critical points for congruence lattices · pdf filedeﬁnitioningeneral lets~= (s p;s p;qjp...

Critical points for congruence lattices

Miroslav Ploščica

P. J. Šafárik University, Košice

June 15, 2017

Congruences

Congruence on algebra A is an equivalence relation θ on the set A,preserved by all basic operations of A, i.e.

(a1, b1) ∈ θ, (a2, b2) ∈ θ, . . . , (an, bn) ∈ θ

implies(f(a1, . . . , an), f(b1, . . . , bn) ∈ θ))

(for f n-ary).Every algebra with more than 1 element has at least twocongruences.

Miroslav Ploščica Critical points for congruence lattices

Importance of congruences

Congruences enable quotients:On the set of θ-classes we define the operations be means ofrepresentatives:

f(a1/θ, . . . , an/θ) = f(a1, . . . , an)/θ.

This gives rise to a new algebra of the same type as A, which is asimplified image of the algebra A.

For instance, Z/(modn) = Zn.


Congruence lattices

Congruences on an algebra A can be ordered by the “refinement"relation (= set inclusion):

ϕ ≤ θ ak (xϕy implies xθy).

We obtain an algebraic lattice ConA.Conversely,

Theorem(G. Grätzer, E. T. Schmidt) A lattice is isomorphic to thecongruence lattice of some algebra if and only if it is algebraic.


Algebraic lattices

An element a of a lattice L is called compact if for every K ⊆ L

a ≤ supK implies a ≤ supK0

for some finite subset K0 ⊆ K.The set of all compact elements is a ∨-subsemilattice of L, denotedby Lc. The lattice L is compact if it is complete and

x = sup{a ∈ Lc | a ≤ x}

for every x ∈ L.Fact: Every algebraic lattice L is isomorphic to Id(Lc), the ideallattice of Lc.


General problem

Problem. For a given class K of algebras describe Con K =alllattices isomorphic to Con A for some A ∈ K.

Very few relevant classes with a satisfactory answer


TheoremL is isomorphic to ConB for some Boolean algebra B iff Lc is aBoolean lattice.

TheoremL is isomorphic to ConD for some distributive lattice D iff Lc is ageneralized Boolean lattice.

TheoremL is isomorphic to ConA for some Stone algebra A iff Lc is a dualStone lattice.


Congruence lattices

One solved problem: Is every distributive algebraic lattice(isomorphic to) the congruence lattice of some lattice?

Partial positive results (R. P. Dilworth, E. T. Schmidt, A. Huhn...),butFinal answer: no (F. Wehrung 2005)

For most common classes of algebras, a full description of ConKseems hopeless.


A more tractable problem:

for given classes K, L determine if Con K = Con L(Con K ⊆ Con L)

and, if Con K * Con L, determine

Crit(K,L) = min{card(Lc) | L ∈ ConK \ ConL}


Some critical points

We are especially interested in the case when K and L arecongruence-distributive varieties (in most results also finitelygenerated). For instance,Crit(N5,M3) = 5,Crit(M3,N5) = Crit(M3,D) = ℵ0,Crit(M4,M3) = ℵ2,Crit(Maj,Lat) = ℵ2.(N5, M3, M4, D are well-known lattice varieties, Lat = alllattices, Maj = all majority algebras.)P. Gillibert: under some reasonable finiteness conditions, the criticalpoint between two varieties cannot be larger than ℵ2.


Why Lc?

ConcA reflects the size of A better.

TheoremIf an infinite algebra A is a subdirect product of finite algebras ofbounded size, then |ConcA| = |A|.


N5 and Mn

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No ℵ3?

Theorem(Gillibert)Let V and W be locally finite varieties. Assume that for any finiteL ∈ ConV there are, up to isomorphism, finitely many B ∈ Wwith L ∼= ConB, and each such B is finite. Then Crit(V,W) ≤ ℵ2or ConV ⊆ ConW.

Any finitely generated congruence-distributive varieties satisfy theassumptions.


Possible cases

For finitely generated congruence-distributive varieties there arefollowing possible cases:

Crit(V,W) is finite;Crit(V,W) = ℵ0;Crit(V,W) = ℵ1;Crit(V,W) = ℵ2;ConV ⊆ ConW.

How to distinguish?


Topological approach

M(L)....completely meet-irreducible elements of a lattice L(a = inf X implies a ∈ X)Fact: if L is algebraic, then every element is a meet of completelymeet-irreducible elements.Topology on M(L): all sets of the form

M(L) ∩ ↑x = {a ∈ M(L) | a ≥ x}

are closed.

TheoremIf L is distributive algebraic, then L ∼= O(M(L)). (The lattice of allopen subsets of M(L).


