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 $3n-2$.
Categories: math.GR