congruence lattices of algebras

PAVOL JOZEF ŠAFÁRIK UNIVERSITY IN KOŠICE

FACULTY OF SCIENCE

Institute of Mathematics

CONGRUENCE LATTICES OF ALGEBRAS

Dissertation Thesis

Košice, 2013 Filip Krajník

PAVOL JOZEF ŠAFÁRIK UNIVERSITY IN KOŠICE

FACULTY OF SCIENCE

CONGRUENCE LATTICES OF ALGEBRAS

Dissertation Thesis

Study Programme: Discrete Mathematics

Field of Study: 9.1.6. Discrete Mathematics

Institute: Institute of Mathematics

Tutor: doc. RNDr. Miroslav Ploščica, CSc.

Košice, 2013 RNDr. Filip Krajník

Acknowledgement

I would like to thank my supervisor, doc. RNDr. Miroslav Ploščica, CSc., for his kind and valuable

advice and help.

3350863510

Univerzita P. J. Šafárika v KošiciachPrírodovedecká fakulta

ZADANIE ZÁVEREČNEJ PRÁCE

Meno a priezvisko študenta: RNDr. Filip KrajníkŠtudijný program: Diskrétna matematika (Jednoodborové štúdium,

doktorandské III. st., denná forma)Študijný odbor: 9.1.6. diskrétna matematikaTyp záverečnej práce: Dizertačná prácaJazyk záverečnej práce: slovenský

Názov: Zväzy kongruencií algebier

Cieľ: Oboznámiť sa s pojmom kongruencie v algebre a s využitím zväzov kongruenciípri skúmaní štruktúry algebier. Získať prehľad o hlavných výsledkocha moderných metódach používaných pri štúdiu kongruencií. Dosiahnuť novévýsledky v oblasti popisu zväzov kongruencií algebier vo vybraných varietách.Zamerať sa predovšetkým na kongruenčne distributívne variety, napríkladzväzy, pseudokomplementárne zväzy, zväzovo usporiadané grupy, mediánovealgebry a podobne. Skúmať kongruenčnú ekvivalentnosť variet a kritické body.

Literatúra: 1. M. Kolibiar a kol.: Algebra a príbuzné disciplíny, Alfa, Bratislava 1992.2. R. McKenzie, R. McNulty, W. Taylor: Algebras, lattices, varieties I.,Wadsworth&Brooks/Cole, Monterey 1987.3. G. Grätzer: General lattice theory (2nd edition), Birkhäuser Verlag, Basel1998.4. M. Ploščica: Separation properties in congruence lattices of lattices, Colloq.Math. 83(2000), 71-84.5. M. Ploščica: Finite congruence lattices in congruence distributive varieties,in: Contributions to General Algebra 14, Verlag Johannes Heyn, Klagenfurt2004, 119-126.

Kľúčovéslová: algebra, kongruencia, zväz, distributívnosť

Školiteľ: doc. RNDr. Miroslav Ploščica, CSc.Ústav: ÚMV - Ústav matematických viedRiaditeľ ústavu: doc. RNDr. Roman Soták, PhD.

Dátum zadania: 02.09.2009

Dátum schválenia: 31.05.2013 prof. RNDr. Stanislav Jendroľ, DrSc.predseda odborovej komisie

Abstract

A variety V of algebras has the Compact Intersection Property (CIP), if the family of compact

congruences of every A ∈ V is closed under finite intersection. This thesis investigates the congru-

ence lattices of algebras in locally finite, congruence-distributive varieties with the CIP. It proves

some general results and obtain a complete characterization for some types of such varieties. It

provides two kinds of description of congruence lattices: via direct limits and via Priestley duality.

Key words: algebra, congruence, lattice, distributivity

Abstrakt

Varieta V algebier má vlastnosť kompaktného prieniku, ak systém kompaktných kongruencií ľu-

bovoľnej algebry A ∈ V je uzavretý na konečné prieniky. Práca sa zaoberá zväzmi kongruencií

algebier v lokálne konečných, kongruenčne-distributívnych varietách s vlastnosťou kompaktného

prieniku. Poskytuje niekoľko všeobecných výsledkov a úplnú charakterizáciu pre niekoľko typov

takých variet. Uvádza dva spôsoby popisu zväzov kongruencií: pomocou direktných limít a pomo-

cou Priestleyovej duality.

Kľúčové slová: algebra, kongruencia, zväz, distributívnosť

Contents

Introduction 6

1 Basic definitions and denotations 13

2 Congruence intersection property 19

2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Description via direct limits 26

3.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Description via Priestley duality 42

4.1 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

References 65

Introduction

Congruences has an extremely important position in algebra. It enables factoring,

i.e. reduction of complex mathematical objects into simpler ones. The set of all

congruences of algebra forms a lattice and suggests a lot about how the given algebra

is constructed from smaller algebras.

Recall that algebraic lattice is a complete lattice L, such that every element x

of L is the supremum of the compact elements below x. G. Birkhoff and O. Frink

in [6](1948) proved that the congruence lattice of an algebra is an algebraic lattice.

They raised the question whether the converse is true. This problem was earlier

raised by G. Birkhoff in a 1945 lecture and again in [5](1948). This problem was

solved in G. Grätzer and E. T. Schmidt [17](1963):

Theorem 0.1. (Congruence Lattice Characterization Theorem)

Let L be an algebraic lattice. Then there exists an algebra A whose congruence lattice

is isomorphic to L.

Since algebraic lattices are determined by their sets of compact elements, an

equivalent formulation is:

Theorem 0.2. Every join-semilattice with zero can be represented as ConcA, the

join-semilattice with zero of compact congruences of an algebra A.

More polished versions of the original proof appeared inW. A. Lampe ([27](1973)),

P. Pudlák ([39](1985)), J. Tůma ([44](1989)) and others. All the proofs yield an in-

finite algebra A, even if L is finite. So naturally, this raises the question whether

Theorem 0.1 is true for finite lattices and finite algebras. Whether for every finite

lattice L there exists a finite algebra with congruence lattice isomorphic to L. This

unsolved problem is known as Finite Lattice Representation Problem, or Finite Con-

gruence Lattice Problem.

Distributive case of this problem is more tractable. For a lattice L, the congruence

lattice of L is distributive - according to a result of N. Funayama and T. Nakayama

[11](1942). Soon after this result, R. P. Dilworth discovered (but did not publish)

the even more remarkable converse:

Theorem 0.3. (The Dilworth Theorem for Finite Congruence Lattices)

Every finite distributive lattice is isomorphic to the congruence lattice of some finite

lattice.

6

The Dilworths result was made into an exercise in [5]. The proof was in R. P. Dil-

worths lecture notes, but copies of his lecture notes were no longer available. G. Grät-

zer and E. T. Schmidt published the proof in [16](1962). In fact, they proved some-

thing much stronger. The solution lattice found in Grätzer and Schmidt’s proof is

sectionally complemented:

Theorem 0.4. Every finite distributive lattice can be represented as the congruence

lattice of a finite sectionally complemented lattice.

In the 60 years since the discovery of this result, a large number of papers have

been published, strengthening and generalizing the Dilworth Theorem. Let’s see

which properties (P) of finite distributive lattices are such that every finite distribu-

tive lattice can be represented as the congruence lattice of a finite lattice with prop-

erty (P). For instance, we already proved that the finite lattice L for the Dilworth

Theorem can be constructed as a sectionally complemented lattice. By Theorem,

0.4 (P) ⇔ (sectionally complemented).

The basic representation theorems are all of the same general type. For exam-

ple V. A. Baranskii and A. Urquhart [4],[45](1979) solved the problem raised by

G. Grätzer in 1978: Whether the congruence lattice and the automorphism group

of a finite lattice are independent.

Theorem 0.5. (The independence theorem)

Let D be a finite distributive lattice with more than one element, and let G be a

finite group. Then there exists a finite lattice L such that the congruence lattice of

L is isomorphic to D and the automorphism group of L is isomorphic to G.

Hence, (P)⇔ (given authomorphism group). Furthermore, there is a congruence-

preserving extension variant for this result due to G. Grätzer and E. T. Schmidt

[18](1995):

Let K be a finite lattice. A finite lattice L is a congruence-preserving extensions

of K, if L is an extension and every congruence ϕ of K has exactly one extension ψ

to L - that is ψ/K = ϕ. Of course, the congruence lattice of K is isomorphic to the

congruence lattice of L.

Theorem 0.6. (Strong independence theorem)

Let K be a finite lattice with more then one element and let G be a finite group.

Then K has a congruence-preserving extension L whose automorphism group is

isomorphic to G.

7

For sectionally complemented lattices, the congruence-preserving extension the-

orem was published in G. Grätzer and E. T. Schmidt [19](1999):

Theorem 0.7. Every finite lattice K has a finite and sectionally complemented,

congruence-preserving extension L.

There exists the basic representation theorem and its congruence-preserving ex-

tension variant for many others (P). A thorough survey of this field is in G. Grätzer

[15](2006). For example

• P ⇔ sectionally complemented,

• P ⇔ given authomorphism group,

• P ⇔ semimodular,

• P ⇔ uniform,

• P ⇔ isoform.

The infinite case is much different. It is called the Congruence Lattice Problem: Is

every distributive algebraic lattice isomorphic to the congruence lattice of a suitable

lattice? We can state the semilattice formulation of this problem. Recall that join-

semilattice with zero S is distributive, if for all a, b, c ∈ S with c ≤ a ∨ b there are

x ≤ a and y ≤ b such that c = x ∨ y. Let us call S representable, if it is isomorphic

to Conc L (the distributive semilattice of compact congruences of the lattice L), for

some lattice L. The semilattice formulation of the Congruence Lattice Problem: Is

every distributive join-semilattice with zero representable?

It was one of the most famous and long-standing open problems in lattice theory.

There are two groups of positive results.

The first group of positive results started with two papers of E. T. Schmidt,

[42](1968), [43](1981). To state Schmidt’s results, we need some concepts. A con-

gruence θ of a join-semilattice with zero S is monomial if any θ-equivalence class

has a largest element. A congruence of S is distributive if it is a join of monomial

congruences.

A generalized Boolean semilattice is defined as the underlying join-semilattice of

a sectionally complemented distributive lattice with zero. A join-semilattice with

zero satisfies Schmidt’s Condition if it is isomorphic to B/θ for some distributive

congruence θ of a generalized Boolean semilattice B. E. T. Schmidt in [42](1968)

8

presents an important sufficient condition, for a distributive join-semilattice with

zero:

Theorem 0.8. Any semilattice with zero satisfying Schmidt’s Condition is repre-

sentable.

As an important consequence, E. T. Schmidt in [43](1981) proves the following

result:

Theorem 0.9. Every distributive lattice with zero is representable.

F. Wehrung in [47](2003) extended Schmidt’s result:

Theorem 0.10. Every direct limit of a countable sequence of distributive lattices

with zero is representable.

The second group of positive results is phrased in terms of the cardinality of

the join-semilattice with zero. H. Bauer around 1980 proved a result (unpublished)

implying the following:

Theorem 0.11. Every countable distributive join-semilattice with zero is repre-

sentable.

An extended version of this result is proved by H. Dobbertin [9](1986):

Theorem 0.12. Every distributive join-semilattice with zero in which any principal

ideal is countable is representable.

A. P. Huhn in [21],[22](1989) proved that:

Theorem 0.13. Every distributive join-semilattice with zero of cardinality at most

ℵ1 is representable.

The Congruence Lattice Problem remained open for over sixty years. F. Wehrung

in [48](2007) gave a negative solution. (He used a structure of M. Ploščica, J. Tůma

[37](1998) as a counterexample.)

Theorem 0.14. There exists a distributive algebraic lattice which is not isomorphic

to the congruence lattice of any lattice. This lattice has ℵω+1 compact elements.

The cardinality ℵω+1 in Wehrung’s result was improved by P. Růžička [41](2008):

Theorem 0.15. There exists a distributive join-semilattice with zero of cardinality

ℵ2 which is not isomorphic to the semilattice of compact congruences of any lattice.

9

Cardinality ℵ2 is, of course, optimal by Huhn’s result. M. Ploščica provides two

more examples of nonrepresentable distributive join-semilattices with zero of cardi-

nality ℵ2 in [36](2008).

Congruence lattice problem is a special case of the general problem: For given

class K of algebras describe ConK (the class of all lattices isomorphic to ConA

for some A ∈ K.) Or, at least: For given class K, L determine if ConK = ConL(ConK ⊆ ConL).

For finitely generated varieties K,L we have an algorithmic problem. Recall that

a variety of algebras is the class of all algebraic structures of a given signature

satisfying a given set of identities. Equivalently, a variety is a class of algebraic

structures of the same signature that is closed under the taking of homomorphic

images, subalgebras and (direct) products. In 1935 G. Birkhoff proved equivalent the

two definitions of variety given above (see [29]), a result of fundamental importance

to universal algebra and known as Birkhoff’s theorem or as the HSP theorem. H, S,

and P stand, respectively, for the closure operations of homomorphism, subalgebra,

and product.

A variety V is congruence-distributive (CD-variety) iff for every A ∈ V , the

congruence lattice of A is distributive. (For example variety of lattices or variety of

lattice ordered algebras.)

We do not know which lattices are isomorphic to congruence lattices of most com-

mon types of algebras, such as groups, rings, modules, lattices. There is a number of

partial results, for example well known is characterization of varieties, all members of

which have distributive congruence lattices (B. Jónsson [23](1968)). However, com-

plete characterization is a far more complex problem. The most significant progress

in this area, which occurred in the last decade, is resolution of Congruence lattice

problem by F. Wehrung.

In recent years M. Ploščica systematically pursued a description of congruences

lattices in locally finite, congruence-distributive varieties (see [31],[32],[33],[34]). He

elaborated an approach, based on the topological representation of algebraic lattices,

which allows "visualization" of some complicated algebraic properties. This has led

to the discovery of new numerical and combinatorial characteristics of lattices associ-

ated with the variety, for example size and shape of nonseparable sets. Nevertheless,

satisfactory characterization is available only for some simplest varieties.

Another important approach has been developed by P. Gillibert. He elaborated

10

a method based on lifting commutative diagrams and introduced the concept of a

critical point between congruence classes of two varieties K,L:

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

(Crit(K,L) =∞ if ConK ⊆ ConL.) Some results of P. Gillibert ([12](2011)):

Theorem 0.16. Let K,L be finitely generated, congruence-distributive varieties.

The following conditions are equivalent.

(1) ConK * ConL.

(2) There exists a diagram of finite join-semilattices with zero indexed by {0, 1}n

(for some n) liftable in K but not in L.

Theorem 0.17. Let K,L be finitely generated, congruence-distributive varieties.

Then (1) implies (2), where

(1) There exists a diagram of finite join-semilattices with zero indexed by a product

of n+ 1 finite chains liftable in K but not in L.

(2) Crit(K,L) ≤ ℵn.

If n = 0 then also (1) implies (2).

The problem of describing ConK seems to be much more tractable in the case of

varieties satisfying the Compact Intersection Property (CIP):

A variety V of algebras has the Compact Intersection Property (CIP), if the

family of compact congruences of every A ∈ V is closed under finite intersection.

This seems quite natural. Algebraic lattices are determined by their sets of com-

pact elements. There is a considerable evidence that the difficulty in describing

congruence lattices is connected with the fact that the compact congruences form

a join-semilattice, which in general is not a lattice. For instance, there are several

refinement properties, that are trivial in lattices, but very nontrivial in semilattices.

(see [46], [38], [35]).

Compact Intersection Property (CIP) was introduced by K. A. Baker in [3](1974).

He claimed that congruence-distributive variety definable by a finite list of identities

has the CIP if and only if any subalgebra of an subdirectly irreducible algebra or

finitely subdirectly irreducible algebra is finitely subdirectly irreducible algebra. It

was proved by W. J. Blok and D. Pigozzi in [8](1986) using the concept of equation-

ally definable principal meets (EDPM):

11

A variety has equationally definable principal meets (EDPM), if the intersection

of any pair of principal congruences is finitely generated.

Theorem 0.18. For any variety V the following are equivalent:

(1) V has the EDPM.

(2) V is congruence-distributive and has the CIP.

(3) V is congruence-distributive and the class of finitely subdirectly irreducible mem-

bers of V is closed under isomorphic images, subalgebras and ultraproducts.

