On certain semigroups of contraction mappings of a finite chain
Let \([n]=\{1,2,\ldots,n\}\) be a finite chain and let \(\mathcal{P}_{n}\) (resp., \(\mathcal{T}_{n}\)) be the semigroup of partial transformations on \([n]\) (resp., full transformations on \([n]\)). Let \(\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (\text{for all }x,y\in \operatorname{Dom}\alpha...
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| Дата: | 2022 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2022
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1816 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543140465344512 |
|---|---|
| author | Umar, A. Zubairu, M. M. |
| author_facet | Umar, A. Zubairu, M. M. |
| author_sort | Umar, A. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2022-03-28T05:34:02Z |
| description | Let \([n]=\{1,2,\ldots,n\}\) be a finite chain and let \(\mathcal{P}_{n}\) (resp., \(\mathcal{T}_{n}\)) be the semigroup of partial transformations on \([n]\) (resp., full transformations on \([n]\)). Let \(\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (\text{for all }x,y\in \operatorname{Dom}\alpha)\ |x\alpha-y\alpha|\leq|x-y|\}\) (resp., \(\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: (\text{for all }x,y\in [n])\ |x\alpha-y\alpha|\leq|x-y|\}\) ) be the subsemigroup of partial contraction mappings on \([n]\) (resp., subsemigroup of full contraction mappings on \([n]\)). We characterize all the starred Green's relations on \(\mathcal{CP}_{n}\) and it subsemigroup of order preserving and/or order reversing and subsemigroup of order preserving partial contractions on \([n]\), respectively. We show that the semigroups \(\mathcal{CP}_{n}\) and \(\mathcal{CT}_{n}\), and some of their subsemigroups are left abundant semigroups for all \(n\) but not right abundant for \(n\geq 4\). We further show that the set of regular elements of the semigroup \(\mathcal{CT}_{n}\) and its subsemigroup of order preserving or order reversing full contractions on \([n]\), each forms a regular subsemigroup and an orthodox semigroup, respectively. |
| first_indexed | 2026-02-08T07:57:46Z |
| format | Article |
| id | admjournalluguniveduua-article-1816 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:46Z |
| publishDate | 2022 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-18162022-03-28T05:34:02Z On certain semigroups of contraction mappings of a finite chain Umar, A. Zubairu, M. M. starred Green's relations, orthodox semigroups, quasi-adequate semigroups, regularity 20M20 Let \([n]=\{1,2,\ldots,n\}\) be a finite chain and let \(\mathcal{P}_{n}\) (resp., \(\mathcal{T}_{n}\)) be the semigroup of partial transformations on \([n]\) (resp., full transformations on \([n]\)). Let \(\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}: (\text{for all }x,y\in \operatorname{Dom}\alpha)\ |x\alpha-y\alpha|\leq|x-y|\}\) (resp., \(\mathcal{CT}_{n}=\{\alpha\in \mathcal{T}_{n}: (\text{for all }x,y\in [n])\ |x\alpha-y\alpha|\leq|x-y|\}\) ) be the subsemigroup of partial contraction mappings on \([n]\) (resp., subsemigroup of full contraction mappings on \([n]\)). We characterize all the starred Green's relations on \(\mathcal{CP}_{n}\) and it subsemigroup of order preserving and/or order reversing and subsemigroup of order preserving partial contractions on \([n]\), respectively. We show that the semigroups \(\mathcal{CP}_{n}\) and \(\mathcal{CT}_{n}\), and some of their subsemigroups are left abundant semigroups for all \(n\) but not right abundant for \(n\geq 4\). We further show that the set of regular elements of the semigroup \(\mathcal{CT}_{n}\) and its subsemigroup of order preserving or order reversing full contractions on \([n]\), each forms a regular subsemigroup and an orthodox semigroup, respectively. Lugansk National Taras Shevchenko University Bayero University and TET Fund Nigeria The Petroleum Institute, Khalifa University of Science and Technology, UAE 2022-03-28 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1816 10.12958/adm1816 Algebra and Discrete Mathematics; Vol 32, No 2 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1816/pdf Copyright (c) 2022 Algebra and Discrete Mathematics |
| spellingShingle | starred Green's relations orthodox semigroups quasi-adequate semigroups regularity 20M20 Umar, A. Zubairu, M. M. On certain semigroups of contraction mappings of a finite chain |
| title | On certain semigroups of contraction mappings of a finite chain |
| title_full | On certain semigroups of contraction mappings of a finite chain |
| title_fullStr | On certain semigroups of contraction mappings of a finite chain |
| title_full_unstemmed | On certain semigroups of contraction mappings of a finite chain |
| title_short | On certain semigroups of contraction mappings of a finite chain |
| title_sort | on certain semigroups of contraction mappings of a finite chain |
| topic | starred Green's relations orthodox semigroups quasi-adequate semigroups regularity 20M20 |
| topic_facet | starred Green's relations orthodox semigroups quasi-adequate semigroups regularity 20M20 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1816 |
| work_keys_str_mv | AT umara oncertainsemigroupsofcontractionmappingsofafinitechain AT zubairumm oncertainsemigroupsofcontractionmappingsofafinitechain |