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 $\varnothing \neq Z\subseteq Y\subseteq X$, 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.