the finite lattice representation problem
TRANSCRIPT
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IntroductionLatticesGroups
MilestonesSummary
The Finite Lattice Representation Problem:intervals in subgroup lattices and
the dawn of tame congruence theory
William DeMeo
University of Hawai‘i at Manoa
Math 613: Group TheoryNovember 2009
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
Outline
1 Introductionalgebras and their congruences
2 Latticessubgroup lattices and congruence latticesthe finite lattice representation problem
3 GroupsG-setsintervals in subgroup lattices
4 Milestonesthe theorem of Pálfy-Pudlákthe seminal lemma of tct
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Background and problem statement
Theorem (Grätzer-Schmidt, 1963)Every algebraic lattice is isomorphic tothe congruence lattice of an algebra.
What if the lattice is finite?Problem: Given a finite lattice L, does there exist
a finite algebra A such that ConA ∼= L?
status: openage: 45+ years
difficulty: what do you think?
W. DeMeo FLRP
![Page 4: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/4.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Background and problem statement
Theorem (Grätzer-Schmidt, 1963)Every algebraic lattice is isomorphic tothe congruence lattice of an algebra.
What if the lattice is finite?Problem: Given a finite lattice L, does there exist
a finite algebra A such that ConA ∼= L?
status: openage: 45+ years
difficulty: what do you think?
W. DeMeo FLRP
![Page 5: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/5.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Background and problem statement
Theorem (Grätzer-Schmidt, 1963)Every algebraic lattice is isomorphic tothe congruence lattice of an algebra.
What if the lattice is finite?Problem: Given a finite lattice L, does there exist
a finite algebra A such that ConA ∼= L?
status: openage: 45+ years
difficulty: what do you think?
W. DeMeo FLRP
![Page 6: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/6.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Background and problem statement
Theorem (Grätzer-Schmidt, 1963)Every algebraic lattice is isomorphic tothe congruence lattice of an algebra.
What if the lattice is finite?Problem: Given a finite lattice L, does there exist
a finite algebra A such that ConA ∼= L?
status: openage: 45+ years
difficulty: what do you think?
W. DeMeo FLRP
![Page 7: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/7.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
What is an algebra?
Definition (algebra)
A (universal) algebra A is an ordered pair A = 〈A,F 〉 whereA is a nonempty set, called the universe of AF is a family of finitary operations on A
An algebra 〈A,F 〉 is finite if |A| is finite.
Definition (arity)The arity of an operation f ∈ F is the number of operands.
f is n-ary if it maps An into Anullary, unary, binary, and ternary operations have arities0, 1, 2, and 3, respectively
W. DeMeo FLRP
![Page 8: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/8.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
What is an algebra?
Definition (algebra)
A (universal) algebra A is an ordered pair A = 〈A,F 〉 whereA is a nonempty set, called the universe of AF is a family of finitary operations on A
An algebra 〈A,F 〉 is finite if |A| is finite.
Definition (arity)The arity of an operation f ∈ F is the number of operands.
f is n-ary if it maps An into Anullary, unary, binary, and ternary operations have arities0, 1, 2, and 3, respectively
W. DeMeo FLRP
![Page 9: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/9.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
What is an algebra?
Definition (algebra)
A (universal) algebra A is an ordered pair A = 〈A,F 〉 whereA is a nonempty set, called the universe of AF is a family of finitary operations on A
An algebra 〈A,F 〉 is finite if |A| is finite.
Definition (arity)The arity of an operation f ∈ F is the number of operands.
f is n-ary if it maps An into Anullary, unary, binary, and ternary operations have arities0, 1, 2, and 3, respectively
W. DeMeo FLRP
![Page 10: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/10.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
What is an algebra?
Definition (algebra)
A (universal) algebra A is an ordered pair A = 〈A,F 〉 whereA is a nonempty set, called the universe of AF is a family of finitary operations on A
An algebra 〈A,F 〉 is finite if |A| is finite.
Definition (arity)The arity of an operation f ∈ F is the number of operands.
f is n-ary if it maps An into Anullary, unary, binary, and ternary operations have arities0, 1, 2, and 3, respectively
W. DeMeo FLRP
![Page 11: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/11.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
What is an algebra?
