (or preferences, or tastes), potentiallyleading to unresolvable conflicts.

• Analogously, in economics intransitivity can occur in a consumer’s preferences. This may lead to consumerbehaviour that does not conform to perfect economic rationality. In recent years, economists and philosophershave questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand(1993)).

9.5 Likelihood

It has been suggested that Condorcet voting tends to eliminate “intransitive loops” when large numbers of votersparticipate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidateson several different units of measure such as by order of social consciousness or by order of most fiscally conservative.In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units ofmeasure in assessing candidates.Such as:

44 CHAPTER 9. INTRANSITIVITY

• 30% favor 60/40 weighting between social consciousness and fiscal conservatism

• 50% favor 50/50 weighting between social consciousness and fiscal conservatism

• 20% favor a 40/60 weighting between social consciousness and fiscal conservatism

While each voter may not assess the units of measure identically, the trend then becomes a single vector on whichthe consensus agrees is a preferred balance of candidate criteria.

9.6 References[1] Wolves do eat grass - see Engel, Cindy (2003). Wild Health: Lessons in Natural Wellness from the Animal Kingdom

(paperback ed.). Houghton Mifflin. p. 141. ISBN 0-618-34068-8..

[2] Guide to Logic, Relations II

[3] IntransitiveRelation

9.7 Further reading• Anand, P (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press..

Chapter 10

Material conditional

“Logical conditional” redirects here. For other related meanings, see Conditional statement.Not to be confused with material inference.Thematerial conditional (also known as "material implication", "material consequence", or simply "implication",

Venn diagram of A → B .If a member of the set described by this diagram (the red areas) is a member of A , it is in the intersection of A and B , and ittherefore is also in B .

"implies" or "conditional") is a logical connective (or a binary operator) that is often symbolized by a forward ar-row "→". The material conditional is used to form statements of the form "p→q" (termed a conditional statement)which is read as “if p then q” or “p only if q” and conventionally compared to the English construction “If...then...”.But unlike the English construction, the material conditional statement "p→q" does not specify a causal relationshipbetween p and q and is to be understood to mean “if p is true, then q is also true” such that the statement "p→q"is false only when p is true and q is false.[1] Intuitively, consider that a given p being true and q being false wouldprove an “if p is true, q is always also true” statement false, even when the “if p then q” does not represent a causal

45

46 CHAPTER 10. MATERIAL CONDITIONAL

relationship between p and q. Instead, the statement describes p and q as each only being true when the other istrue, and makes no claims that p causes q. However, note that such a general and informal way of thinking aboutthe material conditional is not always acceptable, as will be discussed. As such, the material conditional is also to bedistinguished from logical consequence.The material conditional is also symbolized using:

1. p ⊃ q (Although this symbol may be used for the superset symbol in set theory.);2. p ⇒ q (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for

material conditional.)

With respect to the material conditionals above, p is termed the antecedent, and q the consequent of the conditional.Conditional statements may be nested such that either or both of the antecedent or the consequent may themselvesbe conditional statements. In the example "(p→q) → (r→s)" both the antecedent and the consequent are conditionalstatements.In classical logic p→ q is logically equivalent to ¬(p ∧ ¬q) and by De Morgan’s Law logically equivalent to ¬p ∨ q.[2] Whereas, in minimal logic (and therefore also intuitionistic logic) p→ q only logically entails ¬(p∧¬q) ; and inintuitionistic logic (but not minimal logic) ¬p ∨ q entails p→ q .

10.1 Definitions of the material conditional

Logicians have many different views on the nature of material implication and approaches to explain its sense.[3]

10.1.1 As a truth function

In classical logic, the compound p→q is logically equivalent to the negative compound: not both p and not q. Thusthe compound p→q is false if and only if both p is true and q is false. By the same stroke, p→q is true if and only ifeither p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truthvalues of the compound p→q, whose truth value is entirely a function of the truth values of the components. Hence,this interpretation is called truth-functional. The compound p→q is logically equivalent also to ¬p∨q (either not p, orq (or both)), and to ¬q→¬p (if not q then not p). But it is not equivalent to ¬p→¬q, which is equivalent to q→p.

Truth table

The truth table associated with the material conditional p→q is identical to that of ¬p∨q and is also denoted by Cpq.It is as follows:It may also be useful to note that in Boolean algebra, true and false can be denoted as 1 and 0 respectively with anequivalent table.

10.1.2 As a formal connective

The material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying allthe classical inferences involving →, in particular the following characteristic rules:

1. Modus ponens;2. Conditional proof;3. Classical contraposition;4. Classical reductio ad absurdum.

Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identi-cal propositional forms in various logical systems, where somewhat different properties may be demonstrated. Forexample, in intuitionistic logic which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ qis not a propositional theorem, but the material conditional is used to define negation.

10.2. FORMAL PROPERTIES 47

10.2 Formal properties

When studying logic formally, the material conditional is distinguished from the semantic consequence relation |= .We say A |= B if every interpretation that makes A true also makes B true. However, there is a close relationshipbetween the two in most logics, including classical logic. For example, the following principles hold:

• If Γ |= ψ then ∅ |= (φ1 ∧ · · · ∧ φn → ψ) for some φ1, . . . , φn ∈ Γ . (This is a particular form of thededuction theorem. In words, it says that if Γ models ψ this means that ψ can be deduced just from somesubset of the theorems in Γ.)

• The converse of the above

• Both→ and |= are monotonic; i.e., if Γ |= ψ then ∆ ∪ Γ |= ψ , and if φ → ψ then (φ ∧ α) → ψ for any α,Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do theyhold in relevance logics.Other properties of implication (the following expressions are always true, for any logical values of variables):

• distributivity: (s→ (p→ q)) → ((s→ p) → (s→ q))

• transitivity: (a→ b) → ((b→ c) → (a→ c))

• reflexivity: a→ a

• totality: (a→ b) ∨ (b→ a)

• truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces atruth value of 'true' as a result of material implication.

• commutativity of antecedents: (a→ (b→ c)) ≡ (b→ (a→ c))

Note that a → (b → c) is logically equivalent to (a ∧ b) → c ; this property is sometimes called un/currying.Because of these properties, it is convenient to adopt a right-associative notation for → where a → b → c denotesa→ (b→ c) .Comparison of Boolean truth tables shows that a→ b is equivalent to ¬a ∨ b , and one is an equivalent replacementfor the other in classical logic. See material implication (rule of inference).

10.3 Philosophical problems with material conditional

Outside of mathematics, it is a matter of some controversy as to whether the truth function for material implica-tion provides an adequate treatment of conditional statements in English (a sentence in the indicative mood with aconditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., acounterfactual conditional).[4] That is to say, critics argue that in some non-mathematical cases, the truth value ofa compound statement, “if p then q", is not adequately determined by the truth values of p and q.[4] Examples ofnon-truth-functional statements include: "q because p", "p before q" and “it is possible that p".[4] “[Of] the sixteenpossible truth-functions of A and B, material implication is the only serious candidate. First, it is uncontroversial thatwhen A is true and B is false, “If A, B" is false. A basic rule of inference is modus ponens: from “If A, B" and A, wecan infer B. If it were possible to have A true, B false and “If A, B" true, this inference would be invalid. Second, it isuncontroversial that “If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false,false)… Non-truth-functional accounts agree that “If A, B" is false when A is true and B is false; and they agree thatthe conditional is sometimes true for the other three combinations of truth-values for the components; but they denythat the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when Aand B are both true, “If A, B" must be true. Some do not, demanding a further relation between the facts that A andthat B.”[4]

48 CHAPTER 10. MATERIAL CONDITIONAL

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was takenup enthusiastically by Russell (who called it “material implication”), Wittgenstein in the Tractatus, andthe logical positivists, and it is now found in every logic text. It is the first theory of conditionals whichstudents encounter. Typically, it does not strike students as obviously correct. It is logic’s first surprise.Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders.It is a strikingly simple theory: “If A, B" is false when A is true and B is false. In all other cases, “If A,B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "A ⊃ B" has, by stipulation, these truthconditions.

— Dorothy Edgington, The Stanford Encyclopedia of Philosophy, “Conditionals”[4]

The meaning of the material conditional can sometimes be used in the natural language English “if condition thenconsequence" construction (a kind of conditional sentence), where condition and consequence are to be filled withEnglish sentences. However, this construction also implies a “reasonable” connection between the condition (protasis)and consequence (apodosis) (see Connexive logic).The material conditional can yield some unexpected truths when expressed in natural language. For example, anymaterial conditional statement with a false antecedent is true (see vacuous truth). So the statement “if 2 is odd then 2is even” is true. Similarly, any material conditional with a true consequent is true. So the statement “if I have a pennyin my pocket then Paris is in France” is always true, regardless of whether or not there is a penny in my pocket. Theseproblems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense;that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and othernatural languages) are tempted to equivocate between the material conditional and the indicative conditional, or otherconditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that arigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwiseexpressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview ofsome the various analyses, formal and informal, of conditionals, see the “References” section below.

10.4 See also

10.4.1 Conditionals

• Counterfactual conditional

• Indicative conditional

• Corresponding conditional

• Strict conditional

10.5 References[1] Magnus, P.D (January 6, 2012). “forallx: An Introduction to Formal Logic” (PDF). Creative Commons. p. 25. Retrieved

28 May 2013.

[2] Teller, Paul (January 10, 1989). “A Modern Formal Logic Primer: Sentence Logic Volume 1” (PDF). Prentice Hall. p.54. Retrieved 28 May 2013.

[3] Clarke, Matthew C. (March 1996). “A Comparison of Techniques for Introducing Material Implication”. Cornell Univer-sity. Retrieved March 4, 2012.

[4] Edgington, Dorothy (2008). Edward N. Zalta, ed. “Conditionals”. The Stanford Encyclopedia of Philosophy (Winter 2008ed.).

10.6 Further reading• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, KluwerAcademic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

10.7. EXTERNAL LINKS 49

• Edgington, Dorothy (2001), “Conditionals”, in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic,Blackwell.

• Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, HarvardUniversity Press, Cambridge, MA.

• Stalnaker, Robert, “Indicative Conditionals”, Philosophia, 5 (1975): 269–286.

10.7 External links• Conditionals entry by Edgington, Dorothy in the Stanford Encyclopedia of Philosophy

Chapter 11

Matrilineality

Matrilineality is the tracing of descent through the female line. It may also correlate with a societal system in whicheach person is identified with their matriline – their mother’s lineage – and which can involve the inheritance ofproperty and/or titles. A matriline is a line of descent from a female ancestor to a descendant (of either sex) inwhich the individuals in all intervening generations are mothers – in other words, a “mother line”. In a matrilinealdescent system, an individual is considered to belong to the same descent group as her or his mother. This matrilinealdescent pattern is in contrast to the more common pattern of patrilineal descent from which a family name is usuallyderived. The matriline of historical nobility was also called her or his enatic or uterine ancestry (corresponding tothe patrilineal “agnatic” ancestry).In some traditional societies and cultures, membership in their groups was – and, in the following list, still is if shownin italics – inherited matrilineally. Examples include the Cherokee, Choctaw,Gitksan, Haida, Hopi, Iroquois, Lenape,Navajo and Tlingit of North America; the Minangkabau people of West Sumatra, Indonesia and Negeri Sembilan,Malaysia; the Nairs of Kerala and the Bunts and Billava of Karnataka in south India; the Khasi, Jaintia and Garoof Meghalaya in northeast India; Muslims and the Tamils in eastern Sri Lanka; the Mosuo of China; the Basques ofSpain and France; the Akan including the Ashanti of west Africa; virtually all groups across the so-called “matrilinealbelt” of Central Africa; the Tuaregs of west and north Africa; the Kuna people of Panama; the Serer of Senegal, theGambia and Mauritania; and most Jewish communities.

11.1 Early human kinship

In the late nineteenth century, almost all prehistorians and anthropologists believed, following Lewis H. Morgan'sinfluential book Ancient Society, that early human kinship was everywhere matrilineal.[1] This idea was taken up byFriedrich Engels in his book The Origin of the Family, Private Property and the State. The Morgan-Engels thesisthat humanity’s earliest domestic institution was not the family but the matrilineal clan soon became incorporated intocommunist orthodoxy. In reaction, most twentieth-century social anthropologists considered the theory of matrilinealpriority untenable,[2][3] although during the 1970s and 1980s, a range of feminist scholars often attempted to reviveit.[4]

In recent years, evolutionary biologists, geneticists and palaeoanthropologists have been reassessing the issues, manyciting genetic and other evidence that early human kinship may have been matrilineal after all.[5][6][7][8] One crucialpiece of indirect evidence has been genetic data suggesting that over thousands of years, women among sub-SaharanAfrican hunter-gatherers have chosen to reside postmaritally not with their husbands’ family but with their ownmotherand other natal kin.[9][10][11][12][13] Another line of argument is that when sisters and their mothers help each otherwith childcare, the descent line tends to be matrilineal rather than patrilineal.[14] Biological anthropologists are nowwidely agreed that cooperative childcare was a development crucial in making possible the evolution of the unusuallylarge human brain and characteristically human psychology.[15] Putting these two findings together generally supportsthe idea that early human kinship was likely to have been matrilineal.

50

11.2. MATRILINEAL SURNAME 51

11.2 Matrilineal surname

Main article: Matriname

Matrilineal surnames are names transmitted from mother to daughter, in contrast to the more familiar patrilinealsurnames transmitted from father to son, the pattern most common across the world today. See Family name for anin-depth treatment of patrilineal (father-line) family names or surnames. For clarity and for brevity, the scientificterms patrilineal surname and matrilineal surname are usually abbreviated as patriname and matriname.[16]

11.3 Cultural patterns

There appears to be some evidence for the presence of matrilineality in pre-Islamic Arabia, in a very limited numberof the Arabian peoples (first of all among the Amirites of Yemen, and among some strata of Nabateans in NorthernArabia);[17] on the other hand, there does not seem to be any reliable evidence for the presence of matrilineality inIslamic Arabia, although the Fatimid Caliphate claimed succession from the Islamic Prophet Mohammad via hisdaughter Fatima.A modern example from South Africa is the order of succession to the position of the Rain Queen in a culture ofmatrilineal primogeniture: Not only is dynastic descent reckoned through the female line, but only females are eligibleto inherit.

11.3.1 Clan names vs. surnames

Most of the example cultures in this article are based on (matrilineal) clans. Any clan might possibly contain from oneto several or many descent groups or family groups – i.e., any matrilineal clan might be descended from one or severalor many unrelated female ancestors. Also, each such descent group might have its own family name or surname, asone possible cultural pattern. The following two example cultures each follow a different pattern, however:Example 1. Members of the (matrilineal) clan culture Minangkabau do not even have a surname or family name, seethis culture’s own section below. In contrast, members do have a clan name, which is important in their lives althoughnot included in the member’s name. Instead, one’s name is just one’s given name.Example 2. Members of the (matrilineal) clan culture Akan, see its own section below, also do not have matrilinealsurnames and likewise their important clan name is not included in their name. However, members’ names do com-monly include second names which are called surnames but which are not routinely passed down from either fatheror mother to all their children as a family name.[18]

Note well that if a culture did include one’s clan name in one’s name and routinely handed it down to all children inthe descent group then it would automatically be the family name or surname for one’s descent group (as well as forall other descent groups in one’s clan).

11.3.2 Care of children

While a mother normally takes care of her own children in all cultures, in some matrilineal cultures an “uncle-father”will take care of his nieces and nephews instead: in other words social fathers here are uncles. There is a disconnectionbetween the role of father and genitor (who in the general case may be unknown anyway). In such matrilineal cultures,especially where residence is also matrilocal, a man will exercise guardianship rights not over the children he fathersbut exclusively over his sisters’ children, who are viewed as 'his own flesh'. These children’s biological father – unlikean uncle who is their mother’s brother and thus their caregiver – is in some sense a 'stranger' to them, even whenaffectionate and emotionally close.[19] This may be true for the traditional Akan culture below, for example.According to Steven Pinker, attributing to Kristen Hawkes, among foraging groups matrilocal societies are less likelyto commit female infanticide than are patrilocal societies.[20]

52 CHAPTER 11. MATRILINEALITY

11.3.3 A feminist and patriarchal relationship

According to Kanchana N. Ruwanpura, “Sri Lanka .... is highly regarded even among feminist economists forthe relatively favourable position of its women, reflected [in part] in the ... matrilineal and bilateral inheritancepatterns and property rights”,[21][22][lower-alpha 1] although she argued for caution in interpreting Sri Lanka’s “gender-based ... achievements and/or matrilineal communities”[23] and wrote that matrilineality coexists with “patriarchalstructures and ideologies”,[24] which are influenced by “the main religious traditions, Buddhism, Hinduism, andIslam”,[25][lower-alpha 2][lower-alpha 3][lower-alpha 4] even while “repressive cultural practices ... [may not be] pervasive”[26]and “Sri Lankan women are surely not constrained by classical patriarchy”.[27][28] She wrote that “feminists haveclaimed that Sri Lankan women are relatively well positioned in the South Asian region”.[27][28][lower-alpha 5] She alsowrote that, on the other hand, feminists have criticized a romanticized view of women’s lives in Sri Lanka and saidthat, in accordance with “village practices and folklore[,] ... young women raped (usually by a man) are married-off/required to cohabit with the rapists!"[29] She wrote that “female-heads have no legal recourse” from “patriarchalinterests”.[30] According to her, “some female heads possessed” “feminist consciousness”[31] and “the economic wel-fare of female-heads depends upon networks that mediate the patriarchal-ideological nexus, although the distinctionsand similarities of the ethnically-based experiences of female-heads provide a sound basis for a coherent feministperspective.”[32][33] She wrote that “in many cases female-heads are not vociferous feminists ... but rather 'victims’of patriarchal relations and structures that place them in precarious positions.... [while] they have held their ground... [and] provided for their children”.[34] She wrote that in a “shift from economic to non-economic forms of support.... feminists would no doubt wish to observe a significant shift in attitudes reflecting progressive and accommodatingvalues towards female-heads, [but] this is not taking place on any scale in these communities.”[35]

11.4 Matrilineality in specific ethnic groups

11.4.1 In America

Lenape

Main article: Lenape

Occupied for 10,000 years by Native Americans, the land that would become New Jersey was overseen by clans of theLenape or Lenni Lenape or Delaware, who farmed, fished, and hunted upon it. The pattern of their culture was thatof a matrilineal agricultural and mobile hunting society that was sustained with fixed, but not permanent, settlementsin their clan territories.Villages were established and relocated as the clans farmed new sections of the land when soil fertility lessened andwhen they moved among their fishing and hunting grounds by seasons. The area was claimed as a part of the DutchNew Netherland province dating from 1614, where active trading in furs took advantage of the natural pass west, butthe Lenape prevented permanent settlement beyond what is now Jersey City.“Early Europeans who first wrote about these Indians found matrilineal social organization to be unfamiliar andperplexing. ... As a result, the early records are full of 'clues’ about early Lenape society, but were usually written byobservers who did not fully understand what they were seeing.”[36]

Hopi

Main article: Hopi people

The Hopi (in what is now the Hopi Reservation in northeastern Arizona), according to Alice Schlegel, had as its“gender ideology ... one of female superiority, and it operated within a social actuality of sexual equality.”[37] Ac-cording to LeBow (based on Schlegel’s work), in the Hopi, “gender roles ... are egalitarian .... [and] [n]either sex isinferior.”[38] LeBow concluded that Hopi women “participate fully in ... political decision-making.”[39] According toSchlegel, “the Hopi no longer live as they are described here”[40] and “the attitude of female superiority is fading”.[40]Schlegel said the Hopi “were and still are matrilinial”[41] and “the household ... was matrilocal”.[41]

Schlegel explains why there was female superiority as that the Hopi believed in “life as the highest good ... [with] thefemale principle ... activated in women and in Mother Earth ... as its source”[42] and that the Hopi “were not in a state

11.4. MATRILINEALITY IN SPECIFIC ETHNIC GROUPS 53

of continual war with equally matched neighbors”[43] and “had no standing army”[43] so that “the Hopi lacked the spurto masculine superiority”[43] and, within that, as that women were central to institutions of clan and household andpredominated “within the economic and social systems (in contrast to male predominance within the political andceremonial systems)",[43] the Clan Mother, for example, being empowered to overturn land distribution by men if shefelt it was unfair,[42] since there was no “countervailing ... strongly centralized, male-centered political structure”.[42]

Iroquois

Main article: Iroquois

The Iroquois Confederacy or League, combining five to six Native American Haudenosaunee nations or tribes beforethe U.S. became a nation, operated by The Great Binding Law of Peace, a constitution by which women retainedmatrilineal-rights and participated in the League’s political decision-making, including deciding whether to proceed towar,[44] through what may have been a matriarchy[45] or "'gyneocracy'".[46] The dates of this constitution’s operationare unknown: the League was formed in approximately 1000–1450, but the constitution was oral until written inabout 1880.[47] The League still exists.

11.4.2 In Africa

Akan

Main articles: Akan people and Abusua

Some 20 million Akan live in Africa, particularly in Ghana and Côte d'Ivoire. (See as well their subgroup, theAshanti, also calledAsante.) Many but not all of theAkan still (2001)[48] practice their traditionalmatrilineal customs,living in their traditional extended family households, as follows. The traditional Akan economic, political and socialorganization is based on matrilineal lineages, which are the basis of inheritance and succession. A lineage is definedas all those related by matrilineal descent from a particular ancestress. Several lineages are grouped into a politicalunit headed by a chief and a council of elders, each of whom is the elected head of a lineage — which itself mayinclude multiple extended-family households. Public offices are thus vested in the lineage, as are land tenure andother lineage property. In other words, lineage property is inherited only by matrilineal kin.[48][49]

Each lineage controls the lineage land farmed by its members, functions together in the veneration of its ancestors,supervises marriages of its members, and settles internal disputes among its members.[50]

The political units above are likewise grouped into eight larger groups called abusua (similar to clans), namedAduana,Agona, Asakyiri, Asenie, Asona, Bretuo, Ekuona and Oyoko. The members of each abusua are united by their beliefthat they are all descended from the same ancient ancestress. Marriage between members of the same abusua isforbidden. One inherits or is a lifelong member of the lineage, the political unit, and the abusua of one’s mother,regardless of one’s gender and/or marriage. Note that members and their spouses thus belong to different abusuas,mother and children living and working in one household and their husband/father living and working in a differenthousehold.[48][49]

According to this source[51] of further information about the Akan, “A man is strongly related to his mother’s brother(wɔfa) but only weakly related to his father’s brother. This must be viewed in the context of a polygamous societyin which the mother/child bond is likely to be much stronger than the father/child bond. As a result, in inheritance,a man’s nephew (sister’s son) will have priority over his own son. Uncle-nephew relationships therefore assume adominant position.”[51]

“The principles governing inheritance stress sex, generation and age — that is to say, men come before women andseniors before juniors.” When a woman’s brothers are available, a consideration of generational seniority stipulatesthat the line of brothers be exhausted before the right to inherit lineage property passes down to the next seniorgenealogical generation of sisters’ sons. Finally, “it is when all possible male heirs have been exhausted that thefemales” may inherit.[51]

Certain other aspects of the Akan culture are determined patrilineally rather than matrilineally. There are 12 patrilin-eal Ntoro (which means spirit) groups, and everyone belongs to their father’s Ntoro group but not to his (matrilineal)family lineage and abusua. Each patrilineal Ntoro group has its own surnames,[52] taboos, ritual purifications, andetiquette.[49]

54 CHAPTER 11. MATRILINEALITY

A recent (2001) book[48] provides this update on the Akan: Some families are changing from the above abusuastructure to the nuclear family.[53] Housing, childcare, education, daily work, and elder care etc. are then handled bythat individual family rather than by the abusua or clan, especially in the city.[54] The above taboo on marriage withinone’s abusua is sometimes ignored, but “clan membership” is still important,[53] with many people still living in theabusua framework presented above.[48]

Comoros

The Comorians are matrilineal and inheritance is passed to the daughters. The culture is similar to that of theMinangkabau in Indonesia and Malaysia, possibly due to the influence of the Malayo-Polynesian people who mi-grated there two thousand years ago. As in the case of the Minangkabau, the society is patriarchal.

Tuareg

Main article: Tuareg people

The Tuareg (Arabic:طوارق, sometimes spelled Touareg in French, or Twareg in English) are a Berber ethnic groupfound across several nations in north Africa, including Niger, Mali and Algeria. The Tuareg are clan-based,[55] andare (still, in 2007) “largely matrilineal”.[55][56][57] The Tuareg are Muslim, but mixed with a “heavy dose” of theirpre-existing beliefs including matrilineality.[55][57]

Tuareg women enjoy high status within their society, compared with their Arab counterparts and with other Berbertribes: Tuareg social status is transmitted through women, with residence often matrilocal.[56] Most women couldread and write, while most men were illiterate, concerning themselves mainly with herding livestock and other maleactivities.[56] The livestock and other movable property were owned by the women, whereas personal property isowned and inherited regardless of gender.[56] In contrast to most other Muslim cultural groups, men wear veils butwomen do not.[55][57] This custom is discussed in more detail in the Tuareg article’s clothing section, which mentionsit may be the protection needed against the blowing sand while traversing the Sahara desert.[58]

Serer

Main article: Serer maternal clans

The Serer people of Senegal, the Gambia and Mauritania are patrilineal (simanGol in Serer language[59]) as well asmatrilineal (tim[60] ). There are several Serer matriclans and matriarchs. Some of these matriarchs include FatimBeye (1335) and Ndoye Demba (1367) — matriarchs of the Joos matriclan which also became a dynasty in Waalo(Senegal). Somematriclans or maternal clans form part of Serer medieval and dynastic history, such as the Guelowars.The most revered clans tend to be rather ancient and form part of Serer ancient history. These proto-Serer clans holdgreat significance in Serer religion and mythology. Some of these proto-Serer matriclans include the Cegandum andKagaw, whose historical account is enshrined in Serer religion, mythology and traditions.[61]

In Serer culture, inheritance is both matrilineal and patrilineal.[62] It all depends on the asset being inherited — i.e.whether the asset is a paternal asset — requiring paternal inheritance (kucarla[62] ) or a maternal asset — requiringmaternal inheritance (den yaay[60] or ƭeen yaay[62] ). The actual handling of these maternal assets (such as jewelry,land, livestock, equipment or furniture, etc.) is discussed in the subsection Role of the Tokoor of one of the above-listed main articles.

Guanches

Main article: Guanches

The Berber inhabitants of Gran Canaria had developed a matrilineal society by the time the island was conquered bythe Spanish. The power of the Guanarteme (king) was based on the link with Atidamana, the legendary Queen thatunified the island some centuries before the conquest. The women of the family passed down the legitimacy of themonarchy, to the point that when the island surrendered, the natives handed over a young girl to the conquistadors,since she was the sole inheritor of the royal legitimacy.

11.4. MATRILINEALITY IN SPECIFIC ETHNIC GROUPS 55

11.4.3 In Asia

Sri Lanka

On Kerala, see the India section.

Matrilineality among the Muslims and Tamils in the Eastern Province of Sri Lanka arrived from Kerala, India viaMuslim traders before 1200 CE.[63][64][65]Matrilineality here includes kinship and social organization, inheritanceand property rights.[66][28][22]For example, “the mother’s dowry property and/or house is passed on to the eldestdaughter.”[67][68]The Sinhalese people are the third ethnic group in eastern Sri Lanka,[69] and have a kinship systemwhich is “intermediate” between that of matrilineality and that of patrilineality,[70][71] along with “bilateral inheri-tance” (in some sense intermediate between matrilineal and patrilineal inheritance).[72][28] While the first two groupsshare the Tamil language, the third group speaks the Sinhalese language. The Tamils largely identify with Hinduism,the Sinhalese being primarily Buddhist.[73]

Patriarchal social structures apply to all of Sri Lanka, but in the Eastern Province are mixed with the matrilinealfeatures summarized above.[74] This resulting mixture is discussed elsewhere in this article, in the earlier subsection“A feminist and patriarchal relationship”.

Minangkabau people, in Indonesia

Main article: Minangkabau people

In the Minangkabau matrilineal clan culture in Indonesia, a person’s clan name is important in their marriage andtheir other cultural-related events.[75][76][77] Two totally unrelated people who share the same clan name can neverbe married because they are considered to be from the same clan mother (unless they come from distant villages).Likewise, when Minangs meet total strangers who share the same clan name, anywhere in Indonesia, they couldtheoretically expect to feel that they are distant relatives.[78] Minang people do not have a family name or surname;neither is one’s important clan name included in one’s name; instead one’s given name is the only name one has.[79]

The Minangs are one of the world’s largest matrilineal societies/cultures/ethnic groups, with a population of 4 millionin their home province West Sumatra in Indonesia and about 4 million elsewhere, mostly in Indonesia. The Minangpeople are well-known within their country for their tradition of matrilineality and for their “dedication to Islam” —despite Islam being “supposedly patrilineal”.[75] This well-known accommodation, between their traditional complexof customs, called adat, and their religion, was actually worked out to help end the Minangkabau 1821-37 PadriWar.[75] This source is available online.[75]

As further described in the same online source, their (matrilineal) adat and their Islam religion each help the otherto avoid the extremes of some modern global trends: Their strong belief in and practice of adat helps their Islamreligion to not adopt a “simplistic anti-Western” version of Islam, while their strong belief in and practice of bothIslam and adat helps the Minangs to limit or avoid some undesired effects of modern global capitalism.[75]

The Minangkabau are a prime example of a matrilineal culture with female inheritance.

China

Originally, Chinese surnames were derived matrilineally,[80] although by the time of the Shang Dynasty (1600 to 1046BCE) they had become patrilineal.[80][81] The Chinese character for “surname” ( ) still contains a female radical ( ),suggesting its matrilineal etymology.Archaeological data supports the theory that during the Neolithic period (7000 to 2000 BCE) in China, Chinesematrilineal clans evolved into the usual patrilineal families by passing through a transitional patrilineal clan phase.[81]Evidence includes some “richly furnished” tombs for young women in the early Neolithic Yangshao culture, whosemultiple other collective burials imply a matrilineal clan culture.[81] Toward the late Neolithic period, when buri-als were apparently of couples, “a reflection of patriarchy”, an increasing elaboration of presumed chiefs’ burials isreported.[81]

Relatively isolated ethnic minorities such as the Mosuo (Na) in southwestern China are highly matrilineal, and usematrilineal family names, i.e., matrinames. (See the General practice section of the Mosuo article.)

