The monoid of monotone and decreasing partial transformations on a finite chain
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| Format: | Preprint |
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2025
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| _version_ | 1866912792987041792 |
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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 | In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{ρ\in \mathcal{DORP}_{n} : \, |Im \, ρ| \leq p\}$ and the Rees quotient ${RQ}_{p}(n)$ of the ideal $I(n,p)$. We compute the order of the monoid $\mathcal{DORP}_{n}$ and show that for any semigroup $S$ in $\{\mathcal{DORP}_{n}, \, I(n,p), \, {RQ}_{p}(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_{p}(n)$, is a non-regular $0-*$bisimple abundant semigroup. In addition, we compute the ranks of the Rees quotient ${RQ}_{p}(n)$ and the two-sided ideal $I(n,p)$. Finally, the rank of $\mathcal{DORP}_{n}$ is determined to be $3n-2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The monoid of monotone and decreasing partial transformations on a finite chain Zubairu, Muhammad Mansur Umar, Abdullahi Al-Kharousi, Fatma Salim Group Theory 20M20 In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{ρ\in \mathcal{DORP}_{n} : \, |Im \, ρ| \leq p\}$ and the Rees quotient ${RQ}_{p}(n)$ of the ideal $I(n,p)$. We compute the order of the monoid $\mathcal{DORP}_{n}$ and show that for any semigroup $S$ in $\{\mathcal{DORP}_{n}, \, I(n,p), \, {RQ}_{p}(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_{p}(n)$, is a non-regular $0-*$bisimple abundant semigroup. In addition, we compute the ranks of the Rees quotient ${RQ}_{p}(n)$ and the two-sided ideal $I(n,p)$. Finally, the rank of $\mathcal{DORP}_{n}$ is determined to be $3n-2$. |
| title | The monoid of monotone and decreasing partial transformations on a finite chain |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/2512.21134 |