The Finite Lattice Representation Problem
Transcript of The Finite Lattice Representation Problem
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
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
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
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
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
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
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
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
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
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
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
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
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IntroductionLatticesGroups
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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
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
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
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
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
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
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
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
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
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
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
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
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
IntroductionLatticesGroups
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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
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
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
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
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
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
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
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
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
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
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
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
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}
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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}
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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}
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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
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
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
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
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
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]
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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]
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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
IntroductionLatticesGroups
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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
IntroductionLatticesGroups
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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
IntroductionLatticesGroups
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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
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
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.
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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.
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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
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
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
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
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
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
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
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
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