Some other properties of varieties with the CIP was observed by K. A. Baker

and P. Agliano in [2](1999), but the main focus not on a characterization of ConK.(Although the final example describes ConK for the variety generated by the two-

element algebra {0, 1} with the operation p(x, y, z) = x ∨ (y ∧ z).) We say that

a variety has the CIPn, if it has a finite system of congruence intersection terms

without parameters {(pi, qi) : i = 1, . . . , n}. Some results of Baker and Agliano:

Theorem 0.19. Let V be a congruence-distributive variety with the CIP. Then any

system of congruence intersection terms without parameters can be reduced to a finite

one.

Theorem 0.20. A nontrivial variety V is congruence-distributive and has the CIP

if and only if V has the CIPn for some n.

There are nice results using the CIP. One of such result is Theorem 0.9. Another

is the result of P. Růžička ([40](2004)):

Theorem 0.21. Every algebraic distributive lattice in which the compact elements

are closed under finite intersection is isomorphic to the congruence lattice of a locally

matricial algebra.

We investigate the congruence lattices of algebras in locally finite, congruence-

distributive CIP varieties and obtain a complete characterization for several types

of such varieties. It turns out that our descriptions only depends on subdirectly

irreducible algebras in V and embeddings between them. We believe that the strategy

used here can be further developed and used to describe the congruence lattices for

any (locally finite) congruence-distributive CIP variety. We provide two kinds of

description of congruence lattices: via direct limits and via Priestley duality.

12

1 Basic definitions and denotations

If f is a mapping, then dom(f) (rng(f)) stand for its domain (range). By ker f

we denote the binary relation on dom(f) given by (x, y) ∈ ker f iff f(x) = f(y). By

f�X we mean the restriction of f to X.

Let P be a partially ordered set. For every x ∈ P we set ↑x = {y ∈ P | y ≥ x},↓x = {y ∈ P | y ≤ x}. A subset U ⊆ P is called an up-set (a down-set) if ↑x ⊆ U

for every x ∈ U (↓x ⊆ U for every x ∈ U).Let L be a lattice. An element a ∈ L is called strictly meet-irreducible iff a =

∧X

implies that a ∈ X, for every subset X of L. Let M(L) denote the set of all strictly

meet-irreducible elements. The largest element of L is not strictly meet-irreducible.

By adding it to M(L) we obtain the set denoted by M∗(L).

Let A be an algebra. For every a, b ∈ A by Θ(a, b) we denote the congruence

generated by the pair (a, b). The congruence lattice of A will be denoted by ConA.

For α ∈ ConA, the α-class in A/α containing a will be denoted by [a]α.

The set ConcA of all compact (finitely generated) congruences of A is a (0,∨)-

subsemilattice of ConA. The lattice ConA is uniquely determined by the semilattice

Conc A (it is isomorphic to the ideal lattice of ConcA). It is often easier to describe

Conc A instead of ConA.

It is a well known fact that for every θ ∈ ConA the lattice ConA/θ is isomorphic

to ↑θ. Hence θ ∈ M(ConA) if and only if the quotient algebra A/θ is subdirectly

irreducible. Equivalently, θ ∈ M(ConA) if and only if θ = ker f for some surjective

homomorphism f : A → S, with S subdirectly irreducible. This is also true if one

considers one-element algebras as subdirectly irreducible and replace M(ConA) by

M∗(ConA).

Let A,B be an algebras, for every homomorphism f : A → B we define the

mapping

Conc f : ConcA→ ConcB

by the rule that, for every α ∈ ConcA, Conc f(α) is the congruence generated

by the set {(f(x), f(y)) | (x, y) ∈ α}. This mapping is a homomorphism of (0,∨)-

semilattices.

Now let ϕ : K → L be a (0,∨)-homomorphism of finite (0,∨)-semilattices. We

define the map ϕ← : L→ K by

ϕ←(β) =∨{α | ϕ(α) ≤ β}.

13

If K = ConcA, L = ConcB and ϕ = Conc f for some algebras A, B and a

homomorphism f : A → B, then ϕ←(β) = {(x, y) ∈ A | (f(x), f(y)) ∈ β}. If A is a

subalgebra of B and f : A → B is the inclusion, then ϕ←(β) = β�A = β ∩ A2, the

restriction of β ∈ ConB to A. The following three Lemmas are well known.

Lemma 1.1. Let ϕ : K → L be a (0,∨)-homomorphism of finite lattices. Then

(a) ϕ← preserves ∧ and the largest element.

(b) ϕ(α) =∧{β | α ≤ ϕ←(β)}.

(c) ϕ(α) ≤ β ⇔ α ≤ ϕ←(β).

(d) If ψ : L→M is another (0,∨)-homomorphism of finite lattices, then (ψϕ)← =

ϕ←ψ←.

Lemma 1.2. If ϕ : K → L is a 0-preserving homomorphism of finite distributive

lattices, then ϕ←(c) ∈ M∗(K) for every c ∈ M∗(L).

Lemma 1.3. If ϕ : K → L is a 0, 1-preserving homomorphism of finite distributive

lattices, then ϕ←(c) ∈ M(K) for every c ∈ M(L).

Notice that finite (0,∨)-semilattices are, in fact, lattices. Further, for algebras

A,B, by A ≤ B denote that A is isomorphic to a subalgebra of B.

Next, we recall the algebraic constructions of direct and inverse limit. Let P be

an ordered set. Let K be a class of algebras. A P -indexed diagram ~A = (Ap, fp,q)

in K consists of a family (Ap, p ∈ P ) of algebras in K and a family (fp,q, p ≤ q) of

homomorphisms fp,q : Ap → Aq such that fp,p is the identity on Ap and fp,r = fq,rfp,q

for all p ≤ q ≤ r.

If the index set P is directed (for every p, q ∈ P there exists r ∈ P with p, q ≤ r),

then we define the direct limit of ~A as

lim→

~A = lim→Ap = (

⊔p∈P

Ap)/ ∼,

where⊔p∈P Ap is the disjoint union of the family (Ap, p ∈ P ) and the equivalence

relation ∼ is defined (for x ∈ Ap and y ∈ Aq) by

x ∼ y ⇔ ∃r ∈ P : fp,r(x) = fq,r(y).

A special case of the direct limit is the directed union, when all the homomor-

phisms are set inclusions. Note that in the category theory this construction corre-

sponds to the (directed) colimit.

14

The inverse limit of ~A is defined for any partially ordered set P as a subalgebra

of the direct product of∏

p∈P Ap, namely

lim←

~A = lim←Ap = {a ∈

∏p∈P

Ap | aq = fp,q(ap) for every p, q ∈ P, p ≤ q}.

(The elements of∏

p∈P Ap are written in the form a = (ap)p∈P .) A special case of

this construction is the direct product, which arises when P is an antichain. In the

category theory language, this construction is the limit of ~A.

It is well known that any variety V is closed under the formation of direct and

inverse limits.

The direct limit construction will be used to obtain the description of ConcA for

infinite A ∈ V from the description of ConcA for finite A. This is possible due to

the following two facts. First, Conc is a functor preserving the direct limits, which

means that for every directed P -indexed diagram ~A in V we have the P -indexed

diagram Conc~A = (Conc Ap,Conc ϕp,q) in the category of (0,∨)-semilattices and

(0,∨)-homomorphisms, and

Conc lim→

~A ' lim→

Conc~A.

Second, let ~A = (Ap, ϕp,q) and ~B = (Bp, ψp,q) be directed P -indexed diagrams

and let hp : Ap → Bp be an isomorphisms for every p ∈ P , such that the following

diagram commutes for every p, q ∈ P , p ≤ q:

Apϕp,q−−−→ Aq

hp

y hq

yBp

ψp,q−−−→ Bq

Then

lim→

~A ' lim→

~B.

The inverse limits will be used to construct algebras with prescribed finite (dis-

tributive) congruence lattice. For this we need special diagrams called admissible

valuations.

First, let SI(V) denote the class of all subdirectly irreducible members of a variety

V . By 1 we will denote both the one-element algebra in V and its single element. In

this thesis we find it convenient to include 1 into SI(V).

Definition 1.4. Let V be a variety and let M be a partially ordered set, we say that

M-indexed diagram −→v = (v(α), fα,β) is a SI(V)-valuation on M , if v : M → SI(V)

15

such that fα,β : v(α) → v(β) is surjective for every α ≤ β and the assignment

β 7→ ker fα,β is a bijection ↑α→ M∗(Con v(α)).

Lemma 1.5. Let V be a variety, let M be a partially ordered set and let −→v =

(v(α), fα,β) be a SI(V)-valuation on M . For every α ∈ M the bijection ↑α →M∗(Con v(α)) defined by β 7→ ker fα,β, is an isomorphism of ordered sets.

Proof. Let β, γ ∈ ↑α such that β ≤ γ. Thus fα,γ = fβ,γfα,β and hence ker fα,β ≤ker fα,γ.

Conversely, if ker fα,β ≤ ker fα,γ, then there exists a surjective homomorphism

g : v(β)→ v(γ)

such that gfα,β = fα,γ. There is a δ ∈M , δ ≥ β such that ker fβ,δ = ker g. Thus

ker fα,δ = ker fβ,δfα,β = ker gfα,β = ker fα,γ,

so δ = γ, hence β ≤ γ.

Definition 1.6. A P -indexed diagram ~A = (Ap, ϕp,q) in V is called admissible if

the following two conditions are satisfied:

(i) For every p ∈ P and every u ∈ Ap there exists

a ∈ lim←Ap

such that ap = u.

(ii) For every p, q ∈ P , p � q there exist

a, b ∈ lim←Ap

such that ap = bp and aq 6= bq.

Notice that the admissibility is a purely set-theoretical property, depending only

on the sets Ap and maps ϕp,q, and not on the algebraic structure of Ap. Admissible

ordered diagrams arise naturally from systems of equivalences on a given set, as

shown by the following assertion.

Lemma 1.7. ([34] Lemma 2.2) Let P be some set of equivalences on a set A, ordered

by the set inclusion. For every α ∈ P let Aα = A/α. For α ≤ β let fα,β : A/α →A/β be the natural projection map, i.e. fα,β([x]α) = [x]β. Then P -indexed diagram~A = (Aα, ϕα,β) is admissible.

16

The next Theorem follows from [34], Theorem 2.4.

Theorem 1.8. Let V be a locally finite congruence-distributive variety. Let L be a

finite distributive lattice and let M = M∗(L). Let ~A = (v(α), fα,β) be an admissible

SI(V)-valuation on M . Then A = lim← ~A is an algebra whose congruence lattice

is isomorphic to L. The isomorphism h : M∗(L) → M∗(ConA) can be defined by

h(α) = ker πα, where πα is the projection A→ v(α).

By the Birkhoff duality for finite distributive lattices, the isomorphism

h : M∗(L)→ M∗(ConA)

induces an isomorphism

k : ConA→ L

by k(x) =∧{y ∈ M∗(L) | h(y) ≥ x}. For x ∈ M∗(ConA) we have k(x) = h−1(x), so

k(kerπα) = α.

In Chapter 4 we prove a generalization of Theorem 1.8 for infinite M. Let us

recall the Priestley duality for distributive lattices with 0 (but not necessarily with

1). Let L be a distributive lattice with 0. Let P(L) denote the set of all prime ideals

of L (including L itself). For every x ∈ L we define

Ux = {I ∈ P(L) | x ∈ I}, Vx = {I ∈ P(L) | x /∈ I}.

We endow P(L) with the ordering ≤ by the set inclusion and the topology τ gen-

erated by all sets of the form Ux and Vx. The resulting structure (P(L),≤, τ) is

called the dual Priestley space of L. The ordered topological space P(L) determines

L uniquely. In fact, L is isomorphic to the lattice of all proper clopen down subsets

of P(L). As a topological space, P(L) is compact, Hausdorff, zerodimensional. It has

a largest element. The compatibility of the order and the topology can be expressed

by the following condition of compact totally order-disconnectedness:

(CTOD) If y, z ∈ P(L), y 6≤ z, then there exists a clopen up-set U ⊆ P(L) with

y ∈ U , z /∈ U .

Moreover, if F,G are closed sets such that ↑F∩ ↓ G = ∅, then there exists a clopen

up-set U such that F ⊆ U and U ∩G = ∅.Further denote by Id(L) an ideal lattice of a lattice L. Prime ideals of L can be

also characterized as finitely meet-irreducible elements of IdL. The next Lemma is

easy to prove.

17

Lemma 1.9. Let L be a distributive lattice and let I ∈ IdL. Then I is prime if and

only if I is finitely meet-irreducible element of IdL or I = L.

Now let V be a finitely generated, congruence-distributive variety. We prove that,

for every A ∈ V , all finitely meet-irreducible elements of ConA are strictly meet-

irreducible. We use the following concept from [32].

Definition 1.10. A subset P of an algebraic lattice L is called separable, if P ⊆M(L) and there exists a family {xp | p ∈ P} ⊆ L such that

(i) xp � p for every p ∈ P .

(ii)∧{xp | p ∈ P} = 0.

Let s(V) = max{|M(ConB)| | B ≤ A ∈ SI(V)}. Since V is finitely generated,

every subdirectly irreducible algebra is finite and hence s(V) ∈ N.

Lemma 1.11. ([32], Consequence 2.4) If Q ⊆ M(ConA) is non-separable, for some

A ∈ V, then |Q| ≤ s(V).

Lemma 1.12. Let α be a finitely meet-irreducible element of ConA for some A ∈ V.Then α ∈ M(ConA).

Proof. Let α be a finitely meet-irreducible element of ConA for some A ∈ V . Forcontradiction suppose that there exists an infinite R ⊆ M(ConA) such that

α =∧

R, α /∈ R.

Choose finite P ⊆ R with |P | > s(V). By Lemma 1.11, P is separable, so we

have xp � p (hence xp � α) for every p ∈ P and∧{xp | p ∈ P} = 0 ≤ α, which

contradicts the finite meet-irreducibility of α.

Lemma 1.13. For any algebra A ∈ V,

I ∈ P(Conc A)⇔ sup I ∈ M∗(ConA).

Proof. The equivalence follows from Lemma 1.9 and Lemma 1.12.

18

2 Congruence intersection property

We say that a variety V of algebras has the Compact Intersection Property (CIP),

if the family of compact congruences of every A ∈ V is closed under finite intersec-

tion.

Theorem 2.1. Let V be a locally finite congruence-distributive variety. The follow-

ing conditions are equivalent.

(1) The Intersection of any two compact congruences of A is compact for every

A ∈ V.

(2) Every finite subalgebra of a subdirectly irreducible algebra of V is subdirectly

irreducible.

(3) If T is a finite subalgebra of a subdirectly irreducible algebra of V, then the

ordered set M∗(ConT ) has a least element.

(4) For every embedding f : A → B of algebras in V with A finite, the mapping

Conc f preserves meets.

Proof. (2)⇔(3) is well known.

(1)⇒(3): If T = 1, then (3) hold trivially. Let T be a subalgebra of a subdirectly

irreducible S ∈ SI(V) such that T 6= 1, T finite. Since ConT is finite, it suffices

to show that for all β1, β2 ∈ M∗(ConT ) there exists β ∈ M∗(ConT ) such that

β ⊆ β1 ∩ β2. Let A = F (ℵ0) denote the free algebra in V with ℵ0 as free generating

set. Choose a surjective homomorphism h0 : 〈X0〉 → T , where X0 ⊆ ℵ0 is finite and

large enough. Since A is free, h0 can be extended to a homomorphism h : A → T .

Further, we consider the natural homomorphisms g1 : T → T/β1, g2 : T → T/β2.

Then ker(gih0) ∈ M∗(Con〈X0〉).

?

〈X0〉

HHHHHj

T��

��T/β2T/β1

g1 g2

h0

Figure 1

Since Con〈X0〉 is finite and distributive, there is a smallest element γi in the

set {α ∈ Con〈X0〉 | α � ker(gih0)}. Let αi ∈ ConA be the congruence generated

19

by γi. Then αi � 〈X0〉 ⊇ γi. The inverse inclusion follows from the fact that the

projection 〈X0〉 → 〈X0〉/γi can be extended to a homomorphism l : A → 〈X0〉/γi,thus αi ⊆ ker(l) and αi � 〈X0〉 ⊆ ker(l � 〈X0〉) = γi. So, αi � 〈X0〉 = γi.

