On the Congruence Extension Property for Semigroups: Preservation under Homomorphic Images

Abstract

A congruence ρT on a subsemigroup T of S extends to the semigroup S, if there exists a congruence ρ on S such that ρ∣T=ρT. A semigroup S has the congruence extension property (CEP) if each congruence on each subsemigroup extends to S. Previously, it was shown that a semigroup S has CEP if and only if it satisfies the so-called CEP conditions, a construction for semigroups with CEP was given, and we characterized congruences on this kind of semigroup. In the 1970s, the question whether or not CEP for semigroups is preserved in homomorphic images was asked. We give a positive answer for the problem in this paper.