Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 85-92.

Modular Elements in Congruence Lattices of G-sets

Boris M. Vernikov

Department of Mathematics and Mechanics, Ural State University Lenina 51, 620083 Ekaterinburg, Russia e-mail: Boris.Vernikov@usu.ru

Abstract. An element x in a lattice ⟨L; ∨, ∧⟩ is called modular if (x ∨ y) ∧ z = (x ∧ z) ∨ y whenever y, z ∈ L and y ≤ z. We describe modular elements in congruence lattices of G-sets. MSC 1991: 08A60, 08A30 Keywords: G-set, congruence lattice, modular element in a lattice.

Introduction

Let A be a non-empty set, G a group, and φ a homomorphism from G into the full transformation group of A. Then A may be considered as a unary algebra with the set G of operations where an operation g ∈ G is defined by the rule g(a) = (φ(g))(a) for every a ∈ A. These algebras are called G-sets. Some basic information about G-sets and, in particular, about their congruences may be found, for example, in [1]. In [3], we described G-sets with several natural properties of their congruences, in particular, G-sets whose congruence lattice is distributive, modular, arguesian, semimodular, etc.

Studying congruences of G-sets appears to be sufficiently natural per se. Our main motivation, however, comes from a different source. Recent results by M. V. Volkov and the author show that, for some wide classes of semigroup varieties, the structure of subvariety lattices can be described in terms of congruence lattices of certain G-sets (see, in particular, [5, 6]). In particular, it is the case for overcommutative semigroup varieties (that is, varieties containing the variety of all commutative semigroups). These results together with the mentioned results of [3] permit to obtain a series of results concerning identities in lattices of overcommutative varieties and related questions (see [4]).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

Here we consider a new restriction to congruences on G-sets. Recall that an element x in a lattice ⟨L; ∨, ∧⟩ is called modular if

(x ∨ y) ∧ z = (x ∧ z) ∨ y

whenever y, z ∈ L and y ≤ z. In the present paper we describe modular elements in congruence lattices of G-sets. We think that this result certainly may be applied for a description of modular elements in the lattice of overcommutative semigroup varieties.

The paper consists of three sections. Section 1 contains preliminary information from [3] about congruences on G-sets. At the conclusion of this section, we formulate the main result of the work. It is proved in Sections 2 and 3.

1. Preliminaries

The congruence lattice of a G-set A is denoted by Con(A). A G-set A is said to be transitive if, for any pair of elements a, b ∈ A, there exists g ∈ G such that g(a) = b. It is well known (see, for example, [1, Lemma 4.20]) that if A is a transitive G-set then the lattice Con(A) is isomorphic to an interval of the subgroup lattice of G (more precisely, Con(A) ≅ [Stab_{G}(a), G] where a is an arbitrary element in A and Stab_{G}(a) = {g ∈ G | g(a) = a}).

A transitive G-subset of a G-set A is called an orbit of A. Clearly, any G-set is a disjoint union of its orbits. In view of the remark in the previous paragraph, it is natural to investigate the lattice Con(A) modulo the congruence lattices of the orbits of A.

Let A be a G-set and μ a congruence on A. We say that μ isolates an orbit B of A if B is the union of μ-classes. We say that μ connects (collapses) orbits B and C of A if B ≠ C and there are elements b ∈ B and c ∈ C with b μ c (respectively, if B ≠ C and x μ y whenever x, y ∈ B ∪ C). If M is a non-singleton family of orbits of A then we say that μ connects (collapses) M if μ connects (collapses) any pair of different orbits of M. We call μ greedy if it collapses any pair of orbits it connects. By μ^{∗} we denote the following binary relation on the set of all orbits of A: B μ^{∗} C if and only if either B = C or μ connects B and C. Clearly, μ^{∗} is an equivalence relation. Let GCon(A) denote the set of all greedy congruences of A. We denote by Orb(A) the set of all orbits of A and by Eq(X) the lattice of equivalence relations on a set X. The following lemma clarifies the structure of GCon(A).

Lemma 1.1. ([3, Lemma 1.1 and Proposition 1.2]) Let A be a G-set and Orb(A) = {A_{i} | i ∈ I}. The set GCon(A) forms a sublattice of the lattice Con(A). The lattice GCon(A) is isomorphic to a subdirect product of the lattices Eq(Orb(A)) and Con(A_{i}) where i runs over I. The corresponding embedding f of GCon(A) into Eq(Orb(A)) × ∏_{i∈I} Con(A_{i}) is given by the

rule: if α ∈ GCon(A) then f(α) = (α^{∗}; …, α_{i}, …) where α_{i} is the restriction of α to the orbit A_{i}. □

We need also information about non-greedy congruences on a G-set.

