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• Chapter 1

Join (topology)

Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.

In topology, a eld of mathematics, the join of two topological spaces A and B, often denoted by A ? B , is denedto be the quotient space

(AB I)/R;

where I is the interval [0, 1] and R is the equivalence relation generated by

(a; b1; 0) (a; b2; 0) for all a 2 A and b1; b2 2 B;

(a1; b; 1) (a2; b; 1) for all a1; a2 2 A and b 2 B:At the endpoints, this collapses AB f0g to A and AB f1g to B .Intuitively, A?B is formed by taking the disjoint union of the two spaces and attaching a line segment joining everypoint in A to every point in B.

2

• 1.1. PROPERTIES 3

1.1 Properties The join is homeomorphic to sum of cartesian products of cones over spaces and spaces itself, where sum istaken over cartesian product of spaces:

A ? B = C(A)B [AB C(B)A:

Given basepointed CW complexes (A,a0) and (B,b0), the reduced join

A ? B

A ? fb0g [ fa0g ? Bis homeomorphic to the reduced suspension

(A ^B)

of the smash product. Consequently, since A ? fb0g [ fa0g ? B is contractible, there is a homotopy equivalence

A ? B ' (A ^B):

1.2 Examples The join of subsets of n-dimensional Euclidean space A and B is homotopy equivalent to the space of paths inn-dimensional Euclidean space, beginning in A and ending in B.

The join of a space X with a one-point space is called the cone CX of X. The join of a space X with S0 (the 0-dimensional sphere, or, the discrete space with two points) is called thesuspension SX of X.

The join of the spheres Sn and Sm is the sphere Sn+m+1 .

1.4 References Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0

Brown, Ronald, Topology and Groupoids Section 5.7 Joins.

• Chapter 2

Join and meet

a b

a b

a b

This Hasse diagram depicts a partially ordered set with four elements - a, b, the maximal element equal to the join of a and b (a b)and the minimal element equal to the meet of a and b (a b). The join/meet of a maximal/minimal element and another element isthe maximal/minimal element and conversely the meet/join of a maximal/minimal element with another element is the other element.Thus every pair in this poset has both a meet and a join and the poset can be classied as a lattice (order theory).

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• 2.1. PARTIAL ORDER APPROACH 5

In a partially ordered set P, the join and meet of a subset S are respectively the supremum (least upper bound) ofS, denoted S, and inmum (greatest lower bound) of S, denoted S. In general, the join and meet of a subset of apartially ordered set need not exist; when they do exist, they are elements of P.Join and meet can also be dened as a commutative, associative and idempotent partial binary operation on pairs ofelements from P. If a and b are elements from P, the join is denoted as a b and the meet is denoted a b.Join and meet are symmetric duals with respect to order inversion. The join/meet of a subset of a totally ordered setis simply its maximal/minimal element.A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which allpairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilatticeis a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. Itis also possible to dene a partial lattice, in which not all pairs have a meet or join but the operations (when dened)satisfy certain axioms.

2.1 Partial order approachLet A be a set with a partial order , and let x and y be two elements in A. An element z of A is the meet (or greatestlower bound or inmum) of x and y, if the following two conditions are satised:

1. z x and z y (i.e., z is a lower bound of x and y).2. For any w in A, such that w x and w y, we have w z (i.e., z is greater than or equal to any other lower

bound of x and y).

If there is a meet of x and y, then it is unique, since if both z and z are greatest lower bounds of x and y, then z zand z z, and thus z = z. If the meet does exist, it is denoted x y. Some pairs of elements in A may lack a meet,either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If allpairs of elements have meets, then the meet is a binary operation on A, and it is easy to see that this operation fulllsthe following three conditions: For any elements x, y, and z in A,

a. x y = y x (commutativity),b. x (y z) = (x y) z (associativity), andc. x x = x (idempotency).

