Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 1, 97-108.

The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Finite Lattices

G. Grätzer^{1} and E. T. Schmidt^{2}

Department of Mathematics, University of Manitoba Winnipeg, Man. R3T 2N2, Canada e-mail: George Gratzer@umanitoba.ca Department of Mathematics, Transport Engineering Faculty Technical University of Budapest, Műegyetem rkp. 9, 1111 Budapest, Hungary e-mail: h1175sch@ella.hu

Abstract. The Independence Theorem for the congruence lattice and the automorphism group of a finite lattice was proved by V. A. Baranskiĭ and A. Urquhart. Both proofs utilize the characterization theorem of congruence lattices of finite lattices (as finite distributive lattices) and the characterization theorem of automorphism groups of finite lattices (as finite groups).

In this paper, we introduce a new, stronger form of independence.

Let L be a finite lattice. A finite lattice K is a congruence-preserving extension of L, if K is an extension and every congruence of L has exactly one extension to K. Of course, then the congruence lattice of L is isomorphic to the congruence lattice of K.

A finite lattice K is an automorphism-preserving extension of L, if K is an extension and every automorphism of L has exactly one extension to K, and in addition, every automorphism of K is the extension of an automorphism of L. Of course, then the automorphism group of L is isomorphic to the automorphism group of K.

Theorem. Let L_{C} and L_{A} be finite lattices, L_{C} ∩ L_{A} = {0}. Then there exists a finite atomistic lattice K that is a congruence-preserving extension of L_{C} and

1 The research of the first author was supported by the NSERC of Canada. 2 The research of the second author was supported by the Hungarian National Founda-

tion for Scientific Research, under Grant No. 1903.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

an automorphism-preserving extension of L_{A}. In fact, both extensions preserve the zero.

Of course, the congruence lattice of K is isomorphic to congruence lattice of L_{C}, and the automorphism group of K is isomorphic to the automorphism group of L_{A}. MSC 1991: 06B10; Secondary 06D05

1. Introduction

We associate with an algebra a number of related structures: the automorphism group, the congruence lattice, the subalgebra lattice, the endomorphism semigroup, and so on.

In this paper, the algebra we deal with is a finite lattice, and the related structures are the congruence lattice and the automorphism group.

We start by stating two characterization theorems (the reader is referred to [5] and Section 2 for the basic concepts and notation):

Theorem 1. Let D be a finite distributive lattice. Then there exists a finite lattice K such that the congruence lattice of K is isomorphic to D. The lattice K can be chosen as a sectionally complemented lattice.

This result is due to R. P. Dilworth (unpublished); it was first published in G. Grätzer and E.T. Schmidt [7].

Theorem 2. Let G be a finite group. Then there exists a finite lattice K such that the automorphism group of K is isomorphic to G. The lattice K can be chosen as a simple, sectionally complemented lattice of length three.

The first statement is due to G. Birkhoff [2]; the lattice K constructed by Birkhoff is distributive. A simple, sectionally complemented lattice K of length three was constructed in R. Frucht [3] and [4] (see Section 3).

In [7], the first author raised the following question:

Problem. Let L be a lattice, and let G be a group. Does there exist a lattice K such that the congruence lattice of K is isomorphic to the congruence lattice of L and the automorphism group of K is isomorphic to G? If L and G are finite, can K chosen to be finite?

This problem was solved for finite lattices by V. A. Baranskiĭ [1] and A. Urquhart [10], independently. We now state the Baranskiĭ-Urquhart theorem:

Teorem 3. (The Independence Theorem) Let D be a finite distributive lattice, and let G be a finite group. Then there exists a finite lattice K such that the congruence lattice of K is isomorphic to D and the automorphism group of K is isomorphic to G.

Both proofs ([1] and [10]) utilize the characterization theorems stated above.

In this paper, we define a strong form of independence that does not use the characterization theorems. In preparation, we introduce a few concepts.

Definition 1. Let L be a lattice. The lattice K is a congruence-preserving extension of L, if K is an extension and every congruence of L has exactly one extension to K.

Of course, then the congruence lattice of K is isomorphic to the congruence lattice of L.