56 CHAPTER 11. MATRILINEALITY

Cambodia and Việt Nam

Most ethnic groups classified as "(Montagnards , Malayo-Polynesian and Austroasian)" are matrilineal.[82]

OnNorth Vietnam, according to Alessandra Chiricosta, the legend of Âu Cơ is said to be evidence of “the presence ofan original 'matriarchy' ... and [it] led to the double kinship system, which developed there .... [and which] combinedmatrilineal and patrilineal patterns of family structure and assigned equal importance to both lines.”[83][lower-alpha 6]

India

Main articles: Marumakkathayam and Aliyasantana

Of communities recognized in the national Constitution as Scheduled Tribes, “some ... [are]matriarchal andmatrilineal”[84]“and thus have been known to be more egalitarian.”[85] Several communities in South India practiced matrilineality,especially the Tiyyas[86] and Nair[87][88] (or Nayar) in the state of Kerala, and the Bunts and Billava in the statesof Kerala and Karnataka. The system of inheritance was known as Marumakkathayam in the Nair community orAliyasantana in the Bunt and the Billava community, and both communities were subdivided into clans. This systemwas exceptional in the sense that it was one of the few traditional systems in western historical records of India thatgave women some liberty and the right to property.In the matrilineal system, the family lived together in a tharavadu which was composed of a mother, her brothersand younger sisters, and her children. The oldest male member was known as the karanavar and was the head ofthe household, managing the family estate. Lineage was traced through the mother, and the children belonged to themother’s family. All family property was jointly owned. In the event of a partition, the shares of the children wereclubbed with that of the mother. The karanavar’s property was inherited by his sisters’ sons rather than his own sons.(For further information see the articles Nair and Bunts and Billava.) Amitav Ghosh has stated that, although therewere numerous other matrilineal succession systems in communities of the south Indian coast, the Nairs “achievedan unparalleled eminence in the anthropological literature on matrilineality”.[89]

The Marumakkathayam system is not very common in Kerala and Karnataka these days for many reasons. Societyhas become much more cosmopolitan and modern. Men seek jobs away from their hometown and take their wivesand children along with them. In this scenario, a joint-family system is no longer viable. But conceivably, there mightstill be a few tharavads that pay homage to this system.

Malaysia

A culture similar to the Minangkabau’s, above, is present in Negeri Sembilan, Malaysia, ever since West Sumatranssettled there in the 14th century.

11.4.4 In Oceania

Some oceanic societies, such as the Marshallese, the Trobrianders,[90] the Palauans, the Yapese and the Siuai, arecharacterized by matrilineal descent. The sister’s sons or the brothers of the decedent are commonly the successorsin these societies.

11.5 Matrilineal identification within Judaism

Main article: Matrilineality in Judaism

Matrilineality in Judaism is the view that people born of a Jewish mother are themselves Jewish.[91] The conferringof Jewish status through matrilineality is not stated explicitly in the Torah, though Jewish oral tradition maintainsthis was always the rule, and adduces indirect textual evidence. In biblical times, many Israelites married foreignwomen, and their children appear to have been accepted as Israelite without question; the Talmud understands thatthe women in question converted to Judaism. (See the above-mentioned main article for more information. Thissection, “Matrilineal identification within Judaism”, is simply a shortened version of that main article.)

11.6. IN MYTHOLOGY 57

In the Hellenistic period, some evidence indicates that the offspring of intermarriages between Jewish men and non-Jewish women were considered Jewish;[92] as is usual in prerabbinic texts, there is no mention of conversion on thepart of the Gentile spouse. On the other hand, Philo of Alexandria calls the child of a Jew and a non-Jew a nothos(bastard), regardless of whether the non-Jewish parent is the father or the mother.[93]

The Mishnah (Kiddushin 3:12) states that, to be a Jew, one must be either the child of a Jewish mother or a convertto Judaism. The Talmud (Kiddushin 68b) derives this law from the Torah. The relevant Torah passage (Deut. 7:3-4)reads: “Thy daughter thou shalt not give to his son, nor shalt thou take his daughter to thy son. For they will turnaway thy son from following me, that they may serve other gods.”With the emergence of Jewish denominations and the modern rise in Jewish intermarriage in the 20th century, ques-tions about the law of matrilineal descent have assumed greater importance to the Jewish community at large. Theheterogeneous Jewish community is divided on the issue of "Who is a Jew?" via descent; matrilineal descent stillis the rule within Orthodox Judaism, which also holds that anyone with a Jewish mother has an irrevocable Jewishstatus, and matrilineal descent is the norm in the Conservative movement. Since 1983, Reform Judaism in the UnitedStates of America officially adopted a bilineal policy: one is a Jew if either of one’s parents is Jewish, provided thateither (a) one is raised as a Jew, by Reform standards, or (b) one engages in an appropriate act of public identification,formalizing a practice that had been common in Reform synagogues for at least a generation. Karaite Judaism, whichincludes only the Tanakh in its canon, interprets the Torah to indicate that Jewishness passes exclusively through thefather’s line. See the above-mentioned main article Matrilineality in Judaism for more-complete context and sources.

11.6 In mythology

While Indo-European peoples mainly were patriarchal and patrilineal, certain ancient myths have been argued toexpose ancient traces of matrilineal customs that existed before historical records.The ancient historian Herodotus is cited by Robert Graves in his translations of Greek myths as attesting that theLycians[94][95] of their times “still reckoned” by matrilineal descent, or were matrilineal, as were the Carians.[96]

In Greek mythology, while the royal function was a male privilege, power devolution often came through women,and the future king inherited power through marrying the queen heiress. This is illustrated in the Homeric mythswhere all the noblest men in Greece vie for the hand of Helen (and the throne of Sparta), as well as the Oedipiancycle where Oedipus weds the recently widowed queen at the same time he assumes the Theban kingship.This trend also is evident in many Celtic myths, such as the (Welsh) mabinogi stories of Culhwch and Olwen, or the(Irish) Ulster Cycle, most notably the key facts to the Cúchulainn cycle that Cúchulainn gets his final secret trainingwith a warrior woman, Scáthach, and becomes lover both to her and her daughter; and the root of the Táin BóCuailnge, that while Ailill may wear the crown of Connacht, it is his wife Medb who is the real power, and she needsto affirm her equality to her husband by owning chattels as great as he does.A number of other Breton stories also illustrate the motif. Even the King Arthur legends have been interpreted inthis light by some. For example the Round Table, both as a piece of furniture and as concerns the majority of knightsbelonging to it, was a gift to Arthur from Guinevere's father Leodegrance.Arguments also have been made that matrilineality lay behind various fairy tale plots which may contain the vestigesof folk traditions not recorded.For instance, the widespread motif of a father who wishes to marry his own daughter—appearing in such tales asAllerleirauh, Donkeyskin, The King who Wished to Marry His Daughter, and The She-Bear—has been explained ashis wish to prolong his reign, which he would lose after his wife’s death to his son-in-law.[97] More mildly, the hostilityof kings to their daughter’s suitors is explained by hostility to their successors. In such tales as The Three May Peaches,Jesper Who Herded the Hares, or The Griffin, kings set dangerous tasks in an attempt to prevent the marriage.[98]

Fairy tales with hostility between the mother-in-law and the heroine—such as Mary’s Child, The Six Swans, andPerrault’s Sleeping Beauty—have been held to reflect a transition between a matrilineal society, where a man’s loyaltywas to his mother, and a patrilineal one, where his wife could claim it, although this interpretation is predicated onsuch a transition being a normal development in societies.[99]

11.7 See also• Blanca de La Cerda y Lara, matrilineal ancestor (1317–1347) of Queen Victoria and other European royalty.

58 CHAPTER 11. MATRILINEALITY

• Early human kinship was matrilineal.

• Ruth Bré, advocate for matrilineality

• List of matrilineal or matrilocal societies

• Married and maiden names

• Mater semper certa est, “the mother is always certain” – until 1978 and in vitro pregnancies.

• Matrifocal family

11.8 Notes[1] Feminist economics, the study of economics that attempts to overcome androcentrism and focus on women

[2] Buddhism, a religion indigenous to the Indian subcontinent

[3] Hinduism, the dominant religion of the Indian subcontinent

[4] Islam, a monotheistic and Abrahamic religion articulated by the Qur'an

[5] South Asia, the southern region of the Asian continent

[6] Patrilineal, belonging to the father’s lineage, generally for inheritance

11.9 References[1] Murdock, G. P. 1949. Social Structure. London and New York: Macmillan, p. 185.

[2] Malinowski, B. 1956. Marriage: Past and Present. A debate between Robert Briffault and Bronislaw Malinowski, ed. M. F.Ashley Montagu. Boston: Porter Sargent.

[3] Harris, M. 1969. The Rise of Anthropological Theory. London: Routledge, p. 305.

[4] Leacock, E. B. 1981. Myths of Male Dominance. Collected articles on women cross-culturally. New York: Monthly ReviewPress.

[5] Hrdy, S. B. 2009. Mothers and others. The evolutionary origins of mutual understanding. London and Cambridge, MA:Belknap Press of Harvard University Press.

[6] Knight, C. 2008. Early human kinship was matrilineal. In N. J. Allen, H. Callan, R. Dunbar and W. James (eds.), EarlyHuman Kinship. Oxford: Blackwell, pp. 61-82.

[7] Opie, K. and C. Power, 2009. Grandmothering and Female Coalitions. A basis for matrilineal priority? In N. J. Allen, H.Callan, R. Dunbar and W. James (eds.), Early Human Kinship. Oxford: Blackwell, pp. 168-186.

[8] Chris Knight, 2012. Engels was Right: Early Human Kinship was Matriliineal. .

[9] G Destro-Bisol, with F Donati, V Coia, I Boschi, F Verginelli, A Caglia, S Tofanelli, G Spednini and C Capelli, ‘Variationof female and male lineages in sub-Saharan populations: the importance of sociocultural factors’ Molecular Biology andEvolution 21(9) 2004, pp1673-82.

[10] Verdu, P., Becker, N., Froment, A., Georges, M., Grugni, V., Quintana-Murci, L., Hombert, J-M., Van der Veen, L.,Le Bomin, S., Bahuchet, S., Heyer, E. ,Austerlitz, F. (2013) Sociocultural behavior, sex-biased admixture and effectivepopulation sizes in Central African Pygmies and non-Pygmies Mol Biol Evol, first published online January 7, 2013 doi:10.1093/molbev/mss328

[11] Schlebusch, C.M. (2010) Genetic variation in Khoisan-speaking populations from southern Africa. Dissertation, Universityof Witwatersrand this is available online, see pages following p.68, Fig 3.18 and p.180-81, fig 4.23 and p.243, p.287

[12] Hammer MF, Karafet TM, Redd AJ, Jarjanazi H, Santachiara-Benerecetti S, Soodyall H and Zegura SL (2001a). Hierar-chical patterns of global human Y-chromosome diversity. Mol Biol Evol 18: 1189-203

11.9. REFERENCES 59

[13] Wood ET, Stover DA, Ehret C, Destro-Bisol G, Spedini G, McLeod H, Louie L, et al., (2005). Contrasting patterns ofY chromosome and mtDNA variation in Africa: evidence for sex-biased demographic processes. Eur J Hum Genet 13:867-76

[14] Wu J-J, He Q-Q, Deng L-L, Wang S-C, Mace R, Ji T, Tao Y. 2013 Communal breeding promotes a matrilineal socialsystem where husband and wife live apart. Proc R Soc B 280: 20130010. http://dx.doi.org/10.1098/rspb.2013.0010

[15] Burkart, J. M., S. B. Hrdy, and C. P. van Schaik. 2009. Cooperative breeding and human cognitive evolution. EvolutionaryAnthropology 18:175– 186.

[16] Sykes, Bryan (2001). The Seven Daughters of Eve. W.W. Norton. ISBN 0-393-02018-5; pp. 291-2. Bryan Sykes uses“matriname” and states that women adding their own matriname to men’s patriname (or “surname” as Sykes calls it) wouldreally help in future genealogy work and historical record searches. Sykes also states (p. 292) that a woman’s matrinamewill be handed down with her mtDNA, the main topic of his book.

[17] Korotayev, A. V. (1995), “Were There Any Truly Matrilineal Lineages in the Arabian Peninsula?" Proceedings of theSeminar for Arabian Studies 25 (1995); pp. 83-98.

[18] deWitte, Marleen (2001). Long live the dead!: changing funeral celebrations in Asante, Ghana. Published by Het Spinhuis.ISBN 90-5260-003-1, p. 55. Readers may verify this (i.e., that surnames are not passed down as a family name) by inspect-ing an actual family tree on p. 55 via Google Books at http://books.google.com/books?id=Fmf5UqZzbvoC&printsec=frontcover&dq=de+Witte,+Marleen&hl=en&sa=X&ei=L_ihT5_jM4Tg2gX7_-jaCA&ved=0CDYQ6AEwAQ#v=onepage&q=%22Adwoa%20Dufie%22&f=false .

[19] Schneider, D. M. 1961. The distinctive features of matrilineal descent groups. Introduction. In Schneider, D. M. and K.Gough (eds) Matrilineal Kinship. Berkeley: University of California Press, pp. 1-29.

[20] Pinker, Steven, The Better Angels of Our Nature: Why Violence Has Declined (N.Y.: Viking, hardback 2011 (ISBN 978-0-670-02295-3)), p. 421 (author prof. psychology, Harvard Univ.).

[21] Ruwanpura, Kanchana N. (2006). Matrilineal Communities, Patriarchal Realities: A Feminist Nirvana Uncovered (AnnArbor: University of Michigan Press, paperback (ISBN 978-0-472-06977-4)), p. 1 (fieldwork in 1998–'99 during “ethnicconflict”, per p. 45) (author asst. prof., Hobart & William Smith Colleges).

[22] Humphries, Jane (1993). “Gender Inequality and Economic Development,” in Dieter Bos (ed) Economics in a ChangingWorld, Volume 3: Public Policy and Economic Organization. New York: St. Martin’s Press; pp. 218-33.

[23] Ruwanpura, 2006, p. 3 and see pp. 10 (caution), 182 (mootness & not negating), & 186 (only relatively favorable &patriarchal relations).

[24] Ruwanpura, 2006, p. 10 and see p. 6 (“prevalence of patriarchal structures and ideologies”).

[25] Ruwanpura, 2006, pp. 4–5.

[26] Ruwanpura, 2006, p. 182.

[27] Ruwanpura, 2006, p. 4.

[28] Agarwal, Bina (1996). A Field of One’s Own: Gender and Land Rights in South Asia. New Delhi: Cambridge UniversityPress. (First edition was 1994.)

[29] Ruwanpura, 2006, p. 76 n. 7.

[30] Ruwanpura, 2006, p. 182 (both quotations).

[31] Ruwanpura, 2006, p. 142 (both quotations).

[32] Ruwanpura, 2006, pp. 145–146.

[33] Hirschon, Renee (1984)."Introduction: Property, Power and Gender Relations,” pp. 1-22 in Renee Hirschon (ed.) Womenand Property--Women as Property. New York: St. Martin’s Press; p. 5.

[34] Ruwanpura, 2006, p. 37.

[35] Ruwanpura, 2006, p. 159.

[36] This quote is from Lenni-Lenape's Society section.

[37] Schlegel, Alice, Hopi Gender Ideology of Female Superiority, in Quarterly Journal of Ideology: “A Critique of the Conven-tional Wisdom”, vol. VIII, no. 4, 1984, p. 44 and see pp. 44–52 (essay based partly on “seventeen years of fieldworkamong the Hopi”, per p. 44 n. 1) (author of Dep't of Anthropology, Univ. of Ariz., Tucson).

60 CHAPTER 11. MATRILINEALITY

[38] LeBow, Diana, Rethinking Matriliny Among the Hopi, op. cit., p. [8].

[39] LeBow, Diana, Rethinking Matriliny Among the Hopi, op. cit., p. 18.

[40] Schlegel, Alice, Hopi Gender Ideology of Female Superiority, op. cit., p. 44 n. 1.

[41] Schlegel, Alice, Hopi Gender Ideology of Female Superiority, op. cit., p. 45.

[42] Schlegel, Alice, Hopi Gender Ideology of Female Superiority, op. cit., p. 50.

[43] Schlegel, Alice, Hopi Gender Ideology of Female Superiority, op. cit., p. 49.

[44] Jacobs, Renée E., Iroquois Great Law of Peace and the United States Constitution: How the Founding Fathers Ignored theClan Mothers, in American Indian Law Review, vol. 16, no. 2, pp. 497–531, esp. pp. 498–509 (© author 1991).

[45] Jacobs, Renée, Iroquois Great Law of Peace and the United States Constitution, in American Indian Law Review, op. cit.,pp. 506–507.

[46] Jacobs, Renée, Iroquois Great Law of Peace and the United States Constitution, in American Indian Law Review, op. cit.,p. 505 & p. 506 n. 38, quoting Carr, L., The Social and Political Position of Women Among the Huron-Iroquois Tribes,Report of the Peabody Museum of American Archaeology, p. 223 (1884).

[47] Jacobs, Renée, Iroquois Great Law of Peace and the United States Constitution, in American Indian Law Review, op. cit., p.498 & n. 6.

[48] deWitte, Marleen (2001). Long live the dead!: changing funeral celebrations in Asante, Ghana. Published by Het Spinhuis.ISBN 90-5260-003-1. All de Witte (2001) pages referenced below, and many more pages, are available online via GoogleBooks at http://books.google.com/books?id=Fmf5UqZzbvoC&pg=PA52&dq=Abusua&hl=en&ei=iTRaTdj1N8P7lweKm7XfDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwAA#v=onepage&q=Abusua&f=false .

[49] Busia, Kofi Abrefa (1970). Encyclopædia Britannica, 1970. William Benton, publisher, The University of Chicago. ISBN0-85229-135-3, Vol. 1, p. 477. (This Akan article was written by Kofi Abrefa Busia, formerly professor of Sociology andCulture of Africa at the University of Leiden, Netherlands.)

[50] Owusu-Ansah, David (November 1994). http://lcweb2.loc.gov/cgi-bin/query/r?frd/cstdy:@field%28DOCID+gh0048%29, “Ghana: The Akan Group”. This source, “Ghana”, is one of the Country Studies available from the US Library ofCongress. Archived by WebCite® at http://www.webcitation.org/61M7J7JwT on 31Aug11.

[51] ashanti.com.au (before 2010). http://ashanti.com.au/pb/wp_8078438f.html, “Ashanti Home Page: The Ashanti Familyunit” Archived at WebCite http://www.webcitation.org/5xVwnX0ie on 28 March 2011.

[52] de Witte (2001), p. 55 shows such surnames in a family tree, which provides a useful example of names.

[53] de Witte (2001), p. 53.

[54] de Witte (2001), p. 73.

[55] Haven, Cynthia (23 May 07). http://news.stanford.edu/pr/2007/pr-tuareg-052307.html, “New exhibition highlights the'artful' Tuareg of the Sahara,” Stanford University. Archived at WebCite http://www.webcitation.org/5xd6eNYUc on1Apr11.

[56] Spain, Daphne (1992). Gendered Spaces. University of North Carolina Press. ISBN 0-8078-2012-1; p. 57.

[57] Murphy, Robert F. (April 1966). Untitled review of a 1963 major ethnographic study of the Tuareg. American Anthropolo-gist, New Series, 68 (1966), No. 2, 554-556. (The main part of this review is available online at www.jstor.org/pss/669389,a JSTOR-archive Permalink.)

[58] Bradshaw Foundation (2007 or later). http://www.bradshawfoundation.com/tuareg/index.php, “The Tuareg of the Sa-hara”. Archived by WebCite® at http://www.webcitation.org/5zT80I8SJ on 15Jun2011.

[59] (French) Kalis, Simone, “Médecine traditionnelle religion et divination chez les Seereer Sine du Senegal", La connaissancede la nuit, L'Harmattan (1997), p 299, ISBN 2-7384-5196-9

[60] Dupire, Marguerite, “Sagesse sereer : Essais sur la pensée sereer ndut, KARTHALA Editions (1994). For tim and denyaay (see p. 116). The book also deals in depth about the Serer matriclans and means of succession through the matrilinealline. See also pages : 38, 95-99, 104, 119-20, 123, 160, 172-4 (French) ISBN 2865374874 (Retrieved : 4 August 2012)

[61] (French)Gravrand, Henry, “LaCivilisation Sereer - Cosaan”, p 200, Nouvelles Editions africaines (1983), ISBN2723608778

[62] (French) Becker, Charles: “Vestiges historiques, trémoins matériels du passé clans les pays sereer”, Dakar (1993), CNRS- ORS TO M. Excerpt (Retrieved : 4 August 2012)

11.9. REFERENCES 61

[63] Ruwanpura, Kanchana N. (2006). Matrilineal Communities, Patriarchal Realities: A Feminist Nirvana Uncovered. AnnArbor: University of Michigan Press, paperback (ISBN 978-0-472-06977-4); pp. 51-56.

[64] This part of the Ruwanpura book is easily available via Google Books, at Ruwanpura book. Just scroll down to the book’stable of contents and click on Chapter Four, which begins with p. 51.

[65] McGilvray, Dennis B. (1989). “Households in Akkaraipattu: Dowry and Domestic Organization amongMatrilineal Tamilsand Moors of Sri Lanka,” in J. N. Gray and D. J. Mearns (eds.) Society From the Inside Out: Anthropological Perspectiveson the South Asian Household, pp. 192-235. London: Sage Publications.

[66] Ruwanpura, 2006, p. 1. Or click on Ruwanpura book and scroll down to p. 1.

[67] Ruwanpura, 2006, p. 53. Or click on Ruwanpura book and scroll down to p. 53 (as above).

[68] McGilvray, 1989, pp. 201-2.

[69] Ruwanpura, 2006, p. 39.

[70] Ruwanpura, 2006, p. 72.

[71] Yalman, Nur (1971). Under the Bo Tree: Studies in Caste, Kinship, and Marriage in the Interior of Ceylon. Berkeley:University of California Press.

[72] Ruwanpura, 2006, p. 73.

[73] Ruwanpura, 2006, pp. 3-4.

[74] Ruwanpura, 2006, pp. 1-2. Or click on Ruwanpura book and scroll down to pp. 1-2.

[75] Sanday, PeggyReeves (Dec2002). http://www.sas.upenn.edu/~{}psanday/report_02.html, “Report from Indonesia”. Archivedby WebCite® at http://www.webcitation.org/5yuG1WLRW on 23May11.

[76] Sanday, Peggy Reeves (2004). Women at the Center: Life in a Modern Matriarchy. Cornell University Press. ISBN 0-8014-8906-7. Parts of this book are available online at books.google.com .

[77] Fitzsimmons, Caitlin (21Oct09). http://www.roamingtales.com/2009/10/21/a-matrilineal-islamic-society-in-sumatra/,“Amatrilineal, Islamic society in Sumatra”. Archived byWebCite at http://www.webcitation.org/5yuEENZ0B on 23May11.

[78] Sanday 2004, p.67

[79] Sanday 2004, p.241

[80] linguistics.berkeley.edu (2004). http://www.linguistics.berkeley.edu/~{}rosemary/55-2004-names.pdf, “Naming prac-tices”. A PDF file with a section on “Chinese naming practices (Mak et al., 2003)". Archived at WebCite http://www.webcitation.org/5xd5YvhE3 on 1Apr11.

[81] An Zhimin (1988). Archaeological Research on Neolithic China. Current Anthropology, Vol. 29, No. 5 (Dec., 1988), pp.753-759. See p. 755 and p. 758. (The first few sentences are accessible online via JSTOR at http://www.jstor.org/stable/2743616 , i.e., p.753.)

[82] UNHCR document describing that most “Montagnards” are matrilineal

[83] Chiricosta, Alessandra, Following the Trail of the Fairy-Bird: The Search For a Uniquely Vietnamese Women’s Movement,in Roces, Mina, & Louise P. Edwards, eds.,Women’s Movements in Asia: Feminisms and Transnational Activism (Londonor Oxon: Routledge, pbk. 2010 (ISBN 978-0-415-48703-0)), p. 125 and see p. 126 (single quotation marks so in original)(author Chiricosta philosopher & historian of religions, esp. intercultural philosophy, religious & cultural dialogue, gender,& anthropology, & taught at La Sapienza (univ.), Urbaniana (univ.), & Roma Tre (univ.), all in Italy, School of Oriental& African Studies, & Univ. of Ha Noi).

[84] Mukherjee, Sucharita Sinha,Women’s Empowerment and Gender Bias in the Birth and Survival of Girls in Urban India, inFeminist Economics, vol. 19, no. 1 (January, 2013) (doi:10.1080/13545701.2012.752312), p. 9, citing Srinivas, MysoreNarasimhachar, The Cohesive Role of Sanskritization and Other Essays (Delhi: Oxford University Press, 1989), & Agarwal,Bina, A Field of One’s Own: Gender and Land Rights in South Asia (Cambridge: Cambridge Univ. Press, 1994).

[85] Mukherjee, Sucharita Sinha, Women’s Empowerment and Gender Bias in the Birth and Survival of Girls in Urban India,op. cit., p. 9.

[86] Nossiter, Thomas Johnson (1982). Kerala’s Identity: Unity and Diversity. In Communism in Kerala: A Study in PoliticalAdaptation. University of California Press. ISBN 978-0-520-04667-2. Retrieved 2011-06-09. P. 30.

62 CHAPTER 11. MATRILINEALITY

[87] Panikkar, Kavalam Madhava (July–December 1918). “Some Aspects of Nayar Life”. The Journal of the Royal Anthropo-logical Institute of Great Britain and Ireland, 48: 254–293. doi:10.2307/2843423. Retrieved 2011-06-09.

[88] Schneider, David Murray, and Gough, Kathleen (Editors) (1961). Matrilineal Kinship. Berkeley: University of CaliforniaPress. pp. 298–384 is the whole “Nayar: Central Kerala” chapter, for example. ISBN 9780520025295. Accessible here,via GoogleBooks.

[89] Ghosh, Amitav (2003). The Imam and the Indian: prose pieces. Orient Blackswan. p. 193. ISBN 9788175300477. Toaccess it via GoogleBooks, click on book title.

[90] Malinowski, Bronisław. Argonauts Of The Western Pacific, esp. or only chaps. I, II, & VI.

[91] Apple, Raymond (Rabbi Dr.) (2009). http://www.oztorah.com/2009/07/matrilineality-is-still-best-for-jewish-identity/,“Matrilineality is still best for Jewish Identity”. Archived at WebCite http://www.webcitation.org/5xedDM65a on 2Apr11.See this article for the origins of the matrilineality principle in Judaism.

[92] Flavius Josephus, Antiquities of the Jews 16.225, 18.109, 18.139, 18.141, 14.8-10, 14.121, 14.403, or, according to one ofhis statements, "half-Jewish"

[93] On the Life of Moses 2.36.193, On the Virtues 40.224, On the Life of Moses 1.27.147

[94] Herodotus, before 425BCE. http://en.wikisource.org/wiki/History_of_Herodotus/Book_1, “History ofHerodotus”. Graves’snotation is “i.173” meaning in Book 1 – Scroll down to paragraph 173 to find the (matrilineal) Lycians.

[95] Graves, Robert (1955, 1960). The Greek Myths, Vol. 1. Penguin Books. ISBN 0-14-020508-X; p. 296 (myth #88,comment #2).

[96] Graves 1955,1960; p. 256 (myth #75, comment #5).

[97] Schlauch, Margaret (1969). Chaucer’s Constance and Accused Queens. New York: Gordian Press. ISBN 0-87752-097-6;p. 43.

[98] Schlauch 1969, p. 45.

[99] Schlauch 1969, p. 34.

11.10 Further reading• Schlegel, Alice (1972) Male dominance and female autonomy: domestic authority in matrilineal societies.HRAF Press. (review)

• The origin of Matrilineal Descent in Judaism

• Why is Judaism passed on through the mother?

• Louis Jacobs, “There is No Problem of Descent”

• Matrilineal or Patrilineal Descent Lisa Katz

• Professor Shaye J. D. Cohen, “The origin of the Matrilineal rule in Rabbinical Judaism”

• Holden, C. J. & Mace, R. (2003). Spread of cattle led to the loss of matrilineal descent in Africa: a coevolu-tionary analysis. The Royal Society Full text

• Holden, C.J., Sear, R. & Mace, R. (2003) Matriliny as daughter-biased investment. Evolution & HumanBehavior 24: 99-112. Full text

• Knight, C. 2008. Early human kinship was matrilineal. In N. J. Allen, H. Callan, R. Dunbar and W. James(eds.), Early Human Kinship. Oxford: Blackwell, pp. 61–82.Full text

• Sear, R. (2008). Kin and child survival in rural Malawi: Are matrilineal kin always beneficial in a matrilinealsociety? Human Nature, 19, 277-293. Full text

• Mattison, S.M. (2011). Evolutionary contributions to solving the “Matrilineal Puzzle": A test of Holden, Sear,and Mace’s model. Human Nature, 22, 64-88. Full text

Chapter 12

Nontransitive dice

A set of dice is nontransitive if it contains three dice, A, B, and C, with the property that A rolls higher than B morethan half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C morethan half the time. In other words, a set of dice is nontransitive if its “rolls a higher number than more than half thetime” relation is not transitive.It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die thatrolls a higher number than it more than half the time. Using such a set of dice, one can invent games which are biasedin ways that people unused to nontransitive dice might not expect (see Example).

12.1 Example

92

4 81

6 73

5An example of nontransitive dice (opposite sides have the same value as those shown).

Consider the following set of dice.

• Die A has sides 2, 2, 4, 4, 9, 9.

• Die B has sides 1, 1, 6, 6, 8, 8.

• Die C has sides 3, 3, 5, 5, 7, 7.

The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability thatC rolls higher than A are all 5/9, so this set of dice is nontransitive. In fact, it has the even stronger property that, foreach die in the set, there is another die that rolls a higher number than it more than half the time.Now, consider the following game, which is played with a set of dice.

1. The first player chooses a die from the set.

63

64 CHAPTER 12. NONTRANSITIVE DICE

2. The second player chooses one of the remaining dice.

3. Both players roll their dice; the player who rolls the higher number wins.

If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because thefirst player can always find a die that will not be beaten by any other die more than half the time. If it is played withthe set of dice described above, however, the game is biased in favor of the second player, because the second playercan always find a die that will beat the first player’s die with probability 5/9.

12.2 Comment regarding the equivalency of intransitive dice

Though the three intransitive dice A, B, C (first set of dice)

• A: 2, 2, 6, 6, 7, 7

• B: 1, 1, 5, 5, 9, 9

• C: 3, 3, 4, 4, 8, 8

P(A > B) = P(B > C) = P(C > A) = 5/9and the three intransitive dice A', B', C' (second set of dice)

• A': 2, 2, 4, 4, 9, 9

• B': 1, 1, 6, 6, 8, 8

• C': 3, 3, 5, 5, 7, 7

P(A' > B') = P(B' > C') = P(C' > A') = 5/9win against each other with equal probability they are not equivalent. While the first set of dice (A, B, C) has a'highest' die the second set of dice has a 'lowest' die. Rolling the three dice of a set and using always the highest scorefor evaluation will show a different winning pattern for the two sets of dice. With the first set of dice, die B will winwith the highest probability (88/216) and dice A and C will each win with a probability of 64/216. With the secondset of dice, die C' will win with the lowest probability (56/216) and dice A' and B' will each win with a probabilityof 80/216.

12.3 Variations of nontransitive dice

12.3.1 Efron’s dice

Efron’s dice are a set of four nontransitive dice invented by Bradley Efron.

40

4 33

3 62

2 15Efron’s dice.

The four dice A, B, C, D have the following numbers on their six faces:

12.3. VARIATIONS OF NONTRANSITIVE DICE 65

• A: 4, 4, 4, 4, 0, 0

• B: 3, 3, 3, 3, 3, 3

• C: 6, 6, 2, 2, 2, 2

• D: 5, 5, 5, 1, 1, 1

Probabilities

Each die is beaten by the previous die in the list, with a probability of 2/3:

P (A > B) = P (B > C) = P (C > D) = P (D > A) =2

3

B’s value is constant; A beats it on 2/3 rolls because four of its six faces are higher.

