Regularity of the transformation semigroups with invariant sets and restricted by an equivalence relation

Abstract

For a nonempty set X, let T(X) be the full transformation semigroup on X. Let Y be a nonempty subset of X and let ρ be an equivalence relation on X. We define the subset T_Y(X,ρ) of T(X), consisting of all transformations that map elements of Y into Y and send elements within the same ρ-equivalence class into a single common element. It can be shown that T_Y(X,ρ) forms a subsemigroup of T(X). In this paper, we provide a characterization of the regular, left regular, and right regular elements of T_Y(X,ρ). Furthermore, we identify the necessary and sufficient conditions for the semigroup T_Y(X,ρ) itself to be regular, left regular, or right regular.