On products of 3-paths in finite full transformation semigroups
Let \(\operatorname{Sing}_{n}\) denotes the semigroup of all singular self-maps of a finite set \(X_n=\lbrace 1, 2,\ldots, n\rbrace\). A map \(\alpha\in \operatorname{Sing}_{n}\) is called a \(3\)-path if there are \(i, j, k\in X_n\) such that \(i\alpha=j\), \(j\alpha=k\) and \(x\alpha = x\) for all...
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| Дата: | 2022 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2022
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1770 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-17702022-10-14T16:01:17Z On products of 3-paths in finite full transformation semigroups Imam, A. T. Ibrahim, M. J. 3-path, length formular, full transformation 20M20, 20M05 Let \(\operatorname{Sing}_{n}\) denotes the semigroup of all singular self-maps of a finite set \(X_n=\lbrace 1, 2,\ldots, n\rbrace\). A map \(\alpha\in \operatorname{Sing}_{n}\) is called a \(3\)-path if there are \(i, j, k\in X_n\) such that \(i\alpha=j\), \(j\alpha=k\) and \(x\alpha = x\) for all \(x\in X_n\setminus \lbrace i,j\rbrace\). In this paper, we described a procedure to factorise each \(\alpha\in \operatorname{Sing}_{n}\) into a product of \(3\)-paths. The length of each factorisation, that is the number of factors in each factorisation, is obtained to be equal to \(\lceil\frac{1}{2}(g(\alpha)+m(\alpha))\rceil\), where \(g(\alpha)\) is known as the gravity of \(\alpha\) and \(m(\alpha)\) is a parameter introduced in this work and referred to as the measure of \(\alpha\). Moreover, we showed that \(\operatorname{Sing}_n\subseteq P^{[n-1]}\), where \(P\) denotes the set of all \(3\)-paths in \(\operatorname{Sing}_n\) and \(P^{[k]}=P\cup P^2\cup \cdots \cup P^k\). Lugansk National Taras Shevchenko University 2022-10-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1770 10.12958/adm1770 Algebra and Discrete Mathematics; Vol 33, No 2 (2022) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1770/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1770/836 Copyright (c) 2022 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
3-path length formular full transformation 20M20 20M05 |
| spellingShingle |
3-path length formular full transformation 20M20 20M05 Imam, A. T. Ibrahim, M. J. On products of 3-paths in finite full transformation semigroups |
| topic_facet |
3-path length formular full transformation 20M20 20M05 |
| format |
Article |
| author |
Imam, A. T. Ibrahim, M. J. |
| author_facet |
Imam, A. T. Ibrahim, M. J. |
| author_sort |
Imam, A. T. |
| title |
On products of 3-paths in finite full transformation semigroups |
| title_short |
On products of 3-paths in finite full transformation semigroups |
| title_full |
On products of 3-paths in finite full transformation semigroups |
| title_fullStr |
On products of 3-paths in finite full transformation semigroups |
| title_full_unstemmed |
On products of 3-paths in finite full transformation semigroups |
| title_sort |
on products of 3-paths in finite full transformation semigroups |
| description |
Let \(\operatorname{Sing}_{n}\) denotes the semigroup of all singular self-maps of a finite set \(X_n=\lbrace 1, 2,\ldots, n\rbrace\). A map \(\alpha\in \operatorname{Sing}_{n}\) is called a \(3\)-path if there are \(i, j, k\in X_n\) such that \(i\alpha=j\), \(j\alpha=k\) and \(x\alpha = x\) for all \(x\in X_n\setminus \lbrace i,j\rbrace\). In this paper, we described a procedure to factorise each \(\alpha\in \operatorname{Sing}_{n}\) into a product of \(3\)-paths. The length of each factorisation, that is the number of factors in each factorisation, is obtained to be equal to \(\lceil\frac{1}{2}(g(\alpha)+m(\alpha))\rceil\), where \(g(\alpha)\) is known as the gravity of \(\alpha\) and \(m(\alpha)\) is a parameter introduced in this work and referred to as the measure of \(\alpha\). Moreover, we showed that \(\operatorname{Sing}_n\subseteq P^{[n-1]}\), where \(P\) denotes the set of all \(3\)-paths in \(\operatorname{Sing}_n\) and \(P^{[k]}=P\cup P^2\cup \cdots \cup P^k\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2022 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1770 |
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AT imamat onproductsof3pathsinfinitefulltransformationsemigroups AT ibrahimmj onproductsof3pathsinfinitefulltransformationsemigroups |
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2024-04-12T06:25:14Z |
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2024-04-12T06:25:14Z |
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1796109218129903616 |