On injective partial Catalan monoids
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929653822783488 |
|---|---|
| author | Al-Kharousi, F. S. Umar, A. Zubairu, M. M. |
| author_facet | Al-Kharousi, F. S. Umar, A. Zubairu, M. M. |
| contents | Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{IC}_{n}$ be the semigroup consisting of all isotone and order-decreasing injective partial transformations on $[n]$. In addition, let $\mathcal{Q}^{\prime}_{n} = \{α\in \mathcal{IC}_{n} : \, 1\not \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{IC}_{n}$, consisting of all transformations in $\mathcal{IC}_{n}$, each of whose domains does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{IC}_{n} : \, |\text{Im }\, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{Q}^{\prime}_{n} : \, |\text{Im } \, α| \leq p\}$ be the two-sided ideals of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$, respectively. Moreover, let ${RIC}_{p}(n)$ and ${RQ}^{\prime}_{p}(n)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any \( S \in \{ \mathcal{RIC}_{p}(n), K(n,p) \} \), \( S \) is abundant; \( \mathcal{IC}_{n} \) is ample; and for any \( S \in \{ \mathcal{Q}^{\prime}_{n}, \mathcal{RQ}^{\prime}_{p}(n), M(n,p) \} \), \( S \) is right abundant for all values of \( n \), but not left abundant for \( n \geq 2 \). Furthermore, the ranks of the Rees quotients ${RIC}_{p}(n)$ and ${RQ}^{\prime}_{p}(n)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. These ranks are found to be $\binom{n}{p}+(n-1)\binom{n-2}{p-1}$ and $\binom{n}{p}+(n-2)\binom{n-3}{p-1}$, respectively. In addition, the ranks of the semigroups $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$ were found to be $2n$ and $n^{2}-3n+4$, respectively. Finally, we characterize all the maximal subsemigroups of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On injective partial Catalan monoids Al-Kharousi, F. S. Umar, A. Zubairu, M. M. Group Theory 20M20 Let $[n]$ be a finite chain $\{1, 2, \ldots, n\}$, and let $\mathcal{IC}_{n}$ be the semigroup consisting of all isotone and order-decreasing injective partial transformations on $[n]$. In addition, let $\mathcal{Q}^{\prime}_{n} = \{α\in \mathcal{IC}_{n} : \, 1\not \in \text{Dom } α\}$ be the subsemigroup of $\mathcal{IC}_{n}$, consisting of all transformations in $\mathcal{IC}_{n}$, each of whose domains does not contain $1$. For $1 \leq p \leq n$, let $K(n,p) = \{α\in \mathcal{IC}_{n} : \, |\text{Im }\, α| \leq p\}$ and $M(n,p) = \{α\in \mathcal{Q}^{\prime}_{n} : \, |\text{Im } \, α| \leq p\}$ be the two-sided ideals of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$, respectively. Moreover, let ${RIC}_{p}(n)$ and ${RQ}^{\prime}_{p}(n)$ denote the Rees quotients of $K(n,p)$ and $M(n,p)$, respectively. It is shown in this article that for any \( S \in \{ \mathcal{RIC}_{p}(n), K(n,p) \} \), \( S \) is abundant; \( \mathcal{IC}_{n} \) is ample; and for any \( S \in \{ \mathcal{Q}^{\prime}_{n}, \mathcal{RQ}^{\prime}_{p}(n), M(n,p) \} \), \( S \) is right abundant for all values of \( n \), but not left abundant for \( n \geq 2 \). Furthermore, the ranks of the Rees quotients ${RIC}_{p}(n)$ and ${RQ}^{\prime}_{p}(n)$ are shown to be equal to the ranks of the two-sided ideals $K(n,p)$ and $M(n,p)$, respectively. These ranks are found to be $\binom{n}{p}+(n-1)\binom{n-2}{p-1}$ and $\binom{n}{p}+(n-2)\binom{n-3}{p-1}$, respectively. In addition, the ranks of the semigroups $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$ were found to be $2n$ and $n^{2}-3n+4$, respectively. Finally, we characterize all the maximal subsemigroups of $\mathcal{IC}_{n}$ and $\mathcal{Q}^{\prime}_{n}$. |
| title | On injective partial Catalan monoids |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/2501.00285 |