Regularity and Green’s relations on semigroups of transformations with restricted range that preserve an equivalence

Abstract

© 2020, Springer Science+Business Media, LLC, part of Springer Nature. Let Y be a subset of X and T(X, Y) the set of all functions from X into Y. Then, under the operation of composition, T(X, Y) is a subsemigroup of the full transformation semigroup T(X). Let E be an equivalence on X. Define a subsemigroup TE(X, Y) of T(X, Y) by TE(X,Y)={α∈T(X,Y):∀(x,y)∈E,(xα,yα)∈E}.Then TE(X, Y) is the semigroup of all continuous self-maps of the topological space X for which all E-classes form a basis carrying X into a subspace Y. In this paper, we give a necessary and sufficient condition for TE(X, Y) to be regular and characterize Green’s relations on TE(X, Y). Our work extends previous results found in the literature.