Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes

Abstract

Let $X$ be a nonempty set and $E$ an equivalence relation on $X$. The semigroup $T_E(X)$ consists of all transformations on $X$ that preserve $E$. Although the collection of all regular elements of $T_E(X)$ is not, in general, a subsemigroup, we investigate maximal regular subsemigroups of $T_E(X)$ containing the group of units $G_E(X)$. We construct two such maximal regular subsemigroups, denoted by $\mathcal{M}_1$ and $\mathcal{M}_2$, and show that their maximality holds in full generality for an arbitrary set $X$ and an arbitrary equivalence relation $E$ that is neither the identity relation nor the universal relation on $X$ (whether finite or infinite, uniform or non-uniform). Furthermore, when $X$ is finite and $E$ is a uniform equivalence relation with exactly two classes of cardinality at least $2$, we provide a complete classification by proving that they are the only maximal regular subsemigroups containing $G_E(X)$.