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 ${\mathcal{D}\mathcal{O}\mathcal{R}\mathcal{P}}_n$ on an $n$ ordered chain $[n]=\{1,\dots ,n\}$, its two-sided ideal $I(n,p)=\{\rho \in {\mathcal{D}\mathcal{O}\mathcal{R}\mathcal{P}}_n:\mathrm{\mid }Im\rho \mathrm{\mid }\le p\}$ and the Rees quotient ${RQ}_p(n)$ of the ideal $I(n,p)$. We compute the order of the monoid ${\mathcal{D}\mathcal{O}\mathcal{R}\mathcal{P}}_n$ and show that for any semigroup $S$ in $\{{\mathcal{D}\mathcal{O}\mathcal{R}\mathcal{P}}_n,I(n,p),{RQ}_p(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_p(n)$, is a non-regular $0-\ast$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 ${\mathcal{D}\mathcal{O}\mathcal{R}\mathcal{P}}_n$ is determined to be $3n-2$.