Lemma 1.2. ([3, proof of Proposition 1.3]) Let A be a G-set. Suppose that a congruence μ on A connects but does not collapse the orbits A_{i} and A_{j} of A. Put μ_{i} = μ|_{A_{i}} and μ_{j} = μ|_{A_{j}}. Then the G-sets A_{i}/μ_{i} and A_{j}/μ_{j} are isomorphic. More precisely, the following mapping μ_{i,j}^{∗} : A_{i}/μ_{i} ⟼ A_{j}/μ_{j} is an isomorphism: if M_{i} is a μ_{i}-class, then M_{j} = μ_{i,j}^{∗}(M_{i}) is the μ_{j}-class such that x μ y for some x ∈ M_{i} and y ∈ M_{j}. □

It is evident that if, in notation of Lemma 1.2, M_{j} = μ_{i,j}^{∗}(M_{i}) then x μ y whenever x, y ∈ M_{i} ∪ M_{j}. One can note also that the isomorphisms μ_{i,j}^{∗} and μ_{j,i}^{∗} are mutually inverse.

Let μ be a non-greedy congruence on A and M = {M_{i} | i ∈ J} a non-singleton μ^{∗}-class such that μ does not collapse M. For any congruence ξ on A and i ∈ J, let ξ_{i} = ξ|_{M_{i}}. Let us fix some j ∈ J. By Lemma 1.2, M_{i}/μ_{i} ≅ M_{j}/μ_{j} for any i ∈ J. Let μ_{i,j}^{#} be the isomorphism from Con(M_{i}/μ_{i}) onto Con(M_{j}/μ_{j}) induced by the isomorphism μ_{i,j}^{∗} from M_{i}/μ_{i} onto M_{j}/μ_{j}. For a congruence χ on A with χ ⊇ μ_{i}, we put χ_{i,j,μ} = μ_{i,j}^{#}(χ_{i}/μ_{i}). One can note that μ_{j,j}^{∗} and μ_{j,j}^{#} are identical mappings and that χ_{j,j,μ} = χ_{j}/μ_{j}.

The main result of the work is the following

Theorem. Let A be a G-set. A congruence α on A is a modular element in the lattice Con(A) if and only if the following holds:

a) α either isolates every orbit of A or collapses some non-singleton set P of orbits of A and isolates every orbit of A outside P;

b) the restriction of α to any orbit of A is a modular element in the congruence lattice of this orbit;

c) if β and γ are congruences on A with β ⊂ γ and B = {B_{i} | i ∈ J} is a non-singleton β^{∗}-class such that α isolates every orbit in B while each of the congruences β and γ does not collapse B then there is a j ∈ J such that

( ∨_{i∈J} α_{i,j,β}) ∧ γ_{j,j,β} = ∨_{i∈J}(α_{i,j,β} ∧ γ_{j,j,β}).  (1)

It is easy to verify that if α is a modular element in Con(A) then the equality (1) actually holds for every j ∈ J.

Clearly, a lattice L is modular if and only if every x ∈ L is a modular element in L. This permits us to easily deduce from Theorem the description of G-sets with modular congruence lattices given by [3, Corollary 2.4].

We note also that the lattice of equivalence relations on a non-empty set A can be considered as the congruence lattice of some G-set. Indeed, we may consider A as a G-set where G is the singleton group. Clearly, any equivalence relation on A is a congruence of this G-set and the lattices Eq(A) and Con(A) coincide. Thus, our Theorem extends the description of modular elements in lattices of equivalence relations obtained in [2].

Sections 2 and 3 are devoted, respectively, to the proof of necessity and sufficiency of Theorem.

2. Necessity

We start with the following

Lemma 2.1. Let A be a G-set. If a congruence α on A is a modular element in the lattice Con(A) then α is greedy.

Proof. Suppose that α is not greedy. Then it connects but does not collapse some pair of orbits A_{i} and A_{j} of A. Let us consider the following three congruences on A:

α_{i,j}: x α_{i,j} y if and only if either x = y or x, y ∈ A_{i} and x α y or x, y ∈ A_{j} and x α y;

β: x β y if and only if either x = y or x, y ∈ A_{i} and x α y or x, y ∈ A_{j};
γ: x γ y if and only if either x = y or x, y ∈ A_{i} or x, y ∈ A_{j}.

