Semigroups of transformations whose characters belong to a given semigroup

Abstract: Let $X$ be a nonempty set and $\mathcal{P}={X_i\colon i\in I}$ a partition of $X$. Denote by $T(X)$ the full transformation semigroup on $X$, and $T(X, \mathcal{P})$ the subsemigroup of $T(X)$ consisting of all transformations that preserve $\mathcal{P}$. For every subsemigroup $\mathbb{S}(I)$ of $T(I)$, let $T_{\mathbb{S}(I)}(X,\mathcal{P})$ be the semigroup of all transformations $f\in T(X, \mathcal{P})$ such that $\chi^{(f)}\in \mathbb{S}(I)$, where $\chi^{(f)}\in T(I)$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe regular and idempotent elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a regular semigroup [inverse semigroup]. With the assumption that $\mathbb{S}(I)$ contains the identity, we characterize Green's relations on $T_{\mathbb{S}(I)}(X,\mathcal{P})$, describe unit-regular elements in $T_{\mathbb{S}(I)}(X,\mathcal{P})$, and determine when $T_{\mathbb{S}(I)}(X,\mathcal{P})$ is a unit-regular semigroup. We apply these general results to obtain more concrete results for $T(X,\mathcal{P})$.