On the small Schröder semigroup $\mathcal{SS}^{\prime}_{n}$
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911332495785984 |
|---|---|
| author | Zubairu, Muhammad Mansur Umar, Abdullahi Al-Kharousi, Fatma Salim |
| author_facet | Zubairu, Muhammad Mansur Umar, Abdullahi Al-Kharousi, Fatma Salim |
| contents | 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 $\mathcal{SS}^{\prime}_{n} = \{α\in \mathcal{LS}_{n} : \, 1\not\in \text{ Dom } α\}$ 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 $K(n,p) = \{α\in \mathcal{SS}^{\prime}_{n} : \, |Im \, α| \leq p\}$
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 $3n-4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the small Schröder semigroup $\mathcal{SS}^{\prime}_{n}$ Zubairu, Muhammad Mansur Umar, Abdullahi Al-Kharousi, Fatma Salim Group Theory 20M20 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 $\mathcal{SS}^{\prime}_{n} = \{α\in \mathcal{LS}_{n} : \, 1\not\in \text{ Dom } α\}$ 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 $K(n,p) = \{α\in \mathcal{SS}^{\prime}_{n} : \, |Im \, α| \leq p\}$ 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 $3n-4$. |
| title | On the small Schröder semigroup $\mathcal{SS}^{\prime}_{n}$ |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/2512.19422 |