Extensions of completely 0-simple semigroups by completely 0-𝑠𝑖𝑚𝑝𝑙𝑒 𝑠𝑒𝑚𝑖𝑔𝑟𝑜𝑢𝑝𝑠.

Abstract

Let S and T be disjoint semigroups, T having a zero element, O'. A semigroup (V, 0) is called an extension of S by T if it contains S as an ideal and if the Rees factor semigroup V/S is isomorphic to T. We shall say that V is determined by a partial homomorphism if there exists a partial homomorphism A -->: of T\O' into S such that A oB=AB if AB 5?O', A oB=AB if AB=O', A os=As, soA = sA, and s o t = st where s, t in S and the operations in S and T are denoted by juxtaposition. The purpose of this note is to give a necessary and sufficient condition that V be determined by a partial homomorphism when S is a completely 0-simple semigroup and T is a completely 0-simple semigroup. Since these partial homomorphisms are known mod group homomorphisms [1, p. 109, Theorem 3.14], our extensions may be given an explicit form. A corollary to our theorem will include an important theorem due to W. D. M unn [1, p. 143, Theorem 4.22]. Our result should have important applications to the study of finite semigroups and to semigroups with some finiteness condition. If S is any subset of a semigroup, 8(S) will denote the set of idempotents of S and S* will denote the set of nonzero elements of S. 6R, ?, 5D and 3C will denote Green's relations [1, p. 47]. If aES, Ra will denote the (R-class containing a. If e and f are idempotents and ef=fe = e, we say e is under f and write e <f. Basic definitions are given in [1]. Likewise references to the fundamental work of Clifford, Green, and Munn will be found in [1].