1

6 2

5 15

1/3 2/3

1/2 1/21/2 1/2

1/6 1/6 2/6 2/6

C

DC CC D

A conditional probability tree can be used to discern the probability with which C rolls higher than D.

Similarly, B beats C with a 2/3 probability because only two of C’s faces are higher.P(C>D) can be calculated by summing conditional probabilities for two events:

• C rolls 6 (probability 1/3); wins regardless of D (probability 1)

• C rolls 2 (probability 2/3); wins only if D rolls 1 (probability 1/2)

The total probability of win for C is therefore

(1

3× 1

)+

(2

3× 1

2

)=

2

3

66 CHAPTER 12. NONTRANSITIVE DICE

With a similar calculation, the probability of D winning over A is

(1

2× 1

)+

(1

2× 1

3

)=

2

3

Best overall die

The four dice have unequal probabilities of beating a die chosen at random from the remaining three:As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. The probability of dieA beating C is 4/9 (A must roll 4 and C must roll 2). So the likelihood of A beating any other randomly selected dieis:

1

3×(2

3+

1

3+

4

9

)=

13

27

Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. The probability of die Bbeating D is 1/2 (only when D rolls 1). So the likelihood of B beating any other randomly selected die is:

1

3×(2

3+

1

3+

1

2

)=

1

2

Die C beats D two-thirds of the time but beats B only one-third of the time. The probability of die C beating A is5/9. So the likelihood of C beating any other randomly selected die is:

1

3×(2

3+

1

3+

5

9

)=

14

27

Finally, die D beats A two-thirds of the time but beats C only one-third of the time. The probability of die D beatingB is 1/2 (only when D rolls 5). So the likelihood of D beating any other randomly selected die is:

1

3×(2

3+

1

3+

1

2

)=

1

2

Therefore the best overall die is C with a probability of winning of 0.5185. C also rolls the highest average numberin absolute terms, 3 1⁄3. (A’s average is 2 2⁄3, while B’s and D’s are both exactly 3.)

Variants with equal averages

Note that Efron’s dice have different average rolls: the average roll of A is 8/3, while B and D each average 9/3,and C averages 10/3. The non-transitive property depends on which faces are larger or smaller, but does not dependon the absolute magnitude of the faces. Hence one can find variants of Efron’s dice where the odds of winning areunchanged, but all the dice have the same average roll. For example,

• A: 6, 6, 6, 6, 0, 0

• B: 4, 4, 4, 4, 4, 4

• C: 8, 8, 2, 2, 2, 2

• D: 7, 7, 7, 1, 1, 1

or

• A: 7, 7, 7, 7, 1, 1

12.3. VARIATIONS OF NONTRANSITIVE DICE 67

• B: 5, 5, 5, 5, 5, 5

• C: 9, 9, 3, 3, 3, 3

• D: 8, 8, 8, 2, 2, 2

These variant dice are useful, e.g., to introduce students to different ways of comparing random variables (and howonly comparing averages may overlook essential details).

12.3.2 Numbered 1 through 24 dice

A set of four dice using all of the numbers 1 through 24 can be made to be nontransitive. With adjacent pairs, onedie will win approximately 2 out of 3 times.For rolling high number, B beats A, C beats B, D beats C, A beats D.

• A: 1, 2, 16, 17, 18, 19

• B: 3, 4, 5, 20, 21, 22

• C: 6, 7, 8, 9, 23, 24

• D: 10, 11, 12, 13, 14, 15

Relation to Efron’s dice

These dice are basically the same as Efron’s dice, as each number of a series of successive numbers on a single diecan all be replaced by the lowest number of the series and afterwards renumbering them.

• A: 1, 2, 16, 17, 18, 19 -> 1, 1, 16, 16, 16, 16 -> 0, 0, 4, 4, 4, 4

• B: 3, 4, 5, 20, 21, 22 -> 3, 3, 3, 20, 20, 20 -> 1, 1, 1, 5, 5, 5

• C: 6, 7, 8, 9, 23, 24 -> 6, 6, 6, 6, 23, 23 -> 2, 2, 2, 2, 6, 6

• D: 10, 11, 12, 13, 14, 15 -> 10, 10, 10, 10, 10, 10 -> 3, 3, 3, 3, 3, 3

12.3.3 Miwin’s dice

Main article: Miwin’s dice

Miwin’s Dice were invented in 1975 by the physicist Michael Winkelmann.Consider a set of three dice, III, IV and V such that

• die III has sides 1, 2, 5, 6, 7, 9

• die IV has sides 1, 3, 4, 5, 8, 9

• die V has sides 2, 3, 4, 6, 7, 8

Then:

• the probability that III rolls a higher number than IV is 17/36

• the probability that IV rolls a higher number than V is 17/36

• the probability that V rolls a higher number than III is 17/36

68 CHAPTER 12. NONTRANSITIVE DICE

Miwin’s dice

12.3.4 Three-dice set with minimal alterations to standard dice

The following intransitive dice have only a few differences compared to 1 through 6 standard dice:

• as with standard dice, the total number of pips is always 21

• as with standard dice, the sides only carry pip numbers between 1 and 6

• faces with the same number of pips occur a maximum of twice per die

• only two sides on each die have numbers different from standard dice:

• A: 1, 1, 3, 5, 5, 6• B: 2, 3, 3, 4, 4, 5• C: 1, 2, 2, 4, 6, 6

Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. A) is 17/36. The probability of a draw,however, is 4/36, so that only 15 out of 36 rolls lose. So the overall winning expectation is higher.

12.4 Warren Buffett

Warren Buffett is known to be a fan of nontransitive dice. In the book Fortune’s Formula: The Untold Story ofthe Scientific Betting System that Beat the Casinos and Wall Street, a discussion between him and Edward Thorp isdescribed. Buffett and Thorp discussed their shared interest in nontransitive dice. “These are amathematical curiosity,a type of 'trick' dice that confound most people’s ideas about probability.”Buffett once attempted to win a game of dice with Bill Gates using nontransitive dice. “Buffett suggested that each ofthem choose one of the dice, then discard the other two. They would bet on who would roll the highest number most

12.5. NONTRANSITIVE DICE SET FOR MORE THAN TWO PLAYERS 69

often. Buffett offered to let Gates pick his die first. This suggestion instantly aroused Gates’s curiosity. He asked toexamine the dice, after which he demanded that Buffett choose first.”[1]

In 2010, Wall Street Journal magazine quoted Sharon Osberg, Buffett’s bridge partner, saying that when she firstvisited his office 20 years earlier, he tricked her into playing a game with nontransitive dice that could not be won and“thought it was hilarious”.[2]

12.5 Nontransitive dice set for more than two players

A number of people have introduced variations of non-transitive dice where one can compete against more than oneopponent.

12.5.1 Three players

Oskar dice

Oskar van Deventer introduced a set of seven dice (all faces with probability 1/6) as follows:[3]

• A: 2, 2, 14, 14, 17, 17

• B: 7, 7, 10, 10, 16, 16

• C: 5, 5, 13, 13, 15, 15

• D: 3, 3, 9, 9, 21, 21

• E: 1, 1, 12, 12, 20, 20

• F: 6, 6, 8, 8, 19, 19

• G: 4, 4, 11, 11, 18, 18

One can verify that A beats B,C,E; B beats C,D,F; C beats D,E,G; D beats A,E,F; E beats B,F,G; F beats A,C,G; Gbeats A,B,D. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely,

• G beats A,B; F beats A,C; G beats A,D; D beats A,E; D beats A,F; F beats A,G;

• A beats B,C; G beats B,D; A beats B,E; E beats B,F; E beats B,G;

• B beats C,D; A beats C,E; B beats C,F; F beats C,G;

• C beats D,E; B beats D,F; C beats D,G;

• D beats E,F; C beats E,G;

• E beats F,G.

Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents’dice.

Grime dice

Dr James Grime discovered a set of five dice as follows:[4]

• A: 2, 2, 2, 7, 7, 7

• B: 1, 1, 6, 6, 6, 6

• C: 0, 5, 5, 5, 5, 5

70 CHAPTER 12. NONTRANSITIVE DICE

• D: 4, 4, 4, 4, 4, 9

• E: 3, 3, 3, 3, 8, 8

One can verify that, when the game is played with one set of Grime dice:

• A beats B beats C beats D beats E beats A (first chain);

• A beats C beats E beats B beats D beats A (second chain).

However, when the game is played with two such sets, then the first chain remains the same but the second chain isreversed (i.e. A beats D beats B beats E beats C beats A). Consequently, whatever dice the two opponents choose,the third player can always find one of the remaining dice that beats them both (as long as he is then allowed to choosebetween a one-die option and a two-die option):There are two major issues with this set, however. The first one is that in a two-die option of the game, the first chainshould stay exactly the same in order to make the game nontransitive. In practice, though, D actually beats C. Thesecond problem is that the third player would have to be allowed to choose between a one-die option and a two-dieoption – which may be seen as unfair to other players.

12.5.2 Four players

A four-player set has not yet been discovered, but it was proven that such a set would require at least 19 dice.[4]

12.6 Nontransitive dodecahedra

In analogy to the nontransitive six-sided dice, there are also dodecahedra which serve as nontransitive twelve-sideddice. The points on each of the dice result in the sum of 114. There are no repetitive numbers on each of thedodecahedra.The miwin’s dodecahedra (set 1) win cyclically against each other in a ratio of 35:34.The miwin’s dodecahedra (set 2) win cyclically against each other in a ratio of 71:67.Set 1:Set 2:

12.6.1 Nontransitive prime-numbers-dodecahedra

It is also possible to construct sets of nontransitive dodecahedra such that there are no repeated numbers and allnumbers are primes. Miwin’s nontransitive prime-numbers-dodecahedra win cyclically against each other in a ratioof 35:34.Set 1: The numbers add up to 564.Set 2: The numbers add up to 468.

12.7 See also

• Blotto games

• Freivalds’ algorithm

• Nontransitive game

• Condorcet’s voting paradox

12.8. REFERENCES 71

12.8 References[1] Bill Gates speaks: insight from the world’s greatest entrepreneur - Bill Gates, Janet Lowe. Books.google.ie. Retrieved 2011-

11-29.

[2] “like-a-marriage-only-more-enduring: Personal Finance News from Yahoo! Finance”. Finance.yahoo.com. 2010-12-06.Retrieved 2011-11-29.

[3] “Math Games - Tournament Dice by Ed Pegg Jr.”. The Mathematical Association of America. 2005-07-11. Retrieved2012-07-06.

[4] Non-transitive Dice (“Grime Dice”)

• Gardner, Martin. The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: NumberTheory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of RecreationalMathematics. 1st ed. New York: W. W. Norton & Company, 2001. pp. 286–311.

• Spielerische Mathematik mit Miwin’schen Würfeln, Bildungsverlag Lemberger, ISBN 978-3-85221-531-0

12.9 Further reading• Lowe, Janet (January 2001). Bill Gates Speaks: Insight from the World’s Greatest Entrepreneur. New York:Wiley. ISBN 9780471401698. Retrieved 15 March 2013.

12.10 External links• MathWorld page

• Ivars Peterson’s MathTrek - Tricky Dice Revisited (April 15, 2002)

• Jim Loy’s Puzzle Page

• Miwin official site (German)

• Open Source nontransitive dice finder

• Non-transitive Dice by James Grime

• mgf.winkelmann Miwins intransitive Dodekaeder

• Maths Gear

Chapter 13

Partition of a set

For the partition calculus of sets, see infinitary combinatorics.In mathematics, a partition of a set is a grouping of the set’s elements into non-empty subsets, in such a way that

A set of stamps partitioned into bundles: No stamp is in two bundles, and no bundle is empty.

every element is included in one and only one of the subsets.

72

13.1. DEFINITION 73

13.1 Definition

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[1] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[2]

1. P does not contain the empty set.

2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)

3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)

In mathematical notation, these conditions can be represented as

1. ∅ /∈ P

2.∪

A∈P A = X

3. if A,B ∈ P and A ̸= B then A ∩B = ∅ ,

where ∅ is the empty set.The sets in P are called the blocks, parts or cells of the partition.[3]

The rank of P is |X| − |P|, if X is finite.

13.2 Examples• Every singleton set {x} has exactly one partition, namely { {x} }.

• For any nonempty set X, P = {X} is a partition of X, called the trivial partition.

• For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, {A, U−A}.

• The set { 1, 2, 3 } has these five partitions:

• { {1}, {2}, {3} }, sometimes written 1|2|3.• { {1, 2}, {3} }, or 12|3.• { {1, 3}, {2} }, or 13|2.• { {1}, {2, 3} }, or 1|23.• { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).

• The following are not partitions of { 1, 2, 3 }:

• { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set.• { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block.• { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partitionof {1, 2}.

13.3 Partitions and equivalence relations

For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[4]

The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.

74 CHAPTER 13. PARTITION OF A SET

13.4 Refinement of partitions

A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarserthan α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentationof ρ. In that case, it is written that α ≤ ρ.This finer-than relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (forpartitions of a finite set) it is a geometric lattice.[5] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn − 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalenceclasses: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~C has a refinement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.

13.5 Noncrossing partitions

A partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells’ C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.

13.6 Counting partitions

The total number of partitions of an n-element set is the Bell number Bn. The first several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion Bn+1 =

∑nk=0

(nk

)Bk

and have the exponential generating function

∞∑n=0

Bn

n!zn = ee

z−1.

The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by

Cn =1

n+ 1

(2n

n

).

13.7. SEE ALSO 75

13.7 See also• Exact cover

• Cluster analysis

• Weak ordering (ordered set partition)

• Equivalence relation

• Partial equivalence relation

• Partition refinement

• List of partition topics

• Lamination (topology)

• Rhyme schemes by set partition

13.8 Notes[1] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.

[2] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732.

[3] Brualdi, pp. 44–45

[4] Schechter, p. 54

[5] Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.

13.9 References• Brualdi, Richard A. (2004). Introductory Combinatorics (4th edition ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.

• Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.

76 CHAPTER 13. PARTITION OF A SET

The 52 partitions of a set with 5 elements

13.9. REFERENCES 77

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53 54

The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning five elements.

78 CHAPTER 13. PARTITION OF A SET

Partitions of a 4-set ordered by refinement

13.9. REFERENCES 79

Construction of the Bell triangle

Chapter 14

Preorder

Not to be confused with Pre-order.This article is about binary relations. For the graph vertex ordering, see Depth-first search. For other uses, seePreorder (disambiguation).“Quasiorder” redirects here. For irreflexive transitive relations, see strict order.

Inmathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.All partial orders and equivalence relations are preorders, but preorders are more general.The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders,but not quite; they're neither necessarily anti-symmetric nor symmetric. Because a preorder is a binary relation,the symbol ≤ can be used as the notational device for the relation. However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply. On the other hand, a preordercan be used, in a straightforward fashion, to define a partial order and an equivalence relation. Doing so, however, isnot always useful or worthwhile, depending on the problem domain being studied.In words, when a ≤ b, one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, thenotation ← or ≲ is used instead of ≤.To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and theorder relation between pairs of elements corresponding to the directed edges between vertices. The converse is nottrue: most directed graphs are neither reflexive nor transitive. Note that, in general, the corresponding graphs maybe cyclic graphs: preorders may have cycles in them. A preorder that is antisymmetric no longer has cycles; it is apartial order, and corresponds to a directed acyclic graph. A preorder that is symmetric is an equivalence relation;it can be thought of as having lost the direction markers on the edges of the graph. In general, a preorder may havemany disconnected components.

14.1 Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive,i.e., for all a, b and c in P, we have that:

a ≤ a (reflexivity)if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set that is equipped with a preorder is called a preordered set (or proset).[1]

If a preorder is also antisymmetric, that is, a ≤ b and b ≤ a implies a = b, then it is a partial order.On the other hand, if it is symmetric, that is, if a ≤ b implies b ≤ a, then it is an equivalence relation.A preorder which is preserved in all contexts (i.e. respected by all functions on P) is called a precongruence. Aprecongruence which is also symmetric (i.e. is an equivalence relation) is a congruence relation.

80

14.2. EXAMPLES 81

Equivalently, a preordered set P can be defined as a category with objects the elements of P, and each hom-set havingat most one element (one for objects which are related, zero otherwise).Alternately, a preordered set can be understood as an enriched category, enriched over the category 2 = (0→1).A preordered class is a class equipped with a preorder. Every set is a class and so every preordered set is a preorderedclass. Preordered classes can be defined as thin categories, i.e. as categories with at most one morphism from anobject to another.

14.2 Examples

• The reachability relationship in any directed graph (possibly containing cycles) gives rise to a preorder, wherex ≤ y in the preorder if and only if there is a path from x to y in the directed graph. Conversely, every preorderis the reachability relationship of a directed graph (for instance, the graph that has an edge from x to y for everypair (x, y) with x ≤ y). However, many different graphs may have the same reachability preorder as each other.In the same way, reachability of directed acyclic graphs, directed graphs with no cycles, gives rise to partiallyordered sets (preorders satisfying an additional anti-symmetry property).

• Every finite topological space gives rise to a preorder on its points, in which x≤ y if and only if x belongs to everyneighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological spacein this way. That is, there is a 1-to-1 correspondence between finite topologies and finite preorders. However,the relation between infinite topological spaces and their specialization preorders is not 1-to-1.

• A net is a directed preorder, that is, each pair of elements has an upper bound. The definition of convergencevia nets is important in topology, where preorders cannot be replaced by partially ordered sets without losingimportant features.

• The relation defined by x ≤ y if f(x) ≤ f(y) , where f is a function into some preorder.

• The relation defined by x ≤ y if there exists some injection from x to y. Injectionmay be replaced by surjection,or any type of structure-preserving function, such as ring homomorphism, or permutation.

• The embedding relation for countable total orderings.

• The graph-minor relation in graph theory.

• A category with at most one morphism from any object x to any other object y is a preorder. Such categoriesare called thin. In this sense, categories “generalize” preorders by allowing more than one relation betweenobjects: each morphism is a distinct (named) preorder relation.

In computer science, one can find examples of the following preorders.

• Many-one and Turing reductions are preorders on complexity classes.

• The subtyping relations are usually preorders.

• Simulation preorders are preorders (hence the name).

• Reduction relations in abstract rewriting systems.

• The encompassment preorder on the set of terms, defined by s≤t if a subterm of t is a substitution instance ofs.

Example of a total preorder:

• Preference, according to common models.

82 CHAPTER 14. PREORDER

14.3 Uses

Preorders play a pivotal role in several situations:

• Every preorder can be given a topology, the Alexandrov topology; and indeed, every preorder on a set is inone-to-one correspondence with an Alexandrov topology on that set.

• Preorders may be used to define interior algebras.

• Preorders provide the Kripke semantics for certain types of modal logic.

14.4 Constructions

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexiveclosure, R+=. The transitive closure indicates path connection in R: x R+ y if and only if there is an R-path from x toy.Given a preorder ≲ on S one may define an equivalence relation ~ on S such that a ~ b if and only if a ≲ b and b ≲a. (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preordertwice, and symmetric by definition.)Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set ofall equivalence classes of ~. Note that if the preorder is R+=, S / ~ is the set of R-cycle equivalence classes: x ∈ [y]if and only if x = y or x is in an R-cycle with y. In any case, on S / ~ we can define [x] ≤ [y] if and only if x ≲ y.By the construction of ~, this definition is independent of the chosen representatives and the corresponding relationis indeed well-defined. It is readily verified that this yields a partially ordered set.Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 corre-spondence between preorders and pairs (partition, partial order).For a preorder "≲ ", a relation "<" can be defined as a < b if and only if (a≲ b and not b≲ a), or equivalently, usingthe equivalence relation introduced above, (a ≲ b and not a ~ b). It is a strict partial order; every strict partial ordercan be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalenceis equality, so the relation "<" can also be defined as a < b if and only if (a ≤ b and a ≠ b).(Alternatively, for a preorder " ≲ ", a relation "<" can be defined as a < b if and only if (a ≲ b and a ≠ b). The resultis the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive,and if it is, as we have seen, it is the same as before.)Conversely we have a ≲ b if and only if a < b or a ~ b. This is the reason for using the notation " ≲ "; "≤" can beconfusing for a preorder that is not anti-symmetric, it may suggest that a ≤ b implies that a < b or a = b.Note that with this construction multiple preorders "≲ " can give the same relation "<", so without more information,such as the equivalence relation, " ≲ " cannot be reconstructed from "<". Possible preorders include the following:

• Define a ≤ b as a < b or a = b (i.e., take the reflexive closure of the relation). This gives the partial orderassociated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, sowe don't need the notations ≲ and ~.

• Define a≲ b as “not b < a" (i.e., take the inverse complement of the relation), which corresponds to defining a~ b as “neither a < b nor b < a"; these relations ≲ and ~ are in general not transitive; however, if they are, ~ isan equivalence; in that case "<" is a strict weak order. The resulting preorder is total, that is, a total preorder.

14.5 Number of preorders

As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus thenumber of preorders is the sum of the number of partial orders on every partition. For example:

• for n=3:

14.6. INTERVAL 83

• 1 partition of 3, giving 1 preorder• 3 partitions of 2+1, giving 3 × 3 = 9 preorders• 1 partition of 1+1+1, giving 19 preorders

i.e. together 29 preorders.

• for n=4:

• 1 partition of 4, giving 1 preorder• 7 partitions with two classes (4 of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders• 6 partitions of 2+1+1, giving 6 × 19 = 114 preorders• 1 partition of 1+1+1+1, giving 219 preorders

i.e. together 355 preorders.

14.6 Interval

For a ≲ b, the interval [a,b] is the set of points x satisfying a ≲ x and x ≲ b, also written a ≲ x ≲ b. It contains atleast the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a <x and x < b, also written a < x < b. An open interval may be empty even if a < b.Also [a,b) and (a,b] can be defined similarly.

14.7 See also• partial order - preorder that is antisymmetric

• equivalence relation - preorder that is symmetric

• total preorder - preorder that is total

• total order - preorder that is antisymmetric and total

• directed set

• category of preordered sets

• prewellordering

• Well-quasi-ordering

14.8 References[1] For “proset”, see e.g. Eklund, Patrik; Gähler, Werner (1990), “Generalized Cauchy spaces”, Mathematische Nachrichten

147: 219–233, doi:10.1002/mana.19901470123, MR 1127325.

• Schröder, Bernd S. W. (2002), Ordered Sets: An Introduction, Boston: Birkhäuser, ISBN 0-8176-4128-9

Chapter 15

Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. In-formally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept wasintroduced by Sen (1969) to study the consequences of Arrow’s theorem.

15.1 Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

(aT b) ∧ ¬(bT a) ∧ (bT c) ∧ ¬(cT b) ⇒ (aT c) ∧ ¬(cT a).

If the relation is also antisymmetric, T is transitive.Alternately, for a relation T, define the asymmetric or “strict” part P:

(aP b) ⇔ (aT b) ∧ ¬(bT a).

Then T is quasitransitive iff P is transitive.

15.2 Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic exampleis a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but whoprefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity ofcertain relations to quasitransitivity.

15.3 Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case theconverse does not hold in general.

15.4 See also

• Intransitivity

• Reflexive relation

84

15.5. REFERENCES 85

15.5 References• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press.ISBN 0674052994.

• Sen, A. (1969). “Quasi-transitivity, rational choice and collective decisions”. Rev. Econ. Stud. 36: 381–393.doi:10.2307/2296434. Zbl 0181.47302.

Chapter 16

Quotient category

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Thenotion is similar to that of a quotient group or quotient space, but in the categorical setting.

16.1 Definition

Let C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalencerelation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if

f1, f2 : X → Y

are related in Hom(X, Y) and

g1, g2 : Y → Z

are related in Hom(Y, Z) then g1f1, g1f2, g2f1 and g2f2 are related in Hom(X, Z).Given a congruence relation R on C we can define the quotient category C/R as the category whose objects are thoseof C and whose morphisms are equivalence classes of morphisms in C. That is,

HomC/R(X,Y ) = HomC(X,Y )/RX,Y .

Composition of morphisms in C/R is well-defined since R is a congruence relation.There is also a notion of taking the quotient of an Abelian category A by a Serre subcategory B. This is done asfollows. The objects of A/B are the objects of A. Given two objects X and Y of A, we define the set of morphismsfrom X to Y in A/B to be lim−→HomA(X

′, Y /Y ′) where the limit is over subobjects X ′ ⊆ X and Y ′ ⊆ Y such thatX/X ′, Y ′ ∈ B . Then A/B is an Abelian category, and there is a canonical functor Q : A → A/B . This Abelianquotient satisfies the universal property that if C is any other Abelian category, and F : A → C is an exact functorsuch that F(b) is a zero object of C for each b ∈ B , then there is a unique exact functor F : A/B → C such thatF = F ◦Q . (See [Gabriel].)

16.2 Properties

There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functoris bijective on objects and surjective on Hom-sets (i.e. it is a full functor).

86

16.3. EXAMPLES 87

16.3 Examples• Monoids and group may be regarded as categories with one object. In this case the quotient category coincideswith the notion of a quotient monoid or a quotient group.

• The homotopy category of topological spaces hTop is a quotient category of Top, the category of topologicalspaces. The equivalence classes of morphisms are homotopy classes of continuous maps.

16.4 See also• Subobject

16.5 References• Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448.

• Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathe-matics 5). Springer-Verlag.

Chapter 17

Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residueclass ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linearalgebra.[2][3] One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I,whose elements are the cosets of I in R subject to special + and operations.Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well asfrom the more general 'rings of quotients’ obtained by localization.

17.1 Formal quotient ring construction

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

a ~ b if and only if a − b is in I.

Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a andb are congruent modulo I. The equivalence class of the element a in R is given by

[a] = a + I := { a + r : r in I }.

This equivalence class is also sometimes written as a mod I and called the “residue class of a modulo I".The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of Rmodulo I, if one defines

• (a + I) + (b + I) = (a + b) + I;

• (a + I)(b + I) = (a b) + I.

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-elementof R/I is (0 + I) = I, and the multiplicative identity is (1 + I).The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism, sometimes called the naturalquotient map or the canonical homomorphism.

17.2 Examples• The quotient R/{0} is naturally isomorphic to R, and R/R is the zero ring {0}, since, by our definition, for anyr in R , we have that [r]=r +{0}:={r+b : b in {0}} (where {0} is the zero ring), which is isomorphic to R itself. This fits with the general rule of thumb that the larger the ideal I, the smaller the quotient ring R/I. If I is aproper ideal of R, i.e., I ≠ R, then R/I is not the zero ring.

88

17.2. EXAMPLES 89

• Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z/2Zhas only two elements, zero for the even numbers and one for the odd numbers; applying the definition again,[z]=z+2Z:={z+2z: 2z in {2Z}}, where {2Z} is the ideal of even numbers. It is naturally isomorphic to thefinite field with two elements, F2. Intuitively: if you think of all the even numbers as 0, then every integer iseither 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1). Modular arithmetic isessentially arithmetic in the quotient ring Z/nZ (which has n elements).

• Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal I = (X2 + 1)consisting of all multiples of the polynomial X2 + 1. The quotient ring R[X]/(X2 + 1) is naturally isomorphicto the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason: we“forced” X2 + 1 = 0, i.e. X2 = −1, which is the defining property of i.

• Generalizing the previous example, quotient rings are often used to construct field extensions. Suppose K issome field and f is an irreducible polynomial in K[X]. Then L = K[X]/(f) is a field whose minimal polynomialover K is f, which contains K as well as an element x = X + (f).

• One important instance of the previous example is the construction of the finite fields. Consider for instancethe field F3 = Z/3Z with three elements. The polynomial f(X) = X2 + 1 is irreducible over F3 (since it hasno root), and we can construct the quotient ring F3[X]/(f). This is a field with 32=9 elements, denoted by F9.The other finite fields can be constructed in a similar fashion.

• The coordinate rings of algebraic varieties are important examples of quotient rings in algebraic geometry. Asa simple case, consider the real variety V = {(x,y) | x2 = y3 } as a subset of the real plane R2. The ring ofreal-valued polynomial functions defined on V can be identified with the quotient ring R[X,Y]/(X2 − Y3), andthis is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.

• SupposeM is a C∞-manifold, and p is a point ofM. Consider the ring R = C∞(M) of all C∞-functions definedonM and let I be the ideal in R consisting of those functions f which are identically zero in some neighborhoodU of p (where U may depend on f). Then the quotient ring R/I is the ring of germs of C∞-functions on M atp.

• Consider the ring F of finite elements of a hyperreal field *R. It consists of all hyperreal numbers differing froma standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers x for which a standardinteger n with −n < x < n exists. The set I of all infinitesimal numbers in *R, together with 0, is an ideal in F,and the quotient ring F/I is isomorphic to the real numbers R. The isomorphism is induced by associating toevery element x of F the standard part of x, i.e. the unique real number that differs from x by an infinitesimal.In fact, one obtains the same result, namely R, if one starts with the ring F of finite hyperrationals (i.e. ratioof a pair of hyperintegers), see construction of the real numbers.

17.2.1 Alternative complex planes

The quotients R[X]/(X), R[X]/(X + 1), and R[X]/(X − 1) are all isomorphic to R and gain little interest at first.But note that R[X]/(X2) is called the dual number plane in geometric algebra. It consists only of linear binomialsas “remainders” after reducing an element of R[X] by X2. This alternative complex plane arises as a subalgebrawhenever the algebra contains a real line and a nilpotent.Furthermore, the ring quotient R[X]/(X2 − 1) does split into R[X]/(X + 1) and R[X]/(X − 1), so this ring is oftenviewed as the direct sumR⊕R. Nevertheless, an alternative complex number z = x + y j is suggested by j as a root ofX2 − 1, compared to i as root of X2 + 1 = 0. This plane of split-complex numbers normalizes the direct sum R⊕Rby providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With thisbasis a unit hyperbola may be compared to the unit circle of the ordinary complex plane.

17.2.2 Quaternions and alternatives

Suppose X and Y are two, non-commuting, indeterminates and form the free algebra R⟨X,Y ⟩. Then Hamilton’squaternions of 1843 can be cast as

90 CHAPTER 17. QUOTIENT RING

R⟨X,Y ⟩/(X2 + 1, Y 2 + 1, XY + Y X).

If Y2 − 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in boththe quadratic binomials also results in split-quaternions. The anti-commutative property YX = −XY implies that XYhas for its square

(XY)(XY) = X(YX)Y = −X(XY)Y = − XXYY = −1.

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminatesR⟨X,Y,Z⟩ and constructing appropriate ideals.

17.3 Properties

Clearly, if R is a commutative ring, then so is R/I; the converse however is not true in general.The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, wecan state that two-sided ideals are precisely the kernels of ring homomorphisms.The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows:the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish(i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : R→ S whose kernelcontains I, then there exists precisely one ring homomorphism g : R/I → S with gp = f (where p is the natural quotientmap). The map g here is given by the well-defined rule g([a]) = f(a) for all a in R. Indeed, this universal propertycan be used to define quotient rings and their natural quotient maps.As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : R→ S inducesa ring isomorphism between the quotient ring R/ker(f) and the image im(f). (See also: fundamental theorem onhomomorphisms.)The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided idealsof R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). This relationshipbetween two-sided ideal extends to a relationship between the corresponding quotient rings: ifM is a two-sided idealin R that contains I, and we write M/I for the corresponding ideal in R/I (i.e. M/I = p(M)), the quotient rings R/Mand (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M ↦ (a+I) + M/I.In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutativering and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integraldomain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwisecoprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip, p=1,...,k.