Since α connects the orbits A_{i} and A_{j}, if the restriction of α to A_{i} is the universal relation on this orbit then α collapses A_{i} and A_{j}. Hence there are elements x, y ∈ A_{i} such that (x, y) ∉ α. On the other hand, since α connects A_{i} and A_{j}, there is an element x′ ∈ A_{j} such that x α x′. Furthermore, since A_{i} is an orbit, there is g ∈ G with y = g(x). We have g(x) α g(x′) and g(x′) ∈ A_{j}. Hence x α x′ β g(x′) α g(x) = y. Thus, (x, y) ∈ α ∨ β. Furthermore, it is evident that x γ y. Therefore, (x, y) ∈ (α ∨ β) ∧ γ. Since α is a modular element in Con(A) and β < γ, we have (x, y) ∈ (α ∧ γ) ∨ β. But it is evident that (α ∧ γ) ∨ β = α_{i,j} ∨ β = β. We have x β y. Since x, y ∈ A_{i}, this implies x α y. A contradiction. □

Now, let us verify the necessity of Theorem. Let A be a G-set and α a modular element in Con(A). By Lemma 2.1, α is greedy. Since GCon(A) is a sublattice in Con(A) (see Lemma 1.1), α is a modular element in GCon(A) too. It is clear that if a lattice L is a subdirect product of lattices {L_{λ} | λ ∈ Λ} then an element x ∈ L is modular if and only if, for any λ ∈ Λ, the projection of x to L_{λ} is a modular element in L_{λ}. By Lemma 1.1, we immediately have that the condition b) of Theorem holds and that the equivalence relation α^{∗} is a modular element in the lattice Eq(Orb(A)). Results of the article [2] easily imply that an equivalence relation ρ on a set X is a modular element in the lattice Eq(X) if and only if either ρ is the equality relation or ρ has exactly one non-singleton class. We see that the condition a) of Theorem holds. It remains to verify the condition c). Here we need the following

Lemma 2.2. Let A be a G-set, B_{i} and B_{j} different orbits of A and β and γ congruences on A such that β ⊆ γ and each of these congruences connects but does not collapse the orbits B_{i} and B_{j}. Then γ_{i,j,β} = γ_{j,j,β}.

Proof. For μ ∈ {β, γ}, we put μ_{i} = μ|_{B_{i}} and μ_{j} = μ|_{B_{j}}.

Let P γ_{j,j,β} Q, p ∈ P and q ∈ Q. Since γ_{j,j,β} = γ_{j}/β_{j} and β_{j} ⊆ γ_{j}, we have p γ q. Let now P′ = β_{j,i}^{∗}(P), Q′ = β_{j,i}^{∗}(Q), p′ ∈ P′ and q′ ∈ Q′. Then p′ β p γ q β q′ whence p′ γ q′. This implies that (P′, Q′) ∈ γ_{i}/β_{i}. Since P = β_{i,j}^{∗}(P′) and Q = β_{i,j}^{∗}(Q′), we have P γ_{i,j,β} Q. Thus, γ_{j,j,β} ⊆ γ_{i,j,β}.

Let now P γ_{i,j,β} Q. Put P′ = β_{j,i}^{∗}(P) and Q′ = β_{j,i}^{∗}(Q). Then (P′, Q′) ∈ γ_{i}/β_{i}. Let p′ ∈ P′ and q′ ∈ Q′. Since β_{i} ⊆ γ_{i}, we have p′ γ q′. Let now p ∈ P and q ∈ Q. Since P = β_{i,j}^{∗}(P′) and Q = β_{i,j}^{∗}(Q′), we have p β p′ γ q′ β q whence p γ q. This implies that (P, Q) ∈ γ_{j}/β_{j} = γ_{j,j,β}. Thus, γ_{i,j,β} ⊆ γ_{j,j,β} whence γ_{i,j,β} = γ_{j,j,β}. □

Let us return to the proof of necessity. Let β and γ be congruences on A with β ⊂ γ and B = {B_{i} | i ∈ J} a non-singleton β^{∗}-class such that α isolates every orbit of B while each of the congruences β and γ does not collapse B. We need to verify that the equality (1) holds for some j ∈ J. Let us prove that, in fact, this equality is valid for an arbitrary j ∈ J. Indeed, let us fix j ∈ J. Clearly, the right part of the equality (1) is contained in the left one. Thus, it remains to check that

( ∨_{i∈J} α_{i,j,β}) ∧ γ_{j,j,β} ≤ ∨_{i∈J}(α_{i,j,β} ∧ γ_{j,j,β}).

