The semigroups of orientation-preserving transformations with restricted range

Abstract

Let $X_n$ be a chain with $n$ elements ($n\in \N$), and let $\OP_n$ be the monoid of all orientation-preserving transformations of $X_n$. Given a nonempty subset $Y$ of $X_n$, we denote by $\OP_n(Y)$ the subsemigroup of $\OP_n$ of all full orientation-preserving transformations with range contained in $Y$. We also denote by $\OP_Y$ the the monoid of all orientation-preserving transformations of $Y$, by $C_Y$ the cycle group on $Y$ and by $\overline{\OP}_Y$ the subsemigroup $\OP_Y\backslash C_Y$ of $\OP_Y$ of all singular orientation-preserving transformations. In 2016, Fernandes et al.\cite{Fernandes&Honyam:2016} consider the subsemigroup of $\OP_n(Y)$ defined by
$$\FOP_n(Y)=\{\alpha\in \OP_n(Y)\mid \im(\alpha)=Y\alpha\}.$$
The authors described its Green's relations and proved that $\FOP_n(Y)$ is the largest regular subsemigroup of $\OP_n(Y)$. The main aim of this paper is to study the core $\C(\FOP_n(Y))$ of the semigroup $\FOP_n(Y)$. We characterize the connections between the maximal (regular) subsemibands of $\C(\FOP_n(Y))$ and the maximal (regular) subsemibands of $\overline{\OP}_Y$. Moreover, we computed the rank of the semigroup $\FOP_n(Y)$ and characterize the structure of the idempotent generating sets of the semigroup $\C(\FOP_n(Y))$. As applications, we compute the number of distinct minimal idempotent generating sets of $\C(\FOP_n(Y))$. We also determine the maximal subsemibands as well as the maximal regular subsemibans of the semigroup $\C(\FOP_n(Y))$.