The Finite Lattice Representation Problem

IntroductionLatticesGroups

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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

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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

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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?

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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

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What is an algebra?






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What is an algebra?






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What is an algebra?






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What is an algebra?






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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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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).

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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.

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Example: groups!



(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and

(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.

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Example: groups!



(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and

(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.

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Example: groups!



(a,b) ∈ θ ⇒ (a−1,b−1) ∈ θ, and

(ai ,bi) ∈ θ ⇒ (a1 · a2,b1 · b2) ∈ θ.

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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.

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What is a lattice?





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What is a lattice?





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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}

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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}

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Example: Hasse diagram of Sub[D4]

The lattice of subgroups ofthe dihedral group D4,represented as groups ofrotations and reflections of aplane figure.

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...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)

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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?

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ConA is a lattice


ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉


∇ = A× A


∆ = {(x , y) ∈ A× A | x = y}


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ConA is a lattice


ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉


∇ = A× A


∆ = {(x , y) ∈ A× A | x = y}


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ConA is a lattice


ConA = 〈Con(A),⊆〉 = 〈Con(A),∧,∨〉


∇ = A× A


∆ = {(x , y) ∈ A× A | x = y}


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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?

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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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What is a G-set?


Definition (G-set)



Stab(a) = {g ∈ G | g(a) = a}

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What is a G-set?


Definition (G-set)



Stab(a) = {g ∈ G | g(a) = a}

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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.

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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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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.

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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.

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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!

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False!

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False!

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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 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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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]

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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.

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The Finite Lattice Representation Problem