On the small Schröder semigroup ${\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}$

Let $[n]$ be a finite $n$-chain $\{1,2,\dots ,n\}$, and let ${\mathcal{L}\mathcal{S}}_n$ be the large Schröder monoid, consisting of all isotone and order-decreasing partial transformations on $[n]$. Furthermore, let ${\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}=\{\alpha \in {\mathcal{L}\mathcal{S}}_n:$ $1\notin \text{Dom}\alpha \}$ be the subsemigroup of ${\mathcal{L}\mathcal{S}}_n$, consisting of all transformations in ${\mathcal{L}\mathcal{S}}_n$, not containing 1 in their domains. For $1\le p\le k\le n$, let $I(n,k)=\{\alpha \in {\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}:|\text{Im}\alpha |\le k\}$ be the two-sided ideal of ${\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}$, consisting of transformations of height at most $k$, and let ${RSS}_n^{\mathrm{\prime }}(p)$ denote the Rees quotient of $I(n,k)$. It is shown in this article that the object ${\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}$ is a left monoid but not a right monoid. Moreover, it is shown that for any $p\le k$ and any $S\in \{{\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }},I(n,k),{RSS}_n^{\mathrm{\prime }}(p)\}$, $S$ is right abundant for all values of $n$, but not left abundant for all $n\ge 2$. In addition, the rank of the Rees quotient ${RSS}_n^{\mathrm{\prime }}(p)$ is shown to be equal to the rank of the two-sided ideal $I(n,p)$, which is equal to $(\genfrac{}{}{0ex}{}{n-1}{p-1})+\sum _{r=p}^{n-1}(\genfrac{}{}{0ex}{}{n-1}{r})(\genfrac{}{}{0ex}{}{r-1}{p-1})$. Finally, the rank of ${\mathcal{S}\mathcal{S}}_n^{\mathrm{\prime }}$ is determined to be $3n-4$.