Let D be the congruence lattice of L. Since K is a congruence-preserving extension of L, it follows that D, as an abstract lattice, is also the congruence lattice of K. We would like to argue that the “algebraic reasons why D is the congruence lattice of L” have been retained in K. For instance, let a_{i}, b_{i} ∈ L, i = 1, 2, 3, and let

Θ(a_{1}, b_{1}) ∨ Θ(a_{2}, b_{2}) = Θ(a_{3}, b_{3})

in L. Then L has a finite partial sublattice H forcing this equation. Since K is an extension of L, H is a partial sublattice of K, hence this equation holds also in K; and it holds for the same “algebraic reason”—namely, the existence of H.

Definition 2. Let L be a lattice. K is an automorphism-preserving extension of L if K is an extension and every automorphism of L has exactly one extension to K; moreover, every automorphism of K is the extension from an automorphism of L.

Of course, then the automorphism group of L is isomorphic to the automorphism group of K.

Let G be the automorphism group of L. The automorphism group of L is the group of all permutations of L that are not excluded for some algebraic reason. An “algebraic reason” can always be formulated in the form of the existence of a finite partial sublattice H. Thus we can again argue that the “algebraic reasons why G is the automorphism group of L” have been retained in K.

Definition 3. Let L be a lattice with zero. Then K is a {0}-extension of L, if the zero of L is the zero of K.

Now we are ready to state our main result:

Theorem 4. (The Strong Independence Theorem) Let L_{C} and L_{A} be finite lattices with more than one element satisfying L_{C} ∩ L_{A} = ∅. Then there exists a lattice K such that the following conditions hold:

(1) K is a finite atomistic lattice.

(2) K is a congruence-preserving {0}-extension of L_{C}.

(3) K is an automorphism-preserving {0}-extension of L_{A}.

Of course, then the congruence lattice of K is isomorphic to the congruence lattice of L_{C}, and the automorphism group of K is isomorphic to the automorphism group of L_{A}.

Arguing intuitively, as above, this theorem states that any set of algebraic conditions forcing the shape of a congruence lattice are compatible with any other set of algebraic conditions forcing the shape of an automorphism group.

The construction of the lattice K of the Strong Independence Theorem is broken down into a number of steps. In Section 3, a number of preliminary constructions are presented, including a new result of M. Tischendorf, Frucht lattices, and the formation of an ideal lattice of a chopped lattice. In Section 4, we carry out the first major step: an automorphism-preserving extension that destroys all congruences. In Section 4, we carry out the second major step: a congruence-preserving extension that destroys all automorphisms. Finally, in Section 5, we combine these constructions to obtain the Strong Independence Theorem.

We conclude the paper with some general discussion of the Strong Independence Theorem and related problems.

2. Notation

We refer the reader to [7] for the basic concepts and notation.

In this paper all lattices are finite, unless otherwise stated. The zero of a lattice is denoted by 0. Isomorphism is denoted by ≅.

For a finite lattice L, the length of L is |C| − 1, where C is the longest chain in L. We call L atomistic if every element is a join of atoms. L is sectionally complemented if, for all b < a ∈ L, there exists a c ∈ L satisfying

b ∧ c = 0 and b ∨ c = a.

Observe that a finite sectionally complemented lattice is atomistic.

If L is a sublattice of K, then we call K an extension of L. If L has a zero, and it is also the zero of K, then K is {0}-extension of L.

For a, b ∈ L, let Θ(a, b) denote the smallest congruence collapsing a and b. The congruence lattice of L will be denoted by ConL; let ι denote the unit element of this lattice. If ConL is the two-element chain, then L is called simple.

The automorphism group of L will be denoted by AutL. If AutL is the one-element group, then L is called rigid.

2.1 Chopped lattices. A chopped lattice M is a meet-semilattice with zero, 0, in which any two elements having an upper bound have a join. M will be regarded as a partial algebra < M; ∧, ∨, 0 >, 0 is a nullary operation, ∧ is an operation, and ∨ is a partial operation. Equivalently, a chopped lattice M is a lattice L with zero, 0, and unit, 1, with the unit removed: M = L − {1}.

We shall use the concept of extension for chopped lattices; observe that, by definition, an extension of a chopped lattice is a {0}-extension.

An ideal I of M is a subset of M containing 0 with the following two properties for x, y ∈ M:

x ∈ I and y ≤ x imply that y ∈ I;
x, y ∈ I implies that x ∨ y ∈ I.

IdM denotes the lattice of ideals of M.

