Homomorphic images of restrictive star semigroups

Abstract

LetYbe a subspace of a topological spaceX. ThenS(X, Y)denotes the semigroup, under composition, of all continuous selfmaps ofXwhich also carryYintoY. In the special caseY=X, the simpler notationS(X)is used. We have devoted several recent papers ([4], [7] and [8]) to the problem of determining whenS(Z)andS(X, Y)are isomorphic and, more generally, whenS(Z)is a homomorphic image ofS(X, Y). In this paper, we investigate the analogous problem for certain semigroups of functions on spaces which were introduced in [5]. These include semigroups of closed functions which are treated in further detail.