On certain semigroups of transformations with an invariant set

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.