For a ∈ M, we denote by (a] the principal ideal generated by a in M, that is,

(a] = {x ∈ M and x ≤ a}.

By identifying a ∈ M with (a], we regard IdM as an extension of M.

3. Some preliminary constructions

In this section we shall formulate a number of constructions we shall need in the proof of the Strong Independence Theorem.

First, the extension theorem of M. Tischendorf [9]:

Theorem 5. Let L be a finite lattice with more than one element. Then L has an atomistic congruence-preserving {0}-extension K.

The second construction is due to G. Grätzer and H. Lakser; it was published in G. Grätzer [5, Lemma II.3.19].

Theorem 6. Let M be a finite chopped lattice.

(1) IdM is a congruence-preserving extension of M.

(2) If M is atomistic, then so is IdM.

Figure 1

Next an easy construction (see Figure 1):

Lemma 1. Let M_{1} and M_{2} be finite chopped lattices. Let p_{1} be an atom of M_{1} and let p_{2} be an atom of M_{2}. Define the set M as the disjoint union M_{1} ∪ M_{2} with the zeroes identified, and p_{1} identified with p_{2}. Then M is a chopped lattice containing M_{1} and M_{2} as chopped sublattices.

Combining Theorem 6 and Lemma 1, we obtain

Lemma 2. Let M_{1}, M_{2}, p_{1}, p_{2}, and M be given as in Lemma 1. Then IdM is a congruence-preserving {0}-extension of M.

Finally, Frucht’s construction.

By a graph < V, E > we mean a nonempty set V of vertices and a set E ⊂ V^{[2]} of edges (two-element subsets) without loops (that is, {x, x} ∉ E for any x ∈ E).

From a graph < V, E > we construct the Frucht lattice F r(V, E). This lattice is defined on the disjoint union

H = V ∪ E ∪ {0, 1}.

We define a partial ordering < on H as follows: 0 ≤ x ≤ 1 for all x ∈ H; and for all v ∈ V and e ∈ E, the relation v < e in H holds iff v is a vertex of the edge e. Then F r(V, E) is an atomistic lattice of length three.

This construction, and the following result, are basically due to R. Frucht [3] and [4]:

Theorem 7.

(1) Let G be a finite group. Then there exists a finite connected graph < V, E > such that the automorphism group of F r(V, E) is isomorphic to G.

(2) The graph < V, E > in the previous statement can be assumed to satisfy the condition: (Simp) For any v ∈ V, there are e_{0}, e_{1} ∈ E with v ∉ e_{0}, e_{1} and e_{0} ∩ e_{1} = ∅. If this condition is satisfied, then F r(V, E) is a simple lattice.

(3) For every natural number n, there are n finite graphs < V_{i}, E_{i} >, i < n, such that the corresponding lattices F r(V_{i}, E_{i}), i < n, are simple and rigid, and for i ≠ j, i, j < n, F r(V_{i}, E_{i}) has no embedding into F r(V_{j}, E_{j}).

Condition (Simp) is from G. Grätzer and H. Lakser [6]. Theorem 7 (3) goes somewhat further than Frucht did; for the categorical background and results needed for this, we refer the reader to P. Pultr and V. Trnková [8].

4. Automorphism-preserving extension to a simple lattice

To prove the Strong Independence Theorem, we start with the construction of an extension that retains the automorphisms but destroys the congruences.

Theorem 8. Let L be an arbitrary finite lattice with more than one element. Then L has a finite {0}-extension K such that the following conditions are satisfied:

(1) K is atomistic.

(2) K is sectionally complemented.

(3) K is simple.

(4) K is an automorphism-preserving extension of L; so AutK ≅ AutL.

(5) K has an atom u that is fixed under all the automorphisms.

Proof. We apply Theoren 7 (3) with n = 1 to obtain a graph < V, E >, and the corresponding simple rigid atomistic lattice F of length 3. For every a > 0, a ∈ L, we take a copy F_{a} of the lattice F with zero 0_{a} and unit 1_{a}. We form the disjoint union

K̄ = L ∪ ⋃{F_{a} − {0_{a}, 1_{a}} | a ∈ L − {0}}

and we identify 0_{a} with 0 and 1_{a} with a.

For x, y ∈ K̄, we define x ∧ y and x ∨ y as follows:

(i) Let L and all the F_{a}-s be sublattices of K̄.