Definition (algebra)
A (universal) algebra A is an ordered pair A = 〈A,F 〉 whereA is a nonempty set, called the universe of AF is a family of finitary operations on A
An algebra 〈A,F 〉 is finite if |A| is finite.
Definition (arity)The arity of an operation f ∈ F is the number of operands.
f is n-ary if it maps An into Anullary, unary, binary, and ternary operations have arities0, 1, 2, and 3, respectively
W. DeMeo FLRP
![Page 12: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/12.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Example: groups!
Definition (group)
A group G is an algebra 〈G, ·,−1 ,1〉 with a binary, unary, andnullary operation satisfying, ∀x , y , z ∈ G,G1: x · (y · z) ≈ (x · y) · zG2: x · 1 ≈ 1 · x ≈ xG3: x · x−1 ≈ x−1 · x ≈ 1
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Congruence relations defined
Definition (congruence relation)
Given A = 〈A,F 〉, an equivalence relation θ ∈ Eq(A) isa congruence on A if θ “admits” F
i.e., for n-ary f ∈ F , and elements ai ,bi ∈ A,
if (ai ,bi) ∈ θ, then (f (a1, . . . ,an), f (b1, . . . ,bn)) ∈ θ
or “f respects θ” for all f ∈ F .
The set of all congruence relations on A is denoted Con(A).
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Congruence relations defined
Definition (congruence relation)
Given A = 〈A,F 〉, an equivalence relation θ ∈ Eq(A) isa congruence on A if θ “admits” F
i.e., for n-ary f ∈ F , and elements ai ,bi ∈ A,
if (ai ,bi) ∈ θ, then (f (a1, . . . ,an), f (b1, . . . ,bn)) ∈ θ
or “f respects θ” for all f ∈ F .
The set of all congruence relations on A is denoted Con(A).
W. DeMeo FLRP
![Page 15: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/15.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Congruence relations defined
Definition (congruence relation)
Given A = 〈A,F 〉, an equivalence relation θ ∈ Eq(A) isa congruence on A if θ “admits” F
i.e., for n-ary f ∈ F , and elements ai ,bi ∈ A,
if (ai ,bi) ∈ θ, then (f (a1, . . . ,an), f (b1, . . . ,bn)) ∈ θ
or “f respects θ” for all f ∈ F .
The set of all congruence relations on A is denoted Con(A).
W. DeMeo FLRP
![Page 16: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/16.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Congruence relations defined
Definition (congruence relation)
Given A = 〈A,F 〉, an equivalence relation θ ∈ Eq(A) isa congruence on A if θ “admits” F
i.e., for n-ary f ∈ F , and elements ai ,bi ∈ A,
if (ai ,bi) ∈ θ, then (f (a1, . . . ,an), f (b1, . . . ,bn)) ∈ θ
or “f respects θ” for all f ∈ F .
The set of all congruence relations on A is denoted Con(A).
W. DeMeo FLRP
![Page 17: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/17.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Example: groups!
What are the congruences of a group?
For a group G = 〈G, ·,−1 ,1〉, an equivalence θ ∈ Eq(G) isa congruence on G provided, ∀a,b,ai ,bi ∈ G,
(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and
(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.
W. DeMeo FLRP
![Page 18: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/18.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Example: groups!
What are the congruences of a group?
For a group G = 〈G, ·,−1 ,1〉, an equivalence θ ∈ Eq(G) isa congruence on G provided, ∀a,b,ai ,bi ∈ G,
(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and
(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.
W. DeMeo FLRP
![Page 19: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/19.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Example: groups!
What are the congruences of a group?
For a group G = 〈G, ·,−1 ,1〉, an equivalence θ ∈ Eq(G) isa congruence on G provided, ∀a,b,ai ,bi ∈ G,
(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and
(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.
W. DeMeo FLRP
![Page 20: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/20.jpg)
IntroductionLatticesGroups
MilestonesSummary
algebras and their congruences
Example: groups!
What are the congruences of a group?