Let P and Q be β_{j}-classes such that (P, Q) ∈ ( ∨_{i∈J} α_{i,j,β}) ∧ γ_{j,j,β}. Then P γ_{j,j,β} Q and there are β_{j}-classes P_{0}, P_{1}, …, P_{n} such that P = P_{0}, P_{n} = Q and, for any s = 0, 1, …, n − 1, there is i_{s} ∈ J such that P_{s} α_{i_{s},j,β} P_{s+1}. Put P′_{0} = β_{j,i_{0}}^{∗}(P_{0}) and P′_{1} = β_{j,i_{0}}^{∗}(P_{1}). By the definition of the mapping β_{j,i_{0}}^{∗}, we have (P′_{0}, P′_{1}) ∈ α_{i_{0}}/β_{i_{0}}. This means that if p′_{0} ∈ P′_{0} and p′_{1} ∈ P′_{1} then (p′_{0}, p′_{1}) ∈ α_{i_{0}} ∨ β_{i_{0}}. Besides that, if p_{0} ∈ P_{0} and p′_{0} ∈ P′_{0} then p_{0} β p′_{0}. Furthermore, put P′′_{1} = β_{j,i_{1}}^{∗}(P_{1}) and P′_{2} = β_{j,i_{1}}^{∗}(P_{2}). Then (P′′_{1}, P′_{2}) ∈ α_{i_{1}}/β_{i_{1}}, and therefore, (p′′_{1}, p′_{2}) ∈ α_{i_{1}} ∨ β_{i_{1}} whenever p′′_{1} ∈ P′′_{1} and p′_{2} ∈ P′_{2}. Furthermore, since β_{i_{1},j}^{∗}(P′′_{1}) = P_{1} = β_{i_{0},j}^{∗}(P′_{1}), we have that if p_{1} ∈ P_{1}, p′_{1} ∈ P′_{1} and p′′_{1} ∈ P′′_{1} then p′_{1} β p_{1} β p′′_{1} whence p′_{1} β p′′_{1}. Furthermore, put P′′_{2} = β_{j,i_{2}}^{∗}(P_{2}) and P′_{3} = β_{j,i_{2}}^{∗}(P_{3}). Then p′_{2} β p′′_{2} and (p′′_{2}, p′_{3}) ∈ α_{i_{2}} ∨ β_{i_{2}} whenever p′_{2} ∈ P′_{2}, p′′_{2} ∈ P′′_{2} and p′_{3} ∈ P′_{3}. Continuing this process, we construct P′′_{3}, P′_{4}, P′′_{4}, …, P′_{n−1}, P′′_{n−1} and P′_{n} such that p′_{3} β p′′_{3}, (p′′_{3}, p′_{4}) ∈ α_{i_{3}} ∨ β_{i_{3}}, …, p′_{n−1} β p′′_{n−1} and (p′′_{n−1}, p′_{n}) ∈ α_{i_{n−1}} ∨ β_{i_{n−1}} whenever p′_{3} ∈ P′_{3}, p′′_{3} ∈ P′′_{3}, p′_{4} ∈ P′_{4}, …, p′_{n−1} ∈ P′_{n−1}, p′′_{n−1} ∈ P′′_{n−1} and p′_{n} ∈ P′_{n}. Further, if p′_{n} ∈ P′_{n} and p_{n} ∈ P_{n} then p′_{n} β p_{n}. Now let us fix arbitrary elements p ∈ P_{0} and q ∈ P_{n}. The considerations given above show that (p, q) ∈ α ∨ β. In addition, since P_{0} γ_{j,j,β} P_{n} and β ⊆ γ, we have p γ q. Thus, (p, q) ∈ (α ∨ β) ∧ γ.

