On certain Semigroups of Transformations that preserve a partition

Let $X$ be a nonempty set, and let ${\mathcal{T}}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P}=\{X_i\mathrm{\mid }i\in I\}$ of $X$, we consider the semigroup $T(X, $\mathcal{P}$) = {f\in $\mathcal{T}$_X;|; \forall X_i;\exists X_j,; X_i f \subseteq X_j}$,thesubsemigroup$\Sigma(X, $\mathcal{P}$) = {f\in T(X, $\mathcal{P}$);|; Xf \cap X_i \neq \emptyset; \forall X_i}$,andthegroupofunits$S(X, $\mathcal{P}$)$of$T(X, $\mathcal{P}$)$\mathrm{.}Inthispaper,wefirstcharacterizetheelementsof$\Sigma(X, $\mathcal{P}$)$\mathrm{.}Forapermutation$f$offinite$X$, we next observe whether there exists a nontrivial partition $\mathcal{P}$ of $X$ such that $f\in S(X, $\mathcal{P}$)$. We then characterize and enumerate the idempotents in the semigroup $\mathrm{\Sigma }(X,\mathcal{P})$ for arbitrary and finite $X$, respectively. We also characterize the elements of $S(X,\mathcal{P})$. For finite $X$, we finally calculate the cardinality of $T(X,\mathcal{P})$, $\Sigma(X, $\mathcal{P}$)$,and$S(X, $\mathcal{P}$)$.