Topological approach

Sometimes the properties of ConA are more effectively expressedas topological properties of M(ConA). A sample:

If A ∈ D then M(ConA) is Hausdorff.There exists a countable B ∈M3 such that M(ConB) is notHausdorff.Therefore, Crit(M3,D) ≤ ℵ0.

The topological approach was used to establish e.g.Crit(M4,M3) = ℵ2. (But the argument is much morecomplicated.)


How much more complicated?

A topological space T is called n-uniformly separable (n ≥ 3) if forevery discrete subset Q there is a family {Bpq | p, q ∈ Q, p 6= q} ofopen sets such that p ∈ Bpq and

⋂{Bpq | p, q ∈ Q0} = ∅ wheneverQ0 ⊆ Q, |Q0| ≥ n.

M(ConA) is (n+ 1)-uniformly separable for every A ∈Mn.M(ConFn(X)) is not n-uniformly separable.Therefore, Crit(Mn+1,Mn) ≤ ℵ2.

The proof of Crit(Mn+1,Mn) ≥ ℵ2: a tedious transfinite induction


Con functor

The Con functor:

For any homomorphism of algebras f : A→ B we define

Con f : ConA→ ConB

byα 7→ congruence generated by {(f(x), f(y)) | (x, y) ∈ α}.

Fact. Con f preserves ∨ and 0, not necessarily ∧.


Lifting of semilattice morphisms

Letϕ : S → T be a (∨, 0)-homomorphisms of lattices;f : A→ B be a homomorphisms of algebras.

We say that f lifts ϕ, if there are isomorphisms ψ1 : S → ConA,ψ2 : T → ConB such that

Sϕ−−−−→ T

ψ1

y ψ2

yConA

Con f−−−−→ ConB

commutes.A generalization: lifting of semilattice diagrams


Results of P. Gillibert

TheoremLet V, W be finitely generated congruence distributive varieties.TFAE

ConV * ConW;there exists a diagram of finite (∨, 0)-semilattices indexed by an-dimensional cube (for some n) liftable in V but not in W


Results of P. Gillibert

TheoremLet V, W be finitely generated congruence distributive varieties.Then (2) implies (1), where

Crit(V,W) ≤ ℵn;there exists a diagram of finite (∨, 0)-semilattices indexed by aproduct of n+ 1 finite chains liftable in V but not in W

If n = 0 then also (1)=⇒ (2).

Question. What about (1)=⇒(2) for n > 0? (Conjecture: nottrue.)


Example

The semilattice homomorphism

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has a lifting in M3 (the embedding of a 3-element chain into M3

lifts it), but not in D. Therefore, Crit(M3,D) ≤ ℵ0.


Diagram lifting and aleph2

We know that Crit(M4,M3) = ℵ2. Is there a diagram indexed bya product of 3 finite chains liftable in M4 but not in M3?

Yes, it is on the next slide. It is an alternative proof of theinequality Crit(M4,M3) = ℵ2.


M3 versus M4

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Critical point ℵ1

Let C∗4 and N∗6 be the varieties generated by the bounded latticesC4 and N6 with an additional unary operation:on C4 ... f(0) = 0, f(a) = b, f(b) = a, f(1) = 1;on N6 ... 180◦ rotation (f(a) = d...) .

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Critical points ℵ1

Theorem(1) Crit(N∗6,N5) = ℵ1;(2) Crit(N5,N

∗6) = ℵ0.

(3) Crit(N∗6,C∗4) = ℵ1;

(4) Crit(C∗4,N∗6) =∞.

(N5 considered as bouned lattice.)


Question

What is the mechanism behind these examples?


N6 versus N5

Both N5 and N∗6 have the same congruence lattice, but N∗6 has anautomorphism h (the vertical symmetry), such that Conc hinterchanges α and β:

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N6 versus N5

Below: D is the diagram in N∗6 , so that ConD has a lifting in N∗6but - no lifting in N5.

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Possible variations

We can construct more examples by considering semilatticemorphisms with several liftings in one of the classes and combiningthem in square diagrams. But we have to be careful to ensurecritical point not ≤ ℵ0


Looking for a link

What is the connection between liftability of diagrams andtopological properties of dual spaces? We have an answer in thefollowing special case.We say that a variety K has Compact Congruence IntersectionProperty (CCIP) if ConcA is a lattice for every A ∈ K.

Examples: distributive lattices, Stone algebras, vector spaces ...

Theorem(Baker, Blok, Pigozzi) A finitely generated congruence-distributivevariety K has CCIP iff every subalgebra of a subdirectly irreduciblealgebra in K is subdirectly irreducible (or one-element).


CCIP

CCIP greatly simplifies the investigation of congruence lattices.(But it is still difficult.)

