Certain congruences on a completely regular semigroup

Abstract

1. Introduction and summary. Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences p on a completely regular semigroup S such that S/p is a semilattice of groups. We shall call such a congruence an SGcongruence.Since Clifford has shown in [1] that a regular semigroup S is a semilattice of groups if and only if idempotents of S lie in the centre of S, it follows that the problem of rinding group congruences and commutative congruences on a completely regular semigroup S can be solved provided that one can describe the SG-congruences on S.§2 contains definitions and notation used in this paper. In §3 we determine all group congruences on a Rees matrix semigroup and use this in § §4 and 5 to find the smallest SGcongruence, the smallest commutative congruence and the smallest group congruence on a completely regular semigroup.The reader is referred to [2], [5] and [7] for terminology and concepts not presented in this paper.