  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.