Congruences α1, α2 are compact, so by our assumption α1∩α2 is compact too. It

means that there exists a finite set Y ⊆ ℵ0, X0 ⊆ Y such that α1 ∩ α2 is generated

by α1 ∩ α2 � 〈Y 〉.Let f : A → S be a surjective homomorphism such that f�〈Y 〉 = h�〈Y 〉,

then ker(f�〈X0〉) = ker(h�〈X0〉) ⊆ ker(gih0). Thus γi � ker(f�〈X0〉) and hence

αi � ker(f). Since ker(f) ∈ M∗(ConA), then α1 ∩ α2 � ker(f) and thus

α1 ∩ α2�〈Y 〉 � ker(f�〈Y 〉) = ker(h�〈Y 〉).

Therefore, there exists δ ∈ M∗(Con〈Y 〉) such that

δ ≥ ker(h�〈Y 〉), δ � α1 ∩ α2�〈Y 〉.

Let b0 : 〈Y 〉 → 〈Y 〉/δ = W be the natural map, it can be extended to a ho-

momorphism b : A → W . Moreover for all y ∈ Y there exists x0 ∈ X0 such

that (y, x0) ∈ ker(h). Therefore (y, x0) ∈ ker(b0), so b0(y) = b0(x0). It shows that

b0(〈X0〉) = b(〈Y 〉) = W .

Since ker(b0�〈X0〉) = δ�〈X0〉 ⊇ ker(h�〈X0〉), there exists a homomorphism

k : T → W

such that kh�〈X0〉 = b0�〈X0〉. Further, since b0(〈X0〉) = W ∈ SI (V), ker(k) ∈M∗(ConT ). Further, α1 ∩ α2�〈Y 〉 * ker(b0) implies that α1 ∩ α2 * ker(b) and thus

α1, α2 * ker(b).〈X0〉 -h�〈X0〉

b0�〈X0〉

?

T��

��

k

WFigure 2

Since αi are generated by γi for i=1,2, we have γi * ker(b) and thus

γi * ker(b0�〈X0〉).

By definition of γi it means ker(b0�〈X0〉) ⊆ ker(gih0). For every (x, y) ∈ ker(k) we

have x′, y′ ∈ 〈X0〉 such that h(x′) = x and h(y′) = y. Thus (x′, y′) ∈ ker(b0�〈X0〉),

20

so (x′, y′) ∈ ker(gih0). It means that gi(h0(x′)) = gi(h0(y′)), hence gi(x) = gi(y). We

have proved that ker(k) ≤ ker(gi) = βi for i=1,2.

(3)⇒(1): Let A ∈ V and suppose that α1, α2 ∈ ConA are compact, but α1 ∩ α2

is not compact. There exists finite subalgebra Y of A such that αi�Y generates αi(i = 1, 2). Denote γi = αi�Y . Since ConY is a finite distributive lattice, there exist ∨-irreducible δ1, δ2, . . . , δn, ε1, ε2, . . . , εm ∈ ConY such that γ1 =

∨nj=1 δj, γ2 =

∨mk=1 εk.

Let δ̄j ∈ ConA be generated by δj, similarly ε̄j. Since δj ⊆ γ1 ⊆ α1 ∈ ConA, we

have δ̄j ⊆ α1. Moreover,∨nj=1 δ̄j ⊇

∨nj=1 δj = γ1 = α1 � Y , thus α1 =

∨nj=1 δ̄j, and

α2 =∨mk=1 ε̄k similarly. By the distributivity, α1 ∩ α2 =

∨j,k(δ̄j ∩ ε̄k). Since α1 ∩ α2

is not compact, δ̄j ∩ ε̄k is not compact for some j, k.

Let β ∈ ConA be generated by (δ̄j ∩ ε̄k)�Y , thus β ( δ̄j ∩ ε̄k, so there ex-

ists a surjective homomorphism h : A → S for some S ∈ SI(V) such that β ⊆ker(h), δ̄j ∩ ε̄k * ker(h). Let T = h(Y ) ⊆ S, then ConT is isomorphic to L =

{α ∈ ConY | ker(h�Y ) ⊆ α}.Since δj, εk are ∨-irreducible in ConY , there exist

η1 = max{α ∈ ConY | δj � α},

η2 = max{α ∈ ConY | εk � α}.

Clearly η1, η2 ∈ M∗(ConY ). If δj ⊆ ker(h�Y ), then δ̄j ⊆ ker(h), which contradicts

our definition of the homomorphism h and thus δj * ker(h�Y ). Hence ker(h�Y ) ⊆ η1,

thus η1 ∈ L and similarly η2 ∈ L. Since η1, η2 ∈ M∗(ConY ), we have η1, η2 ∈ M∗(L).

For every ρ ∈ M∗(L), we have

ρ ⊇ ker(h�Y ) ⊇ β�Y ⊇ δ̄j ∩ ε̄k�Y ⊇ δj ∩ εk.

Either ρ ⊇ δj or ρ ⊇ εk, by the meet-irreducibility of ρ. In the case ρ ⊇ δj we have

ρ * η1, and from ρ ⊇ εk we deduce ρ * η2. Hence η1 and η2 do not have a common

lower bound in L, so L cannot have a least element

(4)⇒(3): We can assume that A is a finite subalgebra of B ∈ SI(V). Suppose that

M∗(ConA) has not a least element. Let α1, α2, . . . , αn be the minimal elements of

M∗(ConA), n ≥ 2. Denote by f : A→ B the inclusion. Then

Conc f(α1 ∧ α2 ∧ · · · ∧ αn) = Conc f(∆) = ∆.

On the other hand,

Conc f(α1) ∧ Conc f(α2) ∧ Conc f(αn) 6= ∆,

21

since the intersection of nonzero congruences in a subdirectly irreducible algebra

cannot be ∆.

(2)⇒(4): Suppose that Conc f : ConcA → ConcB does not preserve meets. We

can assume that f : A → B is an inclusion. Then Conc f(α ∧ β) < Conc f(α) ∧Conc f(β) for some α, β ∈ Conc A = ConA. Hence, there is γ ∈ M∗(ConB) such

that

γ ≥ Conc f(α ∧ β),

γ � Conc f(α) ∧ Conc f(β).

Thus,

γ � Conc f(α), γ � Conc f(β).

Now, A/γ is a finite subalgebra of the subdirectly irreducible algebra B/γ, whose

congruence lattice is isomorphic to L = {θ ∈ ConA | γ�A ⊆ θ}. To prove that A/γ

is not subdirectly irreducible it suffices to find α′, β′ ∈ ConA with α

′ ∧ β ′ = γ�A

and α′ , β ′ 6= γ�A.

We set α′ = α ∨ γ�A and β ′ = β ∨ γ�A. By the distributivity,

α′ ∧ β ′ = (α ∧ β) ∨ γ�A = γ�A.

If α′ = γ�A, then

Conc f(α) ≤ Conc f(α′) = Conc f(γ�A) ≤ γ.

Hence, α′ 6= γ�A and similarly β ′ 6= γ�A.

The above result is not completely new. The equivalence of the first two conditions

was proved by W. J. Blok and D. Pigozzi in [8] (and claimed by K. A. Baker on

page 139 in [3]), using the concept of equationally definable principal meets. (See

also [2].) We provide a new proof which does not refer to polynomials and, we

believe, provides an insight helpful in describing the congruence lattices of algebras

in congruence-distributive CIP varieties. Our proof follows the lines of reasoning

from [31],[32], which connected CIP to the concept of separable sets in M(ConA)

and to topological properties of M(ConA).

22

2.1 Examples

Stone algebra. A bounded distributive lattice with pseudocomplementation L

is called a Stone algebra if and only if it satisfies the Stone identity:

∀a ∈ L : a∗ ∨ a∗∗ = 1

Let B0 denote the two-element Boolean algebra and B1 denote the three-element

chain 0 < e < 1 as a distributive lattice with pseudocomplementation.

B1 :

s1 = 0∗

ses0 = 1∗ = e∗

B0 :

ss1 = 0∗

0 = 1∗

Con(B1) :

suu

Con(B0) :

su

Figure 3

Let V be the variety of all Stone algebras. Up to isomorphism, B0, B1 and 1 are

the only subdirectly irreducible members of V , so the second condition of Theorem

2.1 is satisfied.

The varietiesM3, N5. LetM3 denote the variety generated by the 5-element

lattice M3 (see Figure 4). Up to isomorphism, the subdirectly irreducible algebras

in M3 are two-element chain, M3 and 1. One of the subalgebras of the lattice M3

is a three-element chain. The congruence lattice of a three-element chain is a four-

element lattice D2 (see Figure 4) and since M∗(D2) has not a least element, there is

A ∈ M3 such that there exist two compact congruences of A, whose intersection is

not compact.

Further let N5 denote the variety generated by the lattice N5 (see Figure 4). Up

to isomorphism, the subdirectly irreducible algebras in N5 are two-element chain,

N5 and 1. One of the subalgebras of the lattice N5 is a three-element chain. Hence,

like M3, there is A ∈ N5 such that there exist two compact congruences of A, which

intersection is not compact.

23

M3:

ss

��

�� s@@@@s

s��

@@

@@

N5:

s�� AAAs

sss��

@@

Figure 4

D2:

su

��

��

@@@@u

s��

@@

@@

Distributive lattices with pseudocomplementation. Let Bω be the variety

of all bounded distributive lattices with pseudocomplementation. By [28] (see also

[14], page 165), the subvarieties of Bω form a chain

B−1 ⊂ B0 ⊂ B1 ⊂ · · · ⊂ Bn ⊂ · · · ⊂ Bω.

Here B−1 is the trivial variety, B0 is the variety of all Boolean algebras and for n ≥ 1

the variety Bn is determined by the identity

(x1 ∧ · · · ∧ xn)∗ ∨ (x∗1 ∧ · · · ∧ xn)∗ ∨ · · · ∨ (x1 ∧ · · · ∧ x∗n)∗ = 1.

(Especially, B1 is the variety of all Stone algebras.) The variety Bn (n ≥ 0) is

generated by the algebra Bn = 2n⊕ 1, that is the power set of a n-element set with

a new top element added.

B1 :

s1 = 0∗

ses0 = 1∗ = e∗

B0 :

ss1 = 0∗

0 = 1∗

Figure 5

Subdirectly irreducible members of Bn are Bn, Bn−1, . . . , B−1. (The congruence

lattice of Bn is, as a lattice, dually isomorphic to Bn, that is ConBn = 1⊕2n.) It is

easy to check that all subalgebras of Bn are isomorphic to one of Bn, Bn−1, . . . , B0.

Hence, every Bn has the Compact Intersection Property.

Finite subdirectly irreducible algebras with constants. There is an easy

way to construct examples of varieties satisfying CIP. Let A be a finite algebra

generating a congruence-distributive variety HSP(A). (For instance A can be any

finite lattice.) Enrich the type of A by defining every element a ∈ A as a constant

(nullary operation). Denote the resulting algebra as A∗. Every subdirectly irreducible

member of V =HSP(A∗) belongs to HS(A∗) (by Jónsson’s lemma). Since A∗ has no

proper subalgebras, we have HS(A∗) =H(A∗). And it is easy to see that members of

24

H(A∗) has not proper subalgebras. Hence, subdirectly irreducible algebras in V have

no proper subalgebras, so the second condition of Theorem 2.1 is trivially satisfied.

25

3 Description via direct limits

In this section we assume that V is a locally finite, congruence-distributive variety

with the CIP. We would like to demonstrate how to use Theorem 2.1 to obtain a

description of congruence lattices of algebras in V .

Theorem 3.1. Let L be a distributive lattice with 0. The following conditions are

equivalent.

(1) L ' ConcA for some A ∈ V.

(2) L is isomorphic to the direct limit of a P -indexed diagram ~B = (Bp, ϕp,q),

where each Bp is a finite distributive lattice and each ϕp,q is a 0-preserving

lattice homomorphism, such that

(a) For every p ∈ P , the ordered set M∗(Bp) has an admissible SI(V)-valuation

(vp(α), fpα,β).

(b) For every p, q ∈ P , p ≤ q and for every α ∈ M∗(Bq) there exists an

embedding

eαp,q : vp(ϕ←p,q(α))→ vq(α) such that

eβp,qfp

α′ ,β′= f qα,βe

αp,q,

for every α ≤ β in M∗(Bq) and α′ = ϕ←p,q(α), β′= ϕ←p,q(β).

Proof. (1)⇒(2): Let L ' ConcA for some A ∈ V . Let P be the family of all finite

subsets of A ordered by set inclusion. Let Ap be the subalgebra of A generated by

p ∈ P . Since V is locally finite, every Ap is finite. For every p, q ∈ P , p ≤ q, we

put Bp = ConcAp and ϕp,q = Conc ep,q, where ep,q is the inclusion Ap → Aq. By

Theorem 2.1, every ϕp,q is 0-homomorphism of finite lattices. Then A ' lim→Ap, so

L ' ConcA ' lim→ConcAp = lim→Bp.

Moreover M∗(Bp) = M∗(ConcAp), hence we can define a map

vp : M∗(Bp)→ SI(V)

by vp(α) = Ap/α for every α ∈ M∗(Bp). Further, for every α, β ∈ M∗(Bp), α ≤ β we

define a homomorphisms

fpα,β : Ap/α→ Ap/β

as the natural projection (fpα,β([x]α) = [x]β). It is easy to see that (vp, fpα,β) is a

SI(V)-valuation on M∗(Bp). By Lemma 1.7, it is admissible.

26

Now, let p, q ∈ P , p ≤ q and let α ∈ M∗(Bq) = M∗(ConcAq). Since Ap is a

subalgebra of Aq, we know (see the remark before Lemma 1.1) that α′ = ϕ←p,q(α) =

α�Ap. We define an embedding

eαp,q : Ap/α′ → Aq/α

naturally as eαp,q([x]α′ ) = [x]α. It is easy to see that the following diagram commutes:

Ap/α′ eαp,q−−−→ Aq/α

fpα′,β′

y fqα,β

yAp/β

′ eβp,q−−−→ Aq/β

(2)⇒(1): For every p ∈ P we have aM∗(Bp)-indexed diagram ~Dp = (vp(α), fpα,β).

By Theorem 1.8, lim← ~Dp = Ap ∈ V , such that M∗(ConcAp) ' M∗(Bp).

Let p, q ∈ P , p ≤ q and let M∗(Bp) = {β1, . . . , βr}, M∗(Bq) = {γ1, . . . , γs}.We consider elements of Ap ≤

∏α∈M∗(Bp) vp(α) in the form a = (a1, . . . , ar) with

aj ∈ vp(βj) and similarly for Aq. Further we write f qi,k and fpj,l instead of f qγi,γk and

fpβj ,βl .

By Lemma 1.2, we can define a map gp,q : Ap → Aq such that

gp,q((a1, . . . , ar)) = (d1, . . . , ds),

where di = eγip,q(aj) such that βj = ϕ←p,q(γi). We have aj ∈ vp(βj) and di ∈ vq(γi). We

need to show that (d1, . . . , ds) ∈ Aq.Let γi ≤ γk, then βj = ϕ←p,q(γi) ≤ ϕ←p,q(γk) = βl. Since Ap is an inverse limit, we

have al = fpj,l(aj). Thus, by the assumption (2)(b), we have

f qi,k(di) = f qi,k(eγip,q(aj)) = eγkp,q(f

pj,l(aj)) = eγkp,q(al) = dk.

So (d1, . . . , ds) ∈ Aq, hence gp,q is well defined and it is a routine to show that gp,qis a homomorphism. Hence, ~A = (Ap, gp,q) is a directed P -indexed diagram in V .Denote A the direct limit of this diagram.

Denote by δk the k-th projection Ap → vP (βk) (k = 1, . . . , r) and by εl the l-th

projection Aq → vq(γl) (l = 1, . . . , s). By Theorem 1.8, we have Conc Ap ' Bp,

where the isomorphism hp : ConcAp → Bp can be defined by hp(ker(δk)) = βk.

Similarly, let hq be the isomorphism ConcAq → Bq defined by hq(ker(εl)) = γl.

Now we claim that the following diagram commutes.

27

ConcApConc gp,q−−−−−→ ConcAq

hp

y hq

yBp

ϕp,q−−−→ Bq

Since hp, hq are isomorphisms, we have h←p = h−1p , h←q = h−1

p . By Lemma 1.1, we

can prove equivalently that h←p ϕ←p,q = (Conc gp,q)←h←q . All these maps preserve ∧, it

suffices to show that h←p ϕ←p,q(γi) = (Conc gp,q)←h←q (γi) for every γi ∈ M∗(Bq).

