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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| Видавець: | Lugansk National Taras Shevchenko University |
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| Дата: | 2023 |
| Автори: | , , , , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2023
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1846 |
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oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
full transformation order-preserving quasi-idempotent generating set 20M20 |
| 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 |
| topic_facet |
full transformation order-preserving quasi-idempotent generating set 20M20 |
| format |
Article |
| 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. |
| title |
Quasi-idempotents in finite semigroup of full order-preserving transformations |
| title_short |
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_sort |
quasi-idempotents in finite semigroup of full order-preserving transformations |
| 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 \(x \leq m < x + 1\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2023 |
| 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 |
| first_indexed |
2024-04-12T06:13:53Z |
| last_indexed |
2024-04-12T06:13:53Z |
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1804810505028632576 |