On certain semigroups of finite oriented and order-decreasing partial transformations

Abstract

Let ${\mathcal{P}\mathcal{O}\mathcal{RD}}_n$ be the semigroup consisting of all oriented and order-decreasing partial transformations on the finite chain $X_n=\{1<\cdots <n\}$. Let ${\mathcal{I}\mathcal{O}\mathcal{RD}}_n$ be the subsemigroup of ${\mathcal{P}\mathcal{O}\mathcal{RD}}_n$ consisting of all injective transformations of ${\mathcal{P}\mathcal{O}\mathcal{RD}}_n$. For $2\le r\le n$, let $\mathcal{P}\mathcal{O}\mathcal{RD}(n,r)=\{\hat{I}\pm \in {\mathcal{P}\mathcal{O}\mathcal{RD}}_n:|\text{im}(\hat{I}\pm )|\le r\}$ and $\mathcal{I}\mathcal{O}\mathcal{RD}(n,r)=\{\hat{I}\pm \in {\mathcal{I}\mathcal{O}\mathcal{RD}}_n:|\text{im}(\hat{I}\pm )|\le r\}$. In this paper, we determine some minimal generating sets and ranks of $\mathcal{P}\mathcal{O}\mathcal{RD}(n,r)$ and $\mathcal{I}\mathcal{O}\mathcal{RD}(n,r)$, and moreover, we characterize the maximal subsemigroups of $\mathcal{P}\mathcal{O}\mathcal{RD}(n,r)$ and $\mathcal{I}\mathcal{O}\mathcal{RD}(n,r)$.