(ii) If x ∈ F_{a} − L, y ∈ F_{b} − L, a ≠ b, then x ∧ y = 0 and x ∨ y = a ∨ b.

(iii) If x ∈ F_{a} − L, y ∈ L − F_{a}, then

x ∧ y = \{ \begin{matrix}x, & if a ≤ y, \\ 0, & otherwise; \end{matrix}

and x ∨ y = a ∨ y. And symmetrically. It is an easy computation to show that K̄ is a lattice containing L and all the F_{a},

a ∈ L − {0}, as sublattices.

Finally, we adjoin an element u to K̄ to obtain K; u is a common complement to all the elements of K̄ − {0, 1}; see Figure 2.

We prove that K satisfies the conditions of the theorem.

Ad (1). Every F_{a} is atomistic. Therefore, so is K.

Ad (2). We distinguish two cases. Case 1. y < x ∈ K. If x, y ∈ F_{a}, for some a ∈ L−{0}, then there is a sectional complement z of y in [0, x], because every F_{a} is sectionally complemented. Case 2. x ∈ L. If y ∈ L, then take an arbitrary element z ∈ F_{x} − {0, x}; then z is obviously a relative complement of y in the interval [0, x]. Otherwise, y ∉ L, so y ∈ F_{x}; then any complement z of y in F_{x} will be a relative complement of y in the interval [0, x].

Ad (3). By (2), K is sectionally complemented. Therefore, every congruence relation is of the form Θ(0, a). So it is enough to prove that Θ(0, a) is the unit congruence ι, for all 0 < a. If a ≠ u, then a ≡ 0 (Θ) implies that u ≡ 1 (Θ). Let b ∈ K − {0, u, 1}; then from the last congruence we obtain that b ≡ 0 (Θ). Since we can write 1 as the join of two such elements b, we conclude that 1 ≡ 0 (Θ), and so Θ = ι. A similar computation shows that u ≡ 0 (Θ) implies that Θ = ι.

Ad (4). Let α be an automorphism of L. We define the extension α̂ to K. If a, b ∈ L and aα=b, then let α̂ map F_{a} onto F_{b} under the unique isomorphism between F_{a} and F_{b}. Finally, define uα̂ = u. Then α̂ is obviously an extension of α to K.

We prove that every automorphism β of K is of the form α̂ for a suitable automorphism α of L. Let a ∈ L − {0}. We are going to define aα.

Let p be an atom of F_{a}; then p is an atom of K not covered by 1. Therefore, pβ is an atom of K not covered by 1. Now observe that every atom of K not covered by 1 is an atom of some F_{b}, b ∈ L − {0}. Therefore, pβ ∈ F_{b} for some b ∈ L − {0}. Let q be an atom of F_{a} such that {p, q} is an edge in the graph < V, E >. Then p ∨ q ∈ F_{a} covers p and q. Therefore, (p ∨ q)β covers pβ and qβ. F_{b} is of length 3, and an atom of F_{b} is covered only by elements of F_{b} by the construction of K. Since pβ is an atom in F_{b}, it follows that (p ∨ q)β ∈ F_{b}, and so qβ ∈ F_{b}. The graph defining F is connected, so we conclude that the image of any atom of F_{a} is in F_{b}. The lattice F_{a} is atomistic, so we obtain that F_{a}β is a sublattice of F_{b}. Since F_{a} and F_{b} are copies of the same finite rigid lattice F, it follows that β maps F_{a} onto F_{b}; in particular, a to b. We define aα = b.

This defines α on L − {0}; adding 0α = 0, we now have α defined on L. It is obvious that α is an automorphism of L, and by the rigidity of F, α̂ = β.

Ad (5). Obviously, u.

Figure 2

5. Congruence-preserving extension to a rigid lattice

We continue with the construction of an extension that retains the congruences but destroys the automorphisms.

Theorem 9. Let L be an arbitrary finite lattice with more than one element. Then L has a finite {0}-extension K such that the following conditions are satisfied:

(1) K is atomistic.

(2) K is rigid.

(3) K is a congruence-preserving extension of L; so ConK ≅ ConL.

Proof. By Tischendorf’s result (Theorem 5), we can assume that L is atomistic.

