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...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2015
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543259459846144 |
|---|---|
| author | Catarino, Paula Higgins, Peter M. Levi, Inessa |
| author_facet | Catarino, Paula Higgins, Peter M. Levi, Inessa |
| author_sort | Catarino, Paula |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2015-09-28T11:22:08Z |
| 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 \(n \leq 3\). |
| first_indexed | 2025-12-02T15:31:40Z |
| format | Article |
| id | admjournalluguniveduua-article-43 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:31:40Z |
| publishDate | 2015 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-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 |
| 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 |
| title | 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_short | 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 |
| topic | semigroup semilattice inverse subsemigroup strong inverse transformation order-preserving transformation orientation-preserving transformation orientation-reversing transformation 20M20 05C25 |
| topic_facet | semigroup semilattice inverse subsemigroup strong inverse transformation order-preserving transformation orientation-preserving transformation orientation-reversing transformation 20M20 05C25 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 |
| work_keys_str_mv | AT catarinopaula oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations AT higginspeterm oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations AT leviinessa oninversesubsemigroupsofthesemigroupoforientationpreservingororientationreversingtransformations |