Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

Abstract

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following