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

TL;DR

This paper investigates the algebraic structure of a specific small Schröder semigroup, analyzing its ideals, quotients, and ranks, revealing properties like abundance and precise rank formulas for various related semigroups.

Contribution

It introduces and studies the properties of the subsemigroup $ ext{SS}'_n$, its ideals, and Rees quotients, providing new rank formulas and structural insights.

Findings

All studied semigroups are right abundant.

None are left abundant for $n \\geq 2$.

The rank of ${RSS}^{ ext{'}}_n(p)$ equals that of $K(n,p)$, given by a specific binomial sum.

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 $3n-4$.