On certain semigroups of finite monotone and order-decreasing partial transformations
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| Format: | Preprint |
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2025
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| author | Ayık, Gonca Ayık, Hayrullah Dimitrova, Ilinka Koppitz, Jörg |
| author_facet | Ayık, Gonca Ayık, Hayrullah Dimitrova, Ilinka Koppitz, Jörg |
| contents | Let $\mathcal{PMD}_{n}$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let $\mathcal{IMD}_{n}$ be the subsemigroup of $\mathcal{PMD}_{n}$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. For $2\leq r\leq n$, let $\mathcal{PMD}(n,r) =\{ α\in \mathcal{PMD}_{n} : |\textrm{im}(α)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ α\in \mathcal{IMD}_{n} :|\textrm{im}(α)| \leq r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$, and moreover, we verify that the semigroups $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$ are non-regular but abundant for any $2\leq r\leq n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06047 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain semigroups of finite monotone and order-decreasing partial transformations Ayık, Gonca Ayık, Hayrullah Dimitrova, Ilinka Koppitz, Jörg Rings and Algebras 20M20, 20M05 Let $\mathcal{PMD}_{n}$ be the semigroup consisting of all monotone and order-decreasing partial transformations, and let $\mathcal{IMD}_{n}$ be the subsemigroup of $\mathcal{PMD}_{n}$ consisting of all injective monotone and order-decreasing transformations on the finite chain $X_{n}=\{ 1<\cdots<n \}$. For $2\leq r\leq n$, let $\mathcal{PMD}(n,r) =\{ α\in \mathcal{PMD}_{n} : |\textrm{im}(α)| \leq r\}$ and $\mathcal{IMD}(n,r)=\{ α\in \mathcal{IMD}_{n} :|\textrm{im}(α)| \leq r\}$. In this paper, we determine the cardinalities, maximal subsemigroups and ranks of $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$, and moreover, we verify that the semigroups $\mathcal{PMD}(n,r)$ and $\mathcal{IMD}(n,r)$ are non-regular but abundant for any $2\leq r\leq n$. |
| title | On certain semigroups of finite monotone and order-decreasing partial transformations |
| topic | Rings and Algebras 20M20, 20M05 |
| url | https://arxiv.org/abs/2507.06047 |