### Green’s Relations and Natural Partial Order on the Regular Subsemigroup of Transformations Preserving an Equivalence Relation and Fixed a Cross-Section

#### Abstract

Let $X$ be an arbitrary nonempty set and $T(X)$ the full transformation semigroup on a set $X$. For an equivalence relation $E$ on $X$ and a cross-section $R$ of the partition $X/E$ induced by $E$, let \[ T_{E}(X, R) = \{\alpha \in T(X) : R\alpha = R \mbox{ and } \forall x,y \in X, (x, y) \in E \Rightarrow (x\alpha, y\alpha) \in E \}. \] Then the set $Reg(T_{E}(X, R))$ of all regular elements of $T_{E}(X, R)$ is a regular subsemigroup of $T(X)$. In this paper, we describe Green's relations for elements of the semigroup $Reg(T_{E}(X, R))$. Also, we discuss the natural partial order on semigroup of $Reg(T_{E}(X, R))$ and characterize when two elements of $Reg(T_{E}(X, R))$ are related under this order.

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