Semigroups of transformations restricted by an equivalence

Abstract

Suppose sigma is an equivalence on a set X and let E(X,sigma) denote the semigroup (under composition) of all alpha:X->X such that alpha C alpha o alpha^-1. Here we characterise Green’s relations and ideals in E(X,sigma). This is analogous to recent work by Sullivan on K(V,W), the semigroup (under composition) of all linear transformations alpha of a vector space V such that W C ker alpha, where W is a fixed subspace of V.