Regular, unit-regular, and idempotent elements of semigroups of transformations that preserve a partition

Let $X$ be a set and $\mathcal{T}_X$ the full transformation semigroup on $X$. For partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, T(X, \mathcal{P})|(\forall X_i \mathcal{P})\; Xf \cap \neq \emptyset\}$, $\Gamma(X, \mathcal{T}_X|(\forall \mathcal{P})(\exists X_j\in X_j\}$. We characterize unit-regular elements both \mathcal{P})$ for finite discuss inclusion between certain transformations preserving $\mathcal{P}$. count regular idempotents \mathcal{P})$. prove that every element is also calculate size