On the Rank of Semigroup of Transformations with Restricted Partial Range

Abstract

Let $\mathcal{T}(X)$ be the full transformation semigroup on a nonempty set $X$. For $\emptyset \neq Z\subseteq Y\subseteq X$, let $\mathcal{T}(X,Y,Z)=\{\alpha \in \mathcal{T}(X): Y\alpha \subseteq Z \}$. It is not difficult to see that it is a generalized form of three well-known semigroups. This paper obtains an isomorphism theorem of $\mathcal{T}(X,Y,Z)$. In addition,  when $X$ is finite and $Z\subset Y\subset X$, the rank of the semigroup $\mathcal{T}(X,Y,Z)$ is calculated.