Quasi-idempotents in finite semigroup of full order-preserving transformations
Let \(X_n\) be the finite set \(\left\lbrace1,2,3\cdots,n\right\rbrace\) and \(\mathcal{O}_n\) defined by \(O_n = \lbrace \alpha\in T_n\colon (\forall x,y \in X_n),\; x\leq y\rightarrow x\alpha \leq y\alpha\rbrace\) be the semigroup of full order-preserving mapping on \(X_n\). A~transformation \(\al...
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| Datum: | 2023 |
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Lugansk National Taras Shevchenko University
2023
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1846 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543195563819008 |
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| author | Imam, A. T. Ibrahim, S. Garba, G. U. Usman, L. Idris, A. |
| author_facet | Imam, A. T. Ibrahim, S. Garba, G. U. Usman, L. Idris, A. |
| author_sort | Imam, A. T. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2023-06-18T17:55:47Z |
| description | Let \(X_n\) be the finite set \(\left\lbrace1,2,3\cdots,n\right\rbrace\) and \(\mathcal{O}_n\) defined by \(O_n = \lbrace \alpha\in T_n\colon (\forall x,y \in X_n),\; x\leq y\rightarrow x\alpha \leq y\alpha\rbrace\) be the semigroup of full order-preserving mapping on \(X_n\). A~transformation \(\alpha\) in \(\mathcal{O}_n\) is called quasi-idempotent if \(\alpha\neq \alpha^2= \alpha^4\). We characterise quasi-idempotent in \(\mathcal{O}_n\) and show that the semigroup \(\mathcal{O}_n\) is quasi-idempotent generated. Moreover, we obtained an upper bound for quasi-idempotents rank of \(\mathcal{O}_n\), that is, we showed that the cardinality of a minimum quasi-idempotents generating set for \(\mathcal{O}_n\) is less than or equal to \(\lceil \frac{3(n-2)}{2}\rceil\) where \(\lceil x\rceil\) denotes the least positive integer \(m\) such that \(x \leq m < x + 1\). |
| first_indexed | 2025-12-02T15:39:20Z |
| format | Article |
| id | admjournalluguniveduua-article-1846 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:39:20Z |
| publishDate | 2023 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-18462023-06-18T17:55:47Z Quasi-idempotents in finite semigroup of full order-preserving transformations Imam, A. T. Ibrahim, S. Garba, G. U. Usman, L. Idris, A. full transformation, order-preserving, quasi-idempotent, generating set 20M20 Let \(X_n\) be the finite set \(\left\lbrace1,2,3\cdots,n\right\rbrace\) and \(\mathcal{O}_n\) defined by \(O_n = \lbrace \alpha\in T_n\colon (\forall x,y \in X_n),\; x\leq y\rightarrow x\alpha \leq y\alpha\rbrace\) be the semigroup of full order-preserving mapping on \(X_n\). A~transformation \(\alpha\) in \(\mathcal{O}_n\) is called quasi-idempotent if \(\alpha\neq \alpha^2= \alpha^4\). We characterise quasi-idempotent in \(\mathcal{O}_n\) and show that the semigroup \(\mathcal{O}_n\) is quasi-idempotent generated. Moreover, we obtained an upper bound for quasi-idempotents rank of \(\mathcal{O}_n\), that is, we showed that the cardinality of a minimum quasi-idempotents generating set for \(\mathcal{O}_n\) is less than or equal to \(\lceil \frac{3(n-2)}{2}\rceil\) where \(\lceil x\rceil\) denotes the least positive integer \(m\) such that \(x \leq m < x + 1\). Lugansk National Taras Shevchenko University 2023-06-18 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1846 10.12958/adm1846 Algebra and Discrete Mathematics; Vol 35, No 1 (2023) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1846/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1846/892 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1846/897 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1846/904 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1846/905 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1846/921 Copyright (c) 2023 Algebra and Discrete Mathematics |
| spellingShingle | full transformation order-preserving quasi-idempotent generating set 20M20 Imam, A. T. Ibrahim, S. Garba, G. U. Usman, L. Idris, A. Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title | Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_full | Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_fullStr | Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_full_unstemmed | Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_short | Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_sort | quasi-idempotents in finite semigroup of full order-preserving transformations |
| topic | full transformation order-preserving quasi-idempotent generating set 20M20 |
| topic_facet | full transformation order-preserving quasi-idempotent generating set 20M20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1846 |
| work_keys_str_mv | AT imamat quasiidempotentsinfinitesemigroupoffullorderpreservingtransformations AT ibrahims quasiidempotentsinfinitesemigroupoffullorderpreservingtransformations AT garbagu quasiidempotentsinfinitesemigroupoffullorderpreservingtransformations AT usmanl quasiidempotentsinfinitesemigroupoffullorderpreservingtransformations AT idrisa quasiidempotentsinfinitesemigroupoffullorderpreservingtransformations |