17.4 See also

• Residue field

• Goldie’s theorem

17.5 Notes

[1] Jacobson, Nathan (1984). Structure of Rings (revised ed.). American Mathematical Soc. ISBN 0-821-87470-5.

[2] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[3] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

17.6. FURTHER REFERENCES 91

17.6 Further references• F. Kasch (1978) Moduln und Ringe, translated by DAR Wallace (1982) Modules and Rings, Academic Press,page 33.

• Neal H. McCoy (1948) Rings and Ideals, §13 Residue class rings, page 61, Carus Mathematical Monographs#8, Mathematical Association of America.

• Joseph Rotman (1998). Galois Theory (2nd edition). Springer. pp. 21–3. ISBN 0-387-98541-7.

• B.L. van der Waerden (1970) Algebra, translated by Fred Blum and John R Schulenberger, Frederick UngarPublishing, New York. See Chapter 3.5, “Ideals. Residue Class Rings”, pages 47 to 51.

17.7 External links• Hazewinkel, Michiel, ed. (2001), “Quotient ring”, Encyclopedia ofMathematics, Springer, ISBN978-1-55608-010-4

• Ideals and factor rings from John Beachy’s Abstract Algebra Online

• Quotient ring at PlanetMath.org.

Chapter 18

Quotient space (linear algebra)

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by “collapsing” N tozero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).

18.1 Definition

Formally, the construction is as follows (Halmos 1974, §21-22). Let V be a vector space over a field K, and let Nbe a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is relatedto y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that anyelement of N is related to the zero vector; more precisely all the vectors in N get mapped into the equivalence classof the zero vector.The equivalence class of x is often denoted

[x] = x + N

since it is given by

[x] = {x + n : n ∈ N}.

The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~. Scalar multiplicationand addition are defined on the equivalence classes by

• α[x] = [αx] for all α ∈ K, and

• [x] + [y] = [x+y].

It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative).These operations turn the quotient space V/N into a vector space over K with N being the zero class, [0].The mapping that associates to v ∈ V the equivalence class [v] is known as the quotient map.

18.2 Examples

Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Ycan be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the setX/Y are lines in X parallel to Y. This gives one way in which to visualize quotient spaces geometrically.Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn

consists of all n-tuples of real numbers (x1,…,xn). The subspace, identified with Rm, consists of all n-tuples such thatthe last n-m entries are zero: (x1,…,xm,0,0,…,0). Two vectors of Rn are in the same congruence class modulo the

92

18.3. PROPERTIES 93

subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic toRn−m in an obvious manner.More generally, if V is an (internal) direct sum of subspaces U andW,

V = U ⊕W

then the quotient space V/U is naturally isomorphic toW (Halmos 1974, Theorem 22.1).An important example of a functional quotient space is a Lp space.

18.3 Properties

There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. Thekernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exactsequence

0 → U → V → V /U → 0.

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may beconstructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, thedimension of V is the sum of the dimensions ofU and V/U. If V is finite-dimensional, it follows that the codimensionof U in V is the difference between the dimensions of V and U (Halmos 1974, Theorem 22.2):

codim(U) = dim(V /U) = dim(V )− dim(U).

Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x ∈ V such that Tx = 0. Thekernel is a subspace of V. The first isomorphism theorem of linear algebra says that the quotient space V/ker(T)is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank-nullitytheorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of theimage (the rank of T).The cokernel of a linear operator T : V →W is defined to be the quotient spaceW/im(T).

18.4 Quotient of a Banach space by a subspace

If X is a Banach space andM is a closed subspace of X, then the quotient X/M is again a Banach space. The quotientspace is already endowed with a vector space structure by the construction of the previous section. We define a normon X/M by

∥[x]∥X/M = infm∈M

∥x−m∥X .

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

18.4.1 Examples

Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm.Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 byM. Then the equivalence class of some function g isdetermined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R.If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

94 CHAPTER 18. QUOTIENT SPACE (LINEAR ALGEBRA)

18.4.2 Generalization to locally convex spaces

The quotient of a locally convex space by a closed subspace is again locally convex (Dieudonné 1970, 12.14.8).Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈A} where A is an index set. Let M be a closed subspace, and define seminorms qα by on X/M

qα([x]) = infx∈[x]

pα(x).

Then X/M is a locally convex space, and the topology on it is the quotient topology.If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M (Dieudonné 1970, 12.11.3).

18.5 See also• quotient set

• quotient group

• quotient module

• quotient space (topology)

18.6 References• Halmos, Paul (1974), Finite dimensional vector spaces, Springer, ISBN 978-0-387-90093-3.

• Dieudonné, Jean (1970), Treatise on analysis, Volume II, Academic Press.

Chapter 19

Quotient space (topology)

For quotient spaces in linear algebra, see quotient space (linear algebra).In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively

Illustration of quotient space, S2, obtained by gluing the boundary (in blue) of the disk D2 together to a single point.

speaking, the result of identifying or “gluing together” certain points of a given topological space. The points to beidentified are specified by an equivalence relation. This is commonly done in order to construct new spaces fromgiven ones. The quotient topology consists of all sets with an open preimage under the canonical projection mapthat maps each element to its equivalence class.

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96 CHAPTER 19. QUOTIENT SPACE (TOPOLOGY)

19.1 Definition

Let (X, τX) be a topological space, and let ~ be an equivalence relation on X. The quotient space, Y = X / ~ is definedto be the set of equivalence classes of elements of X:

Y = {[x] : x ∈ X} = {{v ∈ X : v ∼ x} : x ∈ X},

equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions areopen sets in X:

τY =

U ⊆ Y :∪U =

∪[a]∈U

[a]

∈ τX

.

Equivalently, we can define them to be those sets with an open preimage under the surjective map q : X → X / ~,which sends a point in X to the equivalence class containing it:

τY ={U ⊆ Y : q−1(U) ∈ τX

}.

The quotient topology is the final topology on the quotient space with respect to the map q.

19.2 Quotient map

A map f : X → Y is a quotient map if it is surjective, and a subset U of Y is open if and only if f−1(U) is open.Equivalently, f is a quotient map if it is onto and Y is equipped with the final topology with respect to f .Given an equivalence relation ∼ on X , the canonical map q : X → X/∼ is a quotient map.

19.3 Examples• Gluing. Topologists talk of gluing points together. If X is a topological space and points x, y ∈ X are to be“glued”, then what is meant is that we are to consider the quotient space obtained from the equivalence relationa ~ b if and only if a = b or a = x, b = y (or a = y, b = x).

• Consider the unit square I2 = [0,1] × [0,1] and the equivalence relation ~ generated by the requirement that allboundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then I2/~ ishomeomorphic to the unit sphere S2.

• Adjunction space. More generally, suppose X is a space and A is a subspace of X. One can identify all pointsin A to a single equivalence class and leave points outside of A equivalent only to themselves. The resultingquotient space is denoted X/A. The 2-sphere is then homeomorphic to the unit disc with its boundary identifiedto a single point: D2/∂D2 .

• Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y if and only if x − y isan integer. Then the quotient space X/~ is homeomorphic to the unit circle S1 via the homeomorphism whichsends the equivalence class of x to exp(2πix).

• A generalization of the previous example is the following: Suppose a topological group G acts continuously ona space X. One can form an equivalence relation on X by saying points are equivalent if and only if they lie inthe same orbit. The quotient space under this relation is called the orbit space, denoted X/G. In the previousexample G = Z acts on R by translation. The orbit space R/Z is homeomorphic to S1.

Note: The notation R/Z is somewhat ambiguous. If Z is understood to be a group acting on R then the quotient is thecircle. However, if Z is thought of as a subspace of R, then the quotient is an infinite bouquet of circles joined at asingle point.

19.4. PROPERTIES 97

19.4 Properties

Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topologicalspace and f : Y → Z is any function, then f is continuous if and only if f ∘ q is continuous.

X

q

Y Z

f ∘ q

fCharacteristic property of the quotient topology

The quotient space X/~ together with the quotient map q : X → X/~ is characterized by the following universalproperty: if g : X → Z is a continuous map such that a ~ b implies g(a) = g(b) for all a and b in X, then there exists aunique continuous map f : X/~ → Z such that g = f ∘ q. We say that g descends to the quotient.The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps definedon X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). Thiscriterion is constantly used when studying quotient spaces.Given a continuous surjection q : X → Y it is useful to have criteria by which one can determine if q is a quotientmap. Two sufficient criteria are that q be open or closed. Note that these conditions are only sufficient, not necessary.It is easy to construct examples of quotient maps that are neither open nor closed.

98 CHAPTER 19. QUOTIENT SPACE (TOPOLOGY)

19.5 Compatibility with other topological notions• Separation

• In general, quotient spaces are ill-behaved with respect to separation axioms. The separation propertiesof X need not be inherited by X/~, and X/~ may have separation properties not shared by X.

• X/~ is a T1 space if and only if every equivalence class of ~ is closed in X.• If the quotient map is open, then X/~ is a Hausdorff space if and only if ~ is a closed subset of the productspace X×X.

• Connectedness

• If a space is connected or path connected, then so are all its quotient spaces.• A quotient space of a simply connected or contractible space need not share those properties.

• Compactness

• If a space is compact, then so are all its quotient spaces.• A quotient space of a locally compact space need not be locally compact.

• Dimension

• The topological dimension of a quotient space can be more (as well as less) than the dimension of theoriginal space; space-filling curves provide such examples.

19.6 See also

19.6.1 Topology

• Topological space

• Subspace (topology)

• Product space

• Disjoint union (topology)

• Final topology

• Mapping cone

19.6.2 Algebra

• Quotient group

• Quotient space (linear algebra)

• Quotient category

• Mapping cone (homological algebra)

19.7 References• Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 0-486-43479-6.

• Quotient space at PlanetMath.org.

Chapter 20

Rational choice theory

This article is about a theory of economics. For rational choice theory as applied to criminology, see Rational choicetheory (criminology).

Rational choice theory, also known as choice theory or rational action theory, is a framework for understandingand often formally modeling social and economic behavior.[1] The basic premise of rational choice theory is thataggregate social behavior results from the behavior of individual actors, each of whom is making their individualdecisions. The theory therefore focuses on the determinants of the individual choices (methodological individualism).Rational choice theory then assumes that an individual has preferences among the available choice alternatives thatallow them to state which option they prefer. These preferences are assumed to be complete (the person can alwayssay which of two alternatives they consider preferable or that neither is preferred to the other) and transitive (if optionA is preferred over option B and option B is preferred over option C, then A is preferred over C). The rationalagent is assumed to take account of available information, probabilities of events, and potential costs and benefits indetermining preferences, and to act consistently in choosing the self-determined best choice of action.Rationality is widely used as an assumption of the behavior of individuals in microeconomic models and analyses andappears in almost all economics textbook treatments of human decision-making. It is also central to some of modernpolitical science,[2] sociology,[3] and philosophy. A particular version of rationality is instrumental rationality, whichinvolves seeking the most cost-effective means to achieve a specific goal without reflecting on the worthiness of thatgoal. Gary Becker was an early proponent of applying rational actor models more widely.[4] Becker won the 1992Nobel Memorial Prize in Economic Sciences for his studies of discrimination, crime, and human capital.[5]

20.1 Definition and scope

The concept of rationality used in rational choice theory is different from the colloquial and most philosophical useof the word. Colloquially, “rational” behaviour typically means “sensible”, “predictable”, or “in a thoughtful, clear-headed manner.” Rational choice theory uses a narrower definition of rationality. At its most basic level, behavior isrational if it is goal-oriented, reflective (evaluative), and consistent (across time and different choice situations). Thiscontrasts with behavior that is random, impulsive, conditioned, or adopted by (unevaluative) imitation.Early neoclassical economists writing about rational choice, including William Stanley Jevons, assumed that agentsmake consumption choices so as to maximize their happiness, or utility. Contemporary theory bases rational choiceon a set of choice axioms that need to be satisfied, and typically does not specify where the goal (preferences, de-sires) comes from. It mandates just a consistent ranking of the alternatives.[6]:501 Individuals choose the best actionaccording to their personal preferences and the constraints facing them. E.g., there is nothing irrational in preferringfish to meat the first time, but there is something irrational in preferring fish to meat in one instant and preferringmeat to fish in another, without anything else having changed.Rational choice theorists do not claim that the theory describes the choice process, but rather that it predicts theoutcome and pattern of choices. An assumption often added to the rational choice paradigm is that individual prefer-ences are self-interested, in which case the individual can be referred to as a homo oeconomicus. Such an individualacts as if balancing costs against benefits to arrive at action that maximizes personal advantage.[7] Proponents of suchmodels, particularly those associated with the Chicago school of economics, do not claim that a model’s assumptions

99

100 CHAPTER 20. RATIONAL CHOICE THEORY

are an accurate description of reality, only that they help formulate clear and falsifiable hypotheses. In this view, theonly way to judge the success of a hypothesis is empirical tests.[7] To use an example from Milton Friedman, if atheory that says that the behavior of the leaves of a tree is explained by their rationality passes the empirical test, itis seen as successful.

Daniel Kahneman

Without specifying the individual’s goal or preferences it may not be possible to empirically test, or falsify, therationality assumption, in which case rational choice theory becomes a tautology (true by definition). However, thepredictions made by a specific version of the theory are testable. In recent years, the most prevalent version of rationalchoice theory, expected utility theory, has been challenged by the experimental results of behavioral economics.Economists are learning from other fields, such as psychology, and are enriching their theories of choice in orderto get a more accurate view of human decision-making. For example, the behavioral economist and experimentalpsychologist Daniel Kahneman won the Nobel Memorial Prize in Economic Sciences in 2002 for his work in thisfield.

20.2. ACTIONS, ASSUMPTIONS, AND INDIVIDUAL PREFERENCES 101

Rational choice theory has become increasingly employed in social sciences other than economics, such as sociologyand political science in recent decades.[8] It has had far-reaching impacts on the study of political science, espe-cially in fields like the study of interest groups, elections, behaviour in legislatures, coalitions, and bureaucracy.[9]In these fields, the use of the rational choice paradigm to explain broad social phenomena is the subject of activecontroversy.[10][11]

20.2 Actions, assumptions, and individual preferences

The premise of rational choice theory as a social science methodology is that the aggregate behavior in society reflectsthe sum of the choices made by individuals. Each individual, in turn, makes her choice based on her own preferencesand the constraints (or choice set) they face.At the individual level, rational choice theory stipulates that the agent chooses the action (or outcome) they mostprefer. In the case where actions (or outcomes) can be evaluated in terms of costs and benefits, a rational individualchooses the action (or outcome) that provides the maximum net benefit, i.e., the maximum benefit minus cost.The theory applies to more general settings than those identified by costs and benefit. In general, rational decisionmaking entails choosing among all available alternatives the alternative that the individual most prefers. The “alter-natives” can be a set of actions (“what to do?") or a set of objects (“what to choose/buy”). In the case of actions,what the individual really cares about are the outcomes that results from each possible action. Actions, in this case,are only an instrument for obtaining a particular outcome.

20.2.1 Formal statement

The available alternatives are often expressed as a set of objects, for example a set of j exhaustive and exclusiveactions:

A = {a1, . . . , ai, . . . , aj}

For example, if a person can choose to vote for either Roger or Sara or to abstain, their set of possible alternatives is:

A = {V oteforRoger, V oteforSara,Abstain}

The theory makes two technical assumptions about individuals’ preferences over alternatives:

• Completeness – for any two alternatives ai and aj in the set, either ai is preferred to aj, or aj is preferred toai, or the individual is indifferent between ai and aj. In other words, all pairs of alternatives can be comparedwith each other.

• Transitivity – if alternative a1 is preferred to a2, and alternative a2 is preferred to a3, then a1 is preferred toa3.

Together these two assumptions imply that given a set of exhaustive and exclusive actions to choose from, an individualcan rank the elements of this set in terms of his preferences in an internally consistent way (the ranking constitutes apartial ordering), and the set has at least one maximal element.The preference between two alternatives can be:

• Strict preference occurs when an individual prefers a1 to a2 and does not view them as equally preferred.• Weak preference implies that individual either strictly prefers a1 over a2 or is indifferent between them.• Indifference occurs when an individual neither prefers a1 to a2, nor a2 to a1. Since (by completeness) theindividual does not refuse a comparison, they must therefore be indifferent in this case.

Research that took off in the 1980s sought to develop models which drop these assumptions and argue that suchbehaviour could still be rational, Anand (1993). This work, often conducted by economic theorists and analyticalphilosophers, suggests ultimately that the assumptions or axioms above are not completely general and might at bestbe regarded as approximations.

102 CHAPTER 20. RATIONAL CHOICE THEORY

20.2.2 Additional assumptions

• Perfect information: The simple rational choice model above assumes that the individual has full or perfectinformation about the alternatives, i.e., the ranking between two alternatives involves no uncertainty.

• Choice under uncertainty: In a richer model that involves uncertainty about the how choices (actions) lead toeventual outcomes, the individual effectively chooses between lotteries, where each lottery induces a differentprobability distribution over outcomes. The additional assumption of independence of irrelevant alternativesthen leads to expected utility theory.

• Inter-temporal choice: when decisions affect choices (such as consumption) at different points in time, thestandard method for evaluating alternatives across time involves discounting future payoffs.

• Limited cognitive ability: identifying and weighing each alternative against every other may take time, effort,and mental capacity. Recognising the cost that these impose or cognitive limitations of individuals gives riseto theories of bounded rationality.

Alternative theories of human action include such components as Amos Tversky and Daniel Kahneman's prospecttheory, which reflects the empirical finding that, contrary to standard preferences assumed under neoclassical eco-nomics, individuals attach extra value to items that they already own compared to similar items owned by others.Under standard preferences, the amount that an individual is willing to pay for an item (such as a drinking mug) isassumed to equal the amount he or she is willing to be paid in order to part with it. In experiments, the latter priceis sometimes significantly higher than the former (but see Plott and Zeiler 2005,[12] Plott and Zeiler 2007 [13] andKlass and Zeiler, 2013 [14]). Tversky and Kahneman [15] do not characterize loss aversion as irrational. Behavioraleconomics includes a large number of other amendments to its picture of human behavior that go against neoclassicalassumptions.

20.3 Utility maximization

Often preferences are described by their utility function or payoff function. This is an ordinal number an individualassigns over the available actions, such as:

u (ai) > u (aj)

The individual’s preferences are then expressed as the relation between these ordinal assignments. For example, if anindividual prefers the candidate Sara over Roger over abstaining, their preferences would have the relation:

u (Sara) > u (Roger) > u (abstain)

A preference relation that as above satisfies completeness, transitivity, and, in addition, continuity, can be equivalentlyrepresented by a utility function.

20.4 Criticism

Both the assumptions and the behavioral predictions of rational choice theory have sparked criticism from vari-ous camps. As mentioned above, some economists have developed models of bounded rationality, which hope tobe more psychologically plausible without completely abandoning the idea that reason underlies decision-makingprocesses. Other economists have developed more theories of human decision-making that allow for the roles ofuncertainty, institutions, and determination of individual tastes by their socioeconomic environment (cf. Fernandez-Huerga, 2008).Martin Hollis and Edward J. Nell's 1975 book offers both a philosophical critique of neo-classical economics andan innovation in the field of economic methodology. Further they outlined an alternative vision to neo-classicismbased on a rationalist theory of knowledge. Within neo-classicism, the authors addressed consumer behaviour (in theform of indifference curves and simple versions of revealed preference theory) and marginalist producer behaviour in

20.4. CRITICISM 103

both product and factor markets. Both are based on rational optimizing behaviour. They consider imperfect as wellas perfect markets since neo-classical thinking embraces many market varieties and disposes of a whole system fortheir classification. However, the authors believe that the issues arising from basic maximizing models have extensiveimplications for econometric methodology (Hollis and Nell, 1975, p. 2). In particular it is this class of models –rational behavior as maximizing behaviour – which provide support for specification and identification. And this,they argue, is where the flaw is to be found. Hollis and Nell (1975) argued that positivism (broadly conceived) hasprovided neo-classicism with important support, which they then show to be unfounded. They base their critiqueof neo-classicism not only on their critique of positivism but also on the alternative they propose, rationalism.[16]Indeed, they argue that rationality is central to neo-classical economics – as rational choice – and that this conceptionof rationality is misused. Demands are made of it that it cannot fulfill.[17]

In their 1994 work, Pathologies of Rational Choice Theory, Donald P. Green and Ian Shapiro argue that the empiricaloutputs of rational choice theory have been limited. They contend that much of the applicable literature, at least inpolitical science, was done with weak statistical methods and that when corrected many of the empirical outcomes nolonger hold. When taken in this perspective, rational choice theory has provided very little to the overall understandingof political interaction - and is an amount certainly disproportionately weak relative to its appearance in the literature.Yet, they concede that cutting edge research, by scholars well-versed in the general scholarship of their fields (suchas work on the U.S. Congress by Keith Krehbiel, Gary Cox, and Mat McCubbins) has generated valuable scientificprogress.[18]

Duncan K. Foley (2003, p. 1) has also provided an important criticism of the concept of rationality and its role ineconomics. He argued that

“Rationality” has played a central role in shaping and establishing the hegemony of contemporarymainstream economics. As the specific claims of robust neoclassicism fade into the history of economicthought, an orientation toward situating explanations of economic phenomena in relation to rationality hasincreasingly become the touchstone by which mainstream economists identify themselves and recognizeeach other. This is not so much a question of adherence to any particular conception of rationality, butof taking rationality of individual behavior as the unquestioned starting point of economic analysis.

Foley (2003, p. 9) went on to argue that

The concept of rationality, to use Hegelian language, represents the relations of modern capitalistsociety one-sidedly. The burden of rational-actor theory is the assertion that ‘naturally’ constituted in-dividuals facing existential conflicts over scarce resources would rationally impose on themselves theinstitutional structures of modern capitalist society, or something approximating them. But this wayof looking at matters systematically neglects the ways in which modern capitalist society and its socialrelations in fact constitute the ‘rational’, calculating individual. The well-known limitations of rational-actor theory, its static quality, its logical antinomies, its vulnerability to arguments of infinite regress, itsfailure to develop a progressive concrete research program, can all be traced to this starting-point.

Schram and Caterino (2006) contains a fundamental methodological criticism of rational choice theory for promotingthe view that the natural science model is the only appropriate methodology in social science and that political scienceshould follow this model, with its emphasis on quantification andmathematization. Schram andCaterino argue insteadfor methodological pluralism. The same argument is made by William E. Connolly, who in his work Neuropoliticsshows that advances in neuroscience further illuminate some of the problematic practices of rational choice theory.More recently Edward J. Nell and Karim Errouaki (2011, Ch. 1) argued that:

The DNA of neoclassical economics is defective. Neither the induction problem nor the problemsof methodological individualism can be solved within the framework of neoclassical assumptions. Theneoclassical approach is to call on rational economic man to solve both. Economic relationships thatreflect rational choice should be ‘projectible’. But that attributes a deductive power to ‘rational’ that itcannot have consistently with positivist (or even pragmatist) assumptions (which require deductions to besimply analytic). To make rational calculations projectible, the agents may be assumed to have idealizedabilities, especially foresight; but then the induction problem is out of reach because the agents of theworld do not resemble those of the model. The agents of the model can be abstract, but they cannot beendowed with powers actual agents could not have. This also undermines methodological individualism;if behaviour cannot be reliably predicted on the basis of the ‘rational choices of agents’, a social ordercannot reliably follow from the choices of agents.

104 CHAPTER 20. RATIONAL CHOICE THEORY

Furthermore, Pierre Bourdieu fiercely opposed rational choice theory as grounded in amisunderstanding of how socialagents operate. Bourdieu argued that social agents do not continuously calculate according to explicit rational andeconomic criteria. According to Bourdieu, social agents operate according to an implicit practical logic—a practicalsense—and bodily dispositions. Social agents act according to their “feel for the game” (the “feel” being, roughly,habitus, and the “game” being the field).[19]

Other social scientists, inspired in part by Bourdieu’s thinking have expressed concern about the inappropriate use ofeconomic metaphors in other contexts, suggesting that this may have political implications. The argument they makeis that by treating everything as a kind of “economy” they make a particular vision of the way an economy worksseem more natural. Thus, they suggest, rational choice is as much ideological as it is scientific, which does not in andof itself negate its scientific utility.[20]

An evolutionary psychology perspective is that many of the seeming contradictions and biases regarding rationalchoice can be explained as being rational in the context of maximizing biological fitness in the ancestral environmentbut not necessarily in the current one. Thus, when living at subsistence level where a reduction of resources may havemeant death it may have been rational to place a greater value on losses than on gains. It may also explain differencesbetween groups such as males being less risk-averse than females since males have more variable reproductive successthan females. While unsuccessful risk-seeking may limit reproductive success for both sexes, males may potentiallyincrease their reproductive success much more than females from successful risk-seeking.[21]

20.5 Benefits

The rational choice approach allows preferences to be represented as real-valued utility functions. Economic decisionmaking then becomes a problem of maximizing this utility function, subject to constraints (e.g. a budget). This hasmany advantages. It provides a compact theory that makes empirical predictions with a relatively sparse model - justa description of the agent’s objectives and constraints. Furthermore, optimization theory is a well-developed fieldof mathematics. These two factors make rational choice models tractable compared to other approaches to choice.Most importantly, this approach is strikingly general. It has been used to analyze not only personal and householdchoices about traditional economic matters like consumption and savings, but also choices about education, marriage,child-bearing, migration, crime and so on, as well as business decisions about output, investment, hiring, entry, exit,etc. with varying degrees of success.Despite the empirical shortcomings of rational choice theory, the flexibility and tractability of rational choice models(and the lack of equally powerful alternatives) lead to them still being widely used.[22]

20.6 See also

• Ecological rationality

• Bounded rationality

• Decision theory

• Homo economicus

• Game theory

• Neoclassical economics

• Positive political theory

• Rational expectations

• Social choice theory

• Preference (economics)

• Reasonable Person Model

20.7. NOTES 105

20.7 Notes[1] • Lawrence E. Blume and David Easley (2008). “rationality,” The New Palgrave Dictionary of Economics , 2nd Edition.

Abstract.” by Abstract] & pre-publication copy.• Amartya Sen (2008). “rational behaviour,” The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.

[2] Susanne Lohmann (2008). “rational choice and political science,”TheNewPalgraveDictionary of Economics, 2nd Edition.Abstract.

[3] Peter Hedström and Charlotta Stern (2008). “rational choice and sociology,” The New Palgrave Dictionary of Economics,2nd Edition. Abstract.

[4] Gary S. Becker (1976). The Economic Approach to Human Behavior. Chicago. Description and scroll to chapter-previewlinks.

[5] Nobel Prize Committee press release

[6] Grüne-Yanoff, Till (2012). “Paradoxes of Rational Choice Theory”. In Sabine Roeser, Rafaela Hillerbrand, Per Sandin,Martin Peterson. Handbook of Risk Theory. pp. 499–516. doi:10.1007/978-94-007-1433-5_19. ISBN 978-94-007-1432-8.

[7] Milton Friedman (1953), Essays in Positive Economics, pp. 15, 22, 31.

[8] Scott, John. “Rational Choice Theory”. Retrieved 2008-07-30.

[9] Dunleavy, Patrick (1991). Democracy, Bureaucracy and Public Choice: Economic Models in Political Science. London:Pearson.

[10] Donald P. Green and Ian Shapiro (1994). Pathologies of Rational Choice Theory: A Critique of Applications in PoliticalScience. Yale University Press.

[11] Friedman, Jeffrey (1996). The Rational Choice Controversy. Yale University Press.

[12] Charles R. Plott and Kathryn Zeiler. 2005. ″The Willingness to Pay--Willingness to Accept Gap, the ′Endowment Effect,′Subject Misconceptions, and Experimental Procedures for Eliciting Valuations.″ American Economic Review 95(3):530.

[13] Charles R. Plott and Kathryn Zeiler. 2007. ″Exchange Asymmetries Incorrectly Interpreted as Evidence of EndowmentEffect Theory and Prospect Theory?″ American Economic Review 97(4): 1449.

[14] Gregory Klass and Kathryn Zeiler. 2013. ″Against Endowment Theory: Experimental Economics and Legal Scholarship.″UCLA Law Review 61:2.

[15] Amos Tversky andDaniel Kahneman. 1991. Loss Aversion in Riskless Choice: AReference-DependentModel.” QuarterlyJournal of Economics 106(4):1039-1061 at 1057-58.

[16] For an in-depth examination of rationality and economic complexity see Foley (1998). For an account of rationality,methodology and ideology see Foley (1989, 2003).

[17] Somewhat surprisingly and independently, Hollis and Nell (1975) and Boland (1982) both use a ‘cross sectional approach’to the understanding of neo-classical economic theory and make similar points about the foundations of neo-classicism.For an account see Nell, E.J. and Errouaki, K (2011)

[18] Donald P. Green and Ian Shapiro (1994). Pathologies of Rational Choice Theory: A Critique of Applications in PoliticalScience. Yale University Press.

[19] For an account of Bourdieu work see the wikipedia article on Pierre Bourdieu. See also Pierre Bourdieu (2005) The SocialStructures of the Economy, Polity 2005.

[20] McKinnon, AM. (2013). 'Ideology and theMarket Metaphor in Rational Choice Theory of Religion: A Rhetorical Critiqueof “Religious Economies”'. Critical Sociology, vol 39, no. 4, pp. 529-543.

[21] Paul H. Rubin and C. Monica Capra. The evolutionary psychology of economics. In Roberts, S. C. (2011). Roberts, S.Craig, ed. “Applied Evolutionary Psychology”. Oxford University Press. doi:10.1093/acprof:oso/9780199586073.001.0001. ISBN 9780199586073.

[22] Milgrom, Paul; Levin, Jonathan. “Introduction to Choice Theory” (PDF).web.stanford.edu. Stanford University. Retrieved2015-03-03.

106 CHAPTER 20. RATIONAL CHOICE THEORY

20.8 References

• Abella, Alex (2008). Soldiers of Reason: The RAND Corporation and the Rise of the American Empire. NewYork: Harcourt.

• Allingham, Michael (2002). Choice Theory: A Very Short Introduction, Oxford, ISBN 978-0192803030.

• Anand, P. (1993)."Foundations of Rational Choice Under Risk”, Oxford: Oxford University Press.

• Amadae, S.M.(2003). Rationalizing Capitalist Democracy: The Cold War Origins of Rational Choice Liberal-ism, Chicago: University of Chicago Press.

• Arrow, Kenneth J. ([1987] 1989). “Economic Theory and the Hypothesis of Rationality,” in The New Palgrave:Utility and Probability, pp. 25-39.

• Bicchieri, Cristina (1993). Rationality and Coordination. Cambridge University Press

• Bicchieri, Cristina (2003). “Rationality and Game Theory”, in The Handbook of Rationality, The OxfordReference Library of Philosophy, Oxford University Press.

• Downs, Anthony (1957). “An Economic Theory of Democracy.” Harper.

• Coleman, James S. (1990). Foundations of Social Theory

• Dixon, Huw (2001), Surfing Economics, Pearson. Especially chapters 7 and 8

• Elster, Jon (1979). Ulysses and the Sirens, Cambridge University Press.

• Elster, Jon (1989). Nuts and Bolts for the Social Sciences, Cambridge University Press.

• Elster, Jon (2007). Explaining Social Behavior - more Nuts and Bolts for the Social Sciences, Cambridge Uni-versity Press.