2.2 Universal algebra approachBy denition, a binary operation on a set A is a meet, if it satises the three conditions a, b, and c. The pair (A,)then is a meet-semilattice. Moreover, we then may dene a binary relation on A, by stating that x y if and onlyif x y = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,

x x, since x x = x by c; if x y and y x, then x = x y = y x = y by a; and if x y and y z, then x z, since then x z = (x y) z = x (y z) = x y = x by b.

Note that both meets and joins equally satisfy this denition: a couple of associated meet and join operations yieldpartial orders which are the reverse of each other. When choosing one of these orders as the main ones, one also xeswhich operation is considered a meet (the one giving the same order) and which is considered a join (the other one).

2.3 Equivalence of approachesIf (A,) is a partially ordered set, such that each pair of elements in A has a meet, then indeed x y = x if and onlyif x y, since in the latter case indeed x is a lower bound of x and y, and since clearly x is the greatest lower bound if

• 6 CHAPTER 2. JOIN AND MEET

and only if it is a lower bound. Thus, the partial order dened by the meet in the universal algebra approach coincideswith the original partial order.Conversely, if (A,) is a meet-semilattice, and the partial order is dened as in the universal algebra approach, andz = x y for some elements x and y in A, then z is the greatest lower bound of x and y with respect to , since

z x = x z = x (x y) = (x x) y = x y = z

and therefore z x. Similarly, z y, and if w is another lower bound of x and y, then w x = w y = w, whence

w z = w (x y) = (w x) y = w y = w.

Thus, there is a meet dened by the partial order dened by the original meet, and the two meets coincide.In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relationand a binary operation, such that each one of these structures determines the other, and full the conditions for partialorders or meets, respectively.

2.4 Meets of general subsetsIf (A,) is a meet-semilattice, then the meet may be extended to a well-dened meet of any non-empty nite set,by the technique described in iterated binary operations. Alternatively, if the meet denes or is dened by a partialorder, some subsets of A indeed have inma with respect to this, and it is reasonable to consider such an inmum asthe meet of the subset. For non-empty nite subsets, the two approaches yield the same result, whence either may betaken as a denition of meet. In the case where each subset of A has a meet, in fact (A,) is a complete lattice; fordetails, see completeness (order theory).

2.5 Notes Grtzer 1996, p. 52.

2.6 References Davey, B.A.; Priestley, H.A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge: CambridgeUniversity Press. ISBN 0-521-78451-4. Zbl 1002.06001.

Vickers, Steven (1989). Topology via Logic. Cambridge Tracts in Theoretic Computer Science 5. ISBN0-521-36062-5. Zbl 0668.54001.

• Chapter 3

Lagrange bracket

Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph LouisLagrange in 18081810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poissonbrackets, have fallen out of use.

3.1 DenitionSuppose that (q1, , qn, p1, , pn) is a system of canonical coordinates on a phase space. If each of them isexpressed as a function of two variables, u and v, then the Lagrange bracket of u and v is dened by the formula

[u; v]p;q =nXi=1

@qi@u

@pi@v

@pi@u

@qi@v

:

3.2 Properties Lagrange brackets do not depend on the system of canonical coordinates (q, p). If (Q,P) = (Q1, , Qn, P1,, Pn) is another system of canonical coordinates, so that

Q = Q(q; p); P = P (q; p)

is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in thesense that

[u; v]q;p = [u; v]Q;P

Therefore, the subscripts indicating the canonical coordinates are often omitted.

If is the symplectic form on the 2n-dimensional phase spaceW and u1,,u2n form a system of coordinateson W, then canonical coordinates (q,p) may be expressed as functions of the coordinates u and the matrix ofthe Lagrange brackets

[ui; uj ]p;q; 1 i; j 2n

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• 8 CHAPTER 3. LAGRANGE BRACKET

represents the components of , viewed as a tensor, in the coordinates u. This matrix is the inverse ofthe matrix formed by the Poisson brackets

fui; ujg; 1 i; j 2n

of the coordinates u.

As a corollary of the precedi

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