On the small Schr\"{o}der semigroup $\mathcal{SS}^{\prime}_{n}$

Abstract

Let $[n]$ be a finite $n-$chain $\{1, 2, \dots, n\}$, and let $\mathcal{LS}_{n}$ be the Schr\"{o}der monoid, consisting of all isotone and order-decreasing partial transformations on $[n]$. Furthermore, let $\mathcal{SS}^{\prime}_{n} = \{\alpha \in \mathcal{LS}_{n} : \, 1\not\in \text{ Dom } \alpha\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$, each of whose domain does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{\alpha \in \mathcal{SS}^{\prime}_{n} : \, |Im \, \alpha| \leq p\}$ be the two-sided ideal of $\mathcal{SS}^{\prime}_{n}$. Moreover, let ${RSS}^{\prime}_{n}(p)$ denote the Rees quotient of $K(n,p)$. It is shown in this article that for any $S$ in $\{\mathcal{SS}^{\prime}_{n}, K(n,p), {RSS}^{\prime}_{n}(p)\}$, $S$ is right abundant for all values of $n$, but not left abundant for all $n \geq 2$. In addition, the rank of the Rees quotient ${RSS}^{\prime}_{n}(p)$ is shown to be equal to the rank of the two-sided ideal $K(n,p)$, which is equal to $\binom{n-1}{p-1}+\sum\limits_{k=p}^{n-1}\binom{n-1}{k} \binom{k-1}{p-1}$. Finally, the rank of $\mathcal{SS}^{\prime}_{n}$ is determined to be $3n-4$.