Since α is a modular element in the lattice Con(A), we have (p, q) ∈ (α ∧ γ) ∨ β. Hence there are elements r_{0}, r_{1}, …, r_{m} ∈ A such that r_{0} = p, r_{m} = q and, for any s = 0, 1, …, m−1, either (r_{s}, r_{s+1}) ∈ α∧γ or r_{s} β r_{s+1}. Let A_{i} be the orbit of A such that r_{i} ∈ A_{i} (i = 0, 1, …, m). Recall that r_{0} = p ∈ P_{0} = P and r_{m} = q ∈ P_{n} = Q. Since P and Q are β_{j}-classes, we see that A_{0} = A_{m} = B_{j} ∈ B. By the hypothesis, α isolates any orbit of B, and therefore, α ∧ γ also does. This means that, for any s = 0, 1, …, m − 1, if A_{s} ≠ A_{s+1} then r_{s} β r_{s+1}, and therefore, A_{0}, A_{1}, …, A_{m} ∈ B. Suppose that A_{0} = A_{1} = ⋯ = A_{k_{0}} (all these orbits coincide with B_{j}) and either k_{0} = m or A_{k_{0}} ≠ A_{k_{0}+1}. Then (r_{0}, r_{k_{0}}) ∈ (α_{j} ∧ γ_{j}) ∨ β_{j}. Recall that the β_{j}-class containing r_{0} = p is P. Let us denote by R_{1} the β_{j}-class containing r_{k_{0}}. Clearly, (P, R_{1}) ∈ (α_{j} ∧ γ_{j})/β_{j} = (α_{j}/β_{j}) ∧ (γ_{j}/β_{j}) = α_{j,j,β} ∧ γ_{j,j,β}. Put R′_{0} = P and R′_{1} = R_{1}. Furthermore, suppose that k_{0} < m. Let A_{k_{0}+1} = A_{k_{0}+2} = ⋯ = A_{k_{1}} and either k_{1} = m or A_{k_{1}} ≠ A_{k_{1}+1}. Since A_{k_{0}+1} ∈ B, we have that A_{k_{0}+1} = A_{k_{0}+2} = ⋯ = A_{k_{1}} = B_{i_{1}} for some i_{1} ∈ J. Clearly, (r_{k_{0}+1}, r_{k_{1}}) ∈ (α_{i_{1}} ∧ γ_{i_{1}}) ∨ β_{i_{1}}. We denote by R_{2} the β_{i_{1}}-class containing r_{k_{0}+1} and by R_{3} the β_{i_{1}}-class containing r_{k_{1}}. Then (R_{2}, R_{3}) ∈ (α_{i_{1}} ∧ γ_{i_{1}})/β_{i_{1}} = (α_{i_{1}}β_{i_{1}}) ∧ (γ_{i_{1}}/β_{i_{1}}). Recall that r_{k_{0}} β r_{k_{0}+1}. Therefore, (α ∧ γ)_{i_{1},j}^{∗}(R_{2}) = R′_{1}. Put R′_{2} = (α ∧ γ)_{i_{1},j}^{∗}(R_{3}). Then (R′_{1}, R′_{2}) ∈ α_{i_{1},j,β} ∧ γ_{i_{1},j,β}. Using Lemma 2.2, we have (R′_{1}, R′_{2}) ∈ α_{i_{1},j,β} ∧ γ_{j,j,β}. Continuing this process, we obtain β_{j}-classes R′_{3}, …, R′_{t} such that R′_{t} = Q and (R′_{2}, R′_{3}) ∈ α_{i_{2},j,β} ∧ γ_{j,j,β}, (R′_{3}, R′_{4}) ∈ α_{i_{3},j,β} ∧ γ_{j,j,β}, …, (R′_{t−1}, R′_{t}) ∈ α_{i_{t−1},j,β} ∧ γ_{j,j,β} for some i_{2}, i_{3}, …, i_{t−1} ∈ J. Therefore, (P, Q) ∈ ∨_{i∈J}(α_{i,j,β} ∧ γ_{j,j,β}), and we are done.

