The partially ordered family of all ℛ-classes of S(x)

Abstract

Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the followingConjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.(2) ℛ(S(X)) and (S{Y)) are order isomorphic.(3) The semigroupsS(X) and S(Y) are isomorphic.(4) The spaces X and Y are homeomorphic.