Every V with ConV satisfactorily described has CCIP.Every distributive algebraic lattice whose compact elementsform a lattice is the congruence lattice of(a) a lattice (E. T. Schmidt);(b) a locally matricial algebra (P. Ružička).


Priestley duality

Let A belong to a finitely generated congruence-distributive varietywith CCIP. Then ConcA is a distributive lattice and we canconsider its dual Priestley space X. (We use the version for latticeswith 0 but not necessarily with 1.)

Points of the dual space correspond to (completely)meet-irreducible elements of ConA, so X = M(ConA). (Herewe declare 1 ∈ M(ConA).)The order is inherited from ConA.The topology is generated by the sets

Mx,y = {α | (x, y) ∈ α}

and their complements.


Diagrams versus topology

TheoremLet V be a finitely generated congruence-distributive variety withCCIP. Let F be the free algebra in V with ℵ1 generators. Let ~S bea diagram of finite distributive (0,∨)-semilattices indexed by afinite ordered set P having a smallest element 0 ∈ P . Thefollowing conditions are equivalent.(i) There exists A ∈ V such that X = M(ConA) is

~S-nonseparable;(ii) M(ConF ) is ~S-nonseparable;(iii) ~S has a lifting in V.


What is ~S-separability?

First,a simple example.Let ~S be a diagram of (∨, 0)-semilattices consisting of a singlemorphism ϕ : {0} → {0, 1}.

~S in not liftable in the variety of Boolean algebras.Consequence: the largest element in the dual space(corresponding to 1-element congruence lattice) does notbelong to the topological closure of the rest (pointscorresponding to 2-element lattices).~S is liftable in the variety of distributive lattices. Consequence:the largest element in the dual space (corresponding to1-element congruence lattice) can belong to the topologicalclosure of the rest (points corresponding to 2-element lattices).


Definition in general

Let ~S = (Sp, sp,q | p ≤ q in P ) be a diagram of finite distributivelattices and 0-preserving lattice homomorphisms, indexed by a finiteordered set P . Let s←p,q be the residuated maps,s←p,q(α) = sup{β | s←p,q(β) ≤ α}.Let X be a Priestley space. For every p ∈ P let Kp be the set ofall order embeddings M(Sp)→ X whose range is an upper subsetof X. An open neighbourhood of k ∈ Kp is a family of open setsU = (U(α) | α ∈ M(Sp)) with k(α) ∈ U(α).Definition. We say that X is ~S-nonseparable, if for any system ofopen neighbourhoods Uk (k ∈ ∪Kp) there exists a family(kp ∈ Kp | p ∈ P ) such that kq(α) ∈ Ukp(s←p,q(α)) whenever p < q,α ∈ M(Sq). Otherwise we say that X is ~S-separable.


Critical point for CCIP

TheoremLet V be a finitely generated congruence-distributive variety withCCIP. Then ConV ⊆ ConW or Crit(V,W) ≤ ℵ1.

Proof: Let ConV * ConW. By Gillibert, there exist a diagram ~Sindexed by some n-dimensional cube liftable in V but not in W.Let F be the free algebra in V with ℵ1 generators. By our theorem,M(ConF )) is ~S-nonseparable, while M(ConA) is ~S-separable forevery A ∈ W. Consequently, ConF /∈ ConW and the cardinalityof F is ℵ1.

Upper bound can occur: Crit(N∗6,C∗4) = ℵ1;


How ℵ1 got there

Well known free set theorem:

Theorem(Hajnal) If |X| ≥ ℵ1, then for every function Φ : X → [X]<ω

there is a set Y ⊆ X such that |Y | = |X| and x /∈ Φ(y) wheneverx, y ∈ Y , x 6= y.


Free set theorem for diagrams

Let (Ap | p ∈ P ) be a family of nonempty finite sets, indexed by afinite poset P with a smallest element, such that Ap ⊆ Aqwhenever p ≤ q. For a set X let H(X,Ap) denote the set of allsurjective mappings X → Ap

TheoremIf |X| ≥ ℵ1, then for every function

supp :⋃p∈P

H(X,Ap)→ [X]<ω

there are hp ∈ H(X,Ap) such that

hq� supphp = hp� supphp

for every p < q.

(A diagram version of the free set theorem.)Miroslav Ploščica Critical points for congruence lattices

Open problems

1. See Chapter 9 in G. Grätzer, F. Wehrung (eds.), Lattice Theory:Special Topics ans Applications, Birkhäuser 2014. Especially,

Is every distributive algebraic lattice isomorphic to thecongruence lattice of a majority algebra?

2. Investigate classes Cw K (weak congruence lattices).


Thanks

Thank you for attention.

ploscica.science.upjs.sk


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