For a group G = 〈G, ·,−1 ,1〉, an equivalence θ ∈ Eq(G) isa congruence on G provided, ∀a,b,ai ,bi ∈ G,
(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and
(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
What is a lattice?
Definition (lattice)
A lattice is an algebra L = 〈L,∧,∨〉 with universe L, a partiallyordered set, and binary operations:
x ∧ y = g.l.b.(x , y) the “meet” of x and yx ∨ y = l.u.b.(x , y) the “join” of x and y
ExamplesSubsets of a setClosed subsets of a topology.Subgroups of a group.
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
What is a lattice?
Definition (lattice)
A lattice is an algebra L = 〈L,∧,∨〉 with universe L, a partiallyordered set, and binary operations:
x ∧ y = g.l.b.(x , y) the “meet” of x and yx ∨ y = l.u.b.(x , y) the “join” of x and y
ExamplesSubsets of a setClosed subsets of a topology.Subgroups of a group.
W. DeMeo FLRP
![Page 23: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/23.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
What is a lattice?
Definition (lattice)
A lattice is an algebra L = 〈L,∧,∨〉 with universe L, a partiallyordered set, and binary operations:
x ∧ y = g.l.b.(x , y) the “meet” of x and yx ∨ y = l.u.b.(x , y) the “join” of x and y
ExamplesSubsets of a setClosed subsets of a topology.Subgroups of a group.
W. DeMeo FLRP
![Page 24: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/24.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
Example: Sub[G]
The lattice of subgroups of a group G, denoted
Sub[G] = 〈Sub[G],⊆〉 = 〈Sub[G],∧,∨〉,
has universe Sub[G], the set of subgroups of G.
For subgroups H,K ∈ Sub[G],meet is set intersection:
H ∧ K = H ∩ K
join is the subgroup generated by the union:
H ∨ K =⋂{J ∈ Sub[G] |H ∪ K ⊆ J}
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
Example: Sub[G]
The lattice of subgroups of a group G, denoted
Sub[G] = 〈Sub[G],⊆〉 = 〈Sub[G],∧,∨〉,
has universe Sub[G], the set of subgroups of G.
For subgroups H,K ∈ Sub[G],meet is set intersection:
H ∧ K = H ∩ K
join is the subgroup generated by the union:
H ∨ K =⋂{J ∈ Sub[G] |H ∪ K ⊆ J}
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
Example: Hasse diagram of Sub[D4]
The lattice of subgroups ofthe dihedral group D4,represented as groups ofrotations and reflections of aplane figure.
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
...ok, but is it useful?
Lattice-theoretic information (about Sub[G]) can be used toobtain group-theoretic information (about G).
Examples:G is locally cyclic if and only if Sub[G] is distributive.
Ore, “Structures and group theory,” Duke Math. J. (1937)Similar lattice-theoretic characterizations exist for solvableand perfect groups.
Suzuki, “On the lattice of subgroups of finite groups,”Trans. AMS (1951)
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
...ok, but is it useful?
Lattice-theoretic information (about Sub[G]) can be used toobtain group-theoretic information (about G).
Examples:G is locally cyclic if and only if Sub[G] is distributive.
Ore, “Structures and group theory,” Duke Math. J. (1937)Similar lattice-theoretic characterizations exist for solvableand perfect groups.
Suzuki, “On the lattice of subgroups of finite groups,”Trans. AMS (1951)
W. DeMeo FLRP
![Page 29: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/29.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
...ok, but is it useful?
Lattice-theoretic information (about Sub[G]) can be used toobtain group-theoretic information (about G).
Examples:G is locally cyclic if and only if Sub[G] is distributive.
Ore, “Structures and group theory,” Duke Math. J. (1937)Similar lattice-theoretic characterizations exist for solvableand perfect groups.
Suzuki, “On the lattice of subgroups of finite groups,”Trans. AMS (1951)
W. DeMeo FLRP
![Page 30: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/30.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
...ok, but is it useful?
Lattice-theoretic information (about Sub[G]) can be used toobtain group-theoretic information (about G).
Examples:G is locally cyclic if and only if Sub[G] is distributive.