• Fernandez-Huerga (2008.) The Economic Behavior of Human Beings: The Institutionalist//Post-KeynesianModel” Journal of Economic Issues. vol. 42 no. 3, September.

• Schram, Sanford F. and Brian Caterino, eds. (2006). Making Political Science Matter: Debating Knowledge,Research, and Method. New York and London: New York University Press.

• Walsh, Vivian (1996). Rationality, Allocation, and Reproduction, Oxford. Description and scroll to chapter-preview links.

• Martin Hollis and Edward J. Nell (1975) Rational Economic Man. Cambridge: Cambridge University Press.

• Foley, D. K. (1989) Ideology and Methodology. An unpublished lecture to Berkeley graduate students in 1989discussing personal and collective survival strategies for non-mainstream economists.

• Foley, D.K. (1998). Introduction (chapter 1) in Peter S. Albin, Barriers and Bounds to Rationality: Essays onEconomic Complexity and Dynamics in Interactive Systems. Princeton: Princeton University Press.

• Foley, D. K. (2003) Rationality and Ideology in Economics. lecture in the World Political Economy course atthe Graduate Faculty of New School UM, New School.

• Boland, L. (1982) The Foundations of Economic Method. London: George Allen & Unwin

• Edward J. Nell and Errouaki, K. (2011) Rational Econometric Man. Cheltenham: E. Elgar.

• Pierre Bourdieu (2005) The Social Structures of the Economy, Polity 2005

• Calhoun, C. et al. (1992) “Pierre Bourdieu: Critical Perspectives.” University of Chicago Press.

• Grenfell, M (2011) “Bourdieu, Language and Linguistics” London, Continuum.

• Grenfell, M. (ed) (2008) “Pierre Bourdieu: Key concepts” London, Acumen Press

20.9. EXTERNAL LINKS 107

20.9 External links• Rational Choice Theory at the Stanford Encyclopedia of Philosophy

• Rational Choice Theory - Article by John Scott

• The New Nostradamus - on the use by Bruce Bueno de Mesquita of rational choice theory in political fore-casting

• To See The Future, Use The Logic Of Self-Interest - NPR audio clip

Chapter 21

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In otherwords, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when ∀x∈S: x~x holds.[1][2]An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number isequal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.

21.1 Related terms

A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. Anexample is the “greater than” relation (x>y) on the real numbers. Note that not every relation which is not reflexiveis irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e.,neither all nor none are). For example, the binary relation “the product of x and y is even” is reflexive on the set ofeven numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is also related to itself,formally: if ∀x,y∈S: x~y ⇒ x~x ∧ y~y. An example is the relation “has the same limit as” on the set of sequences ofreal numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the samelimit as some sequence, then it has the same limit as itself.The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~.Equivalently, it is the union of ~ and the identity relation on S, formally: (≃) = (~) ∪ (=). For example, the reflexiveclosure of x<y is x≤y.The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalentto the complement of the identity relation on S with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.

21.2 Examples

Examples of reflexive relations include:

• “is equal to” (equality)

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “is greater than or equal to”

• “is less than or equal to”

Examples of irreflexive relations include:

108

21.3. NUMBER OF REFLEXIVE RELATIONS 109

• “is not equal to”

• “is coprime to” (for the integers>1, since 1 is coprime to itself)

• “is a proper subset of”

• “is greater than”

• “is less than”

21.3 Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2−n.[3]

110 CHAPTER 21. REFLEXIVE RELATION

21.4 Philosophical logic

Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in themathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.[4][5]

21.5 See also

• Binary relation

• Symmetric relation

• Transitive relation

21.6. NOTES 111

• Coreflexive relation

21.6 Notes[1] Levy 1979:74

[2] Relational Mathematics, 2010

[3] On-Line Encyclopedia of Integer Sequences A053763

[4] Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy—AModern Introduction. Wadsworth. ISBN1-133-05000-X. Here: p.327-328

[5] D.S. Clarke, Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory.University Press of America. ISBN 0-7618-0922-8. Here: p.187

21.7 References• Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002,Dover. ISBN 0-486-42079-5

• Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag.ISBN 0-387-98290-6

• Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN0-674-55451-5

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

21.8 External links• Hazewinkel, Michiel, ed. (2001), “Reflexivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 22

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.The binary operation of a semigroup is most often denoted multiplicatively: x·y, or simply xy, denotes the result ofapplying the semigroup operation to the ordered pair (x, y). Associativity is formally expressed as that (x·y)·z = x·(y·z)for all x, y and z in the semigroup.The name “semigroup” originates in the fact that a semigroup generalizes a group by preserving only associativityand closure under the binary operation from the axioms defining a group.[note 1] From the opposite point of view (ofadding rather than removing axioms), a semigroup is an associative magma. As in the case of groups or magmas,the semigroup operation need not be commutative, so x·y is not necessarily equal to y·x; a typical example of asso-ciative but non-commutative operation is matrix multiplication. If the semigroup operation is commutative, then thesemigroup is called a commutative semigroup or (less often than in the analogous case of groups) it may be called anabelian semigroup.Amonoid is an algebraic structure intermediate between groups and semigroups, and is a semigroup having an identityelement, thus obeying all but one of the axioms of a group; existence of inverses is not required of a monoid. A naturalexample is strings with concatenation as the binary operation, and the empty string as the identity element. Restrictingto non-empty strings gives an example of a semigroup that is not a monoid. Positive integers with addition form acommutative semigroup that is not a monoid. A semigroup without an identity element can be easily turned into amonoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups ratherthan in group theory. Semigroups should not be confused with quasigroups, which are a generalization of groups ina different direction; the operation in a quasigroup need not be associative but quasigroups preserve from groups anotion of division. Division in semigroups (or in monoids) is not possible in general.The formal study of semigroups began in the early 20th century. Early results include a Cayley theorem for semi-groups realizing any semigroup as transformation semigroup, in which arbitrary functions replace the role of bijec-tions from group theory. Other fundamental techniques of studying semigroups like Green’s relations do not imitateanything in group theory though. A deep result in the classification of finite semigroups is Krohn–Rhodes theory.The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950sbecause of the natural link between finite semigroups and finite automata via the syntactic monoid. In probabilitytheory, semigroups are associated with Markov processes.[1] In other areas of applied mathematics, semigroups arefundamental models for linear time-invariant systems. In partial differential equations, a semigroup is associated toany equation whose spatial evolution is independent of time. There numerous special classes of semigroups, semi-groups with additional properties, which appear in particular applications. Some of these classes are even closer togroups by exhibiting some additional but not all properties of a group. Of these we mention: regular semigroups,orthodox semigroups, semigroups with involution, inverse semigroups and cancellative semigroups. There also in-teresting classes of semigroups that do not contain any groups except the trivial group; examples of the latter kindare bands and their commutative subclass—semilattices, which are also ordered algebraic structures.

22.1 Definition

A semigroup is a set S together with a binary operation " · " (that is, a function · : S × S → S ) that satisfies theassociative property:

112

22.2. EXAMPLES OF SEMIGROUPS 113

For all a, b, c ∈ S , the equation (a · b) · c = a · (b · c) holds.

More succinctly, a semigroup is an associative magma.

22.2 Examples of semigroups• Empty semigroup: the empty set forms a semigroup with the empty function as the binary operation.

• Semigroup with one element: there is essentially only one, the singleton {a} with operation a · a = a.

• Semigroup with two elements: there are five which are essentially different.

• The set of positive integers with addition. (With 0 included, this becomes a monoid.)

• The set of integers with minimum or maximum. (With positive/negative infinity included, this becomes amonoid.)

• Square nonnegative matrices of a given size with matrix multiplication.

• Any ideal of a ring with the multiplication of the ring.

• The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation— the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the freemonoid over Σ.

• A probability distribution F together with all convolution powers of F, with convolution as the operation. Thisis called a convolution semigroup.

• A monoid is a semigroup with an identity element.

• A group is a monoid in which every element has an inverse element.

• Transformation semigroups and monoids

• The set of continuous functions from a topological space to itself

22.3 Basic concepts

22.3.1 Identity and zero

If it has both a left identity and a right identity, a semigroup (and indeed magma) has at most one identity element,which is then two-sided. A semigroup with identity is called a monoid. A semigroup may have multiple left identitiesbut no right identity,[note 2] or vice versa. A semigroup without identity may be embedded in a monoid formed byadjoining an element e /∈ S to S and defining e · s = s · e = s for all s ∈ S ∪ {e} .[2][3] The notation S1 denotes amonoid obtained from S by adjoining an identity if necessary (S1 = S for a monoid).[3]

Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero. Analogousto the above construction, for every semigroup S, one can define S0, a semigroup with 0 that embeds S.

22.3.2 Subsemigroups and ideals

The semigroup operation induces an operation on the collection of its subsets: given subsets A and B of a semigroupS, their product A · B, written commonly as AB, is the set { ab | a in A and b in B }. (This notion is defined identicallyas it is for groups.) In terms of this operations, a subset A is called

• a subsemigroup if AA is a subset of A,

• a right ideal if AS is a subset of A, and

• a left ideal if SA is a subset of A.

114 CHAPTER 22. SEMIGROUP

If A is both a left ideal and a right ideal then it is called an ideal (or a two-sided ideal).If S is a semigroup, then the intersection of any collection of subsemigroups of S is also a subsemigroup of S. So thesubsemigroups of S form a complete lattice.An example of semigroup with no minimal ideal is the set of positive integers under addition. The minimal ideal ofa commutative semigroup, when it exists, is a group.Green’s relations, a set of five equivalence relations that characterise the elements in terms of the principal ideals theygenerate, are important tools for analysing the ideals of a semigroup and related notions of structure.The subset with the property that its every element commutes with any other element of the semigroup is called thecenter of the semigroup.[4] The center of a semigroup is actually a subsemigroup.[5]

22.3.3 Homomorphisms and congruences

A semigroup homomorphism is a function that preserves semigroup structure. A function f: S → T between twosemigroups is a homomorphism if the equation

f(ab) = f(a)f(b).

holds for all elements a, b in S, i.e. the result is the same when performing the semigroup operation after or beforeapplying the map f.A semigroup homomorphism between monoids preserves identity if it is a monoid homomorphism. But there aresemigroup homomorphisms which are not monoid homomorphisms, e.g. the canonical embedding of a semigroup Swithout identity into S1 . Conditions characterizing monoid homomorphisms are discussed further. Let f : S0 → S1

be a semigroup homomorphism. The image of f is also a semigroup. If S0 is a monoid with an identity element e0, then f(e0) is the identity element in the image of f . If S1 is also a monoid with an identity element e1 and e1belongs to the image of f , then f(e0) = e1 , i.e. f is a monoid homomorphism. Particularly, if f is surjective, thenit is a monoid homomorphism.Two semigroups S and T are said to be isomorphic if there is a bijection f : S ↔ T with the property that, for anyelements a, b in S, f(ab) = f(a)f(b). Isomorphic semigroups have the same structure.A semigroup congruence ∼ is an equivalence relation that is compatible with the semigroup operation. That is, asubset ∼ ⊆ S × S that is an equivalence relation and x ∼ y and u ∼ v implies xu ∼ yv for every x, y, u, v in S.Like any equivalence relation, a semigroup congruence ∼ induces congruence classes

[a]∼ = {x ∈ S| x ∼ a}

and the semigroup operation induces a binary operation ◦ on the congruence classes:

[u]∼ ◦ [v]∼ = [uv]∼

Because ∼ is a congruence, the set of all congruence classes of ∼ forms a semigroup with ◦ , called the quotientsemigroup or factor semigroup, and denoted S/ ∼ . The mapping x 7→ [x]∼ is a semigroup homomorphism, calledthe quotient map, canonical surjection or projection; if S is a monoid then quotient semigroup is a monoid withidentity [1]∼ . Conversely, the kernel of any semigroup homomorphism is a semigroup congruence. These resultsare nothing more than a particularization of the first isomorphism theorem in universal algebra. Congruence classesand factor monoids are the objects of study in string rewriting systems.A nuclear congruence on S is one which is the kernel of an endomorphism of S.[6]

A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered byinclusion, has a maximal element. By Zorn’s lemma, this is equivalent to saying that the ascending chain conditionholds: there is no infinite strictly ascending chain of congruences on S.[7]

Every ideal I of a semigroup induces a subsemigroup, the Rees factor semigroup via the congruence x ρ y ⇔ eitherx = y or both x and y are in I.

22.4. STRUCTURE OF SEMIGROUPS 115

22.4 Structure of semigroups

For any subset A of S there is a smallest subsemigroup T of S which contains A, and we say that A generates T. Asingle element x of S generates the subsemigroup { xn | n is a positive integer }. If this is finite, then x is said to be offinite order, otherwise it is of infinite order. A semigroup is said to be periodic if all of its elements are of finiteorder. A semigroup generated by a single element is said to be monogenic (or cyclic). If a monogenic semigroup isinfinite then it is isomorphic to the semigroup of positive integers with the operation of addition. If it is finite andnonempty, then it must contain at least one idempotent. It follows that every nonempty periodic semigroup has atleast one idempotent.A subsemigroup which is also a group is called a subgroup. There is a close relationship between the subgroups of asemigroup and its idempotents. Each subgroup contains exactly one idempotent, namely the identity element of thesubgroup. For each idempotent e of the semigroup there is a unique maximal subgroup containing e. Each maximalsubgroup arises in this way, so there is a one-to-one correspondence between idempotents and maximal subgroups.Here the term maximal subgroup differs from its standard use in group theory.More can often be said when the order is finite. For example, every nonempty finite semigroup is periodic, and hasa minimal ideal and at least one idempotent. The number of finite semigroups of a given size (greater than 1) is(obviously) larger than the number of groups of the same size. For example, of the sixteen possible “multiplicationtables” for a set of two elements {a, b}, eight form semigroups[note 3] whereas only four of these are monoids and onlytwo form groups. For more on the structure of finite semigroups, see Krohn–Rhodes theory.

22.5 Special classes of semigroups

Main article: Special classes of semigroups

• A monoid is a semigroup with identity.

• A subsemigroup is a subset of a semigroup that is closed under the semigroup operation.

• A band is a semigroup the operation of which is idempotent.

• A cancellative semigroup is one having the cancellation property:[8] a · b = a · c implies b = c and similarly forb · a = c · a.

• A semilattice is a semigroup whose operation is idempotent and commutative.

• 0-simple semigroups.

• Transformation semigroups: any finite semigroup S can be represented by transformations of a (state-) set Qof at most |S| + 1 states. Each element x of S then maps Q into itself x: Q → Q and sequence xy is definedby q(xy) = (qx)y for each q in Q. Sequencing clearly is an associative operation, here equivalent to functioncomposition. This representation is basic for any automaton or finite state machine (FSM).

• The bicyclic semigroup is in fact a monoid, which can be described as the free semigroup on two generators pand q, under the relation pq = 1.

• C0-semigroups.

• Regular semigroups. Every element x has at least one inverse y satisfying xyx=x and yxy=y; the elements x andy are sometimes called “mutually inverse”.

• Inverse semigroups are regular semigroups where every element has exactly one inverse. Alternatively, a regularsemigroup is inverse if and only if any two idempotents commute.

• Affine semigroup: a semigroup that is isomorphic to a finitely-generated subsemigroup of Zd. These semigroupshave applications to commutative algebra.

116 CHAPTER 22. SEMIGROUP

22.6 Structure theorem for commutative semigroups

There is a structure theorem for commutative semigroups in terms of semilattices.[9] A semilattice (or more preciselya meet-semilattice) (L,≤) is a partially ordered set where every pair of elements a, b ∈ L has a greatest lower bound,denoted a ∧ b . The operation ∧ makes L into a semigroup satisfying the additional idempotence law a ∧ a = a .Given a homomorphism f : S → L from an arbitrary semigroup to a semilattice, each inverse image Sa = f−1{a}is a (possibly empty) semigroup. Moreover S becomes graded by L , in the sense thatSaSb ⊆ Sa∧b

If f is onto, the semilattice L is isomorphic to the quotient of S by the equivalence relation ∼ such that x ∼ y ifff(x) = f(y) . This equivalence relation is a semigroup congruence, as defined above.Whenever we take the quotient of a commutative semigroup by a congruence, we get another commutative semigroup.The structure theorem says that for any commutative semigroup S , there is a finest congruence ∼ such that thequotient of S by this equivalence relation is a semilattice. Denoting this semilattice by L , we get a homomorphismf from S onto L . As mentioned, S becomes graded by this semilattice.Furthermore, the components Sa are all Archimedean semigroups. An Archimedean semigroup is one where givenany pair of elements x, y , there exists an element z and n > 0 such that xn = yz .The Archimedean property follows immediately from the ordering in the semilattice L , since with this ordering wehave f(x) ≤ f(y) if and only if xn = yz for some z and n > 0 .

22.7 Group of fractions

The group of fractions or group completion of a semigroup S is the group G = G(S) generated by the elementsof S as generators and all equations xy = z which hold true in S as relations.[10] There is an obvious semigrouphomomorphism j : S → G(S) which sends each element of S to the corresponding generator. This has a universalproperty for morphisms from S to a group:[11] given any groupH and any semigroup homomorphism k : S→H, thereexists a unique group homomorphism f : G → H with k=fj. We may think of G as the “most general” group thatcontains a homomorphic image of S.An important question is to characterize those semigroups for which this map is an embedding. This need not alwaysbe the case: for example, take S to be the semigroup of subsets of some set X with set-theoretic intersection as thebinary operation (this is an example of a semilattice). Since A.A = A holds for all elements of S, this must be truefor all generators of G(S) as well: which is therefore the trivial group. It is clearly necessary for embeddability that Shave the cancellation property. When S is commutative this condition is also sufficient[12] and the Grothendieck groupof the semigroup provides a construction of the group of fractions. The problem for non-commutative semigroupscan be traced to the first substantial paper on semigroups.[13][14] Anatoly Maltsev gave necessary and conditions forembeddability in 1937.[15]

22.8 Semigroup methods in partial differential equations

Further information: C0-semigroup

Semigroup theory can be used to study some problems in the field of partial differential equations. Roughly speaking,the semigroup approach is to regard a time-dependent partial differential equation as an ordinary differential equationon a function space. For example, consider the following initial/boundary value problem for the heat equation on thespatial interval (0, 1) ⊂ R and times t ≥ 0:

∂tu(t, x) = ∂2xu(t, x), x ∈ (0, 1), t > 0;

u(t, x) = 0, x ∈ {0, 1}, t > 0;

u(t, x) = u0(x), x ∈ (0, 1), t = 0.

Let X = L2((0, 1) R) be the Lp space of square-integrable real-valued functions with domain the interval (0, 1) andlet A be the second-derivative operator with domain

22.9. HISTORY 117

D(A) ={u ∈ H2((0, 1);R)

∣∣u(0) = u(1) = 0},

where H2 is a Hardy space. Then the above initial/boundary value problem can be interpreted as an initial valueproblem for an ordinary differential equation on the space X:

{u̇(t) = Au(t);

u(0) = u0.

On an heuristic level, the solution to this problem “ought” to be u(t) = exp(tA)u0. However, for a rigorous treatment,a meaning must be given to the exponential of tA. As a function of t, exp(tA) is a semigroup of operators from X toitself, taking the initial state u0 at time t = 0 to the state u(t) = exp(tA)u0 at time t. The operator A is said to be theinfinitesimal generator of the semigroup.

22.9 History

The study of semigroups trailed behind that of other algebraic structures with more complex axioms such as groupsor rings. A number of sources[16][17] attribute the first use of the term (in French) to J.-A. de Séguier in Élements dela Théorie des Groupes Abstraits (Elements of the Theory of Abstract Groups) in 1904. The term is used in Englishin 1908 in Harold Hinton’s Theory of Groups of Finite Order.Anton Suschkewitsch obtained the first non-trivial results about semigroups. His 1928 paper Über die endlichenGruppen ohne das Gesetz der eindeutigen Umkehrbarkeit (On finite groups without the rule of unique invertibility)determined the structure of finite simple semigroups and showed that the minimal ideal (or Green’s relations J-class)of a finite semigroup is simple.[17] From that point on, the foundations of semigroup theory were further laid by DavidRees, James Alexander Green, Evgenii Sergeevich Lyapin, Alfred H. Clifford and Gordon Preston. The latter twopublished a two-volume monograph on semigroup theory in 1961 and 1967 respectively. In 1970, a new periodicalcalled Semigroup Forum (currently edited by Springer Verlag) became one of the few mathematical journals devotedentirely to semigroup theory.In recent years researchers in the field have become more specialized with dedicated monographs appearing on im-portant classes of semigroups, like inverse semigroups, as well as monographs focusing on applications in algebraicautomata theory, particularly for finite automata, and also in functional analysis.

22.10 Generalizations

If the associativity axiom of a semigroup is dropped, the result is a magma, which is nothing more than a set Mequipped with a binary operation M × M → M.Generalizing in a different direction, an n-ary semigroup (also n-semigroup, polyadic semigroup or multiarysemigroup) is a generalization of a semigroup to a set G with a n-ary operation instead of a binary operation.[18] Theassociative law is generalized as follows: ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcdewith any three adjacent elements bracketed. N-ary associativity is a string of length n + (n − 1) with any n adjacentelements bracketed. A 2-ary semigroup is just a semigroup. Further axioms lead to an n-ary group.A third generalization is the semigroupoid, in which the requirement that the binary relation be total is lifted. Ascategories generalize monoids in the same way, a semigroupoid behaves much like a category but lacks identities.Infinitary generalizations of commutative semigroups have sometimes been considered by various authors. [19]

22.11 See also

• Absorbing element

• Biordered set

118 CHAPTER 22. SEMIGROUP

• Empty semigroup

• Identity element

• Light’s associativity test

• Semigroup ring

• Weak inverse

• Quantum dynamical semigroup

22.12 Notes[1] The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three

laws for a group and only one law (associativity) for semigroup.

[2] For instance, the semigroup of three elements e, f, g with, for any x, ex = x, fx = f, and gx = f has exactly one left identitybut no right identity.

[3] Namely: the trivial semigroup in which (for all x and y) xy = a and its counterpart in which xy = b, the semigroups based onmultiplication modulo 2 (choosing a or b as the identity element 1), the groups equivalent to addition modulo 2 (choosinga or b to be the identity element 0), and the semigroups in which the elements are either both left identities or both rightidentities.

22.13 Citations[1] (Feller 1971)

[2] Jacobson 2009, p. 30, ex. 5

[3] Lawson 1998, p. 20

[4] Kilp, Mati; Knauer, U.; Mikhalev, Aleksandr V. (2000). Monoids, Acts, and Categories: With Applications to WreathProducts and Graphs : a Handbook for Students and Researchers. Walter de Gruyter. p. 25. ISBN 978-3-11-015248-7.Zbl 0945.20036.

[5] Lia͡pin, E. S. (1968). Semigroups. American Mathematical Soc. p. 96. ISBN 978-0-8218-8641-0.

[6] Lothaire 2011, p. 463

[7] Lothaire 2011, p. 465

[8] Clifford & Preston 1967, p. 3

[9] Grillet 2001

[10] Farb, B. (2006), Problems on mapping class groups and related topics, Amer. Math. Soc., p. 357, ISBN 0-8218-3838-5

[11] Auslander, M.; Buchsbaum, D. A. (1974). Groups, rings, modules. Harper & Row. p. 50. ISBN 0-06-040387-X.

[12] Clifford & Preston 1961, p. 34

[13] (Suschkewitsch 1928)

[14] Preston, G. B. (1990), Personal reminiscences of the early history of semigroups, retrieved 2009-05-12

[15] Maltsev, A. (1937), “On the immersion of an algebraic ring into a field”,Math. Annalen 113: 686–691, doi:10.1007/BF01571659.

[16] Earliest Known Uses of Some of the Words of Mathematics

[17] An account of Suschkewitsch’s paper by Christopher Hollings

[18] Dudek, W.A. (2001), “On some old problems in n-ary groups”, Quasigroups and Related Systems 8: 15–36

[19] See references in Udo Hebisch and Hanns Joachim Weinert, Semirings and Semifields, in particular, Section 10, Semiringswith infinite sums, in M. Hazewinkel, Handbook of Algebra, Vol. 1, Elsevier, 1996. Notice that in this context the authorsuse the term semimodule in place of semigroup.

22.14. REFERENCES 119

22.14 ReferencesGeneral references

• Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9, Zbl0835.20077.

• Clifford, A. H.; Preston, G. B. (1961), The Algebraic Theory of Semigroups 1, AmericanMathematical Society,ISBN 978-0-8218-0271-7, Zbl 0111.03403.

• Clifford, A. H.; Preston, G. B. (1967), The Algebraic Theory of Semigroups 2, AmericanMathematical Society,ISBN 978-0-8218-0272-4, Zbl 0178.01203.

• Grillet, Pierre A. (1995), Semigroups: An Introduction to the Structure Theory, Marcel Dekker, ISBN 978-0-8247-9662-4, Zbl 0830.20079.

• Grillet, PierreA. (2001),Commutative Semigroups, Springer Verlag, ISBN978-0-7923-7067-3, Zbl 1040.20048.

• Hollings, Christopher (2014),Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semi-groups, American Mathematical Society, ISBN 978-1-4704-1493-1, Zbl 06329297.

• Petrich, Mario (1973), Introduction to Semigroups, Charles E. Merrill, ISBN 0-675-09062-8, Zbl 0321.20037.

Specific references

• Feller, William (1971), An introduction to probability theory and its applications II (2nd ed.), Wiley, MR0270403.

• Hille, Einar; Phillips, Ralph S. (1974), Functional analysis and semi-groups, American Mathematical Society,ISBN 0821874640, MR 0423094.

• Suschkewitsch, Anton (1928), "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit”,Mathematische Annalen 99 (1): 30–50, doi:10.1007/BF01459084, ISSN 0025-5831, MR 1512437.

• Kantorovitz, Shmuel (2009), Topics in Operator Semigroups, Springer, ISBN978-0-8176-4932-6, Zbl 1187.47003.

• Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

• Lawson, M.V. (1998), Inverse semigroups: the theory of partial symmetries, World Scientific, ISBN 978-981-02-3316-7, Zbl 1079.20505

• Lothaire, M. (2011) [2002], Algebraic combinatorics on words, Encyclopedia of Mathematics and Its Appli-cations 90, Cambridge University Press, ISBN 978-0-521-18071-9, Zbl 1221.68183

Chapter 23

Set (mathematics)

This article is about what mathematicians call “intuitive” or “naive” set theory. For a more detailed account, see Naiveset theory. For a rigorous modern axiomatic treatment of sets, see Set theory.In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example,

A set of polygons in a Venn diagram

the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectivelythey form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics.Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as afoundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics suchas Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

120

23.1. DEFINITION 121

The German word Menge, rendered as “set” in English, was coined by Bernard Bolzano in his work The Paradoxesof the Infinite.

23.1 Definition

Passage with the original set definition of Georg Cantor

A set is a well defined collection of distinct objects. The objects that make up a set (also known as the elements ormembers of a set) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor,the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung dertransfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung]or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the sameelements.[2]

There is the image popular, that sets are like boxes containing their elements. But there is a huge difference betweenboxes and sets. While boxes don't change their identity when objects are removed from or added to them, sets changetheir identity when their elements change. So its better to have the image of a set as the content of an imaginary box:

• A set of polygons

• The same set as a box

• The set as the content of a box

Cantor’s definition turned out to be inadequate for formal mathematics; instead, the notion of a “set” is taken as anundefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. Themost basic properties are that a set has elements, and that two sets are equal (one and the same) if and only if everyelement of each set is an element of the other.

23.2 Describing sets

Main article: Set notation

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using arule or semantic description:

122 CHAPTER 23. SET (MATHEMATICS)

A is the set whose members are the first four positive integers.B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted byenclosing the list of members in curly brackets:

C = {4, 2, 1, 3}D = {blue, white, red}.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance,A = C and B = D.There are two important points to note about sets. First, a set can have two or more members which are identical, forexample, {11, 6, 6}. However, we say that two sets which differ only in that one has duplicate members are in factexactly identical (see Axiom of extensionality). Hence, the set {11, 6, 6} is exactly identical to the set {11, 6}. Thesecond important point is that the order in which the elements of a set are listed is irrelevant (unlike for a sequenceor tuple). We can illustrate these two important points with an example:

{6, 11} = {11, 6} = {11, 6, 6, 11} .

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the firstthousand positive integers may be specified extensionally as

{1, 2, 3, ..., 1000},

where the ellipsis ("...”) indicates that the list continues in the obvious way. Ellipses may also be used where sets haveinfinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }.The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have themeaning “the set of all ...”. So, E = {playing card suits} is the set whose four members are ♠, ♦, ♥, and ♣. A moregeneral form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers thatare four less than perfect squares can be denoted

F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}.

In this notation, the colon (":") means “such that”, and the description can be interpreted as "F is the set of all numbersof the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive.” Sometimes the vertical bar("|") is used instead of the colon.

23.3 Membership

Main article: Element (mathematics)

If B is a set and x is one of the objects of B, this is denoted x ∈ B, and is read as “x belongs to B”, or “x is an elementof B”. If y is not a member of B then this is written as y ∉ B, and is read as “y does not belong to B”.For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer; and 0≤ n ≤ 19} defined above,

4 ∈ A and 12 ∈ F; but9 ∉ F and green ∉ B.

23.3. MEMBERSHIP 123

23.3.1 Subsets

Main article: Subset

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (also pronouncedA is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. Therelationship between sets established by ⊆ is called inclusion or containment.If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B)or B ⊋ A (B is a proper superset of A).Note that the expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to meanthe same as A ⊆ B (respectively B ⊇ A), whereas other use them to mean the same as A ⊊ B (respectively B ⊋ A).

AB

A is a subset of B

Example:

• The set of all men is a proper subset of the set of all people.• {1, 3} ⊆ {1, 2, 3, 4}.• {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

124 CHAPTER 23. SET (MATHEMATICS)

The empty set is a subset of every set and every set is a subset of itself:

• ∅ ⊆ A.• A ⊆ A.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

• A = B if and only if A ⊆ B and B ⊆ A.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

23.3.2 Power sets

Main article: Power set

The power set of a set S is the set of all subsets of S. Note that the power set contains S itself and the empty setbecause these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2,3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).The power set of a finite set with n elements has 2n elements. This relationship is one of the reasons for the terminologypower set. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8elements.The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set ofa set is always strictly “bigger” than the original set in the sense that there is no way to pair every element of S withexactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)Every partition of a set S is a subset of the powerset of S.

23.4 Cardinality

Main article: Cardinality

The cardinality | S | of a set S is “the number of members of S.” For example, if B = {blue, white, red}, | B | = 3.There is a unique set with no members and zero cardinality, which is called the empty set (or the null set) and isdenoted by the symbol ∅ (other notations are used; see empty set). For example, the set of all three-sided squares haszero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is importantin mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalitiesare greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers.However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the sameas the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclideanspace.

23.5 Special sets

There are some sets that hold great mathematical importance and are referred to with such regularity that they haveacquired special names and notational conventions to identify them. One of these is the empty set, denoted {} or ∅.Another is the unit set {x}, which contains exactly one element, namely x.[2] Many of these sets are represented usingblackboard bold or bold typeface. Special sets of numbers include

• P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.

• N or ℕ, denoting the set of all natural numbers: N = {1, 2, 3, . . .} (sometimes defined containing 0).