3. Sufficiency

Let α be a congruence on a G-set A and suppose the conditions a)–c) of Theorem hold. We are going to verify that α is a modular element in the lattice Con(A). Let us fix congruences β and γ on A with β ⊆ γ. It is clear that (α ∧ γ) ∨ β ⊆ (α ∨ β) ∧ γ. It remains to verify that (α ∨ β) ∧ γ ⊆ (α ∧ γ) ∨ β. Evidently, we may (and will) assume that β ⊂ γ. Suppose that x, y ∈ A and (x, y) ∈ (α ∨ β) ∧ γ. We need to check that

(x, y) ∈ (α ∧ γ) ∨ β.  (2)

Since (x, y) ∈ (α ∨ β) ∧ γ, there are elements x_{0}, x_{1}, …, x_{n} ∈ A such that x_{0} = x, x_{n} = y and, for any i = 0, 1, …, n − 1, either x_{i} α x_{i+1} or x_{i} β x_{i+1}. Besides that x γ y. Let us denote by A_{i} the orbit of A containing x_{i} (i = 0, 1, …, n). If A_{0} = A_{1} = ⋯ = A_{n} then (x, y) ∈ (α_{0} ∨ β_{0}) ∧ γ_{0} (where μ_{0} is the restriction of μ to A_{0} for any μ ∈ {α, β, γ}). By the condition b) of Theorem, (x, y) ∈ (α_{0} ∧ γ_{0}) ∨ β_{0} ⊆ (α ∧ γ) ∨ β, and we are done. Thus, we may assume that A_{i} ≠ A_{i+1} for some i ∈ {0, 1, …, n − 1}.

Suppose at first that there is i ∈ {0, 1, …, n − 1} such that A_{i} ≠ A_{i+1} and x_{i} α x_{i+1}. Let

j = min_{0≤i≤n−1}{A_{i} ≠ A_{i+1}, x_{i} α x_{i+1}}

and

k = max_{1≤i≤n}{A_{i−1} ≠ A_{i}, x_{i−1} α x_{i}}.

Both the orbits A_{j} and A_{k} belong to non-singleton α^{∗}-classes. But by the condition a) of Theorem, α^{∗} contains at most one non-singleton class. Hence the orbits A_{j} and A_{k} lie in the same α^{∗}-class. In other words, either A_{j} = A_{k} or α connects A_{j} and A_{k}. The condition a) of Theorem implies that α is greedy. Hence either A_{j} = A_{k} or α collapses A_{j} and A_{k}. The choice of the orbits A_{j} and A_{k} guarantees that either A_{0} = A_{j} or β connects the orbits A_{0} and A_{j} as well as either A_{k} = A_{n} or β connects the orbits A_{k} and A_{n}. Since x ∈ A_{0} and y ∈ A_{n}, there are elements x′ ∈ A_{j} and y′ ∈ A_{k} such that x β x′ and y′ β y. We have x′ β x γ y β y′, and therefore, x′ γ y′. The orbit A_{j} belongs to a non-singleton α^{∗}-class, and therefore, the restriction of α to this orbit is the universal relation on A_{j}. In particular, this means that if A_{j} = A_{k} then x′ α y′. If α collapses A_{j} and A_{k} then, evidently, x′ α y′ too. Hence (x′, y′) ∈ α ∧ γ. Thus, x β x′, (x′, y′) ∈ α ∧ γ and y′ β y. Therefore, (x, y) ∈ (α ∧ γ) ∨ β.

Thus, we may assume that, for any i = 0, 1, …, n − 1, if A_{i} ≠ A_{i+1} then x_{i} β x_{i+1}. In other words, all the orbits A_{0}, A_{1}, …, A_{n} lie in the same β^{∗}-class. Let us denote this β^{∗}-class by B. Let B = {B_{i} | i ∈ J}. Since A_{i} ≠ A_{i+1} for some i ∈ {0, 1, …, n − 1}, the class B is non-singleton. We need the following easy

Lemma 3.1. Let A be a G-set, μ a congruence on A and M a non-singleton μ^{∗}-class. If μ collapses some pair of orbits of M then μ collapses M.

Proof. Let M = {M_{i} | i ∈ J}. Suppose that μ collapses orbits M_{j} and M_{k} where j, k ∈ J and j ≠ k. Let x, y ∈ ∪_{i∈J} M_{i}. Then there are elements x′ ∈ M_{j} and y′ ∈ M_{k} such that x μ x′ and y′ μ y. Furthermore, x′ μ y′ because μ collapses M_{j} and M_{k}. We have x μ y. □

Thus, if β collapses some pair of orbits B_{i}, B_{j} ∈ B then x β y. This immediately implies the inclusion (2). Thus, we may assume that β does not collapse any pair of orbits of B.

