Regular 𝒟-classes in semigroups
D. D. Miller, A. H. Clifford · Transactions of the American Mathematical Society · 1956
Introduction.J. A. Green has introduced [l], in an arbitrary semigroup, certain equivalence relations which we shall denote by £ and (R, their relative product ©, and their intersection 3C.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 multiplication of such classes, restricting our attention for the most part to products of X-classes that lie in a ©-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