Let $Y$ be a fixed subset of a nonempty set $X$ and let $Fix(X,Y)$ be the set of all self maps on $X$ which fix all elements in $Y$. Then under the composition of maps, $Fix(X,Y)$ is a regular monoid. In this paper, we prove that there are only three types of maximal subsemigroups of $Fix\left(X,Y\right)$ and these maximal subsemigroups coincide with the maximal regular subsemigroups when $X\setminus Y$ is a finite set with $ X\setminus Y \geq 2$. We also give necessary and sufficient conditions for $Fix(X,Y)$ to be factorizable, unit-regular, and directly finite.
CHAIYA, YANISA; HONYAM, PREEYANUCH; and SANWONG, JINTANA
"Maximal subsemigroups and finiteness conditions on transformation semigroups with fixed sets,"
Turkish Journal of Mathematics: Vol. 41:
1, Article 6.
Available at: https://journals.tubitak.gov.tr/math/vol41/iss1/6