On the algebraic structure of the Schröder monoid
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| Formato: | Preprint |
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2024
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| _version_ | 1866915069705584640 |
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| 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 chain $\{1, 2, \ldots, n\}$, and let $\mathcal{LS}_{n}$ be the semigroup consisting of all isotone and order-decreasing partial transformations on $[n]$. Moreover, let $\mathcal{SS}_{n} = \{α\in \mathcal{LS}_{n} : \, 1 \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$ each of whose domain contains $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{LS}_{n} : \, |\text{Im } \, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{SS}_{n} : \, |\text{Im } α| \leq p\}$ be the two-sided ideals of $\mathcal{LS}_{n}$ and $\mathcal{SS}_{n}$, respectively. Furthermore, let ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any $S \in \{\mathcal{SS}_{n}, \mathcal{LS}_{n}, {RLS}_{n}(p), {RSS}_{n}(p)\}$, $S$ is abundant and idempotent generated for all values of $n$. Moreover, the ranks of the Rees quotients ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. Finally, these ranks are computed to be $\sum\limits_{k=p}^{n} \binom{n}{k} \binom{k-1}{p-1}$ and $\binom{n-1}{p-1}2^{n-p}$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13675 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the algebraic structure of the Schröder monoid Zubairu, Muhammad Mansur Umar, Abdullahi Al-Kharousi, Fatma Salim Group Theory 20M20 Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{LS}_{n}$ be the semigroup consisting of all isotone and order-decreasing partial transformations on $[n]$. Moreover, let $\mathcal{SS}_{n} = \{α\in \mathcal{LS}_{n} : \, 1 \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{LS}_{n}$, consisting of all transformations in $\mathcal{LS}_{n}$ each of whose domain contains $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{LS}_{n} : \, |\text{Im } \, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{SS}_{n} : \, |\text{Im } α| \leq p\}$ be the two-sided ideals of $\mathcal{LS}_{n}$ and $\mathcal{SS}_{n}$, respectively. Furthermore, let ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any $S \in \{\mathcal{SS}_{n}, \mathcal{LS}_{n}, {RLS}_{n}(p), {RSS}_{n}(p)\}$, $S$ is abundant and idempotent generated for all values of $n$. Moreover, the ranks of the Rees quotients ${RLS}_{n}(p)$ and ${RSS}_{n}(p)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. Finally, these ranks are computed to be $\sum\limits_{k=p}^{n} \binom{n}{k} \binom{k-1}{p-1}$ and $\binom{n-1}{p-1}2^{n-p}$, respectively. |
| title | On the algebraic structure of the Schröder monoid |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/2412.13675 |