Let ϕ←p,q(γi) = βj. Then h←p ϕ←p,q(γi) = ker(δj). Further, h←q (γi) = ker(εi) and

(x, y) ∈ (Conc gp,q)←(ker(εi))⇔ (gp,q(x), gp,q(y)) ∈ ker(εi)⇔

⇔ gp,q(x)i = gp,q(y)i ⇔ eγip,q(xj) = eγip,q(yj)⇔ xj = yj ⇔ (x, y) ∈ ker(δj),

so

(Conc gp,q)←h←q (γi) = ker(δj) = h←p ϕ

←p,q(γi).

This proves that our diagram commutes. Using this commutativity and the fact

that the fuctor Conc preserves direct limits, we have

Conc A = Conc lim→

~A ' lim→

Conc~A ' lim

→~B ' L.

In concrete cases, the general description of the direct limit system in Theorem

3.1 can be specified more closely, which sometimes leads to a nice description of the

class ConV .

3.1 Special cases

Let V be a locally finite, congruence-distributive variety with the CIP. Moreover,

assume that ConS is a chain for every S ∈ SI(V). We denote

Si = {A ∈ SI(V) | ConA is an i-element chain}.

Further, denote Pn the class of all partially ordered sets (C,≤) with the largest

element, such that for every x ∈ C, ↑x is a k-element chain, k ∈ {1, . . . , n}. HenceC ∈ Pn is a disjoint union of an antichains C0, . . . , Cn−1 such that |↑x| = k + 1 for

x ∈ Ck. If L is a lattice such that M∗(L) ∈ Pn, then denote Mk = Mk(L) = (M∗(L))k

for k = 0, . . . , n− 1. Notice that C0 is a one-element set (C0 = {1}).We present a detailed analysis of four special cases.

The first case. We suppose that V satisfies the following additional assumption:

28

(A1) max{j | Sj 6= ∅} = 2.

(A2) If A ≤ B ∈ SI(V), then ConA ' ConB.

As an example of such a variety one can consider the variety of all bounded dis-

tributive lattices.

Theorem 3.2. Let V satisfy the assumptions stated above. Let L be a distributive

lattice with 0. The following conditions are equivalent.


(2) L is isomorphic to the direct limit of a P -indexed diagram ~B = (Bp, ϕp,q), where

each Bp is a finite Boolean lattice and each ϕp,q is a Boolean homomorphism.

(3) L is a Boolean lattice.

Proof. (1)⇒(2): By Theorem 3.1, there exists a P -indexed diagram ~B = (Bp, ϕp,q),

such that L is isomorphic to the direct limit of ~B. Moreover, by Theorem 3.1, for

every p ∈ P , Bp is a finite distributive lattice such that M∗(Bp) has an admissible

SI(V)-valuation (vp(α), fpα,β). By Lemma 1.5, ↑α is isomorphic to Con vp(α) which is

a chain of length at most 1, thus M∗(Bp) ∈ P2 and so Bp is a finite Boolean lattice.

Now, let q ∈ P such that p ≤ q. By Theorem 3.1, ϕp,q is a 0-preserving lattice

homomorphism. Suppose that ϕp,q(1) < 1, then ϕp,q(1) ≤ γ for some γ ∈ M(Bq).

Hence by Lemma 1.1, ϕ←p,q(γ) = 1. Further, by the definition of SI(V)-valuation,

vp(1) ' 1 and vq(γ) ' F for some F ∈ S2. Since γ ∈ M∗(Bq), by Theorem 3.1, there

exists an embedding eγp,q : 1 → F , contradicting our assumption (A2). Therefore,

ϕp,q is a lattice homomorphism, which preserves 0 and 1. It is well known that

such a homomorphism must also preserve the complements, so ϕp,q is a Boolean

homomorphism.

(2)⇒(3): L is the direct limit of a Boolean lattices and all ϕp,q are Boolean

homomorphisms, thus L is a Boolean lattice.

(3)⇒(2): Every Boolean lattice is the direct limit of its finite Boolean sublattices

(with inclusions as homomorphisms).

(2)⇒(1): Let p ∈ P . Since Bp is a finite Boolean lattice, we have M∗(Bp) ∈ P2.

Choose F ∈ S2 and for every α ∈ M∗(Bp) set

vp(α) = 1 if α ∈ M0,

vp(α) = F if α ∈ M1 .

29

Let α ∈ M(Bp) and let β ∈ M0(Bp), we define fα,β as the unique constant map

F → 1. (and, of course, fα,α as the identity for every α ∈ M∗(Bp).) It is easy to see

that (vp(α), fα,β) is a SI(V)-valuation on M∗(Bp). By Lemma 1.7, it is admissible.

Let p, q ∈ P, p ≤ q and let α ∈ M(Bq). Since ϕp,q is a 0, 1-preserving homo-

morphism of finite distributive lattices, by Lemma 1.3, ϕ←(α) ∈ M(Bp). So we can

define an embedding eαp,q : vp(ϕ←(α)) → vq(α) as the identity F → F for every

α ∈ M(Bq) and the identity 1→ 1 for α = 1. Hence, by Theorem 3.1, L ' ConcA

for some A ∈ V .

The second case. Similarly as in the first special case, we assume that V is

a locally finite, congruence-distributive variety with the CIP and ConA is a chain

for every A ∈ SI(V). Instead of (A1), (A2) we consider the following additional

assumptions:

(B1) max{j | Sj 6= ∅} = 2.

(B2) There exists F0 ∈ S2 such that 1 ≤ F0.

Denote f0 an embedding 1→ F0. As an example of such a variety one can consider

the variety of all distributive lattices.

We prove a similar result as in the first case. Recall that a generalized Boolean

lattice B is a distributive lattice with the least element 0 such that for any b ∈ B,

the interval [0, b] is a Boolean lattice.





where each Bp is a finite Boolean lattice and each ϕp,q is a 0-preserving lattice

homomorphism.

(3) L is a generalized Boolean lattice.

Proof. (1)⇒(2): The same as in Theorem 3.2 except that we do not prove ϕp,q(1) = 1.

(2)⇒(3): It is easy to check that the direct limit of a system of generalized Boolean

lattices and 0-preserving lattice homomorphisms is a generalized Boolean lattice.

30

(3)⇒(2): Let B be a generalized Boolean lattice. For every finite G ⊆ B let BG be

the Boolean sublattice of the interval 〈0,∨G〉 generated by G. It is easy to see that

B is the direct limit of the system of all BG with the inclusions as homomorphisms.

(2)⇒(1): We proceed similarly as in Theorem 3.2. Let p ∈ P . Since Bp is a finite

Boolean lattice, we have M∗(Bp) ∈ P2. For every α ∈ M∗(Bp) set

vp(α) = 1 if α ∈ M0,

vp(α) = F0 if α ∈ M1 .

Let α ∈ M(Bp) and let β ∈ M0(Bp), we define fα,β as the unique constant map

F0 → 1. (and, of course, fα,α as the identity for every α ∈ M∗(Bp).) It is easy to see

that (vp(α), fα,β) is a SI(V)-valuation on M∗(Bp). By Lemma 1.7, it is admissible.

Further, let p, q ∈ P, p ≤ q and let α ∈ M(Bq). Since ϕp,q is a 0-preserving

homomorphism of finite distributive lattices, by Lemma 1.2, ϕ←(α) ∈ M∗(Bp). So

we can define an embedding eαp,q : vp(ϕ←(α)) → vq(α) as the identity F0 → F0 or

eαp,q = f0 for every α ∈ M(Bp). (and the identity 1→ 1 for every α = 1.) Hence, by

Theorem 3.1, L ' ConcA for some A ∈ V .

The third case. Similarly as in the first special case, we assume that V is a

locally finite, congruence-distributive variety with the CIP and ConA is a chain


assumptions:

(C1) max{j | Sj 6= ∅} = 3.

(C2) If A ≤ B ∈ SI(V), then ConA ' ConB.

As an example of such a variety one can consider the variety of all principal Stone

algebras. It is the variety generated by the algebra ({0, e, 1},∨,∧, ∗, 0, e, 1), where

0 < e < 1 and ∗ denotes the pseudocomplementation.

For the study of this case we need to recall some basic facts about dual Stone

lattices. A bounded lattice is called dually pseudocomplemented if for every x ∈ Lthere exists its dual pseudocomplement x+ = min{y ∈ L | x∨ y = 1}. The elements

satisfying x+ = 1 are called co-dense and form an ideal of L denoted by D̄(L). A

dual Stone lattice is a distributive dually pseudocomplemented lattice satisfying the

identity x+ ∧ x++ = 0. In a dual Stone lattice L, the set S(L) = {x+ | x ∈ L} is aBoolean subalgebra and is called the skeleton of L.

31

It is easy to see that every finite distributive lattice is a dually pseudocomple-

mented and its largest co-dense element is the meet of its all coatoms.

Lemma 3.4. Let B1, B2 be a finite dual Stone lattices with largest co-dense elements

d1 and d2 respectively. Let ϕ be a 0, 1-preserving lattice homomorphism with ϕ(d1) =

d2. Then ϕ preserves the dual pseudocomplements.

Proof. Every x ∈ B1 satisfies the equality x = x++ ∨ (x ∧ d1). Hence

ϕ(x)+ = (ϕ(x++) ∨ (ϕ(x) ∧ d2))+ = ϕ(x++)+ ∧ (ϕ(x)+ ∨ 1) = ϕ(x++)+.

Since the restriction of ϕ to S(B1) is a homomorphism of Boolean algebras and

x++ is a complement of x+, we obtain that ϕ(x++) is the complement of ϕ(x+), so

ϕ(x++)+ = ϕ(x+).

Lemma 3.5. Let V satisfy the assumptions stated above. For every finite distributive

lattice L the following conditions are equivalent.

(1) L ' ConA for some A ∈ V.

(2) M∗(L) ∈ P3.

(3) L is a dual Stone lattice and its co-dense elements form a Boolean lattice.

Proof. (1)⇔(2): This equivalence follows from [13], Theorem 8.

(2)⇔(3): Equivalence was proved in [25] Theorem 4.5 (in a dual form).

Lemma 3.6. Let B1, B2 be a finite distributive lattices such that M∗(Bi) ∈ P3

(i = 1, 2). Let ϕ : B1 → B2 be a 0, 1-preserving lattice homomorphism. The following

conditions are equivalent.

(1) For every c ∈ M(B2), ϕ←(c) ∈M1(B1) if and only if c ∈ M1(B2).

(2) ϕ preserves the largest co-dense element.

Proof. Denote di =∧

M1(Bi), the largest co-dense element of Bi (i=1,2). Clearly,

by distributivity and meet-irreducibility, d1 ≤ b ∈ M(B1) if and only if b ∈ M1(B1).

(1)⇒(2): Since every element in B2 is a meet of meet-irreducible elements above

it, we have

ϕ(d1) =∧{c ∈ M(B2) | ϕ(d1) ≤ c}.

32

Now, ϕ(d1) ≤ c is equivalent to d1 ≤ ϕ←(c) and hence to ϕ←(c) ∈ M1(B1). By (1),

this is equivalent to c ∈ M1(B2), hence

ϕ(d1) =∧{c ∈ M(B2) | c ∈ M1(B2)} = d2.

So ϕ preserves the largest co-dense element.

(2)⇒(1): Let c ∈ M(B2). Then c ∈ M1(B2) if and only if

c ≥ d2 = ϕ(d1) =∧{ϕ(b) | b ∈ M1(B1)}.

Since c is meet-irreducible, this is equivalent to c ≥ ϕ(b) for some b ∈ M1(B1), hence

to ϕ←(c) ≥ b, which is only possible if ϕ←(c) = b.

Lemma 3.7. Let L be a dual Stone lattice and let its co-dense elements form a

Boolean lattice. For every finite set Y ⊆ L there exists a finite dual Stone lattice

LY such that Y ⊆ LY and LY is a sublattice of L, containing 0, 1 and the largest

co-dense element.

Proof. By the triple representation of Stone algebras and its simplified version due to

Katriňák [24] (see also [1]) we can assume that there exist Boolean latticeB, bounded

distributive lattice D and (0, 1)-preserving lattice homomorphism h : B → D such

that

L = {(b, d) ∈ B ×D | h(b) ≤ d},

with the lattice operations defined componentwise and (b, d)+ = (b′, h(b′)), where b′

denotes the complement of b in B. Moreover S(L) = {(b, h(b)) | b ∈ B} is isomorphic

to B and D̄(L) = {(0, d) | d ∈ D}. Hence, the largest co-dense element of L is (0, 1).

Since D̄(L) is isomorphic to D, we have D a Boolean lattice.

Now, let Y be a finite subset of L. Let X be a finite sublattice of L generated by

Y . Let BY be the Boolean sublattice of B generated by

{b ∈ B | (b, d) ∈ X for some d ∈ D}.

Further, let DY be the Boolean sublattice of D generated by

{d ∈ D | (b, d) ∈ X for some b ∈ B} ∪ {h(b) | b ∈ BY }.

Clearly, BY and DY are finite. Denote

LY = {(b, d) ∈ BY ×DY | h�BY (b) ≤ d}.

33

Clearly, h�BY : BY → DY is a (0, 1)-preserving lattice homomorphism. Then LY is

a finite dual Stone lattice such that Y ≤ LY . It is easy to see that LY is a sublattice

of L containing 0, 1 and the largest co-dense element.





where each Bp is a finite distributive lattice with M∗(Bp) ∈ P3 and each ϕp,q is

a 0, 1-lattice homomorphism, preserving the largest co-dense element.

(3) L is a dual Stone lattice and its co-dense elements form a Boolean lattice.

Proof. (1)⇒(2): By Theorem 3.1, there exists a P -indexed diagram ~B = (Bp, ϕp,q)



SI(V)-valuation (vp(α), fpα,β). By Lemma 1.5, ↑α is isomorphic to Con vp(α) which

is a chain of length at most 2, thus M∗(Bp) ∈ P3.

Now, let q ∈ P such that p ≤ q. By Theorem 3.1, ϕp,q is a 0-preserving lat-

tice homomorphism. Let γ ∈ M∗(Bq). By Theorem 3.1, there exists an embedding

vp(ϕ←p,q(γ)) → vq(γ). By assumption (C2) Con vp(ϕ

←p,q(γ)) ' Con vq(γ). Hence, for

every γ ∈ M∗(Bq), ϕ←p,q(γ) ∈ Mi(Bp) iff γ ∈ Mi(Bq) (i ∈ {0, 1, 2}). Hence, by Lemma

3.6, ϕp,q preserves the largest co-dense element.

Further, suppose that ϕp,q(1) < 1, then ϕp,q(1) ≤ γ for some γ ∈ M1(Bq). Hence

by Lemma 1.1, ϕ←p,q(γ) = 1. Further, by the definition of SI(V)-valuation, vp(1) ' 1

and vq(γ) ' F for some F ∈ S2. Since γ ∈ M∗(Bq), by Theorem 3.1, there exists

an embedding eγp,q : 1→ F , contradicting our assumption (C2). Therefore, ϕp,q is a

0, 1-lattice homomorphism, which preserves 0 and 1, preserving the largest co-dense

element.

(2)⇒(3): By Lemma 3.5, Bp is a dual Stone lattice and its co-dense elements form

a Boolean lattice for every p ∈ P . By Lemma 3.4, ϕp,q preserves the dual pseudo-

complements. Since every ϕp,q is a 0, 1-lattice homomorphism of dual Stone lattices

(preserving ∨,∧,+, 0, 1) and dual Stone lattices form a variety, the direct limit L is

34

a dual Stone lattice. Moreover, restriction of ϕp,q to D̄(Bp) is a homomorphism of a

Boolean lattices, so D̄(L) is a Boolean lattice (the limit of Boolean lattices D̄(Bp)).

(3)⇒(2): Let P be the family of all finite subsets of L ordered by set inclusion.

Using Lemma 3.7 we can see, that L is the direct limit of the P-indexed system

(LX , ϕX,Y ), where ϕX,Y is the set inclusion. As LX is a finite dual Stone lattice such

that LX is a sublattice of L, containing 0, 1 and the largest co-dense element. Hence,

ϕX,Y has the required properties.

