On certain semigroups of transformations with an invariant set

TL;DR

This paper investigates the algebraic structure of a subsemigroup of transformations fixing a subset, characterizing regularity, unit-regularity, Green's relations, and ideals, with results depending on the finiteness of the set and subset.

Contribution

It provides new characterizations of regular and unit-regular elements, Green's relations, and ideals in the semigroup of transformations fixing a subset, extending existing literature.

Findings

Regularity of the semigroup depends on the finiteness of Y.

Unit-regularity of the semigroup depends on the finiteness of X.

Green's relations simplify to equality of D and J when Y is finite.

Abstract

Let $X$ be a nonempty set and let $T(X)$ be the full transformation semigroup on $X$. The main objective of this paper is to study the subsemigroup $\overline{\Omega}(X, Y)$ of $T(X)$ defined by \[\overline{\Omega}(X, Y) = \{f\in T(X)\colon Yf = Y\},\] where $Y$ is a fixed nonempty subset of $X$. We describe regular elements in $\overline{\Omega}(X, Y)$ and show that $\overline{\Omega}(X, Y)$ is regular if and only if $Y$ is finite. We characterize unit-regular elements in $\overline{\Omega}(X, Y)$ and prove that $\overline{\Omega}(X, Y)$ is unit-regular if and only if $X$ is finite. We characterize Green's relations on $\overline{\Omega}(X, Y)$ and prove that $\mathcal{D} =\mathcal{J}$ on $\overline{\Omega}(X, Y)$ if and only if $Y$ is finite. We also determine ideals of $\overline{\Omega}(X, Y)$ and investigate its kernel. This paper extends several results appeared in the literature.