Restrictive Semigroups of Continuous Functions on 0-Dimensional Spaces

Let X be a topological space and Y a nonempty subspace of X. Γ(X, Y) denotes the semigroup under composition of all closed self maps of X which carry Y into Y, and is referred to as a restrictive semigroup of closed functions. Similarly, S(X, Y) is the analogous semigroup of continuous selfmaps of X, and is referred to as a restrictive semigroup of continuous functions. It is immediate that each homeomorphism from X onto U which carries the subspace Y of X onto the subspace V of U induces an isomorphism between Γ(X, Y) and Γ(U, V), and also an isomorphism between S(X, Y) and S(U, V). Indeed, one need only map f onto h o f o h-1. An isomorphism of this form is called representable. In [5, Theorem (3.1), p. 1223] it was shown that in most cases, each isomorphism from Γ(X, Y) onto Γ(U, V) is representable. The analogous problem was discussed for the semigroup S(X, Y) and it was pointed out by means of an example that one could not hope to obtain the same result for these semigroups without some further restrictions.