algebraic structure in a family of nim-like arrays
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Algebraic Structure in a Family of Nim-like Arrays. Lowell Abrams The George Washington University. Dena Cowen-Morton Xavier University. CanaDaM 2009. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A. Combinatorial Games. full information - PowerPoint PPT PresentationTRANSCRIPT
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Algebraic Structure in a Family of Nim-like Arrays
Lowell AbramsThe George Washington University
Dena Cowen-MortonXavier University
CanaDaM 2009
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Combinatorial Games
• full information• no probability• winning strategy
(“1st player win”vs. “2nd player win”)
Basics
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Nim
even piles --- who wins?
Combinatorial Games
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Nim
(this is a “sum” of three individual single-pile games)
Combinatorial Games
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Combinatorial Games
Nimbers and Nim addition
• Nim pile with n stones has nimber n
• nimber 0 means the second player wins
• two side-by-side Nim piles, both with nimber n, have sum 0:
n+n = 0
• if (r+s)+n = 0 (i.e. is a second player win) then r+s=n
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the Nimbers table (G+H)
0 1 2 3 4 5 6 7
1 0 3 2 5 4 7 6
2 3 0 1 6 7 4 5
3 2 1 0 7 6 5 44 5 6 7 0 1 2 35 4 7 6 1 0 3 26 7 4 5 2 3 0 17 6 5 4 3 2 1 0
Rule: Seed with 0.Rule: Enter smallest non-negative integer appearing neither above nor to left.
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Another way to combine games
•Ullman and Stromquist: sequential compound G → H
•misère play: G → 1
•misère nim addition: G+H → 1
•something else: G+H → s for integer s ¸ 2
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the Nimbers table(G+H → 2)
2 0 1 3 4 5 6 7 8 9 0 1 2 4 3 6 5 8 7 101 2 0 5 6 3 4 9 10 73 4 5 0 1 2 7 6 9 84 3 6 1 0 7 2 5 11 125 6 3 2 7 0 1 4 12 116 5 4 7 2 1 0 3 13 147 8 9 6 5 4 3 0 1 28 7 10 9 11 12 13 1 0 39 10 7 8 12 11 14 2 3 0
Rule: Seed with 2.Proceed with same algorithm.
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An algebraic approach...
view array as defining an operation ¤ on N¸0
2 0 1 3 4 5 6
0 1 2 4 3 6 5
1 2 0 5 6 3 4
3 4 5 0 1 2 7
4 3 6 1 0 7 2
5 6 3 2 7 0 1
6 5 4 7 2 1 0
3 ¤ 3 = 0
4 ¤ 5 = 7
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Basic algebraic structure
view array as defining an operation ¤ on N¸0
2 0 1 3 4 5 6
0 1 2 4 3 6 5
1 2 0 5 6 3 4
3 4 5 0 1 2 7
4 3 6 1 0 7 2
5 6 3 2 7 0 1
6 5 4 7 2 1 0
¤ is commutative
2 is the ¤-identity
¤ is not associativee.g. 1 ¤ (1 ¤ 4) = 1 ¤ 6 = 4
(1 ¤ 1) ¤ 4 = 0 ¤ 4 = 3 write: A2 := (N¸0 , ¤) have As , by analogy, for each seed s
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“(Q,¤) is a quasigroup” means: for every i,j2 Q there exist unique p,q 2 Q such that i¤p = j and q¤i = j
“(Q,¤) is a loop” means: (Q,¤) is a quasigroup with a two-sided ¤-identity
Basic algebraic structure, continued...
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Quasigroups
all groups are quasigroups
x 1 2 3 41 1 2 3 42 2 4 1 33 3 1 4 24 4 3 2 1
(units in Z/5Z, under multiplication)
butnot every quasigroup
is a group
(units in Z/5Z, under division)
/ 1 2 3 41 1 2 3 42 3 1 4 23 2 4 1 34 4 3 2 1
note: 2/(3/2) = 2/4 = 2 but (2/3)/2 = 4/2 = 3
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“(Q,¤) is a quasigroup” means: for every i,j2 Q there exist unique p,q 2 Q such that i¤p = j and q¤i = j
“(Q,¤) is a loop” means: (Q,¤) is a quasigroup with a two-sided ¤-identity
Basic algebraic structure, continued...
observe: As is a loop
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Algebraic results
provide a way to encode
combinatorial properties
Take-Home Point:
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Main Results (in brief)
Theorem
For each seed s ≥ 2, As is monogenic.
Theorem
There are no nontrivial homomorphisms As →At
if s ≥ 2 or t ≥ 2.Otherwise, there are a lot of them.
