On the algebraic structure of the Schröder monoid

Abstract: Let $[n]$ be a finite chain ${1, 2, \ldots, n}$, and let $\mathcal{LS}{n}$ be the semigroup consisting of all isotone and order-decreasing partial transformations on $[n]$. Moreover, let $\mathcal{SS}{n} = {\alpha \in \mathcal{LS}{n} : \, 1 \in \text{Dom } \alpha}$ be the subsemigroup of $\mathcal{LS}{n}$, consisting of all transformations in $\mathcal{LS}{n}$ each of whose domain contains $1$. For $1 \leq p \leq n$, let $K(n,p) = {\alpha \in \mathcal{LS}{n} : \, |\text{Im } \, \alpha| \leq p}$ and $M(n,p) = {\alpha \in \mathcal{SS}{n} : \, |\text{Im } \alpha| \leq p}$ be the two-sided ideals of $\mathcal{LS}{n}$ and $\mathcal{SS}{n}$, 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 $S \in {\mathcal{SS}{n}, \mathcal{LS}{n}, {RLS}{n}(p), {RSS}{n}(p)}$, $S$ 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 $\sum\limits{k=p}^{n} \binom{n}{k} \binom{k-1}{p-1}$ and $\binom{n-1}{p-1}2^{n-p}$, respectively.