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No general rule for determining the number N(n) of topologies definable for a finite set of cardinal n is known. In this note we relate N(n) to a function Ft(r1,…, rt+1) defined below which has a simple combinatorial interpretation. This relationship seems useful for the study of N (n). In particular this can be used to calculate N(n) for small values. For n 3, 4, 5, 6 we find N(3) = 29, N(4) = 355, N(5) = 7,181, N(6) = 145,807.

For all positive integers m, n, m ≤ n, let m,n denote the variety of algebras with m n-ary operations ω1, …, ωm and n m-ary operations φ1, …, φn satisfying the system of identities The varieties m,n are considered by Swierczkowski [1] and by Akataeb and Smirnov [2]. Jonsson and Tarski [3] consider m,n in the case m = 1, n = 2. In [2] it is shown that the lattice Lm,n of subvarieties of m,n, is uncountable except when m = 1 in which case Lm,n has a very simple description. In particular, for n > 1 the lattice L1,n is the two element chain which means that 1,n has no proper subvarieties.

We show that if the lattice of subquasivarieties of a quasivariety of Q-algebras has a finite maximal chain then every algebra of the quasivariety is a subcartesian product of a family of certain finitely many finite algebras. For definitions of common concepts related to Q-algebras that are used but not defined here we refer the reader to [1]. Throughout this note we shall assume Q to be arbitrarily fixed. The letter f will denote an arbitrary class of Q-algebras. There are many well-known examples of quasivarieties f of algebras with the following property: (v) There exists a finite set 9' of finite algebras in ft such that every algebra in ft is embeddable in the cartesian product of a family of algebras in W'. Varieties of semnilattices, distributive lattices, abelian groups of exponent n, normal idempotent semigroups [7], and the quasivarieties W.Xm of unary algebras (A ; f) with f satisfying fm(x) = x, f (x) =f (x2) ->Xl =x2 are some of these examples. Every one of these quasivarieties has finitely many subquasivarieties and the purpose of this note is to relate this property (of finiteness of the number of subquasivarieties) to (.). More precisely we shall prove THEOREM 1. For every quasivariety M of Q-algebras the following two properties are equivalent: (i) Every subquasivariety of R satisfies (.). (ii) f is a locally finite quasivariety and the lattice L,q(S) of subquasivarieties of f satisfies the ascending chain condition. In [7] it is shown how results like Theorem 1 can be used to obtain some structural results for a certain type of idempotent semigroups. Theorem 1 can also have consequences like the following: COROLLARY 1. If a locally finite quasivariety f has algebras that are Received by the editors July 16, 1969 and, in revised form, March 3, 1970. A MS 1968 subject classifications. Primary 0830.

By a homomorphic relation over an algebra A we mean a subalgebra of A × A. A variety [1[ of algebras will be called Mal′cevian [2[ if the identities of include two identities of the form f (x, y, y)=x, f(x, x, y)=y. In [3[ many examples and interesting properties of Mal′cevian varieties have been quoted or proved. In [4[ it is noted that every reflexive homomorphic relation over an algebra of a Mal′cevian variety is a congruence. The purpose of this short note is to observe that the property of Mal′cevian varieties noted in [1] is in fact characteristic of such varieties.

Some general theorems concerning residual finiteness of algebras are given that are applied to show that every idempotent semigroup satisfying xyzx = xzyx identically is a subcartesian product of certain simple semigroups of order two and three. Introduction. In this paper we present a technique involving a special type of infinitely long sentence which seems of fairly general applicability in the study of structure of bands (idempotent semigroups). The technique is applied to a special type of band, namely normal bands to obtain a rather complete picture of their structure. Some partial structural results on normal bands were obtained earlier by Kimura [4]. Normal bands are also considered in [8] and [2]. For a more meaningful introduction to the paper we need some definitions. A semigroup equation in a set X of variables is a formula V0 = V1, where V0, V1 are semigroup words in X, that is, finite sequences of variables chosen from X. A semigroup identity or law is a statement of the form Vx1, . . ., xj(e), where s is a semigroup equation. Let R1, R2 be two systems of semigroup equations in X and let VX(R1 -? R2) denote the statement that every solution of the system R1 in the variables of X is also a solution of the system R2. A statement of the form VX(R1 R2) is called (cf. [5]) an identical semigroup implication. The length of VX(R1 R2) by definition is the cardinal number of R1. Identities can be regarded as implications of any given length by taking R1 to be a big enough set of equations of the form V= V. Now a class X of semigroups is called [6] implicationally defined if X can be defined by a set of identical semigroup implications. If X can be defined by implications of finite length it is called a quasivariety. If X can be defined by implications of length one we call it a semivariety. Finally, if X is definable by implications of length zero then X is called a variety. All these concepts are special cases of the more general concept of a quasiprimitive class. Our arbitrary class X is called quasiprimitive if it is closed under the formation of isomorphs, subsemigroups and cartesian products of its semigroups. Let Q(Y) denote the class of those semigroups that are embeddable in cartesian products of Received by the editors June 11, 1969 and, in revised form, September 22, 1969. AMS Subject Classifications. Primary 2093, 0830.