Suppose that $X$ be a nonempty set. Denote by $\mathcal{ T}(X)$ the full transformation semigroup on $X$. For $\varnothing \neq Z\subseteq Y\subseteq X$, let $\mathcal{ T}(X,Y,Z)=\{ \alpha \in \mathcal{ T}(X): Y\alpha \subseteq Z \}$. Then $\mathcal{ T}(X,Y,Z)$ is a subsemigroup of $\mathcal{ T}(X)$. In this paper, we characterize the regular elements of the semigroup $\mathcal{ T}(X,Y,Z)$, and present a necessary and sufficient condition under which $\mathcal{ T}(X,Y,Z)$ is regular. Furthermore, we investigate the abundance of the semigroup $\mathcal{ T}(X,Y,Z)$ for the case $Z\subsetneq Y\subsetneq X$. In addition, we compute the cardinalities of $\mathcal{ T}(X,Y,Z)$, ${ \rm Reg}(\mathcal{ T}(X,Y,Z))$ and ${ \rm E}(\mathcal{ T}(X,Y,Z))$ when $X$ is finite, respectively.
Categories: math.GR