On a Conjecture Concerning Semigroup Homomorphisms

Abstract

In this paper we settle (with a counterexample) the question raised by Clifford and Preston in [2, p. 275], concerning maximal group homomorphic images of semigroups. We also consider the question in a more general context and characterize all such examples. The notation and definitions follow [1; 2].By a type of semigroups we mean a class of semigroups, closed under isomorphisms and containing the one-element semigroup. If S is any semigroup and is a type, then a semigroup S* is defined, in [1, p. 18], to be a maximal homomorphic image of S having type if(i) ,(ii) S* is a homomorphic image of S, and(iii) whenever and T is a homomorphic image of S, then there exists a homomorphism from S* onto T.