Ore, “Structures and group theory,” Duke Math. J. (1937)Similar lattice-theoretic characterizations exist for solvableand perfect groups.
Suzuki, “On the lattice of subgroups of finite groups,”Trans. AMS (1951)
W. DeMeo FLRP
![Page 31: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/31.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
ConA is a lattice
The set Con(A), ordered by set inclusion, is a 0-1 lattice:
ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉
The greatest congruence is the all relation
∇ = A× A
The least congruence is the diagonal
∆ = {(x , y) ∈ A× A | x = y}
What are meet and join?
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
ConA is a lattice
The set Con(A), ordered by set inclusion, is a 0-1 lattice:
ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉
The greatest congruence is the all relation
∇ = A× A
The least congruence is the diagonal
∆ = {(x , y) ∈ A× A | x = y}
What are meet and join?
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
ConA is a lattice
The set Con(A), ordered by set inclusion, is a 0-1 lattice:
ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉
The greatest congruence is the all relation
∇ = A× A
The least congruence is the diagonal
∆ = {(x , y) ∈ A× A | x = y}
What are meet and join?
W. DeMeo FLRP
![Page 34: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/34.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
ConA is a lattice
The set Con(A), ordered by set inclusion, is a 0-1 lattice:
ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉
The greatest congruence is the all relation
∇ = A× A
The least congruence is the diagonal
∆ = {(x , y) ∈ A× A | x = y}
What are meet and join?
W. DeMeo FLRP
![Page 35: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/35.jpg)
IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
The finite lattice representation problem
Definition (representable lattice)Call a finite lattice representable if it is (isomorphic to) thecongruence lattice of a finite algebra.
The (≤ $1m) questionIs every finite lattice representable?
Equivalently, given a finite lattice L, does there exista finite algebra A such that ConA ∼= L?
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
The finite lattice representation problem
Definition (representable lattice)Call a finite lattice representable if it is (isomorphic to) thecongruence lattice of a finite algebra.
The (≤ $1m) questionIs every finite lattice representable?
Equivalently, given a finite lattice L, does there exista finite algebra A such that ConA ∼= L?
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
subgroup lattices and congruence latticesthe finite lattice representation problem
The finite lattice representation problem
Definition (representable lattice)Call a finite lattice representable if it is (isomorphic to) thecongruence lattice of a finite algebra.
The (≤ $1m) questionIs every finite lattice representable?
Equivalently, given a finite lattice L, does there exista finite algebra A such that ConA ∼= L?
W. DeMeo FLRP
![Page 38: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/38.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
What is a G-set?
Let G = 〈G, ·,−1 ,1G〉 be a group, A a set.
Definition (G-set)
A G-set is a unary algebra A = 〈A,G〉, where G = {g : g ∈ G},1G = idA, and g1 ◦ g2 = g1 · g2, for all gi ∈ G.
Definition (stabilizer)For any a ∈ A, the stabilizer of a is the set
Stab(a) = {g ∈ G | g(a) = a}
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
What is a G-set?
Let G = 〈G, ·,−1 ,1G〉 be a group, A a set.
Definition (G-set)
A G-set is a unary algebra A = 〈A,G〉, where G = {g : g ∈ G},1G = idA, and g1 ◦ g2 = g1 · g2, for all gi ∈ G.
Definition (stabilizer)For any a ∈ A, the stabilizer of a is the set
Stab(a) = {g ∈ G | g(a) = a}
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
What is a G-set?
Let G = 〈G, ·,−1 ,1G〉 be a group, A a set.
Definition (G-set)
A G-set is a unary algebra A = 〈A,G〉, where G = {g : g ∈ G},1G = idA, and g1 ◦ g2 = g1 · g2, for all gi ∈ G.
Definition (stabilizer)For any a ∈ A, the stabilizer of a is the set
Stab(a) = {g ∈ G | g(a) = a}
W. DeMeo FLRP
![Page 41: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/41.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
G-sets: basic facts and transitivity
Basic facts about the G-set 〈A,G〉 (efts)
1. Each g ∈ G is a permutation of A.2. If [a] is the subalgebra generated by a ∈ A, then
[a] = {g(a) |g ∈ G} = the orbit of a in A.
