On the algebraic structure of the Schröder monoid

Let $[n]$ be a finite chain $\{1,2,\dots ,n\}$, and let ${\mathcal{L}\mathcal{S}}_n$ be the semigroup consisting of all isotone and order-decreasing partial transformations on $[n]$. Moreover, let ${\mathcal{S}\mathcal{S}}_n=\{\alpha \in {\mathcal{L}\mathcal{S}}_n:1\in \text{Dom }\alpha \}$ be the subsemigroup of ${\mathcal{L}\mathcal{S}}_n$, consisting of all transformations in ${\mathcal{L}\mathcal{S}}_n$ each of whose domain contains $1$. For $1\le p\le n$, let $K(n,p)=\{\alpha \in {\mathcal{L}\mathcal{S}}_n:\mathrm{\mid }\text{Im }\alpha \mathrm{\mid }\le p\}$ and $M(n,p)=\{\alpha \in {\mathcal{S}\mathcal{S}}_n:\mathrm{\mid }\text{Im }\alpha \mathrm{\mid }\le p\}$ be the two-sided ideals of ${\mathcal{L}\mathcal{S}}_n$ and ${\mathcal{S}\mathcal{S}}_n$, respectively. Furthermore, let ${RLS}_n(p)$ and ${RSS}_n(p)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any $S\in \{{\mathcal{S}\mathcal{S}}_n,{\mathcal{L}\mathcal{S}}_n,{RLS}_n(p),{RSS}_n(p)\}$, $S$ is abundant and idempotent generated for all values of $n$. Moreover, the ranks of the Rees quotients ${RLS}_n(p)$ and ${RSS}_n(p)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. Finally, these ranks are computed to be ${\sum }_{k=p}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{k-1}{p-1}\right)$ and $\left(\genfrac{}{}{0px}{}{n-1}{p-1}\right)2^{n-p}$, respectively.