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

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 TY(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 TY(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 TY(X,ρ). Furthermore, we identify the necessary and sufficient conditions for the semigroup TY(X,ρ) itself to be regular, left regular, or right regular.