Figure 3

Now for every atom a in L, we choose a rigid Frucht lattice R_{a} with zero 0_{a} and unit 1_{a}; we fix an atom p_{a} in R_{a} in an arbitrary fashion. We assume that if a and b are distinct atoms of L, then R_{a} has no embedding into R_{b}. The existence of such lattices is guaranteed by Theorem 7 (3).

Let M be the chopped lattice we obtain from L and the lattices R_{a} by inductively applying Lemma 1 to the atoms a of L and the lattices R_{a} (with atom p_{a}); see Figure 3.

Claim 1. M is a rigid chopped lattice.

M is an chopped lattice by Lemma 1. Let α be an automorphism of M, and let a be an atom of L. Then

1_{a}α = 1_{a}.

To prove this let us assume that 1_{a}α ≠ 1_{a}. We distinguish two cases. Case 1. 1_{a}α ∈ R_{b} for an atom b of L, b ≠ a. Then we get an embedding of R_{a} into R_{b}, contradicting Theorem 7 (3). Case 2. If 1_{a}α < 1_{L}, then there exist x, y ∈ L with x ≺ 1_{a}α, x ≺ y. It follows that xα^{−1} ≺ 1_{a} and xα^{−1} ≺ yα^{−1}. The only element covering xα^{−1} is 1_{a}, contradicting that α is one-to-one. So 1_{a}α = 1_{L}. But 1_{L} has the property that every atom (x) below it is contained in an element (1_{x}) incomparable to it (1_{L}). This property fails for 1_{a}, so we conclude that 1_{a}α = 1_{a}.

Since 1_{a}α = 1_{a}, the lattice R_{a} is mapped into itself by α; but R_{a} is rigid, hence α is the identity map on R_{a}. Thus the atom a of L, and consequently, every atom of L, is fixed by α. Since L is atomistic, it follows that α keeps every element of L fixed.

This completes the verification of Claim 1.

Claim 2. M is a congruence-preserving extension of L.

Indeed, let Θ be a congruence relation of L. We define a relation Θ̄ on M as follows: For x, y ∈ L, x ≡ y (Θ̄) iff x ≡ y (Θ). For x ∈ L and y ∈ R_{a} − L, x ≡ y (Θ̄) iff x ≡ a (Θ) and a ≡ 0 (Θ); and symmetrically. For x, y ∈ R_{a}, and x ≠ y, x ≡ y (Θ̄) iff a ≡ 0 (Θ).

For x ∈ R_{a} − L, y ∈ R_{b} − L, and a ≠ b, x ≡ y (Θ̄) iff a ≡ 0 (Θ) and b ≡ 0 (Θ).

It is easy to check that Θ̄ is a congruence of M and that every congruence of M is of this form.

This completes the verification of Claim 2.

By Theorem 6, K = IdM satisfies the conditions of the theorem.

6. Proof of the Strong Independence Theorem

We are given the two finite lattices L_{C} and L_{A}. We construct the lattice K of the Strong Independence Theorem in three steps.

Step 1. Let K_{A} be the lattice provided for L_{A} by Theorem 8; the unit of K_{A} is denoted by 1_{K_{A}}. Observe that K_{A} is simple and

AutK_{A} ≅ AutL_{A}.

Step 2. Let K_{C} be the lattice provided for L_{C} by Theorem 9; the unit of K_{C} is denoted by 1_{K_{C}}. Note that K_{C} is rigid and

ConK_{C} ≅ ConL_{C}.
We can assume that K_{A} ∩ K_{C} = {0}.

We can further assume that K_{A} ≢ K_{C}. Indeed, if K_{A} ≅ K_{C}, then we can take M = K_{A}.

Step 3. Let v be any atom of K_{C}. Let u denote the atom of K_{A} that is fixed by all the automorphisms. We construct the chopped lattice M (see Figure 4) by applying Lemma 1 to K_{C} and K_{A} with the atoms v and u, respectively. Since both K_{C}, K_{A} are atomistic and K_{A} is simple, by imitating (in this easier setup) the proof of Theorem 9 (3), we get that M is a congruence-preserving extension of K_{C}, hence,

ConM ≅ ConK_{C} ≅ ConL_{C}.

An automorphism must take 1_{K_{A}} into 1_{K_{A}} or 1_{K_{C}}. But if 1_{K_{A}} is mapped onto 1_{K_{C}}, then K_{A} ≅ K_{C}, contrary to our assumption. So 1_{K_{A}}, and similarly, 1_{K_{C}} are fixed by all automorphisms.

