Maximal right congruences on semigroups

Abstract

A right congruence p on a semigroup S is a relation on S, but also has an algebraic structure. This is perhaps more evident when one views S/p as an Soperand. In this paper we define the notion of isomorphism of two right congruences, and begin a description and classification of the isomorphism classes for those right congruences which are maximal with respect to the usual partial ordering. In particular we characterize completely the twenty-three nonisomorphic right congruences of order two by means of homomorphisms into a particular semigroup. If p is any right congruence on a semigroup S, we define the index of p, written JpJ, to be JS/pJ, the cardinality of S/p. p is said to be right reductive if (ac)p(bc) for all c in S implies apb. For other definitions, the reader is referred to [I]. i. Maximal riqht conqruences. The following result is known and will be useful in the sequel. It