Zero-free fortifying homomorphisms and semigroups of relations

Abstract

In this paper, the concept of a fitted semigroup of binary relations is introduced. The binary operation for these semigroups is ordinary composition of relations. A particular type of homomorphism is investigated which maps a certain kind of subsemigroup of one fitted semigroup into another. The main result states that these homomorphisms are injective and gives a representation for them. This and several related results are then applied to the semigroup of all binary relations on a topological space which have a certain prescribed type of topological property. Compactness is one of these properties and the semigroup SK[X] of all compact binary relations on a Hausdorff k–space X is given special attention. It is shown that X and Y are homeomorphic if and only if a number of statements are true which relate the semigroups SK[x] and SK[y].For example, in order that X and y be homeomorphic it is both necessary and sufficient that some nonzero ideal of SK[X] be isomorphic to some nonzero ideal of SK[y].