Since β ⊆ γ, all orbits of B lie in the same γ^{∗}-class. Suppose that γ collapses some pair of orbits of B. Then, by Lemma 3.1, γ collapses B. In particular, x_{i} γ x_{i+1} for all i = 0, 1, …, n − 1. Therefore, for any i = 0, 1, …, n − 1, either (x_{i}, x_{i+1}) ∈ α ∧ γ or x_{i} β x_{i+1}. This implies (2). Thus, we may assume that γ does not collapse any pair of orbits of B either.

Suppose now that there is an orbit B_{i} ∈ B such that the restriction of α to B_{i} is the universal relation on this orbit. Since x ∈ A_{0} ∈ B and y ∈ A_{n} ∈ B, there are elements x′, y′ ∈ B_{i} such that x β x′ and y′ β y. Then x′ β x γ y β y′ whence x′ γ y′. The choice of B_{i} implies that x′ α y′. Thus, x β x′, (x′, y′) ∈ α ∧ γ and y′ β y. We have (x, y) ∈ (α ∧ γ) ∨ β. Thus, we may assume that the restriction of α to any orbit of B differs from the universal relation on this orbit. In particular, this means that α isolates any orbit of B.

Thus, it remains to consider the case when the hypothesis of the condition c) of Theorem is fulfilled. As usual, for any i ∈ J and μ ∈ {α, β, γ}, we denote by μ_{i} the restriction of μ to B_{i}. By the condition c), there is j ∈ J such that the equality (1) holds.

Each of the orbits A_{0}, A_{1}, …, A_{n} lies in B. Suppose that A_{0} = A_{1} = ⋯ = A_{k_{0}} = B_{i_{0}} (where i_{0} ∈ J) and A_{k_{0}} ≠ A_{k_{0}+1}. Clearly, (x_{0}, x_{k_{0}}) ∈ α_{i_{0}} ∨ β_{i_{0}}. We denote by R_{0} (respectively, R_{1}) the β_{i_{0}}-class containing x_{0} (respectively, x_{k_{0}}) and put P_{0} = β_{i_{0},j}^{∗}(R_{0}) and P_{1} = β_{i_{0},j}^{∗}(R_{1}). Since x = x_{0} ∈ R_{0}, we have x β p_{0} for any p_{0} ∈ P_{0}. Furthermore, it is clear that P_{0} α_{i_{0},j,β} P_{1}. Let A_{k_{0}+1} = A_{k_{0}+2} = ⋯ = A_{k_{1}} = B_{i_{1}} (where i_{1} ∈ J) and either k_{1} = n or A_{k_{1}} ≠ A_{k_{1}+1}. Clearly, (x_{k_{0}+1}, x_{k_{1}}) ∈ α_{i_{1}} ∨ β_{i_{1}}. We denote by R_{2} (respectively, R_{3}) the β_{i_{1}}-class containing x_{k_{0}+1} (respectively, x_{k_{1}}). Since x_{k_{0}} β x_{k_{0}+1}, we have β_{i_{1},j}^{∗}(R_{2}) = P_{1}. Put P_{2} = β_{i_{1},j}^{∗}(R_{3}). Clearly, P_{1} α_{i_{1},j,β} P_{2}. Continuing this process, we obtain β_{j}-classes P_{3}, …, P_{r} such that P_{2} α_{i_{2},j,β} P_{3} α_{i_{3},j,β} ⋯ α_{i_{r−2},j,β} P_{r−1} α_{i_{r−1},j,β} P_{r} for some i_{2}, i_{3}, …, i_{r−1} ∈ J and the β_{i_{r−1}}-class β_{j,i_{r−1}}^{∗}(P_{r}) contains y. The last means, in particular, that p_{r} β y for any p_{r} ∈ P_{r}. Put P = P_{0} and Q = P_{r}. We have (P, Q) ∈ ∨_{i∈J} α_{i,j,β}. Furthermore, if p ∈ P and q ∈ Q then

p β x γ y β q whence p γ q. This implies that P γ_{j,j,β} Q. Thus, (P, Q) ∈ ( ∨_{i∈J} α_{i,j,β}) ∧ γ_{j,j,β}.

