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

Abstract

Let $[n]$ be a finite $n-$chain $\{1,2,\dots ,n\}$, and let ${\mathcal{LS}}_n$ be the Schröder monoid, consisting of all isotone and order-decreasing partial transformations on $[n]$. Furthermore, let ${\mathcal{SS}}_n^{\prime }=\{\hat{I}\pm \in {\mathcal{LS}}_n:1\notin \text{ Dom }\hat{I}\pm \}$ be the subsemigroup of ${\mathcal{LS}}_n$, consisting of all transformations in ${\mathcal{LS}}_n$, each of whose domain does not contain $1$. For $1\le p\le n$, let $K(n,p)=\{\hat{I}\pm \in {\mathcal{SS}}_n^{\prime }:|Im\hat{I}\pm |\le p\}$ be the two-sided ideal of ${\mathcal{SS}}_n^{\prime }$. Moreover, let ${RSS}_n^{\prime }(p)$ denote the Rees quotient of $K(n,p)$. It is shown in this article that for any $S$ in $\{{\mathcal{SS}}_n^{\prime },K(n,p),{RSS}_n^{\prime }(p)\}$, $S$ 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 ${RSS}_n^{\prime }(p)$ is shown to be equal to the rank of the two-sided ideal $K(n,p)$, which is equal to $\left(\genfrac{}{}{0ex}{}{n-1}{p-1}\right)+\sum _{k=p}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)\left(\genfrac{}{}{0ex}{}{k-1}{p-1}\right)$. Finally, the rank of ${\mathcal{SS}}_n^{\prime }$ is determined to be $3n-4$.

The research was conducted in 2024 at Sultan Qaboos University Oman