The monoid of monotone and decreasing partial transformations on a finite chain

In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{\rho \in \mathcal{DORP}_{n} : \, |Im \, \rho| \leq p\}$ and the Rees quotient ${RQ}_{p}(n)$ of the ideal $I(n,p)$. We compute the order of the monoid $\mathcal{DORP}_{n}$ and show that for any semigroup $S$ in $\{\mathcal{DORP}_{n}, \, I(n,p), \, {RQ}_{p}(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_{p}(n)$, is a non-regular $0-*$bisimple abundant semigroup. In addition, we compute the ranks of the Rees quotient ${RQ}_{p}(n)$ and the two-sided ideal $I(n,p)$. Finally, the rank of $\mathcal{DORP}_{n}$ is determined to be $3n-2$.