Green's Relations for Regular Elements of Semigroups of Endomorphisms

Abstract

X is a set and End X is a semigroup, under composition, of functions, which map X into X. We characterize those elements of End X which are regular and then we completely determine Green's relations for these elements. The conditions we place on End X are sufficiently mild to permit such semigroups as S(X), the semigroup of all continuous self maps of a topological space X and L(V), the semigroup of all linear transformations on a vector space V, to be regarded as special cases.