According to (1), (P, Q) ∈ ∨_{i∈J}(α_{i,j,β} ∧ γ_{j,j,β}). Hence there are β_{j}-classes S_{0}, S_{1}, …, S_{m}

such that S_{0} = P, S_{m} = Q and, for any k = 0, 1, …, m − 1, (S_{k}, S_{k+1}) ∈ α_{i_{k},j,β} ∧ γ_{j,j,β} for some i_{0}, i_{1}, …, i_{m−1} ∈ J. Put S′_{0} = β_{j,i_{0}}^{∗}(S_{0}) and S′_{1} = β_{j,i_{0}}^{∗}(S_{1}). Let s′_{0} ∈ S′_{0} and s′_{1} ∈ S′_{1}. Clearly, (s′_{0}, s′_{1}) ∈ α_{i_{0}}. Furthermore, by Lemma 2.2, γ_{j,j,β} = γ_{i_{0},j,β}, and therefore, (s′_{0}, s′_{1}) ∈ γ_{i_{0}}. Thus, (s′_{0}, s′_{1}) ∈ α_{i_{0}} ∧ γ_{i_{0}} ⊆ α ∧ γ. Besides that, if s_{0} ∈ S_{0} then x β s_{0} β s′_{0} whence x β s′_{0}. Furthermore, put S′′_{1} = β_{j,i_{1}}^{∗}(S_{1}) and S′_{2} = β_{j,i_{1}}^{∗}(S_{2}). Using Lemma 2.2 again, we have that if s′′_{1} ∈ S′′_{1} and s′_{2} ∈ S′_{2} then (s′′_{1}, s′_{2}) ∈ α_{i_{1}} ∧ γ_{i_{1}} ⊆ α ∧ γ. Further, s′_{1} β s′′_{1} because β_{i_{0},j}^{∗}(S′_{1}) = S_{1} = β_{i_{1},j}^{∗}(S′′_{1}). Continuing this process, we obtain elements s′′_{2}, s′_{3}, s′′_{3}, …, s′_{m−1}, s′′_{m−1}, s′_{m} ∈ A such that s′_{2} β s′′_{2}, (s′′_{2}, s′_{3}) ∈ α_{i_{2}} ∧ γ_{i_{2}} ⊆ α ∧ γ, s′_{3} β s′′_{3}, …, s′_{m−1} β s′′_{m−1} and (s′′_{m−1}, s′_{m}) ∈ α_{i_{m−1}} ∧ γ_{i_{m−1}} ⊆ α ∧ γ. Furthermore, by the construction, s′_{m} ∈ β_{j,i_{m−1}}^{∗}(S_{m}). Since S_{m} = Q = P_{r} and y ∈ β_{j,i_{r−1}}^{∗}(P_{r}), we have that if s_{m} ∈ S_{m} then s′_{m} β s_{m} β y whence s′_{m} β y. We find a sequence of elements x, s′_{0}, s′_{1}, s′′_{1}, s′_{2}, s′′_{2}, …, s′_{m−1}, s′′_{m−1}, s′_{m}, y such that x β s′_{0}, (s′_{0}, s′_{1}) ∈ α ∧ γ, s′_{k} β s′′_{k} and (s′′_{k}, s′_{k+1}) ∈ α ∧ γ for all k = 1, 2, …, m − 1 and s′_{m} β y. Thus, (x, y) ∈ (α ∧ γ) ∨ β.

We see that the inclusion (2) always holds. Thus, we have proved the Theorem.

Acknowledgement. The author sincerely thanks Professor M. V. Volkov for helpful discussions.

References

[1] McKenzie, R. N.; McNulty, G. F.; Taylor, W. F.: Algebras. Lattices. Varieties. Vol. I. Wadsworth&Brooks/Cole, Monterey 1987.

[2] Ore, O.: Theory of equivalence relations. Duke Math. J. 9 (1942), 573–627.

[3] Vernikov, B. M.: On congruences of G-sets. Comment. Math. Univ. Carolin. 38 (1997), 603–613.

[4] Vernikov, B. M.: Quasiidentities implying modularity and related conditions in lattices of overcommutative semigroup varieties. Semigroup Forum 57 (1998), 142–150.

[5] Vernikov, B. M.; Volkov, M. V.: Lattices of nilpotent semigroup varieties II. Izv. Ural State Univ. Ser. Matem., Mechan. 10 (1) (1998), 13–33 [Russian].

[6] Volkov, M. V.: Young diagrams and the structure of the lattice of overcommutative semigroup varieties. In P. M. Higgins (ed.): Transformation Semigroups. Proc. Int. Conf. held at the Univ. Essex. Univ. Essex, Colchester, (1994), 99–110.

Received August 4, 1998; revised version January 5, 1999