(2)⇒(1): We proceed similarly as in Theorem 3.2. Choose F ∈ S3, so ConF is a

3-element chain α2 < α1 < α0. For every i ∈ {0, 1, 2} denote Fi = F/αi, so ConFi

is an i+ 1-element chain. Let p ∈ P , for every α ∈ M∗(Bp) set

vp(α) = F0 = 1 if α ∈ M0,

vp(α) = F1 if α ∈ M1,

vp(α) = F2 = F if α ∈ M2 .

For every j ≤ i we define a map gi,j : Fi → Fj as the natural projection and for

every γ ∈ M∗(Bp) denote i(γ) = |↑γ| − 1. Further, for every α, β ∈ M∗(Bp) such

that α < β we define fα,β = gi(α),i(β). (and, of course, fα,α is the identity for every

α ∈ M∗(Bp).) It is easy to see that (vp(α), fα,β) is a SI(V)-valuation on M∗(Bp). By

Lemma 1.7, it is admissible.

Further, let p, q ∈ P (p ≤ q) and let α ∈ M∗(Bq). Let i ∈ {0, 1, 2}, by Lemma

3.6, we have ϕ←(α) ∈ Mi(Bp) iff α ∈ Mi(Bp). So we can define an embedding

eαp,q : vp(ϕ←(α)) → vq(α) as the identity Fi → Fi for every α ∈ Mi(Bq). Hence, by


The fourth case. Similarly as in the first special case, we assume that V is

a locally finite, congruence-distributive variety with the CIP and ConA is a chain


assumptions:

(D1) max{j | Sj 6= ∅} = n > 1.

(D2) If A ≤ B ∈ SI(V), then ConA ' ConB.

Recall [25] or [20] for definition of Stone algebra of order ≤ n (n ≥ 2). By

result of the paper [25], for given n (n ≥ 2), Stone algebras of order ≤ n form a

variety V generated by a chain 0 = e0 ≤ e1 ≤ · · · ≤ en−1 = 1. Further, let L∗ =

35

{0 = a0 < a1 < · · · < am−1 = 1} (1 ≤ m ≤ n) be an m-element chain. Obviously, L∗

is a Stone algebra of order ≤ n if one puts ei = ai for i = 0, . . . ,m−1 and ei = 1 for

i = m, . . . , n − 1. This algebra is usually denoted by Sn(m). The only subdirectly

irreducible algebras of order ≤ n are (up to isomorphism) the algebras Sn(m) where

1 ≤ m ≤ n.

So V is finitely generated with the CIP. Hence, as an example of a locally finite,

congruence-distributive variety with the CIP such that satisfied (D1) and (D2) one

can consider the variety of all Stone algebras of order ≤ n.

In this case we generalize results of previous cases. For this we need recall the

definition of dual Stone lattices of order n (see [25] or [1]):

Let L be a dual Stone lattice. L is a dual Stone lattice of order 1 if L is a trivial

one-element lattice. L is a dual Stone lattice of order n (n ≥ 2), if D̄(L) is a dual

Stone lattice of order n− 1. Further, we denote

D̄0(L) = L,

D̄i(L) = D̄(D̄i−1(L)), i = 1, . . . , n− 1.

We shall denote the largest element of D̄i(L) by di (i = 0, 1, . . . , n − 1). The chain

1 = d0 ≥ d1 ≥ · · · ≥ dn−1 = 0 is said to be the chain of largest co-dense elements of

L. It is clear that 1 = d0 > d1 > · · · > dk−1 = dk = · · · = dn−1 = 0 for some k ∈ N.

Lemma 3.9. Let L be a finite distributive lattice and let b ∈ L. Then the map

ϕ : {x ∈ M(L) | b � x} → M(↓ b)

defined by ϕ(x) = x ∧ b is an isomorphism of ordered sets.

Proof. Let x ∈ M(L), b � x and let x ∧ b = b1 ∧ b2, for some b1, b2 ∈ ↓b. Then by

distributivity

x = x ∨ (x ∧ b) = x ∨ (b1 ∧ b2) = (x ∨ b1) ∧ (x ∨ b2).

Since x is a meet irreducible, we have x ≥ bi for some i ∈ {1, 2}, hence x ∧ b = bi.

So ϕ is well defined. Furthermore if x ∧ b ≤ y ∧ b ≤ y, with x, y ∈ M(L) \ ↑b, thenx ≤ y. Hence ϕ is an embedding.

Further, for every u ∈ L denote M(u) = M(L)∩ ↑u, hence u =∧{x | x ∈ M(u)}.

Let u ∈ M(↓ b). Since u ≤ b, we have u =∧{b ∧ x | x ∈ M(u)}, so there exists

x ∈ M(L) such that u = b ∧ x. Since u 6= b, we have b � x. (If b < x, then

u = b ∧ x = b, contradicting u < b.) Hence, ϕ is surjective.

36

Lemma 3.10. Let V satisfy the assumptions stated above. For every finite distribu-

tive lattice L the following conditions are equivalent.

(1) L ' ConA for some A ∈ V.

(2) M∗(L) ∈ Pn.

(3) L is a dual Stone lattice of order n.

Proof. (1)⇔ (2): Equivalence was proved in [13] Theorem 8.

(2)⇔ (3): Equivalence was proved in [25] Theorem 4.5 (In a dual form).

Lemma 3.11. Let L be a finite dual Stone lattice of order n. Then

di =i∧

k=0

(∧

Mk(L)),

for every i = 0, 1, . . . , n− 1.

Proof. We proceed by induction on i.

It is easy to see that 1 = d0 =∧M0(L). Further, it is easy to see that a ∈ D̄1(L) if

and only if a ≤ x, for every x ∈ M1(L). Hence d1 =∧

M1(L) =∧

M1(L)∧∧

M0(L).

Now, assume that

di =i∧

k=0

(∧

Mk(L)),

for some i ∈ {1, . . . , n− 2}.Let y be a maximal element of {x ∈ M(L) | di � x}. By Lemma 3.9, if x ∈M1(↓

di) = M1(D̄i(L)), then x = di ∧ y. So y ∈Mi+1(L) and by induction assumption

di+1 =∧

M1(D̄i(L)) =∧

Mi+1(L) ∧ di =∧

Mi+1(L) ∧ · · · ∧∧

M0(L).

Lemma 3.12. Let ϕ : B1 → B2 be a lattice homomorphism of finite dual Stone

lattices of order n. The following conditions are equivalent.

(1) For every i ∈ 0, 1, . . . , n− 1 and for every c ∈M∗(B2), ϕ←(c) ∈Mi(B1) if and

only if c ∈Mi(B2).

(2) ϕ preserves the chain of largest co-dense elements.

37

Proof. (1)⇒(2): For every h ∈ {0, 1, . . . , n− 1} denote d1h, d2

h the largest co-dense

elements of D̄h(B1), D̄h(B2).

Let i ∈ {0, 1, . . . , n− 1}, by Lemma 1.1,

ϕ(d1i ) =

∧{c ∈M∗(B2) | d1

i ≤ ϕ←(c)}.

Further, by Lemma 3.11, d1i =

∧ik=0(

∧Mk(B1)). For every c ∈ M∗(B2) with

d1i ≤ ϕ←(c) we have d1

i =∧ik=0(

∧Mk(B1)) ≤ ϕ←(c) ∈ M∗(B1). Hence, ϕ←(c) ≥ u

for some u ∈⋃ik=0 Mk(B1). So ϕ←(c) ∈ Mj(B1) for some j = 0, 1, . . . , i. By (1),

c ∈ Mj(B2). Hence ϕ(d1i ) =

∧ik=0(

∧Mk(B2)) = d2

i . So ϕ preserves the chain of

largest co-dense elements.

(2)⇒(1): Let i ∈ {0, 1, . . . , n− 1}. Denote M′

i(L) =⋃ik=0 Mk(L). Let c ∈ M(B2).

c ∈ M′

i(B2)⇔ c ≥ d2i = ϕ(d1

i ) =∧{ϕ(b) | b ∈ M

′

i(B1)}

⇔ c ≥ ϕ(b) for some b ∈ M′

i(B1)

⇔ ϕ←(c) ≥ b for some b ∈ M′

i(B1)

⇔ ϕ←(c) ∈ M′

i(B1).

Hence

c ∈ M0(B2) = M′

0(B2)⇔ ϕ←(c) ∈ M′

0(B1) = M0(B1)

and for every i ∈ {1, . . . , n− 1}:

c ∈ Mi(B2)⇔ c ∈ M′

i(B2) \M′

i−1(B2)

⇔ ϕ←(c) ∈ M′

i(B1) \M′

i−1(B1) = Mi(B1).

Lemma 3.13. Let ϕ : B1 → B2 be a lattice homomorphism of finite dual Stone

lattices of order n. If ϕ preserves the chain of largest co-dense elements, then ϕ

preserves the dual pseudocomplements in every D̄i(B1) (i ∈ {0, 1, . . . , n− 1}).

Proof. For i = 0 it is Lemma 3.4, for others i it is a simple consequence.

Lemma 3.14. Let L be a dual Stone lattice of order n. For every finite set Y ⊆ L

there exists a finite dual Stone lattice LY of order n such that Y ⊆ LY and LY is a

sublattice of L, containing the chain of largest co-dense elements.

38

Proof. By [24] (see also [1]) a dual Stone lattice L is of order n if and only if there is

a finite sequence of Boolean lattices B1, B2, . . . , Bn−1 and Boolean homomorphisms

hk : Bk → Bk+1 (1 ≤ k < n− 1), such that, up to isomorphism,

L = {(b1, . . . , bn−1) ∈ B1 × · · · ×Bn−1 | h1(b1) ≤ b2, . . . , hn−2(bn−2) ≤ bn−1}.

Moreover, for every i = 1, . . . , n− 1:

D̄i(L) = {(0, . . . , 0, bi+1, . . . , bn−1) ∈ L | bi+1 ∈ Bi+1, . . . , bn−1 ∈ Bn−1}.

So the chain of largest co-dense elements of L is (1, 1, . . . , 1), (0, 1, . . . , 1), . . . ,

(0, 0, . . . , 0). Further, let Y be a finite subset of L and let X be the finite sublattice

of L generated by Y . Let BY1 be a Boolean sublattice of B1 generated by

{β ∈ B1 | (β, b2, . . . , bn−1) ∈ X for some b2 ∈ B2, . . . , bn−1 ∈ Bn−1}.

For every k ∈ {2, . . . , n− 1} let BYk be a Boolean sublattice of Bk generated by

{β ∈ Bk | (b1 . . . , bk−1, β, bk+1, . . . , bn−1) ∈ X for some

b1 ∈ B1, . . . , bn−1 ∈ Bn−1} ∪ hk−1(BYk−1)

Clearly, BYk is finite for every k ∈ {1, . . . , n− 1}.

Further, for every i ∈ {1, . . . , n− 2} denote hYi = hi�BYi . Clearly, hYi : BY

i →BYi+1 is a Boolean homomorphism. Denote

LY = {(b1, . . . , bn−1) ∈ BY1 ×BY

n−1 | hY1 (b1) ≤ b2, . . . , hYn−2(bn−2) ≤ bn−1}.

Then LY is a finite dual Stone lattice of order n such that Y ⊆ LY . It is easy to

see that LY is a sublattice of L containing the chain of largest co-dense elements of

L. (Note that if Bi is a one-element for some i, then Bj is a one-element for every

i ≤ j ≤ n− 1 and 0, 1 coincide in every such Bj.)





where each Bp is a finite distributive lattice with M∗(Bp) ∈ Pn and each ϕp,q is

a lattice homomorphism, preserving the chain of the largest co-dense elements.


39

Proof. (1)⇒(2): By Theorem 3.1, there exists a P -indexed diagram ~B = (Bp, ϕp,q),



SI(V)-valuation (vp(α), fpα,β). By Lemma 1.5, ↑α is isomorphic to Con vp(α) which

is a chain of length at most n− 1, thus M∗(Bp) ∈ Pn.Now, let q ∈ P such that p ≤ q. By Theorem 3.1, ϕp,q is a 0-preserving lat-

tice homomorphism. Let γ ∈ M∗(Bq). By Theorem 3.1, there exists an embedding

vp(ϕ←p,q(γ)) → vq(γ). By assumption (D2) Con vp(ϕ

←p,q(γ)) ' Con vq(γ), hence for

every γ ∈ M(Bq), ϕ←p,q(γ) ∈ Mi(Bp) iff γ ∈ Mi(Bq), i ∈ {0, . . . , n− 1}. Thus, byLemma 3.12, ϕp,q preserves the chain of largest co-dense elements.

(2)⇒(3): By induction we prove that the direct limit of a diagram satisfying (2) is

a dual Stone lattice of order n. The cases n = 2 and n = 3 were shown in Theorems

3.2 and 3.8.

Let n > 3. By Lemma 3.10, Bp is a dual Stone lattice of order n for every p ∈ P .By Lemma 3.13, ϕp,q preserves the dual pseudocomplements. Since every ϕp,q is

a homomorphism of dual Stone lattices (preserving ∨,∧,+, 0, 1) and dual Stone

lattices form a variety, the direct limit L is a dual Stone lattice.

Further, D̄(L) is the limit of the system (D̄(Bp), ϕp,q�D̄(Bp)). Each D̄(Bp) is a

dual Stone lattice of order n − 1 and each ϕp,q�D̄(Bp) is a lattice homomorphism

preserving the chain of the largest co-dense elements. By the induction hypothesis,

D̄(L) is a dual Stone lattice of order n− 1.

(3)⇒(2): Let P be the family of all finite subsets of L ordered by set inclusion.

Using Lemma 3.14 we can see, that L is the direct limit of the P -indexed system

(LX , ϕX,Y ), where ϕX,Y is the set inclusion. As LX is a finite dual Stone lattice of

order n such that LX is a sublattice of L, containing the chain of largest co-dense

elements. Hence, ϕX,Y has the required properties.

(2)⇒(1): We proceed similarly as in Theorem 3.2. Choose F ∈ Sn, so ConF

is a n-element chain α0 > α1 > · · · > αn−1. For every i ∈ {0, . . . , n− 1} denote

Fi = F/αi, so Conc Fi is i+ 1-element chain. Let p ∈ P , for every α ∈ M∗(Bp) set

vp(α) = Fi if α ∈ Mi .

For every j ≤ i we define a map gi,j : Fi → Fj as the natural projection and for

every γ ∈ M∗(Bp) denote i(γ) = |↑γ| − 1. Further, for every α, β ∈ M∗(Bp) such

that α < β we define fα,β = gi(α),i(β). (and, of course, fα,α is the identity for every

40

α ∈ M∗(Bp).) It is easy to see that (vp(α), fα,β) is a SI(V)-valuation on M∗(Bp). By

Lemma 1.7, it is admissible.

Further, let p, q ∈ P, p ≤ q and let α ∈ M∗(Bq). Let i ∈ {0, 1, . . . , n− 1}, byLemma 3.12, ϕ←(α) ∈ Mi(Bp) iff α ∈ Mi(Bp). So we can define an embedding

eαp,q : vp(ϕ←(α)) → vq(α) as the identity Fi → Fi for every α ∈ Mi(Bq). Hence, by


In many cases the description provided by Theorem 3.1 is not quite satisfactory.

That’s why we try another approach.

41

4 Description via Priestley duality

Let V be a finitely generated, congruence-distributive variety with the CIP. Hence

Conc A is a distributive lattice with 0 for every A ∈ V . So it is natural to describe

these lattices by means of Priestley duality.

Let L be a distributive lattice with 0 and let (P(L),≤, τ) be its dual Priestley

space. Consider the following conditions on (P(L),≤, τ):

(Pr1) P(L) has an admissible SI(V)-valuation (v(I), fI,J).

(Pr2) For every I ∈ P(L) there exists an open set U such that I ∈ U and for every

J ∈ U the algebra v(I) ≤ v(J).

Theorem 4.1. If L ' ConcA for some A ∈ V, then the dual Priestley space

(P(L),≤, τ) satisfies (Pr1) and (Pr2).

Proof. Let L = ConcA for some A ∈ V . By Lemma 1.13, we have sup I ∈ M∗(ConA)

for every I ∈ P(L). So we can define a map v : P(L)→ SI(V) such that

v(I) = A/ sup I.

Since for every I, J ∈ P(L) (I ≤ J) we have sup I ≤ sup J , we can define a surjective

homomorphism

fI,J : A/ sup I → A/ sup J

as natural projection fI,J([x]sup I) = [x]sup J . It is easy to see that (v(I), fI,J) is a

SI(V)-valuation on P(L). The admissibility follows from Lemma 1.7.

