Semigroup of transformations with restricted partial range: Regularity, abundance and some combinatorial results

Suppose that $X$ be a nonempty set. Denote by $\mathcal{T}(X)$ the full transformation semigroup on $X$. For $\mathrm{\varnothing }\mathrm{\ne }Z\subseteq Y\subseteq X$, 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 $Z⊊Y⊊X$. In addition, we compute the cardinalities of $\mathcal{T}(X,Y,Z)$, $\mathrm{R}\mathrm{e}\mathrm{g}(\mathcal{T}(X,Y,Z))$ and $\mathrm{E}(\mathcal{T}(X,Y,Z))$ when $X$ is finite, respectively.