A note on finite generated subsemigroups of T(X,Y)

Abstract

It is well known that a countable set of transformations on an innite set X is contained in a two-generated subsemigroup of the full transformation semigroup on X. If Y X, then T(X; Y ), the set of all transformations on X with an image in Y , forms a semigroup of transformations with restricted range, as shown in 1975 by Symons [10]. In this paper, we give a sucient and necessary condition for a countable subset of T(X; Y ) to be contained in a three-generated subsemigroup of T(X; Y ).