To prove (Pr2), let I ∈ P(L). Since the quotient algebra A/ sup I is finite, there

are n ∈ N and x1, . . . , xn ∈ A such that for every y ∈ A there exists i ∈ {1, . . . , n}with xi ∈ [y]sup I .

Let B be the subalgebra of A generated by x1, . . . , xn. Hence B is a finite and

B/ sup I�B is isomorphic to A/ sup I. Denote by U the intersection⋂x,y∈B,Θ(x,y)∈I

{J ∈ P(L) | Θ(x, y) ∈ J} ∩⋂

x,y∈B,Θ(x,y)/∈I

{J ∈ P(L) | Θ(x, y) /∈ J}.

Since U is an intersection of finitely many clopen sets, it is a clopen set. Moreover,

it is easy to see that I ∈ U . For every J ∈ U , we have sup I�B = sup J�B. Indeed,

42

the compactness of Θ(x, y) implies that Θ(x, y) ≤ sup I iff Θ(x, y) ∈ I. Hence

sup I�B = {(x, y) ∈ B2 | (x, y) ∈ sup I} =

= {(x, y) ∈ B2 | Θ(x, y) ≤ sup I} =

= {(x, y) ∈ B2 | Θ(x, y) ∈ I} =

= {(x, y) ∈ B2 | Θ(x, y) ∈ J} = sup J�B.

So, v(J) = A/ sup J ≥ B/ sup J�B = B/ sup I�B ' A/ sup I = v(I).

Now, we prove that the converse to Theorem 4.1 does not hold in general. First,

let A be a four element chain 0 < a < b < 1, let ∗ be a unary operation defined on

A such that 0∗ = 0, 1∗ = 1, a∗ = b, b∗ = a. Further we set 0, 1 as constants (nullary

operations) and consider the algebraic structure (A,∨,∧, ∗, 0, 1). So the only proper

subalgebra of A is 2 = {0, 1}. Let V be a variety generated by A. Since

ConA = {∆,∇, λ = {(ab)(0)(1)}, φ = {(ab0)(1)}, ψ = {(ab1)(0)}},

the algebras A,2 and 1 are the only (up to isomorphism) subdirectly irreducible

members in V . Hence V is a finitely generated, congruence-distributive variety with

the CIP. Denote f0, f1 the two homomorphism A→ 2. Concretely

f0(0) = 0, f0(a) = 0, f0(b) = 0, f0(1) = 1,

f1(0) = 0, f1(a) = 1, f1(b) = 1, f1(1) = 1.

(Notice that f0 and f1 are comparable (f0 ≤ f1) in the pointwise ordering of func-

tions.)

Further, denote by P the class of all partially ordered sets (P,≤) with the largest

element, such that P ∈ P is a disjoint union of a three antichains P0, P1, P2 such

that P0 = {1}, ↑x = {x, 1} for every x ∈ P1 is a two element chain x < 1 and for

every x ∈ P2 there exist exactly two element y, z ∈ P1 such that y, z > x. Let L be

a lattice such that P(L) ∈ P , then denote Pk = (P(L))k for k = 0, 1, 2.

Lemma 4.2. Let V be a variety generated by A. Let L be a distributive lattice

with 0 and let (P(L),≤, τ) be its dual Priestley space. The following conditions are

equivalent.

(1) (P(L),≤, τ) satisfies (Pr1) and (Pr2).

(2) P(L) ∈ P, P0 is clopen and P2 is open.

43

Proof. (1)⇒ (2): By (Pr1), P(L) has an admissible SI(V)-valuation (v(I), fI,J). By

Lemma 1.5, ↑I is isomorphic to Con v(I), so P(L) ∈ P . Moreover, v(I) ' A for

I ∈ P2, v(I) ' 2 for I ∈ P1 and v(1) = 1. Let i ∈ {0, 2} and let I ∈ Pi(L), by (Pr2)

there exists an open set U with I ∈ U and for every J ∈ U we have v(I) ≤ v(J).

It is easy to see that 1 and A are not isomorphic to a proper subalgebra of any

B ∈ SI(V). Thus J ∈ Pi(L). Hence U ⊆ Pi(L), so Pi(L) is open. Since P0(L) is

one-point, it is also closed.

(2)⇒ (1): For every I ∈ P(L) denote

v(I) = A if I ∈ P2,

v(I) = 2 if I ∈ P1,

v(I) = 1 if I ∈ P0.

Let I ∈ P2, J1, J2 ∈ P1 such that J1 6= J2, I ≤ J1, J2. (We choose arbitrarily a

denotation J1, J2.) We set fI,J1 = f0 and fI,J2 = f1. Let I ∈ P1 ∪ P2 and J ∈ P0, we

define fI,J as the unique constant map v(I)→ 1. (and, of course, fI,I as the identity

for every I ∈ P(L).) Hence, it is easy to check that (v(I), fI,J) is an admissible

SI(V)-valuation on P(L).

To prove (Pr2) we set

U = P0 if I ∈ P0,

U = P1 ∪ P2 if I ∈ P1,

U = P2 if I ∈ P2.

It is easy to see that U is open, I ∈ U and for every J ∈ U we have v(I) ≤ v(J).

Now, we construct a Priestley space P satisfies the condition (2) of Lemma 4.2.

Let X be any set, denote P = {0, 1, c1, c2}X ∪ {∞}. For every x ∈ X we define

Ax = {f ∈ P \ {∞} | f(x) = 0}, Bx = {f ∈ P \ {∞} | f(x) = 1},

Cx = {f ∈ P \ {∞} | f(x) = c1}, Dx = {f ∈ P \ {∞} | f(x) = c2}.

We endow P with topology τ generated by all sets of the form Ax, Bx, Cx, Dx

and {∞}. So ∞ is a discrete point and the topology on {0, 1, c1, c2}X is the usual

product topology. (with the discrete topology on {0, 1, c1, c2}.)Further, define an order ≤ on P by

44

(i) f ≤ ∞ for every f ∈ P .

(ii) For f, g ∈ P \ {∞}, f ≤ g iff f = g or g = h0f or g = h1f ,

where h0, h1 : {0, 1, c1, c2} → {0, 1} are maps defined by

h0(0) = 0, h0(c1) = 0, h0(c2) = 1, h0(1) = 1,

h1(0) = 0, h1(c1) = 1, h1(c2) = 0, h1(1) = 1.

It is easy to see that P ∈ P , with

P0 = {∞},

P1 = {f | c1, c2 /∈ rng(f)} = {0, 1}X ,

P2 = {f | c1 ∈ rng(f) or c2 ∈ rng(f)}.

It is easy to see that P0 is clopen set. Moreover, for f ∈ P2, there exists x ∈ Xsuch that f(x) = c1 or f(x) = c2. So f ∈ Cx or f ∈ Dx and since Cx, Dx are clopen

and Cx, Dx ⊆ P2, we have P2 open.

Now suppose that P ' P(ConcB) for some B ∈ V .

Lemma 4.3. Denote ϕ an isomorphism P → P(Conc B). For every f ∈ P1 there

exists a surjective homomorphism pf : B → 2 such that

ϕ(f) = {α ∈ Conc B | α ≤ ker pf}.

Proof. By Lemma 1.13, ϕ(f) = {α ∈ ConcB | α ≤ γ} for some γ ∈ M∗(ConB). So

γ = ker p for some surjective homomorphism p : B → S ∈ SI(V). Since f ∈ P1, we

have S = 2.

The assigment f 7→ pf defines a mapping t : {0, 1}X → {0, 1}B.

Lemma 4.4. The mapping t is a topological embedding (with respect to the product

topologies on {0, 1}X and {0, 1}B).

Proof. To prove the continuity we check that t−1(U) is clopen set for every U from

the subbase of {0, 1}B. Let a ∈ B, let U = {p ∈ {0, 1}B | p(a) = 0} and let W =

{I ∈ P(Conc B) | Θ(a, 0) ∈ I}. The set U is clopen in {0, 1}B and the setW is clopen

in P(ConcB). Let f ∈ P1, then

f ∈ t−1(U)⇔ pf ∈ U ⇔ pf (a) = 0⇔ pf (a) = pf (0)⇔

⇔ (a, 0) ∈ ker pf ⇔ Θ(a, 0) ≤ ker pf ⇔

⇔ Θ(a, 0) ∈ ϕ(f).

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Moreover

f ∈ ϕ−1(W )⇔ ϕ(f) ∈ W ⇔ Θ(a, 0) ∈ ϕ(f).

Hence, t−1(U) = ϕ−1(W ) ∩ P1. It is easy to see that if M is clopen in P , then

M ∩ P1 is clopen in product topology on P1. Hence, t−1(U) is clopen in product

topology on P1.

If U = {p ∈ {0, 1}B | p(a) = 1}, then we set W = {I ∈ P(Conc B) | Θ(a, 1) ∈ I}and similarly as above we can show that t−1(U) is clopen in product topology on

P1.

We have proved that t is continuous. Since both spaces are compact Hausdorff,

closed subsets are exactly compact subsets. Since the compactness is preserved by

continuous maps, we obtain that the images of closed subsets of {0, 1}X are closed

in {0, 1}B and t is an embedding.

We have proved that Z = {pf | f ∈ P1} is a compact subspace of {0, 1}B, so it is

a Boolean space.

Lemma 4.5. If f, g ∈ P1, f 6= g, then pf , pg are comparable in the pointwise ordering

on {0, 1}B.

Proof. Without loss of generality let there exists u ∈ P1 such that f(u) = 0, g(u) = 1.

Define h : X → {0, 1, c1, c2} as follows:

h(x) =

0 if f(x) = g(x) = 0,

1 if f(x) = g(x) = 1,

c1 if f(x) = 0, g(x) = 1,

c2 if f(x) = 1, g(x) = 0.

Then f = h0h, g = h1h. Moreover, ϕ(h) = {α ∈ Conc B | α ≤ ker ph} for some

surjective homomorphism ph : B → A. Since h ≤ f, g, then ϕ(h) ⊆ ϕ(f), ϕ(g).

Hence ker ph ⊆ ker pf , ker pg.

There exists a homomorphisms g0, g1 : A → 2 such that pf = g0ph, pg = g1ph.

This is only possible if {g0, g1} = {f0, f1}. Since f0 and f1 are comparable, pf and

pg are comparable too.

So Z is a chain in the pointwise order.

Lemma 4.6. The Boolean algebra B(Z) of all clopen subsets of Z is generated by a

chain.

46

Proof. For every b ∈ B denote Ub = {g ∈ Z | g(b) = 1}. This set is clopen and also

up-set with respect to pointwise order (g ∈ Ub and g ≤ h imply h ∈ Ub). Since any

two up-sets on a chain are comparable (with respect to the set inclusion), the family

{Ub | b ∈ A} forms a linearly ordered subset of B(Z).

It remains to show that this family generates B(Z). Every clopen subset of Z is

a union of the sets of the form

{g ∈ Z | g(x1) = 0, . . . , g(xn) = 0, g(y1) = 1, . . . , g(ym) = 1}

for some x1, . . . , xn, y1, . . . , ym ∈ B. Every set of the above type is a finite intersection

of the sets Ub and their complements, which completes the proof.

Theorem 4.7. If X is uncountable, then P 6' P(ConcB) for any B ∈ V.

Proof. The Boolean algebra B(Z) is isomorphic to B(P1), the Boolean algebra of all

clopen subsets of P1. By the Stone duality, B(P1) is the free Boolean algebra with

|X| generators (see [26] Theorem 9.5). However, B(P1) is not generated by a chain,

as all chains in free Boolean algebras are countable (see [26] Corollary 9.17) and an

uncountable algebra cannot be generated by a countable subset. This contradiction

completes the proof.

This proves that there exists a Priestley space P satisfying the conditions (Pr1)

and (Pr2) such that P 6' P(Conc B) for any B ∈ V . We are only able to prove the

sufficiency of the conditions (Pr1) and (Pr2) in some special cases. We will present

three such special cases. First, we prove a generalization of Theorem 1.8.

Theorem 4.8. Let L be a distributive lattice with 0 and let (P(L), τ,≤) be its dual

Priestley space. Let (v(I), fI,J) be a SI(V)-valuation on P(L). Let A be a subalgebra

of∏

I∈P(L) v(I) such that

(a) For every a ∈ A and for every I, J ∈ P(L), I ≤ J ,

aJ = fI,J(aI).

(b) For every I ∈ P(L) and for every u ∈ v(I) there exists a ∈ A such that

aI = u.

(c) For every I, J ∈ P(L), I � J there exist a, b ∈ A such that

aI = bI , aJ 6= bJ .

47

(d) For every a, b ∈ A, the set Ua,b = {I ∈ P(L) | aI = bI} is clopen.

Then the Priestley spaces P(L) and P(Conc A) are isomorphic (and hence L and

Conc A are isomorphic) and the isomorphism ϕ : P(L)→ P(Conc A) can be defined

by ϕ(I) = {α ∈ ConcA | α ≤ ker pI}, where pI : A→ v(I) is the projection.

Proof. Let I ∈ P(L), by (b) pI is surjective, hence ker pI ∈ M∗(ConA). By Lemma

1.13, we have

{α ∈ ConcA | α ≤ ker pI} ∈ P(Conc A).

Thus the map ϕ : P(L)→ P(Conc A) is defined correctly.

We prove that ϕ is an isomorphism of ordered topological spaces. By Lemma 1.13,

if K ∈ P(ConcA), then K = {α ∈ ConcA | α ≤ γ} for some γ ∈ M∗(ConA). We

claim that ker pI ≤ γ for some I ∈ P(L). For contradiction suppose that ker pI � γ

for every I ∈ P(L). Our assumption means that⋃(a,b)∈A2\γ

Ua,b = P(L).

Since P(L) is compact, there exists n ∈ N and elements ai, bi ∈ A (i = 1, . . . , n) such

that (ai, bi) /∈ γ and for every J ∈ P(L) there exists j ∈ {1, . . . , n} with ajJ = bjJ .

Hence, Θ(aj, bj) ≤ ker pJ , thus⋂1≤i≤n

Θ(ai, bi) ≤∧

J∈P(L)

ker pJ = 0 ≤ γ.

This contradicts the meet-irreducibility of γ. (Note that if (a, b) /∈ γ, then Θ(a, b) �γ.) Hence, there exists I ∈ P(L) such that ker pI ≤ γ. Since pI : A → v(I) is

surjective, the lattice Con v(I) is isomorphic to the filter ↑ ker pI of ConA. The

congruence γ ∈ ↑ ker pI corresponds to the congruence γ′ ∈ Con v(I) given by

γ′= {(xI , yI) | (x, y) ∈ γ}. By definition 1.4, γ′ = ker fI,J for some J ≥ I, so

(x, y) ∈ γ ⇔ (xI , yI) ∈ γ′= ker fI,J ⇔

⇔ fI,J(xI) = fI,J(yI)⇔ xJ = yJ ⇔

⇔ (x, y) ∈ ker pJ ,

hence γ = ker pJ . Thus for every K ∈ P(ConcA) there exists J ∈ P(L), such that

ϕ(J) = K, so ϕ is surjective. Moreover, by (c), ϕ(I) ≤ ϕ(J) if and only if I ≤ J .

Hence ϕ is bijective and both ϕ and ϕ−1 preserve the order.

48

It remains to show that ϕ is a topological homeomorphism. We check that ϕ−1(U)

is open set for every U from the subbase of P(ConcA). Let α ∈ Conc A, so α =⋃ki=1 Θ(ai, bi) for some ai, bi ∈ A, i ∈ {1, . . . , k}. Let

U = {I ∈ P(ConcA) | α ∈ I},

then

I ∈ ϕ−1(U)⇔ ϕ(I) ∈ U ⇔ α ∈ ϕ(I)⇔ α ≤ ker pI ,

so

ϕ−1(U) = {I | α ≤ ker pI} =⋂

1≤i≤k

{I | Θ(ai, bi) ≤ ker pI} =⋂

1≤i≤k

{I | aiI = biI}.

Hence, by (d), ϕ−1(U) =⋂

1≤i≤k Uai,bi is clopen. Now, let

V = {I ∈ P(ConcA) | α /∈ I}.

It is easy to see that ϕ−1(V ) is a complement of ϕ−1(U), so it is also clopen.

