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

Let $\mathcal{PORD}_{n}$ be the semigroup consisting of all oriented and order-decreasing partial transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. Let $\mathcal{IORD}_{n}$ be the subsemigroup of $\mathcal{PORD}_{n}$ consisting of all injective transformations of $\mathcal{PORD}_{n}$. For $2\leq r\leq n$, let $\mathcal{PORD}(n,r) =\{ \alpha\in \mathcal{PORD}_{n} :\lvert \text{im}(\alpha) \rvert \leq r\}$ and $\mathcal{IORD}(n,r)=\{ \alpha \in \mathcal{IORD}_{n} :\lvert \text{im}(\alpha )\rvert \leq r\}$. In this paper, we determine some minimal generating sets and ranks of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$, and moreover, we characterize the maximal subsemigroups of $\mathcal{PORD}(n,r)$ and $\mathcal{IORD}(n,r)$.