Regular 𝒟-classes in semigroups

Abstract

0. Introduction. J. A. Green has introduced [1], in an arbitrary semigroup, certain equivalence relations which we shall denote by S and (R, their relative product D, and their intersection SC. Since none of the relations is a congruence relation, the product of equivalence classes is not generally contained in an equivalence class. In ?1 of this paper we obtain some information about the multiplicat,ion of such classes, restricting our attention for the most part to products of XC-classes that lie in a O-class D containing an idempotent element, a restriction equivalent, as we show, to requiring that all elements of D be regular in the sense of von Neumann [4]. In ?2 we use these results to obtain a theorem on matrix representations of semigroups which reduces, in the case of completely simple semigroups, to the Rees-Suschkewitsch Theorem, [5] and [6]. 1. Idempotents, inverses, and products. Throughout the paper, S will denote an arbitrary semigroup, i.e., a set closed under an associative binary operation: a(bc) = (ab)c for all a, b, c in S. Green [1] has defined in S the equivalence relations S and (R as follows: