Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition

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