3. The stabilizer Stab(a) is a subgroup of G.
Definition (trasitive G-set)
If A = 〈A,G〉 has only one orbit, we say G acts transitively on A,or A is a transitive G-set.
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
G-sets: basic facts and transitivity
Basic facts about the G-set 〈A,G〉 (efts)
1. Each g ∈ G is a permutation of A.2. If [a] is the subalgebra generated by a ∈ A, then
[a] = {g(a) |g ∈ G} = the orbit of a in A.
3. The stabilizer Stab(a) is a subgroup of G.
Definition (trasitive G-set)
If A = 〈A,G〉 has only one orbit, we say G acts transitively on A,or A is a transitive G-set.
W. DeMeo FLRP
![Page 43: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/43.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
G-sets: basic facts and transitivity
Basic facts about the G-set 〈A,G〉 (efts)
1. Each g ∈ G is a permutation of A.2. If [a] is the subalgebra generated by a ∈ A, then
[a] = {g(a) |g ∈ G} = the orbit of a in A.
3. The stabilizer Stab(a) is a subgroup of G.
Definition (trasitive G-set)
If A = 〈A,G〉 has only one orbit, we say G acts transitively on A,or A is a transitive G-set.
W. DeMeo FLRP
![Page 44: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/44.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
G-sets: basic facts and transitivity
Basic facts about the G-set 〈A,G〉 (efts)
1. Each g ∈ G is a permutation of A.2. If [a] is the subalgebra generated by a ∈ A, then
[a] = {g(a) |g ∈ G} = the orbit of a in A.
3. The stabilizer Stab(a) is a subgroup of G.
Definition (trasitive G-set)
If A = 〈A,G〉 has only one orbit, we say G acts transitively on A,or A is a transitive G-set.
W. DeMeo FLRP
![Page 45: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/45.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
G-sets: basic facts and transitivity
Basic facts about the G-set 〈A,G〉 (efts)
1. Each g ∈ G is a permutation of A.2. If [a] is the subalgebra generated by a ∈ A, then
[a] = {g(a) |g ∈ G} = the orbit of a in A.
3. The stabilizer Stab(a) is a subgroup of G.
Definition (trasitive G-set)
If A = 〈A,G〉 has only one orbit, we say G acts transitively on A,or A is a transitive G-set.
W. DeMeo FLRP
![Page 46: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/46.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
Fundamental theorem of transitive G-sets
Definition (interval in a subgroup lattice)
If G is a group and H ∈ Sub[G] is a subgroup, define
[H,G] = 〈{K ∈ Sub[G] |H ⊆ K},⊆〉
Call [H,G] an (upper) interval in the lattice Sub[G].
Theorem
If A = 〈A,G〉 is a transitive G-set, then for any a ∈ A,
ConA ∼= [Stab(a),G]
W. DeMeo FLRP
![Page 47: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/47.jpg)
IntroductionLatticesGroups
MilestonesSummary
G-setsintervals in subgroup lattices
Fundamental theorem of transitive G-sets
Definition (interval in a subgroup lattice)
If G is a group and H ∈ Sub[G] is a subgroup, define
[H,G] = 〈{K ∈ Sub[G] |H ⊆ K},⊆〉
Call [H,G] an (upper) interval in the lattice Sub[G].
Theorem
If A = 〈A,G〉 is a transitive G-set, then for any a ∈ A,
ConA ∼= [Stab(a),G]
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
A pair of groundbreaking results
Theorem (Pudlák and Tuma, AU 10, 1980)
A finite lattice can be embedded in Eq(X ), for some finite X.
Theorem (Pálfy and Pudlák, AU 11, 1980)The following statements are equivalent:
(i) Any finite lattice is isomorphic tothe congruence lattice of a finite algebra.
(ii) Any finite lattice is isomorphic toan interval in the subgroup lattice of a finite group.
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
A pair of groundbreaking results
Theorem (Pudlák and Tuma, AU 10, 1980)
A finite lattice can be embedded in Eq(X ), for some finite X.