23.6. BASIC OPERATIONS 125

• Z or ℤ, denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.

• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b: a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can beexpressed as the fraction a/1 (Z ⊊ Q).

• R or ℝ, denoting the set of all real numbers. This set includes all rational numbers, together with all irrationalnumbers (that is, numbers that cannot be rewritten as fractions, such as √2, as well as transcendental numberssuch as π, e and numbers that cannot be defined).

• C or ℂ, denoting the set of all complex numbers: C = {a + bi : a, b ∈ R}. For example, 1 + 2i ∈ C.

• H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk : a, b, c, d ∈ R}. For example, 1 + i + 2j −k ∈ H.

Positive and negative sets are denoted by a superscript - or +. For example ℚ+ represents the set of positive rationalnumbers.Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a propersubset of the sets listed below it. The primes are used less frequently than the others outside of number theory andrelated fields.

23.6 Basic operations

There are several fundamental operations for constructing new sets from given sets.

23.6.1 Unions

The union of A and B, denoted A ∪ B

126 CHAPTER 23. SET (MATHEMATICS)

Main article: Union (set theory)

Two sets can be “added” together. The union of A and B, denoted by A ∪ B, is the set of all things that are membersof either A or B.Examples:

• {1, 2} ∪ {1, 2} = {1, 2}.• {1, 2} ∪ {2, 3} = {1, 2, 3}.• {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

• A ∪ B = B ∪ A.• A ∪ (B ∪ C) = (A ∪ B) ∪ C.• A ⊆ (A ∪ B).• A ∪ A = A.• A ∪ ∅ = A.• A ⊆ B if and only if A ∪ B = B.

23.6.2 Intersections

Main article: Intersection (set theory)

A new set can also be constructed by determining which members two sets have “in common”. The intersection of Aand B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B aresaid to be disjoint.Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.• {1, 2} ∩ {2, 3} = {2}.

Some basic properties of intersections:

• A ∩ B = B ∩ A.• A ∩ (B ∩ C) = (A ∩ B) ∩ C.• A ∩ B ⊆ A.• A ∩ A = A.• A ∩ ∅ = ∅.• A ⊆ B if and only if A ∩ B = A.

23.6.3 Complements

Main article: Complement (set theory)

Two sets can also be “subtracted”. The relative complement of B in A (also called the set-theoretic difference of A andB), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. Note that itis valid to “subtract” members of a set that are not in the set, such as removing the element green from the set {1, 2,3}; doing so has no effect.In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \A is called the absolute complement or simply complement of A, and is denoted by A′.Examples:

23.6. BASIC OPERATIONS 127

The intersection of A and B, denoted A ∩ B.

• {1, 2} \ {1, 2} = ∅.

• {1, 2, 3, 4} \ {1, 3} = {2, 4}.

• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E= E′ = O.

Some basic properties of complements:

• A \ B ≠ B \ A for A ≠ B.

• A ∪ A′ = U.

• A ∩ A′ = ∅.

• (A′)′ = A.

• A \ A = ∅.

• U′ = ∅ and ∅′ = U.

• A \ B = A ∩ B′.

An extension of the complement is the symmetric difference, defined for sets A, B as

A∆B = (A \B) ∪ (B \A).

For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set {7,8,11,12}.

128 CHAPTER 23. SET (MATHEMATICS)

The relative complementof B in A

23.6.4 Cartesian product

Main article: Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesianproduct of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A andb is a member of B.Examples:

• {1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.• {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green) }.• {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of cartesian products:

• A × ∅ = ∅.• A × (B ∪ C) = (A × B) ∪ (A × C).• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets. Then

• | A × B | = | B × A | = | A | × | B |.

For example,

• {a,b,c}×{d,e,f}={(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}.

23.7. APPLICATIONS 129

The complement of A in U

23.7 Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structuresin abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomainB is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs(x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of allsquares is subset of the set of all reals. If placed in functional notation, this relation becomes f(x) = x2. The reasonthese two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y,y2) is a member of the set F.

23.8 Axiomatic set theory

Main article: Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, itsoon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

• Russell’s paradox—It shows that the “set of all sets that do not contain themselves,” i.e. the “set” { x : x is a setand x ∉ x } does not exist.

• Cantor’s paradox—It shows that “the set of all sets” cannot exist.

The reason is that the phrase well-defined is not very well defined. It was important to free set theory of theseparadoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid theseparadoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.

130 CHAPTER 23. SET (MATHEMATICS)

The symmetric difference of A and B

For most purposes however, naive set theory is still useful.

23.9 Principle of inclusion and exclusion

Main article: Inclusion-exclusion principle

This principle gives us the cardinality of the union of sets.

|A1 ∪A2 ∪A3 ∪ . . . ∪An| =(|A1|+ |A2|+ |A3|+ . . . |An|)−(|A1 ∩A2|+ |A1 ∩A3|+ . . . |An−1 ∩An|)+. . .+

(−1)n−1

(|A1 ∩A2 ∩A3 ∩ . . . ∩An|)

23.10 De Morgan’s Law

De Morgan stated two laws about Sets.If A and B are any two Sets then,

• (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

23.11. SEE ALSO 131

23.11 See also• Set notation

• Mathematical object

• Alternative set theory

• Axiomatic set theory

• Category of sets

• Class (set theory)

• Dense set

• Family of sets

• Fuzzy set

• Internal set

• Mereology

• Multiset

• Naive set theory

• Principia Mathematica

• Rough set

• Russell’s paradox

• Sequence (mathematics)

• Taxonomy

• Tuple

23.12 Notes[1] “EineMenge, ist die Zusammenfassung bestimmter, wohlunterschiedenerObjekte unserer Anschauung oder unseresDenkens

– welche Elemente der Menge genannt werden – zu einem Ganzen.”

[2] Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.

23.13 References• Dauben, JosephW., Georg Cantor: His Mathematics and Philosophy of the Infinite, Boston: Harvard UniversityPress (1979) ISBN 978-0-691-02447-9.

• Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN 0-387-90092-6.

• Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4.

• Velleman, Daniel, How To Prove It: A Structured Approach, Cambridge University Press (2006) ISBN 978-0-521-67599-4

23.14 External links• C2 Wiki – Examples of set operations using English operators.

• Mathematical Sets: Elements, Intersections & Unions, Education Portal Academy

Chapter 24

Subset

“Superset” redirects here. For other uses, see Superset (disambiguation).In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is

AB

Euler diagram showingA is a proper subset of B and conversely B is a proper superset of A

132

24.1. DEFINITIONS 133

“contained” inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of oneset being a subset of another is called inclusion or sometimes containment.The subset relation defines a partial order on sets.The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

24.1 Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by A ⊆ B ,or equivalently

• B is a superset of (or includes) A, denoted by B ⊇ A.

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A),then

• A is also a proper (or strict) subset of B; this is written as A ⊊ B.

or equivalently

• B is a proper superset of A; this is written as B ⊋ A.

For any set S, the inclusion relation ⊆ is a partial order on the set P(S) of all subsets of S (the power set of S).When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.[1]

24.2 ⊂ and ⊃ symbols

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaningand instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may ormay not equal y, but if x < y, then x may not equal y, and is less than y. Similarly, using the convention that ⊂ isproper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B.

24.3 Examples

• The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions A ⊆ B and A ⊊ B are true.

• The set D = {1, 2, 3} is a subset of E = {1, 2, 3}, thus D ⊆ E is true, and D ⊊ E is not true (false).

• Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)

• The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any setexcept itself.

• The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}

• The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a linesegment is a proper subset of the set of points in a line. These are two examples in which both the subset andthe whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is,the number of elements, of a finite set) as the whole; such cases can run counter to one’s initial intuition.

• The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinitebut the latter set has a larger cardinality (or power) than the former set.

134 CHAPTER 24. SUBSET

polygonsregular

polygons

The regular polygons form a subset of the polygons

Another example in an Euler diagram:

• A is a proper subset of B

• C is a subset but no proper subset of B

24.4 Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ⪯ ) is isomorphic to somecollection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identifiedwith the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b].For the power set P(S) of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product ofk = |S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumeratingS = {s1, s2, …, sk} and associating with each subset T ⊆ S (which is to say with each element of 2S) the k-tuple from{0,1}k of which the ith coordinate is 1 if and only if si is a member of T.

24.5 See also• Containment order

24.6 References[1] Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN

978-0-07-338309-5.

[2] Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1,MR 924157

24.7. EXTERNAL LINKS 135

C B A

A B and B C imply A C

[3] Subsets and Proper Subsets (PDF), retrieved 2012-09-07

• Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

24.7 External links• Weisstein, Eric W., “Subset”, MathWorld.

Chapter 25

Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that if ais related to b then b is related to a.In mathematical notation, this is:

∀a, b ∈ X, aRb⇒ bRa.

25.1 Examples

25.1.1 In mathematics

• “is equal to” (equality) (whereas “is less than” is not symmetric)

• “is comparable to”, for elements of a partially ordered set

• "... and ... are odd":

136

25.2. RELATIONSHIP TO ASYMMETRIC AND ANTISYMMETRIC RELATIONS 137

25.1.2 Outside mathematics

• “is married to” (in most legal systems)

• “is a fully biological sibling of”

• “is a homophone of”

25.2 Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be relatedto a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for “is lessthan or equal to” and “preys on”).Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actuallyindependent of each other, as these examples show.

25.3 Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge’stwo vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatoriallyequivalent objects.

138 CHAPTER 25. SYMMETRIC RELATION

25.4 See also• Symmetry in mathematics

• Symmetry

• Asymmetric relation

• Antisymmetric relation

Chapter 26

Total order

Inmathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denotedby infix ≤) on some setX which is transitive, antisymmetric, and total. A set paired with a total order is called a totallyordered set, a linearly ordered set, a simply ordered set, or a chain.If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If a ≤ b and b ≤ a then a = b (antisymmetry);If a ≤ b and b ≤ c then a ≤ c (transitivity);a ≤ b or b ≤ a (totality).

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a.[1] A relation having the property of“totality” means that any pair of elements in the set of the relation are comparable under the relation. This also meansthat the set can be diagrammed as a line of elements, giving it the name linear.[2] Totality also implies reflexivity, i.e.,a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (Itrequires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extensionof that partial order.

26.1 Strict total order

For each (non-strict) total order ≤ there is an associated asymmetric (hence irreflexive) relation <, called a strict totalorder, which can equivalently be defined in two ways:

• a < b if and only if a ≤ b and a ≠ b

• a < b if and only if not b ≤ a (i.e., < is the inverse of the complement of ≤)

Properties:

• The relation is transitive: a < b and b < c implies a < c.• The relation is trichotomous: exactly one of a < b, b < a and a = b is true.• The relation is a strict weak order, where the associated equivalence is equality.

We can work the other way and start by choosing < as a transitive trichotomous binary relation; then a total order ≤can equivalently be defined in two ways:

• a ≤ b if and only if a < b or a = b

• a ≤ b if and only if not b < a

Two more associated orders are the complements ≥ and >, completing the quadruple {<, >, ≤, ≥}.We can define or explain the way a set is totally ordered by any of these four relations; the notation implies whetherwe are talking about the non-strict or the strict total order.

139

140 CHAPTER 26. TOTAL ORDER

26.2 Examples

• The letters of the alphabet ordered by the standard dictionary order, e.g., A < B < C etc.

• Any subset of a totally ordered set, with the restriction of the order on the whole set.

• Any set of cardinal numbers or ordinal numbers (more strongly, these are well-orders).

• If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on Xby setting x1 < x2 if and only if f(x1) < f(x2).

• The lexicographical order on the Cartesian product of a set of totally ordered sets indexed by an ordinal, isitself a total order. For example, any set of words ordered alphabetically is a totally ordered set, viewed as asubset of a Cartesian product of a countable number of copies of a set formed by adding the space symbol tothe alphabet (and defining a space to be less than any letter).

• The set of real numbers ordered by the usual less than (<) or greater than (>) relations is totally ordered, hencealso the subsets of natural numbers, integers, and rational numbers. Each of these can be shown to be the unique(to within isomorphism) smallest example of a totally ordered set with a certain property, (a total order A isthe smallest with a certain property if whenever B has the property, there is an order isomorphism from A to asubset of B):

• The natural numbers comprise the smallest totally ordered set with no upper bound.• The integers comprise the smallest totally ordered set with neither an upper nor a lower bound.• The rational numbers comprise the smallest totally ordered set which is dense in the real numbers. Thedefinition of density used here says that for every 'a' and 'b' in the real numbers such that 'a' < 'b', there isa 'q' in the rational numbers such that 'a' < 'q' < 'b'.

• The real numbers comprise the smallest unbounded totally ordered set that is connected in the ordertopology (defined below).

• Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers.

26.3 Further concepts

26.3.1 Chains

While chain is sometimes merely a synonym for totally ordered set, it can also refer to a totally ordered subset ofsome partially ordered set. The latter definition has a crucial role in Zorn’s lemma.For example, consider the set of all subsets of the integers partially ordered by inclusion. Then the set { In : n is anatural number}, where In is the set of natural numbers below n, is a chain in this ordering, as it is totally orderedunder inclusion: If n≤k, then In is a subset of Ik.

26.3.2 Lattice theory

One may define a totally ordered set as a particular kind of lattice, namely one in which we have

{a ∨ b, a ∧ b} = {a, b} for all a, b.

We then write a ≤ b if and only if a = a ∧ b . Hence a totally ordered set is a distributive lattice.

26.3. FURTHER CONCEPTS 141

26.3.3 Finite total orders

A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subsetthereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observingthat every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphicto an initial segment of the natural numbers ordered by <. In other words a total order on a set with k elements inducesa bijection with the first k natural numbers. Hence it is common to index finite total orders or well orders with ordertype ω by natural numbers in a fashion which respects the ordering (either starting with zero or with one).

26.3.4 Category theory

Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being mapswhich respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b).A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.

26.3.5 Order topology

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b},(a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, theorder topology.When more than one order is being used on a set one talks about the order topology induced by a particular order.For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology onN induced by < and the order topology on N induced by > (in this case they happen to be identical but will not ingeneral).The order topology induced by a total order may be shown to be hereditarily normal.

26.3.6 Completeness

A totally ordered set is said to be complete if every nonempty subset that has an upper bound, has a least upperbound. For example, the set of real numbers R is complete but the set of rational numbers Q is not.There are a number of results relating properties of the order topology to the completeness of X:

• If the order topology on X is connected, X is complete.

• X is connected under the order topology if and only if it is complete and there is no gap in X (a gap is twopoints a and b in X with a < b such that no c satisfies a < c < b.)

• X is complete if and only if every bounded set that is closed in the order topology is compact.

A totally ordered set (with its order topology) which is a complete lattice is compact. Examples are the closed intervalsof real numbers, e.g. the unit interval [0,1], and the affinely extended real number system (extended real number line).There are order-preserving homeomorphisms between these examples.

26.3.7 Sums of orders

For any two disjoint total orders (A1,≤1) and (A2,≤2) , there is a natural order ≤+ on the set A1 ∪ A2 , which iscalled the sum of the two orders or sometimes just A1 +A2 :

For x, y ∈ A1 ∪A2 , x ≤+ y holds if and only if one of the following holds:

1. x, y ∈ A1 and x ≤1 y

2. x, y ∈ A2 and x ≤2 y

3. x ∈ A1 and y ∈ A2

142 CHAPTER 26. TOTAL ORDER

Intutitively, this means that the elements of the second set are added on top of the elements of the first set.More generally, if (I,≤) is a totally ordered index set, and for each i ∈ I the structure (Ai,≤i) is a linear order,where the sets Ai are pairwise disjoint, then the natural total order on

∪iAi is defined by

For x, y ∈∪

i∈I Ai , x ≤ y holds if:1. Either there is some i ∈ I with x ≤i y

2. or there are some i < j in I with x ∈ Ai , y ∈ Aj

26.4 Orders on the Cartesian product of totally ordered sets

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product oftwo totally ordered sets are:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order.• (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order). This is a partial order.• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product ofthe corresponding strict total orders). This is also a partial order.

All three can similarly be defined for the Cartesian product of more than two sets.Applied to the vector space Rn, each of these make it an ordered vector space.See also examples of partially ordered sets.A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding totalpreorder on that subset.

26.5 Related structures

A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order.A group with a compatible total order is a totally ordered group.There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientationresults in a betweenness relation. Forgetting the location of the ends results in a cyclic order. Forgetting both dataresults in a separation relation.[3]

26.6 See also• Order theory• Well-order• Suslin’s problem• Countryman line

26.7 Notes[1] Nederpelt, Rob (2004). “Chapter 20.2: Ordered Sets. Orderings”. Logical Reasoning: A First Course. Texts in Computing

3 (3rd, Revised ed.). King’s College Publications. p. 325. ISBN 0-9543006-7-X.

[2] Nederpelt, Rob (2004). “Chapter 20.3: Ordered Sets. Linear orderings”. Logical Reasoning: A First Course. Texts inComputing 3 (3rd, Revisied ed.). King’s College Publications. p. 330. ISBN 0-9543006-7-X.

[3] Macpherson, H. Dugald (2011), “A survey of homogeneous structures” (PDF),DiscreteMathematics, doi:10.1016/j.disc.2011.01.024,retrieved 28 April 2011

26.8. REFERENCES 143

26.8 References• George Grätzer (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman and Co. ISBN0-7167-0442-0

• John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. ISBN 0-486-65676-4

Chapter 27

Total relation

In mathematics, a binary relation R over a set X is total or complete if for all a and b in X, a is related to b or b isrelated to a (or both).In mathematical notation, this is

∀a, b ∈ X, aRb ∨ bRa.

Total relations are sometimes said to have comparability.

27.1 Examples

For example, “is less than or equal to” is a total relation over the set of real numbers, because for two numbers eitherthe first is less than or equal to the second, or the second is less than or equal to the first. On the other hand, “is lessthan” is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, noris the second less than the first. (But note that “is less than” is a weak order which gives rise to a total order, namely“is less than or equal to”. The relationship between strict orders and weak orders is discussed at partially ordered set.)The relation “is a subset of” is also not total because, for example, neither of the sets {1,2} and {3,4} is a subset ofthe other.

27.2 Properties and related notions

Totality implies reflexivity.If a transitive relation is also total, it is a total preorder. If a partial order is also total, it is a total order.A binary relation R over X is called connex if for all a and b in X such that a ≠ b, a is related to b or b is related to a(or both):[1]

∀a, b ∈ X, aRb ∨ bRa ∨ (a = b).

Connexity does not imply reflexivity. A strict partial order is a strict total order if and only if it is connex.

27.3 See also

• Total order

144

27.4. REFERENCES 145

27.4 References[1] Rautenberg,Wolfgang (2010),AConcise Introduction toMathematical Logic (3rd ed.), NewYork: Springer Science+Business

Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6

Chapter 28

Transitive closure

For other uses, see Closure (disambiguation).This article is about the transitive closure of a binary relation. For the transitive closure of a set, see transitiveset#Transitive closure.

In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such thatR+ contains R and R+ is minimal (Lidl and Pilz 1998:337). If the binary relation itself is transitive, then the transitiveclosure is that same binary relation; otherwise, the transitive closure is a different relation. For example, if X is a setof airports and x R y means “there is a direct flight from airport x to airport y", then the transitive closure of R on Xis the relation R+: “it is possible to fly from x to y in one or more flights.”

28.1 Transitive relations and examples

A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitiverelations include the equality relation on any set, the “less than or equal” relation on any linearly ordered set, and therelation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x < y and y < z then x< z.One example of a non-transitive relation is “city x can be reached via a direct flight from city y" on the set of all cities.Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to thethird, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is adifferent relation, namely “there is a sequence of direct flights that begins at city x and ends at city y". Every relationcan be extended in a similar way to a transitive relation.An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".The transitive closure of this relation is “some day x comes after a day y on the calendar”, which is trivially true for alldays of the week x and y (and thus equivalent to the Cartesian square, which is "x and y are both days of the week”).

28.2 Existence and description

For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family oftransitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namelythe trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containingR.For finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges. Thisgives the intuition for a general construction. For any set X, we can prove that transitive closure is given by thefollowing expression

R+ =∪

i∈{1,2,3,...}

Ri.

146

28.3. PROPERTIES 147

where Ri is the i-th power of R, defined inductively by

R1 = R

and, for i > 0 ,

Ri+1 = R ◦Ri

where ◦ denotes composition of relations.To show that the above definition of R+ is the least transitive relation containing R, we show that it contains R, that itis transitive, and that it is the smallest set with both of those characteristics.

• R ⊆ R+ : R+ contains all of the Ri , so in particular R+ contains R .

• R+ is transitive: every element ofR+ is in one of theRi , soR+ must be transitive by the following reasoning:if (s1, s2) ∈ Rj and (s2, s3) ∈ Rk , then from composition’s associativity, (s1, s3) ∈ Rj+k (and thus in R+

) because of the definition of Ri .

• R+ is minimal: LetG be any transitive relation containingR , we want to show thatR+ ⊆ G . It is sufficient toshow that for every i > 0 ,Ri ⊆ G . Well, sinceG containsR ,R1 ⊆ G . And sinceG is transitive, wheneverRi ⊆ G , Ri+1 ⊆ G according to the construction of Ri and what it means to be transitive. Therefore, byinduction, G contains every Ri , and thus also R+ .

28.3 Properties

The intersection of two transitive relations is transitive.The union of two transitive relations need not be transitive. To preserve transitivity, one must take the transitiveclosure. This occurs, for example, when taking the union of two equivalence relations or two preorders. To obtain anew equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case ofequivalence relations—are automatic).

28.4 In graph theory

In computer science, the concept of transitive closure can be thought of as constructing a data structure that makesit possible to answer reachability questions. That is, can one get from node a to node d in one or more hops? Abinary relation tells you only that node a is connected to node b, and that node b is connected to node c, etc. Afterthe transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine thatnode d is reachable from node a. The data structure is typically stored as a matrix, so if matrix[1][4] = 1, then it isthe case that node 1 can reach node 4 through one or more hops.The transitive closure of a directed acyclic graph (DAG) is the reachability relation of the DAG and a strict partialorder.

28.5 In logic and computational complexity

The transitive closure of a binary relation cannot, in general, be expressed in first-order logic (FO). This means thatone cannot write a formula using predicate symbols R and T that will be satisfied in any model if and only if T isthe transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operatoris usually called transitive closure logic, and abbreviated FO(TC) or just TC. TC is a sub-type of fixpoint logics.The fact that FO(TC) is strictly more expressive than FO was discovered by Ronald Fagin in 1974; the result wasthen rediscovered by Alfred Aho and Jeffrey Ullman in 1979, who proposed to use fixpoint logic as a database query

148 CHAPTER 28. TRANSITIVE CLOSURE

Output

Input

Transitive closure constructs the output graph from the input graph.

language (Libkin 2004:vii). With more recent concepts of finite model theory, proof that FO(TC) is strictly moreexpressive than FO follows immediately from the fact that FO(TC) is not Gaifman-local (Libkin 2004:49).In computational complexity theory, the complexity class NL corresponds precisely to the set of logical sentencesexpressible in TC. This is because the transitive closure property has a close relationship with the NL-completeproblem STCON for finding directed paths in a graph. Similarly, the class L is first-order logic with the commutative,transitive closure. When transitive closure is added to second-order logic instead, we obtain PSPACE.

28.6 In database query languages

Further information: Hierarchical and recursive queries in SQL

Since the 1980s Oracle Database has implemented a proprietary SQL extension CONNECT BY... START WITHthat allows the computation of a transitive closure as part of a declarative query. The SQL 3 (1999) standard addeda more general WITH RECURSIVE construct also allowing transitive closures to be computed inside the queryprocessor; as of 2011 the latter is implemented in IBM DB2, Microsoft SQL Server, and PostgreSQL, although notin MySQL (Benedikt and Senellart 2011:189).Datalog also implements transitive closure computations (Silberschatz et al. 2010:C.3.6).

28.7 Algorithms

Efficient algorithms for computing the transitive closure of a graph can be found in Nuutila (1995). The fastest worst-case methods, which are not practical, reduce the problem to matrix multiplication. The problem can also be solvedby the Floyd–Warshall algorithm, or by repeated breadth-first search or depth-first search starting from each node ofthe graph.More recent research has explored efficient ways of computing transitive closure on distributed systems based on theMapReduce paradigm (Afrati et al. 2011).

28.8 See also• Deductive closure

28.9. REFERENCES 149

• Transitive reduction (a smallest relation having the transitive closure of R as its transitive closure)

• Symmetric closure

• Reflexive closure

• Ancestral relation

28.9 References• Lidl, R. and Pilz, G., 1998, Applied abstract algebra, 2nd edition, Undergraduate Texts in Mathematics,Springer, ISBN 0-387-98290-6

• Keller, U., 2004, Some Remarks on the Definability of Transitive Closure in First-order Logic and Datalog(unpublished manuscript)

• ErichGrädel; PhokionG. Kolaitis; Leonid Libkin; MaartenMarx; Joel Spencer; Moshe Y. Vardi; YdeVenema;Scott Weinstein (2007). Finite Model Theory and Its Applications. Springer. pp. 151–152. ISBN 978-3-540-68804-4.

• Libkin, Leonid (2004), Elements of Finite Model Theory, Springer, ISBN 978-3-540-21202-7

• Heinz-Dieter Ebbinghaus; Jörg Flum (1999). FiniteModel Theory (2nd ed.). Springer. pp. 123–124, 151–161,220–235. ISBN 978-3-540-28787-2.

• Aho, A. V.; Ullman, J. D. (1979). “Universality of data retrieval languages”. Proceedings of the 6th ACMSIGACT-SIGPLAN Symposium on Principles of programming languages - POPL '79. p. 110. doi:10.1145/567752.567763.

• Benedikt, M.; Senellart, P. (2011). “Databases”. In Blum, Edward K.; Aho, Alfred V. Computer Science. TheHardware, Software and Heart of It. pp. 169–229. doi:10.1007/978-1-4614-1168-0_10. ISBN 978-1-4614-1167-3.

• Nuutila, E., Efficient Transitive Closure Computation in Large Digraphs. Acta Polytechnica Scandinavica,Mathematics and Computing in Engineering Series No. 74, Helsinki 1995, 124 pages. Published by theFinnish Academy of Technology. ISBN 951-666-451-2, ISSN 1237-2404, UDC 681.3.

• Abraham Silberschatz; Henry Korth; S. Sudarshan (2010). Database System Concepts (6th ed.). McGraw-Hill.ISBN 978-0-07-352332-3. Appendix C (online only)

• Foto N. Afrati, Vinayak Borkar, Michael Carey, Neoklis Polyzotis, Jeffrey D. Ullman, Map-Reduce Extensionsand Recursive Queries, EDBT 2011, March 22–24, 2011, Uppsala, Sweden, ISBN 978-1-4503-0528-0

28.10 External links• "Transitive closure and reduction", The Stony Brook Algorithm Repository, Steven Skiena .

• "Apti Algoritmi", An example and someC++ implementations of algorithms that calculate the transitive closureof a given binary relation, Vreda Pieterse.

Chapter 29

Transitive reduction

In mathematics, a transitive reduction of a directed graph is a graph with as few edges as possible that has thesame reachability relation as the given graph. Equivalently, the given graph and its transitive reduction should havethe same transitive closure as each other, and its transitive reduction should have as few edges as possible among allgraphs with this property. Transitive reductions were introduced by Aho, Garey & Ullman (1972), who providedtight bounds on the computational complexity of constructing them.If a given graph is a finite directed acyclic graph, its transitive reduction is unique, and is a subgraph of the givengraph. However, uniqueness is not guaranteed for graphs with cycles, and for infinite graphs not even existence isguaranteed. The closely related concept of a minimum equivalent graph is a subgraph of the given graph thathas the same reachability relation and as few edges as possible.[1] For finite directed acyclic graphs, the minimumequivalent graph is the same as the transitive reduction. However, for graphs that may contain cycles, minimumequivalent graphs are NP-hard to construct, while transitive reductions can still be constructed in polynomial time.Transitive reductions can also be defined for more abstract binary relations on sets, by interpreting the pairs of therelation as arcs in a graph.

29.1 In directed acyclic graphs

The transitive reduction of a finite directed graph G is a graph with the fewest possible edges that has the samereachability relation as the original graph. That is, if there is a path from a vertex x to a vertex y in graph G, theremust also be a path from x to y in the transitive reduction ofG, and vice versa. The following image displays drawingsof graphs corresponding to a non-transitive binary relation (on the left) and its transitive reduction (on the right).The transitive reduction of a finite directed acyclic graphG is unique, and consists of the edges ofG that form the onlypath between their endpoints. In particular, it is always a subgraph of the given graph. For this reason, the transitivereduction coincides with the minimum equivalent graph in this case.In the mathematical theory of binary relations, any relation R on a set Xmay be thought of as a directed graph that hasthe set X as its vertex set and that has an arc xy for every ordered pair of elements that are related in R. In particular,this method lets partially ordered sets be reinterpreted as directed acyclic graphs, in which there is an arc xy in thegraph whenever there is an order relation x < y between the given pair of elements of the partial order. When thetransitive reduction operation is applied to a directed acyclic graph that has been constructed in this way, it generatesthe covering relation of the partial order, which is frequently given visual expression by means of a Hasse diagram.Transitive reduction has been used on networks which can be represented as directed acyclic graphs (e.g. citationnetworks) to reveal structural differences between networks.[2]

29.2 In graphs with cycles

In a finite graph that may have cycles, the transitive reduction is not uniquely defined: there may be more thanone graph on the same vertex set that has a minimal number of edges and has the same reachability relation as thegiven graph. Additionally, it may be the case that none of these minimal graphs is a subgraph of the given graph.Nevertheless, it is straightforward to characterize the minimal graphs with the same reachability relation as the given

150

29.3. COMPUTATIONAL COMPLEXITY 151

graph G.[3] If G is an arbitrary directed graph, and H is a graph with the minimum possible number of edges havingthe same reachability relation as G, then H consists of

• A directed cycle for each strongly connected component of G, connecting together the vertices in this compo-nent

• An edge xy for each edge XY of the transitive reduction of the condensation of G, where X and Y are twostrongly connected components of G that are connected by an edge in the condensation, x is any vertex incomponent X, and y is any vertex in component Y. The condensation of G is a directed acyclic graph that has avertex for every strongly connected component of G and an edge for every two components that are connectedby an edge in G. In particular, because it is acyclic, its transitive reduction can be defined as in the previoussection.