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Monogenicity
Notation: «x; ◊» is the free unital groupoid on generator x with operation ◊
Note, e.g. : (x◊x) ◊ (x◊x) ≠ x ◊ (x ◊ (x◊x) )
Write xn for x ◊ (… ◊ (x◊x) )
n times
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Monogenicity
define L is monogenic: there is n ∈ L such that φn is surjective
loop L , element n ∈ L
define φn : «x; ◊» → L • operation-preserving • φn(e◊ ) = eL • φn(x) = n
note: this differs a little from the standard definition...
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Monogenicity
Theorem (A. and Cowen-Morton)
As is monogenic iff s ≥ 2
For s=2, every element n>2 is a generator.
For s>2, every element n ≠ s is a generator.
apparently, a novelty in the literature
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Homomorphisms
Theorem (A. and Cowen-Morton)
The only loop homomorphismf:As →At
for s ≠ t and either s ≥ 2 or t ≥ 2 (or both)is the trivial map As →{t}.
For s=t ≥ 2, homomorphism f is either the trivial map As →{s}
or the identity map.
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Homomorphisms
Terri Evans (1953):
description of homomorphismsof finitely presented monogenic loops
Theorem (A. and Cowen-Morton) For any seed s,
the loop As is not finitely presented.
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Homomorphisms
Essence of proof
● monogenicity
● commutativity of this diagram:
«x; ◊»
As At
φn ψf(n)
f
ψ is the appropriate evaluation map
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Homomorphisms
case: s = t = 2 and f(3) > 6
set
α= (x2)2 ◊ [x2 ◊ ((x2)2 ◊ x)]
β = ((x2)2 ◊ x ) ◊ (x ◊ [x2 ◊ ((x2)2 ◊ x)] )
α β
4
«x; ◊»
A2 A2
φ3 φf(3)
fφf(3)(α) ≠ φf(3)(β)
?
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Homomorphisms
case: s = t = 2 and f(3) = 4, 5, or 6
set
α= (x2 ◊ x) ◊ [(x2)2 ◊ [x2 ◊ (x ◊ [(x2)2 ◊ (x2 ◊ x)])] ]
β = [(x2)2 ◊ (x2 ◊ x)] ◊ [x2 ◊ (x ◊ [(x2)2 ◊ (x2 ◊ x)]) ]
α β
13
«x; ◊»
A2 A2
φ3 φf(3)
fφf(3)(α) ≠ φf(3)(β)
?
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Homomorphisms
case: s = 2, t = 0
for δ ∈ «x; ◊» define |δ| = number of x’s in δ
«x; ◊»
A2 A0
φn ψf(n)
f
for δ ∈ «x; ◊»,
f○φn(δ) = ψf(n)(x|δ|)
=
f(n) if |δ| ≡ 1 (mod 2)0 otherwise
A0 is asociative
in A0, m2=0for all m
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Homomorphisms
case: s = 2, t = 0«x; ◊»
A2 A0
φ3 ψf(3)
f
set
α= (x2)2 ◊ [x ◊ (x3 ◊ (x2)2)]
β = x ◊ (x2 ◊ [x ◊ (x3 ◊ (x2)2)] )
then we have 0 = f○φ3(α) = f○φ3(β) = f(3)
|α| = 12φ3(α) = 9 = φ3(β)
|β| = 11
since 3 generates A2
and 0 is the identity in A0,
f is trivial
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Homomorphisms
Hom(A0,A0) = ∏≥0 A0
Hom(A0,A1) = ∏≥0 Ζ/2Z
Hom(A1,A0) = ∏≥0 A0
Hom(A1,A1) = ∏≥1 Ζ/2Z [ [Inj(A0,A0) £ {0,1}N]
Theorem (A. and Cowen-Morton)
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Homomorphisms
Each element 2i in A0 (i≥0)generates a subgroup Hi isomorphic to Z/2Z.
A0 is the weak product of the Hi
since its operation is bitwise XOR.
Each element 2i in A1 (i≥1) generates a subgroup Gi = {2i, 0, 2i+1, 1}
isomorphic to Z/4Z.
A1 is not the weak product of the Gi
but the Gi stay out of each other’s way.
behind the proof...
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Homomorphisms
Let Q1 denote the loop quotient of A1
by the relation 0 ≡ 1.
Let Q2 denote the loop quotient of A1
by the relations {2k ≡ 2k+1 | k = 1, 2, ...}.
Let Q3 denote the loop quotient of A1
by all relations enforcing associativity.
Then each of these quotients is isomorphic to A0
under an isom’m sending Gi to Hi-1 for each i,for which all three quotient maps are the same.
Theorem (A. and Cowen-Morton)behind the proof...
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Homomorphisms
Hom(A0,A0) = ∏≥0 A0
Hom(A0,A1) = ∏≥0 Ζ/2Z
Hom(A1,A0) = ∏≥0 A0
Hom(A1,A1) = ∏≥1 Ζ/2Z [ [Inj(A0,A0) £ {0,1}N]
Theorem (A. and Cowen-Morton)
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