Theorem (Pálfy and Pudlák, AU 11, 1980)The following statements are equivalent:
(i) Any finite lattice is isomorphic tothe congruence lattice of a finite algebra.
(ii) Any finite lattice is isomorphic toan interval in the subgroup lattice of a finite group.
W. DeMeo FLRP
![Page 50: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/50.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
The Pálfy-Pudlák theorem: what does it (not) say?
A quote from MathSciNet reviews“In AU 11, Pálfy and Pudlák proved that...a finite lattice isrepresentable if and only if it occurs as an interval in thesubgroup lattice of a finite group.”
False!
W. DeMeo FLRP
![Page 51: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/51.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
The Pálfy-Pudlák theorem: what does it (not) say?
A quote from MathSciNet reviews“In AU 11, Pálfy and Pudlák proved that...a finite lattice isrepresentable if and only if it occurs as an interval in thesubgroup lattice of a finite group.”
False!
W. DeMeo FLRP
![Page 52: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/52.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
The Pálfy-Pudlák theorem: what does it (not) say?
A quote from MathSciNet reviews“In AU 11, Pálfy and Pudlák proved that...a finite lattice isrepresentable if and only if it occurs as an interval in thesubgroup lattice of a finite group.”
False!
W. DeMeo FLRP
![Page 53: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/53.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
The seminal lemma of tct
Lemma
Let A = 〈A,F 〉 be a unary algebra with e2 = e ∈ F.
Define B = 〈B,G〉 with
B = e(A) and G = {ef |B : f ∈ F}
ThenCon(A) 3 θ 7→ θ ∩ (B × B) ∈ Con(B)
is a lattice epimorphism.
W. DeMeo FLRP
![Page 54: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/54.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
The seminal lemma of tct
Lemma
Let A = 〈A,F 〉 be a unary algebra with e2 = e ∈ F.
Define B = 〈B,G〉 with
B = e(A) and G = {ef |B : f ∈ F}
ThenCon(A) 3 θ 7→ θ ∩ (B × B) ∈ Con(B)
is a lattice epimorphism.
W. DeMeo FLRP
![Page 55: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/55.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
Consequence of the seminal lemma
TheoremLet L be a finite lattice satisfying conditions (A), (B), (C).
Let A = 〈A,F 〉 be a finite unary algebra of minimal cardinalitysuch that ConA ∼= L.
Then A is a transitive G-set.
∴ L ∼= ConA ∼= [Stab(a),G]
W. DeMeo FLRP
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IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
Consequence of the seminal lemma
TheoremLet L be a finite lattice satisfying conditions (A), (B), (C).
Let A = 〈A,F 〉 be a finite unary algebra of minimal cardinalitysuch that ConA ∼= L.
Then A is a transitive G-set.
∴ L ∼= ConA ∼= [Stab(a),G]
W. DeMeo FLRP
![Page 57: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/57.jpg)
IntroductionLatticesGroups
MilestonesSummary
the theorem of Pálfy-Pudlákthe seminal lemma of tct
Consequence of the seminal lemma
TheoremLet L be a finite lattice satisfying conditions (A), (B), (C).
Let A = 〈A,F 〉 be a finite unary algebra of minimal cardinalitysuch that ConA ∼= L.
Then A is a transitive G-set.
∴ L ∼= ConA ∼= [Stab(a),G]
W. DeMeo FLRP
![Page 58: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/58.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP
![Page 59: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/59.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP
![Page 60: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/60.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP
![Page 61: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/61.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP
![Page 62: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/62.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP
![Page 63: The Finite Lattice Representation Problem](https://reader036.vdocuments.us/reader036/viewer/2022071602/613d6c95736caf36b75d2adf/html5/thumbnails/63.jpg)
IntroductionLatticesGroups
MilestonesSummary
Summary
Problem: Given a finite lattice L, does there exist a finitealgebra A such that L ∼= ConA?
It is generally believed the answer is no.About 30 years ago Pálfy and Pudlák translated it into onefor the group theorists, but still no answer...
Outlook: bleakSomething you haven’t solved.Something else you haven’t solved.
W. DeMeo FLRP