The total number of edges in this type of transitive reduction is then equal to the number of edges in the transitivereduction of the condensation, plus the number of vertices in nontrivial strongly connected components (componentswith more than one vertex).The edges of the transitive reduction that correspond to condensation edges can always be chosen to be a subgraph ofthe given graphG. However, the cycle within each strongly connected component can only be chosen to be a subgraphofG if that component has a Hamiltonian cycle, something that is not always true and is difficult to check. Because ofthis difficulty, it is NP-hard to find the smallest subgraph of a given graph G with the same reachability (its minimumequivalent graph).[3]

29.3 Computational complexity

As Aho et al. show,[3] when the time complexity of graph algorithms is measured only as a function of the number nof vertices in the graph, and not as a function of the number of edges, transitive closure and transitive reduction havethe same complexity. It had already been shown that transitive closure and multiplication of Boolean matrices of sizen × n had the same complexity as each other,[4] so this result put transitive reduction into the same class. The fastestknown algorithms for matrix multiplication, as of 2013, take time O(n2.3727),[5] and this same time bound applies totransitive reduction as well.To prove that transitive reduction is as hard as transitive closure, Aho et al. construct from a given directed acyclicgraph G another graph H, in which each vertex of G is replaced by a path of three vertices, and each edge of Gcorresponds to an edge in H connecting the corresponding middle vertices of these paths. In addition, in the graphH, Aho et al. add an edge from every path start to every path end. In the transitive reduction of H, there is an edgefrom the path start for u to the path end for v, if and only if edge uv does not belong to the transitive closure of G.Therefore, if the transitive reduction of H can be computed efficiently, the transitive closure of G can be read offdirectly from it.To prove that transitive reduction is as easy as transitive closure, Aho et al. rely on the already-known equivalencewith Boolean matrix multiplication. They let A be the adjacency matrix of the given graph, and B be the adjacencymatrix of its transitive closure (computed using any standard transitive closure algorithm). Then an edge uv belongsto the transitive reduction if and only if there is a nonzero entry in row u and column v of matrix A, and there is not anonzero entry in the same position of the matrix product AB. In this construction, the nonzero elements of the matrixAB represent pairs of vertices connected by paths of length two or more.When measured both in terms of the number n of vertices and the number m of edges in a directed acyclic graph,transitive reductions can also be found in time O(nm), a bound that may be faster than the matrix multiplicationmethods for sparse graphs. To do so, collect edges (u,v) such that the longest-path distance from u to v is one,calculating those distances by linear-time search from each possible starting vertex, u. This O(nm) time boundmatches the complexity of constructing transitive closures by using depth first search or breadth first search to findthe vertices reachable from every choice of starting vertex, so again with these assumptions transitive closures andtransitive reductions can be found in the same amount of time.

29.4 Notes[1] Moyles & Thompson (1969).

152 CHAPTER 29. TRANSITIVE REDUCTION

[2] http://arxiv.org/abs/1310.8224

[3] Aho, Garey & Ullman (1972)

[4] Aho et al. credit this result to an unpublished 1971 manuscript of Ian Munro, and to a 1970 Russian-language paper by M.E. Furman.

[5] Williams (2012).

29.5 References• Aho, A. V.; Garey, M. R.; Ullman, J. D. (1972), “The transitive reduction of a directed graph”, SIAM Journal

on Computing 1 (2): 131–137, doi:10.1137/0201008, MR 0306032.

• Moyles, Dennis M.; Thompson, Gerald L. (1969), “An Algorithm for Finding a Minimum Equivalent Graphof a Digraph”, Journal of the ACM 16 (3): 455–460, doi:10.1145/321526.321534.

• Williams, Virginia Vassilevska (2012), “Multiplying matrices faster than Coppersmith–Winograd”, Proc. 44thACM Symposium on Theory of Computing (STOC '12), pp. 887–898, doi:10.1145/2213977.2214056.

29.6 External links• Weisstein, Eric W., “Transitive Reduction”, MathWorld.

Chapter 30

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b,and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial orderrelations and equivalence relations.

30.1 Formal definition

In terms of set theory, the transitive relation can be defined as:

∀a, b, c ∈ X : (aRb ∧ bRc) ⇒ aRc

30.2 Examples

For example, “is greater than,” “is at least as great as,” and “is equal to” (equality) are transitive relations:

whenever A > B and B > C, then also A > Cwhenever A ≥ B and B ≥ C, then also A ≥ Cwhenever A = B and B = C, then also A = C.

On the other hand, “is the mother of” is not a transitive relation, because if Alice is the mother of Brenda, and Brendais the mother of Claire, then Alice is not the mother of Claire. What is more, it is antitransitive: Alice can never bethe mother of Claire.Then again, in biology we often need to consider motherhood over an arbitrary number of generations: the relation“is a matrilinear ancestor of”. This is a transitive relation. More precisely, it is the transitive closure of the relation“is the mother of”.More examples of transitive relations:

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “implies” (implication)

30.3 Properties

153

154 CHAPTER 30. TRANSITIVE RELATION

30.3.1 Closure properties

The converse of a transitive relation is always transitive: e.g. knowing that “is a subset of” is transitive and “is asuperset of” is its converse, we can conclude that the latter is transitive as well.The intersection of two transitive relations is always transitive: knowing that “was born before” and “has the same firstname as” are transitive, we can conclude that “was born before and also has the same first name as” is also transitive.The union of two transitive relations is not always transitive. For instance “was born before or has the same first nameas” is not generally a transitive relation.The complement of a transitive relation is not always transitive. For instance, while “equal to” is transitive, “not equalto” is only transitive on sets with at most one element.

30.3.2 Other properties

A transitive relation is asymmetric if and only if it is irreflexive.[1]

30.3.3 Properties that require transitivity

• Preorder – a reflexive transitive relation

• partial order – an antisymmetric preorder

• Total preorder – a total preorder

• Equivalence relation – a symmetric preorder

• Strict weak ordering – a strict partial order in which incomparability is an equivalence relation

• Total ordering – a total, antisymmetric transitive relation

30.4 Counting transitive relations

No general formula that counts the number of transitive relations on a finite set (sequence A006905 in OEIS) isknown.[2] However, there is a formula for finding the number of relations that are simultaneously reflexive, symmet-ric, and transitive – in other words, equivalence relations – (sequence A000110 in OEIS), those that are symmetricand transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and an-tisymmetric. Pfeiffer[3] has made some progress in this direction, expressing relations with combinations of theseproperties in terms of each other, but still calculating any one is difficult. See also.[4]

30.5 See also

• Transitive closure

• Transitive reduction

• Intransitivity

• Reflexive relation

• Symmetric relation

• Quasitransitive relation

• Nontransitive dice

• Rational choice theory

30.6. SOURCES 155

30.6 Sources

30.6.1 References[1] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: School

of Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[2] Steven R. Finch, “Transitive relations, topologies and partial orders”, 2003.

[3] Götz Pfeiffer, "Counting Transitive Relations", Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

[4] Gunnar Brinkmann and Brendan D. McKay,”Counting unlabelled topologies and transitive relations"

30.6.2 Bibliography

• Ralph P. Grimaldi, Discrete and Combinatorial Mathematics, ISBN 0-201-19912-2.

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

30.7 External links• Hazewinkel, Michiel, ed. (2001), “Transitivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Transitivity in Action at cut-the-knot

Chapter 31

Weak ordering

Not to be confused with weak order of permutations.In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of

a<bc<b

a<ba<c

b<ac<a

b<ab<c

c<ac<b

a<cb<ca,b,c

c<b<a

b<c<a

b<a<c

c<a<b a<b<c

a<c<b

The 13 possible strict weak orderings on a set of three elements {a, b, c}. The only partially ordered sets are coloured, while totallyordered ones are in black. Two orderings are shown as connected by an edge if they differ by a single dichotomy.

156

31.1. EXAMPLES 157

a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totallyordered sets (rankings without ties) and are in turn generalized by partially ordered sets and preorders.[1]

There are several common ways of formalizing weak orderings, that are different from each other but cryptomorphic(interconvertable with no loss of information): they may be axiomatized as strict weak orderings (partially orderedsets in which incomparability is a transitive relation), as total preorders (transitive binary relations in which at leastone of the two possible relations exists between every pair of elements), or as ordered partitions (partitions of theelements into disjoint subsets, together with a total order on the subsets). In many cases another representation calleda preferential arrangement based on a utility function is also possible.Weak orderings are counted by the ordered Bell numbers. They are used in computer science as part of partitionrefinement algorithms, and in the C++ Standard Library.

31.1 Examples

In horse racing, the use of photo finishes has eliminated some, but not all, ties or (as they are called in this context)dead heats, so the outcome of a horse race may be modeled by a weak ordering.[2] In an example from the MarylandHunt Cup steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied forsecond place, with the remaining horses farther back; three horses did not finish.[3] In the weak ordering describingthis outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before allthe other horses that finished, and the three horses that did not finish would be placed last in the order but tied witheach other.The points of the Euclidean plane may be ordered by their distance from the origin, giving another example of a weakordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to acommon circle centered at the origin), and infinitely many points within these subsets. Although this ordering has asmallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.Opinion polling in political elections provides an example of a type of ordering that resembles weak orderings, but isbetter modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another,or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they arewithin the margin of error of each other. However, if candidate x is statistically tied with y, and y is statistically tiedwith z, it might still be possible for x to be clearly better than z, so being tied is not in this case a transitive relation.Because of this possibility, rankings of this type are better modeled as semiorders than as weak orderings.[4]

31.2 Axiomatizations

31.2.1 Strict weak orderings

A strict weak ordering is a binary relation < on a set S that is a strict partial order (a transitive relation that isirreflexive, or equivalently,[5] that is asymmetric) in which the relation “neither a < b nor b < a" is transitive.[1]

The equivalence classes of this “incomparability relation” partition the elements of S, and are totally ordered by <.Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x < y if and only if thereexists sets A and B in the partition with x in A, y in B, and A < B in the total order.As a non-example, consider the partial order in the set {a, b, c} defined by the relationship b < c. The pairs a,b and a,care incomparable but b and c are related, so incomparability does not form an equivalence relation and this exampleis not a strict weak ordering.A strict weak ordering has the following properties. For all x and y in S,

• For all x, it is not the case that x < x (irreflexivity).

• For all x, y, if x < y then it is not the case that y < x (asymmetry).

• For all x, y, and z, if x < y and y < z then x < z (transitivity).

• For all x, y, and z, if x is incomparable with y, and y is incomparable with z, then x is incomparable with z(transitivity of incomparability).

158 CHAPTER 31. WEAK ORDERING

This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.Transitivity of incomparability (together with transitivity) can also be stated in the following forms:

• If x < y, then for all z, either x < z or z < y or both.

Or:

• If x is incomparable with y, then for all z ≠ x, z ≠ y, either (x < z and y < z) or (z < x and z < y) or (z isincomparable with x and z is incomparable with y).

31.2.2 Total preorders

Strict weak orders are very closely related to total preorders or (non-strict) weak orders, and the samemathematicalconcepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A totalpreorder or weak order is a preorder that is total; that is, no pair of items is incomparable. A total preorder≲ satisfiesthe following properties:

• For all x, y, and z, if x ≲ y and y ≲ z then x ≲ z (transitivity).

• For all x and y, x ≲ y or y ≲ x (totality).

• Hence, for all x, x ≲ x (reflexivity).

A total order is a total preorder which is antisymmetric, in other words, which is also a partial order. Total preordersare sometimes also called preference relations.The complement of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strictweak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we takethe inverse of the complement: for a strict weak ordering <, define a total preorder ≲ by setting x ≲ y whenever it isnot the case that y < x. In the other direction, to define a strict weak ordering < from a total preorder ≲ , set x < ywhenever it is not the case that y ≲ x.[6]

In any preorder there is a corresponding equivalence relation where two elements x and y are defined as equivalent ifx ≲ y and y ≲ x. In the case of a total preorder the corresponding partial order on the set of equivalence classes is atotal order. Two elements are equivalent in a total preorder if and only if they are incomparable in the correspondingstrict weak ordering.

31.2.3 Ordered partitions

A partition of a set S is a family of disjoint subsets of S that have S as their union. A partition, together with atotal order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition[7] and byTheodore Motzkin a list of sets.[8] An ordered partition of a finite set may be written as a finite sequence of the setsin the partition: for instance, the three ordered partitions of the set {a, b} are

{a}, {b},{b}, {a}, and{a, b}.

In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit atotal ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partitiongives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in thepartition, and otherwise inherit the order of the sets that contain them.

31.3. RELATED TYPES OF ORDERING 159

31.2.4 Representation by functions

For sets of sufficiently small cardinality, a third axiomatization is possible, based on real-valued functions. If X is anyset and f a real-valued function on X then f induces a strict weak order on X by setting a < b if and only if f(a) < f(b).The associated total preorder is given by setting a ≲ b if and only if f(a) ≤ f(b), and the associated equivalence bysetting a ∼ b if and only if f(a) = f(b).The relations do not change when f is replaced by g o f (composite function), where g is a strictly increasing real-valued function defined on at least the range of f. Thus e.g. a utility function defines a preference relation. In thiscontext, weak orderings are also known as preferential arrangements.[9]

If X is finite or countable, every weak order on X can be represented by a function in this way.[10] However, thereexist strict weak orders that have no corresponding real function. For example, there is no such function for thelexicographic order on Rn. Thus, while in most preference relation models the relation defines a utility function up toorder-preserving transformations, there is no such function for lexicographic preferences.More generally, if X is a set, and Y is a set with a strict weak ordering "<", and f a function from X to Y, then finduces a strict weak ordering on X by setting a < b if and only if f(a) < f(b). As before, the associated total preorderis given by setting a ≲ b if and only if f(a) ≲ f(b), and the associated equivalence by setting a ∼ b if and onlyif f(a) ∼ f(b). It is not assumed here that f is an injective function, so a class of two equivalent elements on Ymay induce a larger class of equivalent elements on X. Also, f is not assumed to be an surjective function, so a classof equivalent elements on Y may induce a smaller or empty class on X. However, the function f induces an injectivefunction that maps the partition on X to that on Y. Thus, in the case of finite partitions, the number of classes in X isless than or equal to the number of classes on Y.

31.3 Related types of ordering

Semiorders generalize strict weak orderings, but do not assume transitivity of incomparability.[11] A strict weak orderthat is trichotomous is called a strict total order.[12] The total preorder which is the inverse of its complement is inthis case a total order.For a strict weak order "<" another associated reflexive relation is its reflexive closure, a (non-strict) partial order "≤".The two associated reflexive relations differ with regard to different a and b for which neither a < b nor b < a: in thetotal preorder corresponding to a strict weak order we get a ≲ b and b ≲ a, while in the partial order given by thereflexive closure we get neither a ≤ b nor b ≤ a. For strict total orders these two associated reflexive relations arethe same: the corresponding (non-strict) total order.[12] The reflexive closure of a strict weak ordering is a type ofseries-parallel partial order.

31.4 All weak orders on a finite set

31.4.1 Combinatorial enumeration

Main article: ordered Bell number

The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an n-elementset is given by the following sequence (sequence A000670 in OEIS):These numbers are also called the Fubini numbers or ordered Bell numbers.For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are threeways of partitioning the items into one singleton set and one group of two tied items, and each of these partitionsgives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering isreversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons,which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.

160 CHAPTER 31. WEAK ORDERING

(4,1,2,3)(4,2,1,3)

(3,2,1,4)

(3,1,2,4)

(2,1,3,4)

(1,2,3,4)

(1,2,4,3)

(1,3,2,4)

(2,1,4,3)

(2,3,1,4)

(3,1,4,2)

(4,1,3,2)

(4,2,3,1)

(3,2,4,1)(2,4,1,3)

(1,4,2,3)

(1,3,4,2)

(2,3,4,1)

(1,4,3,2)

(2,4,3,1)

(3,4,2,1)

(4,3,2,1)

(4,3,1,2)

(3,4,1,2)

The permutohedron on four elements, a three-dimensional convex polyhedron. It has 24 vertices, 36 edges, and 14 two-dimensionalfaces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.

31.4.2 Adjacency structure

Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves thatadd or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which allthree elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strictweak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weakorderings on a set are more highly connected. Define a dichotomy to be a weak ordering with two equivalence classes,and define a dichotomy to be compatible with a given weak ordering if every two elements that are related in theordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as aDedekind cut for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies.For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of movesthat add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the undirected graph thathas the weak orderings as its vertices, and these moves as its edges, forms a partial cube.[13]

Geometrically, the total orderings of a given finite set may be represented as the vertices of a permutohedron, and thedichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderingson the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedronitself, but not the empty set, as a face). The codimension of a face gives the number of equivalence classes in thecorresponding weak ordering.[14] In this geometric representation the partial cube of moves on weak orderings is thegraph describing the covering relation of the face lattice of the permutohedron.For instance, for n = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon(again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon,six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie,and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.

31.5. APPLICATIONS 161

31.5 Applications

As mentioned above, weak orders have applications in utility theory.[10] In linear programming and other types ofcombinatorial optimization problem, the prioritization of solutions or of bases is often given by a weak order, de-termined by a real-valued objective function; the phenomenon of ties in these orderings is called “degeneracy”, andseveral types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to preventproblems caused by degeneracy.[15]

Weak orders have also been used in computer science, in partition refinement based algorithms for lexicographicbreadth-first search and lexicographic topological ordering. In these algorithms, a weak ordering on the vertices of agraph (represented as a family of sets that partition the vertices, together with a doubly linked list providing a totalorder on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that isthe output of the algorithm.[16]

In the Standard Library for the C++ programming language, the set and multiset data types sort their input by acomparison function that is specified at the time of template instantiation, and that is assumed to implement a strictweak ordering.[17]

31.6 References[1] Roberts, Fred; Tesman, Barry (2011), Applied Combinatorics (2nd ed.), CRC Press, Section 4.2.4 Weak Orders, pp. 254–

256, ISBN 9781420099836.

[2] de Koninck, J. M. (2009), Those Fascinating Numbers, American Mathematical Society, p. 4, ISBN 9780821886311.

[3] Baker, Kent (April 29, 2007), “The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat forsecond”, The Baltimore Sun, (subscription required (help)).

[4] Regenwetter, Michel (2006), Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications, Cam-bridge University Press, pp. 113ff, ISBN 9780521536660.

[5] Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). Transitive Closures of Binary Relations I (PDF). Prague: Schoolof Mathematics - Physics Charles University. p. 1. Lemma 1.1 (iv). Note that this source refers to asymmetric relationsas “strictly antisymmetric”.

[6] Ehrgott, Matthias (2005), Multicriteria Optimization, Springer, Proposition 1.9, p. 10, ISBN 9783540276593.

[7] Stanley, Richard P. (1997), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cam-bridge University Press, p. 297.

[8] Motzkin, Theodore S. (1971), “Sorting numbers for cylinders and other classification numbers”, Combinatorics (Proc.Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968), Providence, R.I.: Amer. Math. Soc., pp.167–176, MR 0332508.

[9] Gross, O. A. (1962), “Preferential arrangements”, The American Mathematical Monthly 69: 4–8, doi:10.2307/2312725,MR 0130837.

[10] Roberts, Fred S. (1979),Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences, Ency-clopedia of Mathematics and its Applications 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0.

[11] Luce, R. Duncan (1956), “Semiorders and a theory of utility discrimination”, Econometrica 24: 178–191, JSTOR 1905751,MR 0078632.

[12] Velleman, Daniel J. (2006),How to Prove It: A Structured Approach, CambridgeUniversity Press, p. 204, ISBN9780521675994.

[13] Eppstein, David; Falmagne, Jean-Claude; Ovchinnikov, Sergei (2008),Media Theory: Interdisciplinary Applied Mathemat-ics, Springer, Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196.

[14] Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer, p. 18.

[15] Chvátal, Vašek (1983), Linear Programming, Macmillan, pp. 29–38, ISBN 9780716715870.

[16] Habib, Michel; Paul, Christophe; Viennot, Laurent (1999), “Partition refinement techniques: an interesting algorithmictool kit”, International Journal of Foundations of Computer Science 10 (2): 147–170, doi:10.1142/S0129054199000125,MR 1759929.

[17] Josuttis, Nicolai M. (2012), The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, p. 469, ISBN9780132977739.

162 CHAPTER 31. WEAK ORDERING

31.7 Text and image sources, contributors, and licenses

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• Intransitivity Source: https://en.wikipedia.org/wiki/Intransitivity?oldid=665745469 Contributors: Michael Hardy, Booyabazooka, Rp,6birc, Radicalsubversiv, Andres, Charles Matthews, Ruakh, Giftlite, Andris, Quickwik, D6, Xezbeth, Paul August, Jnestorius, Spoon!,Polluks, SmackBot, Melchoir, Mauls, MisterHand, Lambiam, Iamagloworm, Dinkumator, CRGreathouse, CmdrObot, CBM, Thomas-meeks, Ael 2, Thijs!bot, VoABot II, Cnilep, Ddxc, Anchor Link Bot, PixelBot, DumZiBoT, Pa68, Addbot, Forich, WissensDürster,Undsoweiter, HRoestBot, ChronoKinetic, RjwilmsiBot, Chharvey and Anonymous: 9

• Material conditional Source: https://en.wikipedia.org/wiki/Material_conditional?oldid=665659334 Contributors: William Avery, Dcljr,AugPi, Charles Matthews, Dcoetzee, Doradus, Cholling, Giftlite, Jason Quinn, Nayuki, TedPavlic, Elwikipedista~enwiki, Nortexoid,Vesal, Eric Kvaalen, BD2412, Kbdank71, Martin von Gagern, Joel D. Reid, Fresheneesz, Vonkje, NevilleDNZ, RussBot, KSchutte,NawlinWiki, Trovatore, Avraham, Closedmouth, Arthur Rubin, SyntaxPC, Fctk~enwiki, SmackBot, Amcbride, Incnis Mrsi, Pokipsy76,BiT, Mhss, Jaymay, Tisthammerw, Sholto Maud, Robma, Cybercobra, Jon Awbrey, Oceanofperceptions, Byelf2007, Grumpyyoung-man01, Clark Mobarry, Beefyt, Rory O'Kane, Dreftymac, Eassin, JRSpriggs, Gregbard, FilipeS, Cydebot, Julian Mendez, Thijs!bot,Egriffin, Jojan, Escarbot, Applemeister, WinBot, Salgueiro~enwiki, JAnDbot, Olaf, Alastair Haines, Arno Matthias, JaGa, SantiagoSaint James, Pharaoh of the Wizards, Pyrospirit, SFinside, Anonymous Dissident, The Tetrast, Cnilep, Radagast3, Newbyguesses, Light-breather, Paradoctor, Iamthedeus, Soler97, Francvs, Classicalecon, Josang, Ruy thompson, Watchduck, Hans Adler, Djk3, Marc vanLeeuwen, Tbsdy lives, Addbot, Melab-1, Fyrael, Morriswa, SpellingBot, CarsracBot, Chzz, Jarble, Meisam, Luckas-bot, AnomieBOT,Sonia, Pnq, Bearnfæder, FrescoBot, Greyfriars, Machine Elf 1735, RedBot, MoreNet, Beyond My Ken, John of Reading, Hgetnet, Hi-bou57, ClueBot NG, Movses-bot, Jiri 1984, Masssly, Dooooot, Noobnubcakes, Hanlon1755, Leif Czerny, CarrieVS, Jochen Burghardt,Lukekfreeman, Lingzhi, NickDragonRyder, Indomitavis, Rathkirani, AnotherPseudonym, Xerula, Matthew Kastor, Mathematical Truthand Anonymous: 73

• Matrilineality Source: https://en.wikipedia.org/wiki/Matrilineality?oldid=670959043 Contributors: Vicki Rosenzweig, -- April, Si-monP, Youandme, Tbarron, Edward, Llywrch, MartinHarper, IZAK, Hashar, Hyacinth, Chl, Vardion, Astronautics~enwiki, AshleyY, Saforrest, Cecropia, Jfdwolff, Sidar, Antandrus, Robin klein, Epimetreus, Jayjg, Rich Farmbrough, Florian Blaschke, Dbachmann,Pavel Vozenilek, Kaisershatner, *drew, Jpallan, Reinyday, Giraffedata, Alansohn, Sushil~enwiki, Babajobu, Mysdaao, Svartalf, Has-drubal~enwiki, TShilo12, Woohookitty, Spettro9, Lofor, Miss Madeline, Mandarax, Dpv, Rjwilmsi, Cribbswh, FlaBot, Alex Kapra-noff, Ewlyahoocom, DaGizza, Carolynparrishfan, RussBot, Pigman, CambridgeBayWeather, Tfine80, Welsh, Dmoss, Mysid, M arpal-mane, Tullie, Petri Krohn, For7thGen, Peyna, Greatal386, Sardanaphalus, SmackBot, Politono, Kintetsubuffalo, Chris the speller, Blue-bot, Gruzd, EPM, Caniago, Ewright12, Parrish Smith, Sammy1339, Bdiscoe, Manojn, Eliyak, Gloriamarie, James.S, AB, The Manin Question, Gdeyoe, RekishiEJ, Penbat, Johnchampe, Jedonnelley, Sirmylesnagopaleentheda, SpK, Rosser1954, Thijs!bot, Missvain,Amity150, Agye, Signeco, Jazzeur, Alastair Haines, .anacondabot, Naval Scene, Magioladitis, RebekahThorn, Dayaanjali, Indon, Ikon-icDeath, Damuna, Walter Breitzke, R'n'B, Bentaguayre, Naniwako, Gabr-el, M-le-mot-dit, 83d40m, Joshua Issac, Medicineman84,28bytes, Kyle the bot, Lradrama, ^demonBot2, Nick Levinson, FrinkMan, Yrichon, Tombomp, Fratrep, FredrikLähnn, Dcattell, BCR123,Balajikartha, ClueBot, SummerWithMorons, EoGuy, Mild Bill Hiccup, Asgreen, Vvmundakkal, Davyisadavy, Half-jewish, JCDen-ton2052, SweetNightmares, Surtsicna, Hijjins, Gbyitc, Addbot, Martindo, AndersBot, Nostri Fahkhoul Gʰålaɪ̯jme, Legobot, Yobot,Catmandu1, Denispir, Yngvadottir, AnomieBOT, Rubinbot, Ulric1313, Materialscientist, Hunnjazal, Joys choice, Saktoth, Wet dog fur,Altg20April2nd, Nacho Insular, The Legal Limit, DevilsAdvocate987, Tamsier, Newsit, Tomyhoi, Brakoholic, Skamecrazy123, Emaus-Bot, K6ka, Djembayz, Midas02, Erianna, Mcc1789, Pernoctator, Mjbmrbot, ClueBot NG, Frietjes, Widr, Jrs4c, Helpful Pixie Bot,Qowieury, Eladynnus, PhnomPencil, BattyBot, Mdann52, Neshmick, ChrisGualtieri, Svensson1, Dlsantos69,Mendsetting, Gabriel.hassler,Schrauwers, Melonkelon, Ginsuloft, Anrnusna, That:guy245, KasparBot and Anonymous: 123

• Nontransitive dice Source: https://en.wikipedia.org/wiki/Nontransitive_dice?oldid=671705991 Contributors: Michael Hardy, Booy-abazooka, AnonMoos, Bkell, Tobias Bergemann, Giftlite, Gro-Tsen, JasonQuinn, Noe, Achituv~enwiki, Discospinster, Cretog8,M3tainfo,

164 CHAPTER 31. WEAK ORDERING

INic, Joriki, DurtyWilly, Ewlyahoocom, Schoen, Poulpy, SmackBot, McGeddon, Gilliam, Amatulic, PrimeHunter, Kjetil1001, Cyber rig-ger, Korako, Robma, Cybercobra, Hgilbert, Byelf2007, Focomoso, Skapur, Lenoxus, Amniarix, CRGreathouse, Doctormatt, Thijs!bot,GuyMacon, Nikolas Karalis, David Eppstein, Markdettinger, RaitisMath, CommonsDelinker, Leon math, LokiClock, Suhayb00, Iamred-dave, TorbjörnAxelsson, PipepBot, Dagron12345, GreekHouse, Petersburg,Mild Bill Hiccup, Takeaway, Smeacock, Addbot, Bogus.Koszalka,Luckas-bot, Ptbotgourou, AnomieBOT,Mlpearc, Srich32977, BenzolBot, Devrand, RedBot, Vectornaut, Gfs6grade, Barninator, Fivealiveprize,Rinolds, Dmitry123456, Zauberworter, LVZee, ChrisGualtieri, Bozoki~enwiki, UNOwenNYC, Dr.mgf.winkelmann, Dehkan, Cgather-cole, Monkbot, Mvpo666, Loraof, Iwilsonp and Anonymous: 74

• Partition of a set Source: https://en.wikipedia.org/wiki/Partition_of_a_set?oldid=671530023 Contributors: AxelBoldt, Tomo, Patrick,Michael Hardy, Wshun, Kku, Revolver, Charles Matthews, Zero0000, Robbot, MathMartin, Ruakh, Tobias Bergemann, Giftlite, Smjg,Arved, Fropuff, Gubbubu, Mennucc, Sam Hocevar, Tsemii, TedPavlic, Paul August, Zaslav, Elwikipedista~enwiki, El C, PhilHibbs,Corvi42, Oleg Alexandrov, Stemonitis, Bobrayner, Kelly Martin, Linas, MFH, Mayumashu, Salix alba, R.e.b., FlaBot, Mathbot, Yurik-Bot, Laurentius, Gaius Cornelius, StevenL, Pred, Finell, Capitalist, That Guy, From That Show!, Adam majewski, Mcld, Mhss, Tsca.bot,Mhym, Armend, Jon Awbrey, Rob Zako, CRGreathouse, Sopoforic, Escarbot, Magioladitis, Jiejunkong, David Eppstein, Lantonov,Elenseel, TXiKiBoT, Anonymous Dissident, PaulTanenbaum, Jamelan, Skippydo, PipepBot, DragonBot, Watchduck, Addbot, AkhtaBot,Legobot, Luckas-bot, Yobot, Bunnyhop11, Calle, AnomieBOT, ArthurBot, Stereospan, MastiBot, EmausBot, MartinThoma, ZéroBot,D.Lazard, Orange Suede Sofa, ClueBot NG, Mesoderm, Helpful Pixie Bot, BG19bot, MinatureCookie, Mark viking, Cepphus, Beth-Naught and Anonymous: 45

• Preorder Source: https://en.wikipedia.org/wiki/Preorder?oldid=611521876 Contributors: AxelBoldt, Patrick, Repton, Delirium, An-dres, Dysprosia, Greenrd, Big Bob the Finder, BenRG, Tobias Bergemann, Giftlite, Markus Krötzsch, Lethe, Fropuff, Vadmium, De-fLog~enwiki, Zzo38, Jh51681, Barnaby dawson, Paul August, EmilJ, Msh210, Melaen, Joriki, Linas, Dionyziz, Mandarax, Salix alba,Cjoev, VKokielov, Mathbot, Jrtayloriv, YurikBot, Laurentius, Hairy Dude, WikidSmaht, Trovatore, Modify, Netrapt, Wasseralm, Smack-Bot, XudongGuan~enwiki, DCary, Jdthood, Mets501, PaulGS, Stotr~enwiki, Zero sharp, CRGreathouse, Michael A. White, DavidEppstein, Jwuthe2, PaulTanenbaum, SieBot, Thehotelambush, Functor salad, He7d3r, Sun Creator, Cenarium, 1ForTheMoney, Palnot,Мыша, Legobot, Luckas-bot, AnomieBOT, DannyAsher, Xqbot, VladimirReshetnikov, ComputScientist, BrideOfKripkenstein, Noted-grant, WikitanvirBot, Lclem, Dfabera, SporkBot, RichardMills65, Khazar2, Lerutit, Jochen Burghardt, Reatlas, Damonamc and Anony-mous: 26

• Quasitransitive relation Source: https://en.wikipedia.org/wiki/Quasitransitive_relation?oldid=625289154Contributors: Patrick, CharlesMatthews, Giftlite, Btyner, Salix alba, SmackBot, Zahid Abdassabur, CRGreathouse, CBM, Gregbard, Sam Staton, Erik9bot, Deltahe-dron, Jochen Burghardt and Anonymous: 1

• Quotient category Source: https://en.wikipedia.org/wiki/Quotient_category?oldid=613452510 Contributors: Fropuff, BD2412, Masn-evets, JoeBot and Anonymous: 6

