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

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