We have proved that ϕ is continuous. Since both spaces are compact Hausdorff,

and ϕ is bijective, it must be a homeomorphism. Hence, P(L) ' P(Conc A), so

L ' Conc A.

Note that if L is finite, then the topology is discrete. Hence, Theorem 1.8 is a

special case of Theorem 4.8.

4.1 Special cases

Let V be a finitely generated, congruence-distributive variety with the CIP. More-

over, assume that ConS is a chain for every S ∈ SI(V). We denote

Si = {A ∈ SI(V) | ConA is an i-element chain}.

Further, denote by Pn the class of all partially ordered sets (C,≤) with the largest

element, such that for every x ∈ C, ↑x is a k-element chain, k ∈ {1, . . . , n}. HenceC ∈ Pn is a disjoint union of an antichains C0, . . . , Cn−1 such that |↑x| = k + 1 for

x ∈ Ck. Let L be a lattice such that P(L) ∈ Pn, then denote Pk = Pk(L) = (P(L))k

for k = 0, . . . , n− 1. Notice that P0 is a one-element set.

We present a detailed analysis of three special cases.

The first case. We suppose that V satisfies the following additional assumption:

(A1’) max{j | Sj 6= ∅} = n > 1.

49

(A2’) If A ≤ B ∈ SI(V), then ConA ' ConB.

Lemma 4.9. Let L be a distributive lattice with 0 such that its dual Priestley space

(P(L),≤, τ) satisfies (Pr1) and (Pr2). Then

(i) P(L) ∈ Pn.

(ii) For every k ∈ {0, . . . , n− 1}, the set Pk(L) is clopen.

Proof. By (Pr1), P(L) has an admissible SI(V)-valuation (v(I), fI,J). By Lemma

1.5, ↑I is isomorphic to Con v(I), which is a chain of length at most n− 1 for every

I ∈ P(L), so P(L) ∈ Pn.Further, let k ∈ {0, . . . , n− 1}. By (Pr2), for every I ∈ Pk(L) there exists an

open set U such that I ∈ U and v(I) is isomorphic to a subalgebra of v(J) for every

J ∈ U . By the assumption (A2’) we have Con v(I) ' Con v(J), thus J ∈ Pk(L).

Hence U ⊆ Pk(L), so Pk(L) is open. Since the sets P0(L), . . . , Pn(L) are mutually

disjoint, they must also be closed.


lattice with 0 and let (P(L),≤, τ) be its dual Priestley space. The following conditions

are equivalent.



(3) P(L) ∈ Pn and for every k ∈ {0, 1, . . . , n− 1}, the set Pk(L) is clopen.

Proof. We have already proved (1)⇒ (2)⇒ (3).

(3) ⇒ (1): By (A1’) there exists F ∈ SI(V) such that ConF is an n-element

chain αn−1 < αn−2 < · · · < α0. For every i ∈ {0, . . . , n− 1} denote Fi = F/αi, so

ConFi is an (i + 1)-element chain. For every j ≤ i we define a map fi,j : Fi → Fj

as the natural projection. For every I, J ∈ P(L) (I ≤ J) denote v(I) = F|↑I|−1 and

fI,J = f|↑I|−1,|↑J |−1. We define an algebra

A ≤∏

I∈P(L)

v(I)

such that a ∈ A, if

(i) aJ = fI,J(aI), whenever I ≤ J .

50

(ii) For every i ∈ {0, . . . , n− 1} and every u ∈ Fi, the set {I ∈ Pi(L) | aI = u} is

open.

We can see that (v(I), fI,J) is a SI(V)-valuation on P(L) and it is easy to

check that A is subalgebra of∏

I∈P(L) v(I). Moreover, since Fi is finite, all the sets

{I ∈ Pi | aI = u} are clopen. For a, b ∈ A, the set Ua,b = {I | aI = bI} is a union of

sets {I ∈ Pi | aI = u} ∩ {I ∈ Pi | bI = u} for every u ∈ Fi, (i = 0, . . . , n− 1), hence

Ua,b is clopen. It remains to check the conditions (b) and (c) of Theorem 4.8.

To prove (b), let j ∈ {0, . . . , n− 1,}, let I ∈ Pj(L) and let k ∈ v(I). Hence,

k = [v]αj for some v ∈ F . Let a = (aK)K∈P(L), where aK = [v]αi for K ∈ Pi(L).

We claim that a ∈ A. Condition (i) hold trivially. Let i ∈ {0, . . . , n− 1}, for everyu ∈ Fi we have u = [w]αi for some w ∈ F . Hence the set

{I ∈ Pi | aI = [w]αi} = {I ∈ Pi | [v]αi = [w]αi} =

∅ if [v]αi 6= [w]αi ,

Pi if [v]αi = [w]αi ,

is in each case clopen. So a ∈ A and aI = k.

To prove (c), let I, J ∈ P(L), such that I � J . Denote j = |↑J | − 1. Since

j ≥ 1, there exists u, v ∈ v(J), u 6= v such that (u, v) ∈ ker fj,j−1. Hence, there exist

t1, t2 ∈ F such that

u = [t1]αj 6= [t2]αj = v

and

[t1]αs = [t2]αs

for every s < j. For every K ∈ P(L) denote

aK = [t1]αl if K ∈ Pl(L).

We have already shown that every element of the form a = (aK)K∈P(L) belongs to A.

Further, by CTOD, there exists a clopen up-set V ⊆ P(L) such that I ∈ V , J /∈ V .

Denote

U =↓ (Pj(L) \ V ).

Both Pj \ V and Pj ∩ V are clopen, so ↓ (Pj \ V ) and ↓ (Pj ∩ V ) are disjoint closed

sets and their union is equal to the clopen set Pj ∪ Pj+1 ∪ · · · ∪ Pn−1. Hence U is a

clopen set. For every l ∈ {0, . . . , n− 1} and every K ∈ Pl we denote

bK =

[t1]αl if K /∈ U,

[t2]αl if K ∈ U.

51

Now, denote b = (bK)K∈P(L) and we prove that b ∈ A. Let K,M ∈ P(L), K ≤ M .

If K,M ∈ U or K,M /∈ U , then clearly fK,M(bK) = bM . If K ∈ U and M /∈ U , then

r = |↑K| − 1 ≥ j,

s = |↑M | − 1 < j

and fK,M(bK) = fr,s([t2]αr) = [t2]αs = [t1]αs = bM .

Further, let i ∈ {0, . . . , n− 1} and w ∈ Fi. The set

{I ∈ Pi | bI = w} =

Pi if w = [t2]αi = [t1]αi ,

Pi ∩ U if w = [t2]αi 6= [t1]αi ,

Pi \ U if w = [t1]αi 6= [t2]αi ,

∅ otherwise,

is in each case clopen. Hence, a, b ∈ A and aI = [t1]α|↑I|−1= bI , aJ = [t1]αj 6= [t2]αj =

bJ .

By Theorem 4.8, we have P(L) ' P(Conc A), so L ' ConcA.

Thus, in our special case we have proved the converse to Theorem 4.1. Thanks

to the result of T. Katriňák and A. Mitschke, we can go even further. Recall the

fourth special case in Chapter 3 for the definition of dual Stone lattice of order n.



are equivalent.


(2) P(L) ∈ Pn and the set Pk(L) is clopen for every k ∈ {0, 1, . . . , n− 1}.

(3) P(L) ∈ Pn and for every i ∈ {0, . . . , n− 2}, there exists an element ei ∈⋂{I | I ∈ P0(L) ∪ · · · ∪ Pi(L)} such that ei /∈ J for every J ∈ Pj(L) (for every

j > i).


Proof. We have already proved the equivalence (1)⇔ (2). The equivalence (3)⇔ (4)

was proved in [25], Theorem 4.5 (in a dual form).

(2) ⇒ (3): Let i, j ∈ {0, . . . , n− 1}, i < j, let I ∈ P0 ∪ · · · ∪ Pi, J ∈ Pj. SinceI � J , there exists αI,J ∈ I \ J . Denote

UI,J = {K ∈ P(L) | αI,J /∈ K},

52

UI = {UI,J | J ∈ Pj for some j > i}.

It is easy to see that I /∈ UI,J , J ∈ UI,J . Moreover since UI is an open cover of

the closed (and hence compact) set Qi =⋃n−1≥j>i Pj, there exist finitely many

J1, . . . , Jm ∈ P(L) such that

Qi ⊆ {K | αI,J1 /∈ K or . . . or αI,Jm /∈ K} = {K | αI,J1 ∨ · · · ∨ αI,Jm /∈ K}.

Denote βI = αI,J1 ∨ · · ·∨αI,Jm . Hence, for every I ∈ P0∪ · · ·∪Pi there exists βI ∈ Lsuch that

(i) βI ∈ I,

(ii) βI /∈ J for every J ∈ Pj, j > i.

Further, denote UI = {K ∈ P(L) | βI ∈ K}. The collection of sets UI , (I ∈ P0 ∪· · · ∪ Pi) covers the compact set P0 ∪ · · · ∪ Pi. By the compactness, there exist

I1, . . . , Iq ∈ P0 ∪ · · · ∪ Pi such that

P0 ∪ · · · ∪ Pi ⊆ {K | βI1 ∈ K or . . . or βIq ∈ K}.

Using the fact that ideals K ∈ P(L) are prime we obtain

P0 ∪ · · · ∪ Pi ⊆ {K | βI1 ∧ · · · ∧ βIq ∈ K}.

Denote ei = βI1 ∧ · · · ∧ βIq . Hence, for every I ∈ P0 ∪ · · · ∪ Pi and for every J ∈ Pj(for every j > i) we have ei ∈ I and ei /∈ J .

(3)⇒ (2): Let i ∈ {0, . . . , n− 2}. By (3),

Pi+1 ∪ · · · ∪ Pn−1 = Qi = {I ∈ P(L) | ei /∈ I},

which is a clopen set. Then also Pj = Qj−1 \ Qj is clopen for j = 1, . . . , n − 2.

Moreover, P0 is the complement of Q0 and Pn−1 = Qn−2.

The second case. Similarly as in the first special case, we assume that V is a

finitely generated, congruence-distributive variety with the CIP and ConA is a chain

for every A ∈ SI(V). Instead of (A1′), (A2′) we consider the following additional

assumptions:

(B1’) max{j | Sj 6= ∅} = 3.

(B2’) For every G ≤ F ∈ SI(V) either ConG ' ConF or G ∈ S2, F ∈ S3.

53

(B3’) There exists F 0 ∈ S3 such that F 0/α ≤ F 0, where α is the only nontrivial

congruence on F 0.

Lemma 4.12. Let G ≤ F ∈ SI(V), such that ConG is a 2-element chain 0 < 1

and ConF is a 3-element chain 0 < α < 1. Let h be an embedding G → F , then

Conh(1) = 1.

Proof. We have Conh(1) 6= 0 because h is injective. For contradiction suppose that

Conh(1) = α. Hence, h(G) is contained in one α-class, so F/α has a one-element

subalgebra. We have a contradiction with the assumption (B2’).

Lemma 4.13. F 0/α is isomorphic to a retract of F 0.

Proof. Let e : F 0/α → F 0 be an embedding and f : F 0 → F 0/α be a natural

projection. Then by Lemma 4.12,

Con fe(1) = Con f(Con e(1)) = Con f(1) = 1,

so Con fe is an isomorphism {0, 1} → {0, 1}, thus fe is injective and since F 0 is

finite, fe is an automorphism. Hence, G = e(F 0/α) is a retract of F 0 isomorphic to

F 0/α (with e(fe)−1f as the retraction).

Lemma 4.14. Let L be a distributive lattice with 0 such that its dual Priestley space

(P(L),≤, τ) satisfies (Pr1) and (Pr2). Then

(i) P(L) ∈ P3.

(ii) P0(L) is clopen, P2(L) is open.

Proof. By (Pr1), P(L) has an admissible SI(V)-valuation (v(I), fI,J). By Lemma 1.5,

↑I is isomorphic to Con v(I) which is a chain of length at most 2, so P(L) ∈ P3.

Further, let i ∈ {0, 2} and let I ∈ Pi(L). By (Pr2) there exists an open set U

with I ∈ U and for every J ∈ U we have v(I) ≤ v(J). Thus, by assumption (B2’),

J ∈ Pi(L). Hence U ⊆ Pi(L), so Pi(L) is open. Since P0(L) is a one-element set, it

is also closed.



are equivalent.

54



(3) P(L) ∈ P3, P0(L) is clopen and P2(L) is open.


(3) ⇒ (1): By (B3’), there exists F = F 0 ∈ S3 such that F/α ≤ F , where α is

the only nontrivial congruence on F . Let G be a retract of F such that G ' F/α.

For every I ∈ P(L) set

v(I) = F if I ∈ P2,

v(I) = G if I ∈ P1,

v(I) = 1 if I ∈ P0.

By Lemma 4.13, there exists a surjective homomorphism f : F → G such that

f�G = idG. For every I, J ∈ P(L), I < J we define a map fI,J = v(I)→ v(J) such

that

fI,J(a) = f(a) if I ∈ P2, J ∈ P1,

fI,J(a) = 1 if J ∈ P0.

(and, of course, fI,I is the identity for every I ∈ P(L).) We define an algebra

A ≤∏

I∈P(L)

v(I)



(ii) For every u ∈ F , the set {I ∈ P(L) | aI = u} is clopen.

(Note that the set {I ∈ P(L) | aI = u} may contain elements from both P1 and P2.)

It is easy to see that (v(I), fI,J) is a SI(V)-valuation on P(L) and it is easy to check

that A is subalgebra of∏

I∈P(L) v(I). Let a, b ∈ A. Since Ua,b = {I | aI = bI} is a

union of sets {I | aI = u} ∩ {I | bI = u} for every possible u, we have that Ua,b is

clopen. It remains to check the conditions (b) and (c) of Theorem 4.8.

First, let U ⊆ P2(L) and V ⊆ P(L) be clopen sets. Let v ∈ F , v1, v2 ∈ G, v1 6= v2.

55

For every K ∈ P(L) denote

a(U, v)K =

1 if K ∈ P0,

v if K ∈ U,

f(v) if K ∈ P1 ∪ (P2 \ U),

b(V, v1, v2)K =

1 if K ∈ P0,

v1 if K ∈ ↓(P1 \ V ),

v2 if K ∈ ↓(P1 ∩ V ).

Since P1 is closed, we have P1 ∩ V and P1 \ V closed, hence both ↓(P1 ∩ V ) and

↓(P1 \V ) are closed. These sets are disjoint and their union P1∪P2 is clopen. Hence,

both ↓(P1∩V ) and ↓(P1\V ) are clopen sets. Denote a = (a(U, v)K)K∈P(L), we prove

that a ∈ A. Let I ∈ P2, J ∈ P1, I < J . Then

fI,J(aI) = f(v) = aJ .

For every u ∈ F the set

{I ∈ P(L) | aI = u} =

U if u = v 6= f(v),

P1 ∪ (P2 \ U) if u = f(v) 6= v,

P1 ∪ P2 if u = v = f(v),

∅ otherwise,

is in each case clopen. Thus, a ∈ A. Denote b = (b(V, v1, v2)K)K∈P(L), we prove that

b ∈ A. Let I ∈ P2, J ∈ P1, I < J . Then

fI,J(bI) =

f(v1) = v1 = bJ if I ∈↓ (P1 \ V ) ∩ P2,

f(v2) = v2 = bJ if I ∈↓ (P1 ∩ V ) ∩ P2.

For every u ∈ F the set

{I ∈ P(L) | bI = u} =

↓ (P1 ∩ V ) if u = v2 = f(v2),

↓ (P1 \ V ) if u = v1 = f(v1),

∅ otherwise,

is in each case clopen, so b ∈ A.Now, we can deal with the conditions (b) and (c) of Theorem 4.8.

First, let I ∈ P0, then the condition (b) of Theorem 4.8 is easy to prove.

Next, let J ∈ P2 and let v ∈ v(J) = F . By CTOD there exists a clopen down-set

U such that J ∈ U and (P1 ∪ P0) ∩ U = ∅. Denote a = (a(U, v)K)K∈P(L). We have

a ∈ A and aJ = v.

56

Next, let J ∈ P1 and let v ∈ v(J) = G, so f(v) = v. Denote a = (a(∅, v)K)K∈P(L).

We have a ∈ A and aJ = f(v).

Further, let I, J ∈ P(L) such that I � J .

