### Regularity of a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

#### Abstract

For a fixed nonempty subset $Y$ of $X$, let $T(X,Y)$ be the semigroup consisting of all transformations from $X$ into $Y$. Let $\rho$ be an equivalence relation on $X$, $\hat{\rho}$ the restriction of $\rho$ on $Y$ and $R$ a cross-section of the partition $Y/\hat{\rho}$. We define

$$T(X,Y,\rho,R) = \{\alpha\in T(X,Y) : R\alpha\subseteq R~\text{and}~(a,b)\in \rho \Rightarrow (a\alpha,b\alpha)\in\rho\}.$$

Then $T(X,Y,\rho,R)$ is a subsemigroup of $T(X,Y)$. In this paper, we describe regular elements in $T(X,Y,\rho,R)$, characterize when $T(X,Y,\rho,R)$ is a regular semigroup and investigate some classes of $T(X,Y,\rho,R)$ such as completely regular and inverse from which the results on $T(X,\rho,R)$ and $T(X,Y)$ can be recaptured easily when taking $Y=X$ and $\rho$ to be the identity relation, respectively. Moreover, the description of unit-regularity on $T(X,\rho,R)$ is obtained.

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