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

Let $\mathcal{PMD}_{n}$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let $\mathcal{IMD}_{n}$ be the subsemigroup of $\mathcal{PMD}_{n}$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. For $2\leq r\leq n$, let $\mathcal{PMD}(n,r) =\{ \alpha\in \mathcal{PMD}_{n} : |\textrm{im}(\alpha)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ \alpha \in \mathcal{IMD}_{n} :|\textrm{im}(\alpha)| \leq r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$, and moreover, we verify that the semigroups $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$ are non-regular but abundant for any $2\leq r\leq n$.