• Quotient ring Source: https://en.wikipedia.org/wiki/Quotient_ring?oldid=666100333 Contributors: AxelBoldt, Patrick, Michael Hardy,Ciphergoth, MathMartin, Nikitadanilov, Giftlite, Waltpohl, Jorge Stolfi, Rgdboer, Linas, Marudubshinki, Chobot, Algebraist, YurikBot,Rsrikanth05, That Guy, From That Show!, Reedy, MalafayaBot, Jim.belk, Rschwieb, CRGreathouse, CBM, Thijs!bot, Albmont, Policron,DemonicInfluence, VolkovBot, Arcfrk, Katzmik, SieBot, Addbot, Jarble, Luckas-bot, Ark11, AnomieBOT, Measles, Ebony Jackson,RjwilmsiBot, EmausBot, ZéroBot, Otaria, Noahc66260, ChrisGualtieri, Mathedu and Anonymous: 19

• Quotient space (linear algebra) Source: https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra)?oldid=655998606 Contribu-tors: Zundark, Michael Hardy, Giftlite, Fropuff, Varuna, Oleg Alexandrov, Joriki, Igny, Frigoris, Bluebot, Silly rabbit, Repliedthemock-turtle, Thijs!bot, RobHar, Englebert, VolkovBot, AlleborgoBot, MikeRumex, SieBot, Ideal gas equation, MystBot, Addbot, Ronhjones,Yobot, RibotBOT, Qm2008q, Sławomir Biały, BenzolBot, Quondum, Helpful Pixie Bot and Anonymous: 11

• Quotient space (topology) Source: https://en.wikipedia.org/wiki/Quotient_space_(topology)?oldid=660820903Contributors: AxelBoldt,The Anome, Dominus, Charles Matthews, Dysprosia, Jitse Niesen, Tobias Bergemann, Tosha, Giftlite, Fropuff, Bobblewik, Bender235,Oleg Alexandrov, Linas, Marudubshinki, Salix alba, Eubot, YurikBot, Zwobot, Lunch, Schizobullet, Bluebot, Nbarth, Mets501, CBM,Tdvance, Thijs!bot, Reminiscenza, Thomasda, Camrn86, LokiClock, Rei-bot, Anonymous Dissident, Subh83, Kromsson, Prashantva,Alexbot, DumZiBoT, ManDay, Addbot, Luckas-bot, Yobot, Xqbot, Dega180, Magmalex, RjwilmsiBot, EmausBot, WikitanvirBot, Con-jugado, Chricho, Quondum, D.Lazard, Behrooz Abazari, Nuermann, Doublethink1984, Majesty of Knowledge, CrazyDewgong, Aritropand Anonymous: 35

• Rational choice theory Source: https://en.wikipedia.org/wiki/Rational_choice_theory?oldid=670602364Contributors: Ryguasu, Patrick,Michael Hardy, Andres, Clausen, Taxman, Robbot, Altenmann, Academic Challenger, Ghormax, Aetheling, Andries, Abu badali, Jdevine,Pgreenfinch, Fintor, Vsb, Absinf, Lucidish, Mormegil, Carlon, Lycurgus, Bobo192, Cretog8, Doctorvee, Tmh, Gary, John Quiggin,Blaxthos, Mindmatrix, Kzollman, Graham87, John Deas, Kspence, Lmatt, David91, Volunteer Marek, FrankTobia, YurikBot, Wave-length, KSchutte, Ziel, Farmanesh, Igiffin, Open2universe, Palthrow, Anclation~enwiki, Erudy, C mon, Sardanaphalus, SmackBot, Red-House18, Commander Keane bot, Bluebot, Thegn, DroEsperanto, MartinPoulter, Mikker, Shagifier, Alieseraj, Lifestream87, Yohan euano4, Tazmaniacs, Aroundthewayboy, RobotDjuret, Doczilla, Levineps, JHP, ScierGuy, CalebNoble, Bliggin, CmdrObot, Thomasmeeks,Moreschi, Llort, Thijs!bot, Hannoscholtz, Juanholanda, TTwist, X96lee15, Dhh28, Lfstevens, Spinmeister, JAnDbot, Epeefleche, AdrianJ. Hunter, Quinceps, Falcor84, Dionysiaca, DarwinPeacock, DadaNeem, Vbreed7, DavidCBryant, Squids and Chips, VolkovBot, Barneca,Philip Trueman, CUBJONES83, P.dunleavy, TheSix, Etmueller, Amd628, Larklight, Lova Falk, Amritasenray, Brenont, Paradoctor, Nat-bengold, Revent, Hedgehogfox, Dans, Precious Roy, Cborn13, ClueBot, SkpRCT, Shavonna88, Redthoreau, Behavioralethics, Poli08,Phoebe13, Pa68, Human fella, Addbot, Katzeiler, Forich, Tassedethe, Legobot, Luckas-bot, Yobot, TaBOT-zerem, Carleas, Eduen,AnomieBOT, Rubinbot, Xqbot, NOrbeck, GrouchoBot, Omnipaedista, Djenbelle, Et bravo, Okcrounders, FrescoBot, Diablotin, Al-dousari, Hpainter, LittleWink, Caspian Rehbinder, TenPhil, Kopernagel, Libby norman, EmausBot, WikitanvirBot, Magneman, L Kens-ington, Phronetic, Ratpow, ClueBot NG, JRheic, Jj1236, PolySciJoe, Hapli, BG19bot, Rapn21, Karim errouaki, BattyBot, AcadēmicaOrientālis, Uday.gautam6, Sangaylodae, Jpmarchant, Rothbardanswer, Byronmercury, Cerabot~enwiki, Lugia2453, Jonthawk, Thenar-rators666a666, Pranavrk, Star767, WeakTrain, Someone not using his real name, Aubreybardo, Jacapsicum, Scarlettail, Republic683,Hardboiledwunderkind, Loraof, Treeedit, Social Theory, KasparBot and Anonymous: 120

• Reflexive relation Source: https://en.wikipedia.org/wiki/Reflexive_relation?oldid=645017228Contributors: AxelBoldt, DavidSJ, Patrick,Wshun, TakuyaMurata, Looxix~enwiki, William M. Connolley, Charles Matthews, Josh Cherry, MathMartin, Henrygb, Tobias Berge-mann, Giftlite, Jason Quinn, Gubbubu, Urhixidur, Ascánder, Paul August, BenjBot, Spayrard, Spoon!, Jet57, LavosBacons, Wtmitchell,Bookandcoffee, Oleg Alexandrov, Joriki, Mel Etitis, LOL, MFH, Isnow, Audiovideo, Margosbot~enwiki, Fresheneesz, Chobot, Yurik-Bot, Laurentius, Maelin, Mathlaura, KarlHeg, Arthur Rubin, Isaac Dupree, Jdthood, Mhym, Ceosion, Mike Fikes, Fjbex, CRGreathouse,

31.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 165

CBM, Gregbard, Farzaneh, Wikid77, JAnDbot, Policron, Joshua Issac, VolkovBot, Jackfork, Jamelan, Ocsenave, SieBot, Davidellerman,Henry Delforn (old), Hello71, Cuyaken, ClueBot, Ywanne, Da rulz07, SoxBot III, WikHead, Addbot, Download, Favonian, Luckas-bot, Yobot, Renato sr, Pkukiss, Galoubet, ArthurBot, Xqbot, Z0973, I dream of horses, RedBot, MastiBot, Gamewizard71, EmausBot,Dmayank, DimitriC, ClueBot NG, Kasirbot, Joel B. Lewis, BG19bot, ChrisGualtieri, Eptified, Lerutit, Jochen Burghardt, Seanhalle andAnonymous: 37

• Semigroup Source: https://en.wikipedia.org/wiki/Semigroup?oldid=669686909Contributors: AxelBoldt, BryanDerksen, Zundark, Fnielsen,XJaM, Youandme, Michael Hardy, TakuyaMurata, LittleDan, Ideyal, Charles Matthews, Dysprosia, Wile E. Heresiarch, Tobias Berge-mann, Giftlite, Markus Krötzsch, Pretzelpaws, Jason Quinn, Cambyses, Beland, MarkSweep, Vina, Chris Howard, AlexG, Foolip, EmilJ,3mta3, Obradovic Goran, Benschop, Msh210, Eric Kvaalen, Oleg Alexandrov, Natalya, Simetrical, Linas, MFH, Graham87, Chenxlee,Salix alba, Magidin, FlaBot, John Baez, Chobot, CDH, Mikeblas, Jess Riedel, Kompik, SmackBot, Sxtr, Bluebot, RDBrown, Nbarth,Kjetil1001, Michael Kinyon, Mathsci, Rschwieb, JMK, James pic, CRGreathouse, W.F.Galway, Pascal.Tesson, Headbomb, Shlomi Hillel,Salgueiro~enwiki, JAnDbot, Magioladitis, Mathematrucker, Sweetjazz3, Sullivan.t.j, Xanthir, R'n'B, Krishnachandranvn, GaborLajos,Theowoll, Alan U. Kennington, LokiClock, Geometry guy, Maarten van Emden, SieBot, Thehotelambush, Skippydo, Classicalecon,PipepBot, Output~enwiki, He7d3r, Beroal, G20071221, SilvonenBot, Addbot, Tjlaxs, AndersBot, FiriBot, Ozob, Lightbot, Zorrobot,Luckas-bot, Yobot, Rdancer, Pcap, AnomieBOT, Citation bot, Nishantjr, Anne Bauval, GrouchoBot, Point-set topologist, Charvest,Sławomir Biały, Negi(afk), Citation bot 1, Lars Washington, Quotient group, Dewey process, John of Reading, H3llBot, Quondum,Vladimirdx, ChuispastonBot, Booradleyp, Halflingr, Helpful Pixie Bot, Paolo Lipparini, Minsbot, Nadapez~enwiki, IkamusumeFan,Deltahedron, Mark viking, CsDix, Haminoon, JMP EAX, Physalis longifolia and Anonymous: 61

• Set (mathematics) Source: https://en.wikipedia.org/wiki/Set_(mathematics)?oldid=671842839 Contributors: Damian Yerrick, Axel-Boldt, Lee Daniel Crocker, Archibald Fitzchesterfield, Uriyan, Bryan Derksen, Zundark, The Anome, -- April, LA2, XJaM, Toby~enwiki,Toby Bartels, Pb~enwiki, Waveguy, LapoLuchini, Youandme, Olivier, Paul Ebermann, Frecklefoot, Patrick, TeunSpaans, Michael Hardy,Wshun, Booyabazooka, Fred Bauder, Nixdorf, Wapcaplet, TakuyaMurata, CesarB, Mkweise, Iulianu, Docu, Den fjättrade ankan~enwiki,Александър, UserGoogol, Rob Hooft, Jonik, Mxn, Etaoin, Vargenau, Pizza Puzzle, Schneelocke, Ideyal, Charles Matthews, Timwi,Dcoetzee, Ike9898, Dysprosia, Jitse Niesen, Greenrd, Prumpf, Furrykef, Hyacinth, Saltine, Stormie, RealLink, JorgeGG, Phil Boswell,Robbot, Astronautics~enwiki, Fredrik, Altenmann, Peak, Romanm, Mfc, Tobias Bergemann, Centrx, Giftlite, Smjg, Ævar ArnfjörðBjarmason, Lethe, Dissident, Waltpohl, Ezhiki, Prosfilaes, Python eggs, Chameleon, Gubbubu, Leonard Vertighel, Utcursch, Gdr, Knu-tux, Antandrus, BozMo, Mustafaa, Rousearts, Joseph Myers, Crawdaddio, Maximaximax, Wiml, Tothebarricades.tk, Zfr, Sam Hoce-var, Urhixidur, Vivacissamamente, Porges, Corti, PhotoBox, Shahab, Brianjd, Dissipate, Vinoir, EugeneZelenko, Discospinster, Mani1,Paul August, Demaag, Rgdboer, Crisófilax, Aaronbrick, Bobo192, Army1987, C S, Johnteslade, Blotwell, ריינהארט ,לערי Jumbuck,Msh210, Gary, Tablizer, Eric Kvaalen, Andrewpmk, Richard Fannin, EvenT, Arag0rn, CloudNine, Dirac1933, Spambit, Oleg Alexan-drov, Kendrick Hang, Scndlbrkrttmc, Ott, OwenX, Prashanthns, Marudubshinki, LimoWreck, Graham87, BD2412, Dpr, Josh Par-ris, Mayumashu, MarSch, Trlovejoy, Salix alba, Heah, Bubba73, VKokielov, Mathbot, Crazycomputers, Jrtayloriv, Fresheneesz, Kri,Chobot, Algebraist, YurikBot, Wavelength, Xcelerate, Charles Gaudette, RussBot, Lucinos~enwiki, Thane, Trovatore, Srinivasasha,BOT-Superzerocool, Bota47, RyanJones, Ms2ger, Hirak 99, Lt-wiki-bot, Arthur Rubin, Gulliveig, Reyk, ArielGold, Finell, Dudzcom,RupertMillard, SmackBot, Unyoyega, Bomac, Brick Thrower, BiT, Gilliam, Kaiserb, @modi, Trebor, MartinPoulter, Jerome CharlesPotts, Octahedron80, DHN-bot~enwiki, Bob K, Cybercobra, Nakon, Jiddisch~enwiki, Richard001, MathStatWoman, Just plain Bill,RayGates, SashatoBot, Dfass, Loadmaster, Squigglet, Rosejn, IvanLanin, Tawkerbot2, JRSpriggs, KNM, CRGreathouse, Benjistern,Ale jrb, Makeemlighter, CBM, Anakata, Except, Sax Russell, ShelfSkewed, WeggeBot, Asztal, Gregbard, Danman3459, MKil, Tick-etMan, Julian Mendez, He Who Is, Viridae, Xantharius, Lindsay658, Daniel Olsen, Malleus Fatuorum, Thijs!bot, Epbr123, Knakts,RobHar, Escarbot, EJR~enwiki, Seaphoto, Tchakra, JAnDbot, Leuko, Tomst, VoABot II, JNW, Kajasudhakarababu, Echoback, DavidEppstein, JoergenB, Sammi84, MartinBot, Vigyani, J.delanoy, Maurice Carbonaro, Cpiral, Utkwes, NewEnglandYankee, DavidCBryant,Djr13, Idioma-bot, VolkovBot, Paul.w.bennett, Am Fiosaigear~enwiki, Philip Trueman, TXiKiBoT, Anonymous Dissident, Ocolon, Jhs-Bot, Wowzavan, PaulTanenbaum, LBehounek, Lerdthenerd, Synthebot, A Raider Like Indiana, Dmcq, Symane, EmxBot, PaddyLeahy,YohanN7, Dogah, SieBot, BotMultichill, Jauerback, Yintan, Happysailor, Paolo.dL, Allmightyduck, Oxymoron83, Kumioko (renamed),Cyfal, DEMcAdams, WikiBotas, Damien Karras, ClueBot, Arcsecant, PipepBot, Cliff, DionysosProteus, Unbuttered Parsnip, Gog-amoga, Drmies, Asdasdasda, Liempt, Jusdafax, Watchduck, Jotterbot, Hans Adler, Apparition11, Gerhardvalentin, Mosaick~enwiki,Ahsanlassi, Multipundit, Addbot, Cxz111, Mnmazur, Protonk, The world deserves the truth, Ingeniosus, Omnipedian, LinkFA-Bot, Har-sha6raju, Tide rolls, Lightbot, Zorrobot, Jarble, Yobot, Fraggle81, TonyFlury, AnomieBOT, Materialscientist, Kimsey0, Twri, Arthur-Bot, Xqbot, Taffer9, Vxk08u, Medoshalaby, Pmlineditor, GrouchoBot, RibotBOT, Mathonius, Fangncurl, Laelele, SD5, FrescoBot, Sła-womir Biały, Mfwitten, DivineAlpha, Tkuvho, Pinethicket, NearSetAccount, 10metreh, SkyMachine, Dashed, Tgv8925, Declan Clam,Miracle Pen, Bluefist, Theo10011, Jesse V., DARTH SIDIOUS 2, EmausBot, Acather96, RenamedUser01302013, John Cline, Bol-lyjeff, Akerans, Hgetnet, Tolly4bolly, L Kensington, Zephyrus Tavvier, Donner60, Chewings72, Puffin, Gwestheimer, Petrb, ClueBotNG, Gareth Griffith-Jones, Wcherowi, Satellizer, SusikMkr, Frietjes, O.Koslowski, Widr, WikiPuppies, MerlIwBot, Helpful Pixie Bot,BG19bot, Saulpila2000, Shashank rathore, Frze, AvocatoBot, AmieKim, Shyamli rao, Cliff12345, Eduardofeld, Krothgar, Sivarama.prsd,Ejazahmed007, MadGuy7023, Volvens, Mark L MacDonald, Stephan Kulla, Jfanderson68, Epicgenius, Jodosma, Tentinator, GlenBehrend, Bg9989, DavidLeighEllis, Bvraamaraaju, Ugog Nizdast, Ginsuloft, Quenhitran, AddWittyNameHere, Wikireadya, Mahusha,IvanZhilin, Degenerate prodigy, KasparBot, RJ raghava, Mohammad Saad Ifrahim Khan and Anonymous: 438

• Subset Source: https://en.wikipedia.org/wiki/Subset?oldid=671828052 Contributors: Damian Yerrick, AxelBoldt, Youssefsan, XJaM,Toby Bartels, StefanRybo~enwiki, Edward, Patrick, TeunSpaans, Michael Hardy, Wshun, Booyabazooka, Ellywa, Oddegg, Andres,Charles Matthews, Timwi, Hyacinth, Finlay McWalter, Robbot, Romanm, Bkell, 75th Trombone, Tobias Bergemann, Tosha, Giftlite,Fropuff, Waltpohl, Macrakis, Tyler McHenry, SatyrEyes, Rgrg, Vivacissamamente, Mormegil, EugeneZelenko, Noisy, Deh, Paul Au-gust, Engmark, Spoon!, SpeedyGonsales, Obradovic Goran, Nsaa, Jumbuck, Raboof, ABCD, Sligocki, Mac Davis, Aquae, LFaraone,Chamaeleon, Firsfron, Isnow, Salix alba, VKokielov, Mathbot, Harmil, BMF81, Chobot, Roboto de Ajvol, YurikBot, Alpt, Dmharvey,KSmrq, NawlinWiki, Trovatore, Nick, Szhaider, Wasseralm, Sardanaphalus, Jacek Kendysz, BiT, Gilliam, Buck Mulligan, SMP, Or-angeDog, Bob K, Dreadstar, Bjankuloski06en~enwiki, Loadmaster, Vedexent, Amitch, Madmath789, Newone, CBM, Jokes Free4Me,345Kai, SuperMidget, Gregbard, WillowW, MC10, Thijs!bot, Headbomb, Marek69, RobHar, WikiSlasher, Salgueiro~enwiki, JAnDbot,.anacondabot, Pixel ;-), Pawl Kennedy, Emw, ANONYMOUS COWARD0xC0DE, RaitisMath, JCraw, Tgeairn, Ttwo, Maurice Car-bonaro, Acalamari, Gombang, NewEnglandYankee, Liatd41, VolkovBot, CSumit, Deleet, Rei-bot, AnonymousDissident, James.Spudeman,PaulTanenbaum, InformationSpace, Falcon8765, AlleborgoBot, P3d4nt, NHRHS2010, Garde, Paolo.dL, OKBot, Brennie8, Jons63,Loren.wilton, ClueBot, GorillaWarfare, PipepBot, The Thing That Should Not Be, DragonBot, Watchduck, Hans Adler, Computer97,Noosentaal, Versus22, PCHS-NJROTC, Andrew.Flock, Reverb123, Addbot, , Fyrael, PranksterTurtle, Numbo3-bot, Zorrobot, Jar-ble, JakobVoss, Luckas-bot, Yobot, Synchronism, AnomieBOT, Jim1138, Materialscientist, Citation bot, Martnym, NFD9001, Char-vest, 78.26, XQYZ, Egmontbot, Rapsar, HRoestBot, Suffusion of Yellow, Agent Smith (The Matrix), RenamedUser01302013, ZéroBot,

166 CHAPTER 31. WEAK ORDERING

Alexey.kudinkin, Chharvey, Quondum, Chewings72, 28bot, ClueBot NG, Wcherowi, Matthiaspaul, Bethre, Mesoderm, O.Koslowski,AwamerT, Minsbot, Pratyya Ghosh, YFdyh-bot, Ldfleur, ChalkboardCowboy, Saehry, Stephan Kulla, , Ilya23Ezhov, Sandshark23,Quenhitran, Neemasri, Prince Gull, Maranuel123, Alterseemann, Rahulmr.17, Johnkennethcfamero and Anonymous: 183

• Symmetric relation Source: https://en.wikipedia.org/wiki/Symmetric_relation?oldid=623361613 Contributors: Patrick, Looxix~enwiki,William M. Connolley, Charles Matthews, MathMartin, Tobias Bergemann, Giftlite, Elektron, Ascánder, Paul August, Syp, Joriki, Is-now, Salix alba, Margosbot~enwiki, Fresheneesz, Roboto de Ajvol, Laurentius, Bota47, Arthur Rubin, Incnis Mrsi, Unyoyega, Jdthood,Vina-iwbot~enwiki, Gregbard, Thijs!bot, Mouchoir le Souris, David Eppstein, Jamelan, Henry Delforn (old), ClueBot, Libcub, Addbot,Luckas-bot, ArthurBot, Xqbot, Adavis444, RedBot, EmausBot, ZéroBot, Zap Rowsdower, EdoBot, DASHBotAV, 28bot, Kasirbot, No-suchforever and Anonymous: 18

• Total order Source: https://en.wikipedia.org/wiki/Total_order?oldid=666410207 Contributors: Damian Yerrick, AxelBoldt, Zundark,XJaM, Fritzlein, Patrick, Michael Hardy, Dori, AugPi, Dysprosia, Jitse Niesen, Greenrd, Zoicon5, Hyacinth, VeryVerily, Fibonacci,McKay, Aleph4, Gandalf61, MathMartin, Rursus, Tobias Bergemann, Giftlite, Mshonle~enwiki, Markus Krötzsch, Lethe, Waltpohl,DefLog~enwiki, Alberto da Calvairate~enwiki, Quarl, Elroch, Paul August, Susvolans, Army1987, Func, Cmdrjameson, Msh210, Pion,Joriki, MattGiuca, Yurik, OneWeirdDude, Salix alba, VKokielov, Mathbot, Margosbot~enwiki, Wastingmytime, Chobot, YurikBot,Hede2000, Tetracube, Rdore, Melchoir, Gelingvistoj, Mhss, Chris the speller, Bazonka, Jdthood, Javalenok, Michael Kinyon, Loadmas-ter, Mets501, George100, CRGreathouse, CBM, Thomasmeeks, Oryanw~enwiki, VectorPosse, JAnDbot, David Eppstein, Infovarius,Osquar F, PaulTanenbaum, SieBot, Ceroklis, Anchor Link Bot, Heinzi.at, WurmWoode, Universityuser, Palnot, Marc van Leeuwen,Addbot, Tanhabot, AsphyxiateDrake, Luckas-bot, Yobot, Charlatino, White gecko, 1exec1, Infvwl, GrouchoBot, Jsjunkie, Quondum,D.Lazard, SporkBot, CocuBot, BG19bot, YumOooze, YFdyh-bot, Austinfeller, Mark viking, नितीश् चन्द्र and Anonymous: 47

• Total relation Source: https://en.wikipedia.org/wiki/Total_relation?oldid=608974105 Contributors: Patrick, Charles Matthews, Dcoet-zee, Jitse Niesen, Tobias Bergemann, Lethe, Alberto da Calvairate~enwiki, Paul August, Ntmatter, Oleg Alexandrov, Joriki, Salix alba,Nneonneo, Mathbot, Fresheneesz, Bota47, Jdthood, Stotr~enwiki, JAnDbot, TXiKiBoT, Jamelan, Hans Adler, Erodium, Addbot, Fres-coBot, SporkBot, Helpful Pixie Bot, Deltahedron, Kephir and Anonymous: 11

• Transitive closure Source: https://en.wikipedia.org/wiki/Transitive_closure?oldid=666646250 Contributors: Awaterl, Vkuncak, Patrick,Michael Hardy, Charles Matthews, Timwi, Dcoetzee, Populus, Borislav, Tobias Bergemann, Giftlite, Fropuff, Matt Crypto, Alexf,Quickwik, Creidieki, Obradovic Goran, Oleg Alexandrov, Joriki, Neonfreon, Salix alba, AL SAM, Bgwhite, Ott2, Arthur Rubin, Plas-ticphilosopher, KnightRider~enwiki, Mhss, Plustgarten, Dreadstar, NeilFraser, Lyonsam, Loadmaster, JRSpriggs, CRGreathouse, CBM,ShelfSkewed, Gregbard, Girlwithglasses, Kirtag Hratiba, Thijs!bot, JAnDbot, A3nm, David Eppstein, Yavoh, Cometstyles, VolkovBot,Sdrucker, PaulTanenbaum, Jamelan, Tomaxer, LungZeno, Henry Delforn (old), DuaneLAnderson, CBM2, Classicalecon, ClueBot, Ben-der2k14, PixelBot, AmirOnWiki, MountainGoat8, Tayste, Addbot, Luckas-bot, AnomieBOT, BenzolBot, RedBot, MastiBot, Trappistthe monk, Wizeguytristram, Quondum, Tijfo098, ClueBot NG, BG19bot, Danwizard208, Dmitri L. Slabk., Vpieterse~enwiki, Seahen,Artdadamo and Anonymous: 26

• Transitive reduction Source: https://en.wikipedia.org/wiki/Transitive_reduction?oldid=655809786 Contributors: Michael Hardy, Rp,Charles Matthews, Dcoetzee, Greenrd, Giftlite, EmilJ, Rjwilmsi, Salix alba, Algebraist, Rwalker, Ott2, Arthur Rubin, SmackBot, Bluebot,Nbarth, Ligulembot, Lyonsam, CRGreathouse, David Cooke, A3nm, David Eppstein, PaulTanenbaum, Addbot, Cuaxdon, Luckas-bot,John of Reading, ZéroBot, W1r3d2 and Anonymous: 10

• Transitive relation Source: https://en.wikipedia.org/wiki/Transitive_relation?oldid=670393145Contributors: Zundark, Patrick,MichaelHardy, Rp, Looxix~enwiki, Andres, Charles Matthews, Dcoetzee, Jitse Niesen, Fredrik, MathMartin, Tobias Bergemann, Giftlite, Ben-FrantzDale, Gubbubu, Chowbok, Paul August, MyNameIsNotBob, Spoon!, Polluks, DanShearer, Woohookitty, Linas, LOL, Isnow,Palica, Jérémie Lumbroso~enwiki, Salix alba, Mathbot, Fresheneesz, YurikBot, Laurentius, Sasuke Sarutobi, 48v, Bota47, Arthur Ru-bin, MaratL, Wasseralm, JJL, SmackBot, InverseHypercube, Nbarth, Robma, Cybercobra, Jóna Þórunn, Lambiam, Coredesat, Lyon-sam, Cbuckley, CRGreathouse, Aggarwal kshitij, CBM, Thomasmeeks, Gogo Dodo, Tawkerbot4, AntiVandalBot, Mhaitham.shammaa,MER-C, .anacondabot, Magioladitis, Albmont, David Eppstein, Edward321, MartinBot, Extransit, Tomaz.slivnik, Policron, VolkovBot,AThomas203, Jamelan, Cnilep, SieBot, Paradoctor, Henry Delforn (old), Anchor Link Bot, ClueBot, Tomvanderweide, Sarbogard, Ot-tawahitech, Alexbot, Wikibojopayne, Pa68, SilvonenBot, Addbot, Luckas-bot, Yobot, Ptbotgourou, GrouchoBot, Undsoweiter, RedBot,Katovatzschyn, EmausBot, Slightsmile, IGeMiNix, ChuispastonBot, ClueBot NG, Pars99, Sourabh.khot, Justincheng12345-bot, Lerutit,Loraof and Anonymous: 60

• Weak ordering Source: https://en.wikipedia.org/wiki/Weak_ordering?oldid=640088882 Contributors: Patrick, Michael Hardy, Chinju,Dcoetzee, MathMartin, Tobias Bergemann, Pretzelpaws, AlphaEtaPi, Zaslav, Aisaac, YurikBot, Gadget850, Modify, Oli Filth, Jdthood,Chlewbot, Radiant chains, Jafet, CRGreathouse, Sdorrance, Gregbard, Widefox, Medinoc, Zeitlupe, David Eppstein, Jonathanrcoxhead,Watchduck, Addbot, Kne1p, Forich, Citation bot, ArthurBot, Howard McCay, Citation bot 1, SporkBot, Joel B. Lewis, Helpful Pixie Bot,Jochen Burghardt, JustBerry and Anonymous: 12

31.7.2 Images• File:13-Weak-Orders.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3c/13-Weak-Orders.svg License: Public do-

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31.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 167

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168 CHAPTER 31. WEAK ORDERING

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• File:Primzahlen-Dodekaeder-PD_1bf.gif Source: https://upload.wikimedia.org/wikipedia/commons/f/fb/Primzahlen-Dodekaeder-PD_1bf.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Dr.mgf.winkelmann

• File:Primzahlen-Dodekaeder-PD_2bf.gif Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Primzahlen-Dodekaeder-PD_2bf.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Dr.mgf.winkelmann

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• File:Standard-Dodekaeder-D_VI.gif Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Standard-Dodekaeder-D_VI.gifLicense: CC BY-SA 3.0 Contributors: Own work Original artist: Dr.mgf.winkelmann

• File:Standard-Dodekaeder-D_VII.gif Source: https://upload.wikimedia.org/wikipedia/commons/3/3f/Standard-Dodekaeder-D_VII.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Dr.mgf.winkelmann

• File:Standard-Dodekaeder-D_VIII.gif Source: https://upload.wikimedia.org/wikipedia/commons/8/8d/Standard-Dodekaeder-D_VIII.gif License: CC BY-SA 3.0 Contributors: Own work Original artist: Dr.mgf.winkelmann

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• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

• File:Torus.png Source: https://upload.wikimedia.org/wikipedia/commons/1/17/Torus.png License: Public domain Contributors: ? Orig-inal artist: ?

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• File:Venn0100.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e6/Venn0100.svg License: Public domain Contributors:? Original artist: ?

• File:Venn0110.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/46/Venn0110.svg License: Public domain Contributors:? Original artist: ?

• File:Venn0111.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/30/Venn0111.svg License: Public domain Contributors:? Original artist: ?

• File:Venn1010.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/eb/Venn1010.svg License: Public domain Contributors:? Original artist: ?

• File:Venn1011.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1e/Venn1011.svg License: Public domain Contributors:? Original artist: ?

• File:Venn_A_intersect_B.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Venn_A_intersect_B.svg License: Pub-lic domain Contributors: Own work Original artist: Cepheus

31.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 169

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• File:Wiki_letter_w_cropped.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/Wiki_letter_w_cropped.svg License:CC-BY-SA-3.0 Contributors:

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SA 3.0 Contributors: Own work Original artist: Snorky• File:Woman-power_emblem.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/41/Woman-power_emblem.svgLicense:

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31.7.3 Content license• Creative Commons Attribution-Share Alike 3.0