Critical points for congruence lattices · PDF fileDefinitioningeneral LetS~= (S p;s p;qjp...
Transcript of 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.
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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}
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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|.
Miroslav Ploščica Critical points for congruence lattices
N5 and Mn
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Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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?
Miroslav Ploščica Critical points for congruence lattices
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).
Miroslav Ploščica Critical points for congruence lattices
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.)
Miroslav Ploščica Critical points for congruence lattices
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
Miroslav Ploščica Critical points for congruence lattices
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 ∧.
Miroslav Ploščica Critical points for congruence lattices
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
Miroslav Ploščica Critical points for congruence lattices
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
Miroslav Ploščica Critical points for congruence lattices
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.)
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
M3 versus M4
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Miroslav Ploščica Critical points for congruence lattices
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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Miroslav Ploščica Critical points for congruence lattices
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.)
Miroslav Ploščica Critical points for congruence lattices
Question
What is the mechanism behind these examples?
Miroslav Ploščica Critical points for congruence lattices
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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Miroslav Ploščica Critical points for congruence lattices
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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Miroslav Ploščica Critical points for congruence lattices
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
Miroslav Ploščica Critical points for congruence lattices
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).
Miroslav Ploščica Critical points for congruence lattices
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).
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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).
Miroslav Ploščica Critical points for congruence 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.
Miroslav Ploščica Critical points for congruence lattices
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;
Miroslav Ploščica Critical points for congruence lattices
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.
Miroslav Ploščica Critical points for congruence lattices
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).
Miroslav Ploščica Critical points for congruence lattices