First, let J ∈ P2, then there exists v1, v2 ∈ v(J) = F , v1 6= v2 such that f(v1) =

f(v2). By CTOD there exists a clopen down-set U ⊆ P2 such that J ∈ U and

(P1 ∪ P0 ∪ {I}) ∩ U = ∅. Denote a = (a(U, v1)K)K∈P(L), b = (a(U, v2)K)K∈P(L). We

have a, b ∈ A and aI = f(v1) = f(v2) = bI , aJ = v1 6= v2 = bJ .

Next, let J ∈ P1 and let v1, v2 ∈ v(J) = G such that v1 6= v2. By CTOD

there exists a clopen up-set V ⊆ P(L) such that I ∈ V and J /∈ V . Denote

a = (a(∅, v2)K)K∈P(L), b = (b(V, v1, v2)K)K∈P(L). We have a, b ∈ A. Moreover aI =

f(v2) = v2 = bI , aJ = v2 6= v1 = bJ .

(Notice that if I ∈ P0 or J ∈ P0, then the condition (c) of Theorem 4.8 is easy

to prove.)

By Theorem 4.8, P(L) ' P(Conc A), so L ' ConcA.

Similarly as in the first case we can go even further. Recall the third special case

in Chapter 3 for the definition of dual Stone lattice. The next Lemma follows from

results of Katriňák and Mitschke (see [25]).

Lemma 4.16. Let L be a dual Stone lattice. Denote max(P(L)) the set of all max-

imal elements of P(L)\{L}. Then

(i) I ∈ max(P(L)) if and only if D̄(L) ∈ I.

(ii) For every I ∈ P(L) there exists exactly one J ∈ max(P(L)) such that I ⊆ J .



are equivalent.


(2) P(L) ∈ P3 and P0(L) is clopen, P2(L) is open.

(3) L is a dual Stone lattice and its co-dense elements form a generalized Boolean

lattice.

Proof. We have already proved the equivalence (1)⇔ (2).

(2) ⇒ (3): By Priestley duality we know that L is isomorphic to the lattice of

all proper clopen down-subsets of P(L), so ∅ is the least and P1 ∪ P2 is the largest

57

element of L. Further, let U be proper clopen down-set of P(L). It is easy to see

that its dual pseudocomplement is U+ =↓ (P1 \ U). Then U++ =↓ (P1 ∩ U), so

U+ ∩ U++ = ∅. Hence L is a dual Stone lattice. Clearly, U+ = 1 if and only if

U ⊆ P2 and thus

D̄(L) = {U | U ⊆ P2, U clopen}.

Obviously, clopen subsets of P2 form a generalized Boolean lattice. This generalized

Boolean lattice is not necessarily a Boolean lattice, since P2 itself need not be clopen.

(3) ⇒ (2): It is easy to see that P0 = {I | 1 ∈ I} = {L} is clopen. Denote

P1 = max(P(L)), by Lemma 4.16(i), for every I /∈ P1 ∪ P0 there exists x ∈ D̄(L)

such that x /∈ I. Hence I ∈ Vx = {J ∈ P(L) | x /∈ J} and since Vx is open and

P1 ∩ Vx = ∅, we have P1 closed.

Further, we prove that P(L)\ (P1∪P0) is an antichain. For contradiction suppose

that there exists I, J ∈ P(L) \ (P1 ∪ P0) such that I < J . By CTOD there exists a

clopen down-set V such that J ∈ V and V ∩ (P1 ∪ P0) = ∅. Also by CTOD, there

exists a clopen down-set U ⊆ V such that I ∈ U and J /∈ U . Identifying L with the

lattice of all proper clopen down-sets of P(L), we have V, U ∈ D̄(L). However, U

has no complement in the interval [∅, V ]. Indeed, let W ⊆ V be a clopen down-set.

Now

• if J ∈ W , then I ∈ W , so U ∩W 6= ∅.

• if J /∈ W , then J /∈ U ∪W , so U ∪W 6= V .

It is a contradiction with the fact that D̄(L) is a generalized Boolean lattice. Thus,

P2 = P(L) \ (P1 ∪ P0) is an antichain.

By Lemma 4.16(ii), for every I ∈ P2 the set ↑I is a 3-element chain. So, P(L) ∈P3.

The third case. Similarly as in the first special case, we assume that V is a

finitely generated, congruence-distributive variety with the CIP and ConA is a chain

for every A ∈ SI(V). Instead of (A1′), (A2′) we consider the following additional

assumptions:

(C1’) max{j | Sj 6= ∅} = 3.

(C2’) For every G ≤ F ∈ SI(V) either ConG ' ConF or G ∈ S2, F ∈ S3.

(C3’) There exists G0, F 0 ∈ SI(V) such that G0 ≤ F 0 and G0 ∈ S2, F 0 ∈ S3.

58

(C4’) For every F1, F2 ∈ S3, F1/α � F2, where α is the only nontrivial congruence

on F1.

Lemma 4.18. Let L be a lattice with 0 such that its dual Priestley space (P(L),≤, τ)

satisfies (Pr1) and (Pr2). Then

(i) P(L) ∈ P3,

(ii) (a) P0(L) is clopen and P2(L) is open.

(b) There exists an open set P ′1 ⊆ P(L) such that

P1 ∩ ↑P2 ⊆ P′

1 ⊆ P1.

Proof. By (Pr1), P(L) has an admissible SI(V)-valuation (v(I), fI,J). By Lemma 1.5,

↑I is isomorphic to Con v(I) which is a chain of length at most 2, so P(L) ∈ P3.

Further, let i ∈ {0, 2} and let I ∈ Pi(L), by (Pr2) there exists an open set U with

I ∈ U and for every J ∈ U we have v(I) ≤ v(J), thus J ∈ Pi(L) (by assumption

(C2’)). Hence U ⊆ Pi(L), so Pi(L) is open. Since P0(L) is a one-element set, it is

also closed.

Further, we set

P′

1 = {I ∈ P1 | v(I) � F for every F ∈ S3}.

Clearly, P ′1 ⊆ P1. Let I ∈ ↑P2 ∩ P1. There exists J ∈ P2 such that J < I. By the

definition of SI(V)-valuation, there exists a surjective homomorphism fJ,I : v(J)→v(I) such that ker fJ,I = α, where α is the only nontrivial congruence of v(J). Hence,

by (C4’), I ∈ P ′1 and so P1 ∩ ↑P2 ⊆ P′1.

Now, let I ∈ P ′1. By (Pr2) there exists an open set U with I ∈ U and for every

J ∈ U we have v(I) ≤ v(J). By the definition of P ′1, we have v(J) /∈ S3, so J /∈ P2.

Thus, by (C2’), J ∈ P1. If J ∈ P1 \ P′1, then v(I) ≤ v(J) ≤ F for some F ∈ S3. We

have a contradiction with I ∈ P ′1, so J ∈ P′1. Hence, U ⊆ P

′1, thus P

′1 is open.



are equivalent.



59

(3) (a) P(L) ∈ P3 and P0(L) is clopen, P2(L) is open.

(b) There exists an open set P ′1 ⊆ P(L) such that

P1 ∩ ↑P2 ⊆ P′

1 ⊆ P1.


(3) ⇒ (1): By (C3’) there exists G,F ∈ SI(V) such that G is a subalgebra

of F and G ∈ S2, F ∈ S3. Let α be the only nontrivial congruence on F . If G

is contained in one α-class, then F/α has a one-element subalgebra. We have a

contradiction with the assumption (C2’). Further, if there exist x, y ∈ G (x 6= y)

such that (x, y) ∈ α, then α�G is a nontrivial congruence, contradicting G ∈ S2.

Hence, G ' G′= {[u]α | u ∈ G} ≤ F/α.

Denote P ′′1 = P1 \ P′1 and for every I ∈ P(L) set

v(I) = F if I ∈ P2,

v(I) = G if I ∈ P ′′1 ,

v(I) = F/α if I ∈ P ′1,

v(I) = 1 if I ∈ P0.

Denote f : F → F/α a natural projection f(x) = [x]α. For every I, J ∈ P(L),

I < J we define a map fI,J = v(I)→ v(J) such that

fI,J(a) = f(a) if I ∈ P2, J ∈ P′

1,

fI,J(a) = 1 if J ∈ P0.

(and, of course, fI,I is the identity for every I ∈ P(L).) We can define an algebra

A ≤∏

I∈P(L)

v(I)



(ii) For every u ∈ G, the set {I ∈ P ′′1 ∪ P2 | aI = u}∪{I ∈ P ′1 | aI = f(u)} is open.

(iii) For every u ∈ F \ G, the sets {I ∈ P2 | aI = u} and {I ∈ P ′1 | aI = f(u)} are

open.

60

Since all sets in (ii) and (iii) (for all possible u) are mutually disjoint and their union

is the clopen set P(L) \P0, they are also closed. It is easy to see that (v(I), fI,J) is a

SI(V)-valuation on P(L) and it is easy to check that A is subalgebra of∏

I∈P(L) v(I).

Let a, b ∈ A, we check the condition (d) of Theorem 4.8.

Ua,b = {I ∈ P(L) | aI = bI} =⋃

w∈F∪F/α∪1

{I | aI = w = bI}.

If w ∈ F \G, then

{I | aI = w = bI} = {I ∈ P2 | aI = w} ∩ {I ∈ P2 | bI = w}

is clopen. If w ∈ F/α \G′, then

{I | aI = w = bI} = {I ∈ P ′1 | aI = f(u)} ∩ {I ∈ P ′1 | bI = f(u)}

is clopen. Further,⋃w∈G∪G′

{I | aI = w = bI} =⋃u∈G

({I | aI = u = bI} ∪ {I | aI = f(u) = bI}).

Let u ∈ G, denote

A = {I ∈ P ′′1 ∪ P2 | aI = u},

B = {I ∈ P ′′1 ∪ P2 | bI = u},

C = {I ∈ P ′1 | aI = f(u)},

D = {I ∈ P ′1 | bI = f(u)}.

Hence,

{I | aI = u = bI} ∪ {I | aI = f(u) = bI} = (A ∩B) ∪ (C ∩D).

Since B ∩ C = ∅ and A ∩D = ∅,

{I | aI = u = bI} ∪ {I | aI = f(u) = bI} = (A ∪ C) ∩ (B ∪D).

Since a, b ∈ A, the sets A ∪ C, B ∪D are clopen. So Ua,b is clopen.

It remains to check the conditions (b) and (c) of Theorem 4.8.

First, let I0 ∈ P0, then the condition (b) of Theorem 4.8 is easy to prove.

Next, let I0 ∈ P′′1 ∪ P2 and let w ∈ G. For every K ∈ P(L) set

aK =

1 if K ∈ P0,

w if K ∈ P ′′1 ∪ P2,

f(w) if K ∈ P ′1.

61

Denote a = (aK)K∈P(L), we prove that a ∈ A. Let I ∈ P2, J ∈ P1 such that I < J ,

then

fI,J(aI) = f(w) = aJ .

For every u ∈ G, the set

{I ∈ P ′′1 ∪ P2 | aI = u} ∪ {I ∈ P ′1 | aI = f(u)} =

P(L) \ P0 if u = w,

∅ otherwise,

is in each case open. For every u ∈ F \ G, the set {I ∈ P2 | aI = u} = ∅ and also

{I ∈ P ′1 | aI = f(u)} = ∅. Thus a ∈ A and aI0 = w.

Next, let I0 ∈ P′1 and let v ∈ G′ . There exists w ∈ G such that f(w) = v. Define

a = (aK)K∈P(L) as above. Hence aI0 = f(w) = v.

Next, let I0 ∈ P′1 and let w ∈ F/α \ G′ . There exists a clopen set U ⊆ P

′1 such

that I0 ∈ U and U ∩P ′′1 = ∅. Denote V = P1 \U . Since U, V are closed, we have ↓Uand ↓V closed, disjoint and their union P1 ∪ P2 is clopen. Hence, both ↓U, ↓V are

clopen sets. There exists u ∈ F \ G such that f(u) = w. Choose v ∈ G arbitrarily.

For every K ∈ P(L) set

bK =

1 if K ∈ P0,

u if K ∈ P2 ∩ ↓U,

f(u) if K ∈ P ′1 ∩ ↓U = U,

v if K ∈ P ′′1 ∪ (P2 ∩ ↓V ),

f(v) if K ∈ P ′1 ∩ V = P′1 \ U.

Denote b = (bK)K∈P(L), we prove that b ∈ A. Let I ∈ P2 ∩ ↓U , J ∈ P1 such that

I < J , then

fI,J(bI) = f(u) = bJ .

Let I ∈ P2 ∩ ↓V , J ∈ P1 such that I < J , then

fI,J(bI) = f(v) = bJ .

For every x ∈ G, the set

{I ∈ P ′′1 ∪ P2 | bI = x} ∪ {I ∈ P ′1 | bI = f(x)} =

↓V if x = v,

∅ otherwise,

is in each case clopen. For every x ∈ F \G, the sets

{I ∈ P2 | bI = x} =

P2 ∩ ↓U if x = u,

∅ otherwise,

62

{I ∈ P ′1 | bI = f(x)} =

U if x = u,

∅ otherwise,

are in each case clopen. (P2 ∩ ↓U is a complement to U ∪ ↓V ∪ P0.) Hence, b ∈ Aand bI0 = f(u) = w.

Next, let J ∈ P2 and let u ∈ F \G. There exists I0 ∈ P′1 such that J < I0. Define

b = (bK)K∈P(L) as above. Hence, bJ = u.

This proves (b) of Theorem 4.8. To prove (c) let I, J ∈ P(L) such that I � J .

First, let J ∈ P2. There is I0 ∈ P′1 such that J < I0. By CTOD there exists

a clopen down-set U ⊆ P(L) such that J ∈ U and U ∩ (P0 ∪ P1 ∪ {I}) = ∅.There exist v1, v2 ∈ F (v1 6= v2) such that f(v1) = f(v2). Choose a ∈ A such that

aI0 = f(v1) = f(aJ), so there exists x 6= aJ such that f(x) = f(aJ). For every

K ∈ P(L) set

bK =

aK if K ∈ P(L) \ U,

x if K ∈ U.

Denote b = (bK)K∈P(L). It is easy to check that b ∈ A and aI = bI , aJ 6= bJ .

Next, let J ∈ P1. Let I ∈ P1 ∪ P2. By CTOD there exists a clopen up-set

W ⊆ P(L) such that I ∈ W , J /∈ W . Then ↓U = ↓(P1 ∩W ), ↓V = ↓(P1 \W ) are

clopen. Let v1, v2 ∈ G (v1 6= v2), for every K ∈ P(L) denote

aK =

1 if K ∈ P0,

v1 if K ∈ P2 ∩ ↓U,

f(v1) if K ∈ P ′1 ∩ ↓U,

v2 if K ∈ P ′′1 ∪ (P2 ∩ ↓V ),

f(v2) if K ∈ P ′1 ∩ ↓V = P′1 \ ↓U.

bK =

1 if K ∈ P0,

v1 if K ∈ P2 ∪ P′′1 ,

f(v1) if K ∈ P ′1,

Denote a = (aK)K∈P(L) and b = (bK)K∈P(L). It is easy to check that a, b ∈ A. SinceI ∈ ↓U and J ∈ ↓V , it is easy to see that aI = bI and aJ 6= bJ .

(Notice that if I ∈ P0 or J ∈ P0, then the condition (c) of Theorem 4.8 is easy

to prove.)

By Theorem 4.8, P(L) ' P(Conc A), so L ' Conc A.

63

There exist infinitely many other cases of varieties V , when

SI(V) = S1 ∪ S2 ∪ S3.

For example, assume that there exist Fi ∈ S3 (i = I) such that Fj/α is not isomor-

phic to a subalgebra of Fk (for every k ≤ j) and

(G e−→)F0f0−→F0/α

e0−→F1f1−→F1/α

e1−→. . .fn−→Fn/α,

where ei are embeddings and fi are natural projections. (Note that, if Fi/α ≤ Fi,

then also Fi ≤ Fj for every j > i.)

All these cases lead to different classes ConV . (It follows from Gillibert’s results.)

We have investigated just two extreme cases. In our second special case we have an

infinite sequence

F/α = G→ F→F/α→F→F/α→. . .

and in our third special case we have only the shortest possible sequence of this form

G0 → F0→F0/α � F ∈ S3.

More other examples follows from the fact, that there are varieties with F ≤ G

for some F ∈ S3, G ∈ S2.

64

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