On the structure of certain idempotent semigroups

Abstract

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.