On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations
It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of or...
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| Дата: | 2015 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2015
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-432015-09-28T11:22:08Z On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations Catarino, Paula Higgins, Peter M. Levi, Inessa semigroup, semilattice, inverse subsemigroup, strong inverse, transformation, order-preserving transformation, orientation-preserving transformation, orientation-reversing transformation 20M20, 05C25 It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain \(X_n\) is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of \(X_n\) is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain \(X_n\) are covered by their inverse subsemigroups precisely when \(n \leq 3\). Lugansk National Taras Shevchenko University 2015-09-28 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 Algebra and Discrete Mathematics; Vol 19, No 2 (2015) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43/10 Copyright (c) 2015 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
semigroup semilattice inverse subsemigroup strong inverse transformation order-preserving transformation orientation-preserving transformation orientation-reversing transformation 20M20 05C25 |
| spellingShingle |
semigroup semilattice inverse subsemigroup strong inverse transformation order-preserving transformation orientation-preserving transformation orientation-reversing transformation 20M20 05C25 Catarino, Paula Higgins, Peter M. Levi, Inessa On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| topic_facet |
semigroup semilattice inverse subsemigroup strong inverse transformation order-preserving transformation orientation-preserving transformation orientation-reversing transformation 20M20 05C25 |
| format |
Article |
| author |
Catarino, Paula Higgins, Peter M. Levi, Inessa |
| author_facet |
Catarino, Paula Higgins, Peter M. Levi, Inessa |
| author_sort |
Catarino, Paula |
| title |
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| title_short |
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| title_full |
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| title_fullStr |
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| title_full_unstemmed |
On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| title_sort |
on inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations |
| description |
It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain \(X_n\) is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of \(X_n\) is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain \(X_n\) are covered by their inverse subsemigroups precisely when \(n \leq 3\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2015 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 |
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AT catarinopaula oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations AT higginspeterm oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations AT leviinessa oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations |
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2024-04-12T06:27:41Z |
| last_indexed |
2024-04-12T06:27:41Z |
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