The semigroups of order-preserving transformations with restricted range

Abstract

Let X be a chain and let $\mathscr {O}(X)$ be the monoid of all full order-preserving transformations of X. Given a nonempty subset Y of X, we denote by $\mathscr {O}(X,Y)$ the subsemigroup of $\mathscr {O}(X)$ of all full order-preserving transformations with range contained in Y and by $\mathscr {O}(Y)$ the monoid of all order-preserving transformations of Y. In 2015, Sommanee and Sanwong investigated the subsemigroup $\mathscr{O}\mathscr{F}(X,Y)=\{\alpha \in \mathscr {O}(X,Y)\mid \mathop {\textrm{im}}\nolimits (\alpha )=Y\alpha \}$ of $\mathscr {O}(X,Y)$. We characterize the connections between the maximal (regular) subsemibands of $\mathscr{O}\mathscr{F}(X,Y)$ and the maximal (regular) subsemibands of $\mathscr {O}\mathbf {_r}=\mathscr {O}(Y)\backslash \{1_Y\}$. Moreover, we characterize the structure of the idempotent generating sets of $\mathscr{O}\mathscr{F}(X,Y)$ when Y is a finite subset of X. As applications, we compute the number of distinct minimal idempotent generating sets of $\mathscr{O}\mathscr{F}(X,Y)$. We also determine the maximal subsemibands as well as the maximal regular subsemibands of $\mathscr{O}\mathscr{F}(X,Y)$.