So 1_{K_{C}} is fixed; since K_{C} is rigid, all automorphisms of M fix u, and both K_{C} and K_{A} are atomistic, it follows that

AutM ≅ AutK_{A} ≅ AutL_{A}.

Finally, we form K = IdM. We claim that K satisfies the conditions of the Strong Independence Theorem. Indeed, by Theorem 6, K is a congruence-preserving extension of M, hence, it is a congruence-preserving extension of L_{C}. Moreover, since K_{C} is rigid and

Figure 4

both K_{C}, K_{A} are atomistic, K is an automorphism-preserving extension of K_{A}, hence, K is an automorphism-preserving extension of L_{A}.

This completes the proof of the Strong Independence Theorem.

7. Concluding remarks

There are several known proofs of the Independence Theorem. In addition to V. A. Baranskiĭ [1] and A. Urquhart [10], see also G. Grätzer and H. Lakser [6] for a proof. All these proofs use some form of Theorem 1, and utilize the special properties of the lattice K whose congruence lattice D is given. For instance, K is sectionally complemented; the algebraic reasons for an equation such as

Θ(a_{1}, b_{1}) ∨ Θ(a_{2}, b_{2}) = Θ(a_{3}, b_{3})

are easily given in a sectionally complemented lattice.

In one respect, the Strong Independence Theorem seems weaker than the Independence Theorem: the Independence Theorem implies the representation theorems (Theorems 1 and 2), while the Strong Independence Theorem does not. Of course, no proof of the Independence Theorem is known that does not utilize the representation theorems.

The Strong Independence Theorem should focus attention to the study of lattice extensions.

We hope that with this new approach to independence theorems, we shall succeed in (1) proving stronger forms of existing independence theorems;

(2) generalizing existing independence theorems;

(3) proving independence theorems for classes of lattices for which the characterization theorems are not known. The present paper is an illustration of this new method with a theorem of type (1).

We hope to illustrate the other types in future papers.

The most fundamental open problem is whether the Strong Independence Theorem holds for arbitrary lattices. We formulate this in two ways:

Problem 1. Let L_{C} and L_{A} be disjoint lattices with more than one element. Does there exist a lattice K that is a congruence-preserving extension of L_{C} and an automorphismpreserving extension of L_{A}?

Problem 2. Let L_{C} and L_{A} be lattices with zero and with more than one element satisfying L_{C} ∩ L_{A} = {0}. Does there exist a lattice K that is a congruence-preserving {0}-extension of L_{C} and an automorphism-preserving {0}-extension of L_{A}?

The level of our ignorance can be best illustrated with the following problem:

Problem 3. Is it true that every lattice with more than one element has a proper congruence-preserving extension?

References

[1] V. A. Baranskiĭ: On the independence of the automorphism group and the congruence lattice for lattices, Abstracts of lectures of the 15th All-Soviet Algebraic Conference, Krasnojarsk, July 1979, vol. 1, p. 11.

[2] G. Birkhoff: On groups of automorphisms. (Spanish) Rev. Un. Math. Argentina 11 (1946), 155–157.

[3] R. Frucht: Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compos. Math. 6 (1938), 239–250.

[4] R. Frucht: Lattices with a given group of automorphisms, Canad. J. Math. 2 (1950), 417–419.

[5] G. Grätzer: General lattice theory, Academic Press, New York, N. Y.; Birkhäuser Verlag, Basel; Akademie Verlag, Berlin, 1978.

[6] G. Grätzer and H. Lakser: Homomorphisms of distributive lattices as restrictions of congruences. II. Planarity and automorphisms, Canadian J. Math. 46 (1) (1994), 3–54.

[7] G. Grätzer and E. T. Schmidt: On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 179–185.

[8] A. Pultr and V. Trnková: Combinatorial, algebraic and topological representations of groups, semigroups and categories, North-Holland Mathematical Library, vol. 22, North-Holland, Amsterdam-New York-Oxford, 1980.

[9] M. Tischendorf: The representation problem for algebraic distributive lattices, Ph.D. thesis, Darmstadt, 1992.

[10] A. Urquhart, A topological representation theory for lattices, Algebra Universalis 8 (1978), 45–58.

Received November 1, 1993; revised version October 2, 1994.