Regular semigroups, fundamental semigroups and groups

Abstract In this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ:S→Tbetween regular semigroups. This is a mapping such that (ab)θ ≦ aθ bθin the natural partial order onT.