Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain
Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2023-09-01
|
| Series: | Scientific African |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2468227623003459 |
| _version_ | 1797676248572362752 |
|---|---|
| author | Bashir Ali Abdullahi Umar Muhammad Mansur Zubairu |
| author_facet | Bashir Ali Abdullahi Umar Muhammad Mansur Zubairu |
| author_sort | Bashir Ali |
| collection | DOAJ |
| description | Let [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In the subsequent fourth and fifth sections we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Finally, in the last section, we characterize all the maximal subgroups of these semigroups. |
| first_indexed | 2024-03-11T22:26:15Z |
| format | Article |
| id | doaj.art-d1abd88ed7d84272bb35db0ea4be0a9b |
| institution | Directory Open Access Journal |
| issn | 2468-2276 |
| language | English |
| last_indexed | 2024-03-11T22:26:15Z |
| publishDate | 2023-09-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Scientific African |
| spelling | doaj.art-d1abd88ed7d84272bb35db0ea4be0a9b2023-09-24T05:16:31ZengElsevierScientific African2468-22762023-09-0121e01890Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chainBashir Ali0Abdullahi Umar1Muhammad Mansur Zubairu2Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna, Nigeria; Department of Mathematical Sciences, Khalifa University, P.O. Box 127788, Sas al Nakhl, Abu Dhabi, United Arab Emirates; Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011, Kano, NigeriaDepartment of Mathematical Sciences, Nigerian Defence Academy, Kaduna, Nigeria; Department of Mathematical Sciences, Khalifa University, P.O. Box 127788, Sas al Nakhl, Abu Dhabi, United Arab Emirates; Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011, Kano, NigeriaCorresponding author at: Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011, Kano, Nigeria.; Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna, Nigeria; Department of Mathematical Sciences, Khalifa University, P.O. Box 127788, Sas al Nakhl, Abu Dhabi, United Arab Emirates; Department of Mathematical Sciences, Bayero University Kano, P. M. B. 3011, Kano, NigeriaLet [n]={1,2,…,n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn={α∈Pn:(for allx,y∈Domα)|xα−yα|≤|x−y|} be the semigroup of partial contractions on [n]. Then CPn is a subsemigroup of Pn. In this paper, in the second and third sections we give a necessary and sufficient condition for an element in CPn to be regular and characterize all the Green’s relations in the semigroup CPn, respectively. In the subsequent fourth and fifth sections we obtain analogous results for the subsemigroups of order preserving and/or order reversing contractions, and that of full contractions, respectively. Finally, in the last section, we characterize all the maximal subgroups of these semigroups.http://www.sciencedirect.com/science/article/pii/S2468227623003459Partial contraction mappingsRegularityGreen’s relations |
| spellingShingle | Bashir Ali Abdullahi Umar Muhammad Mansur Zubairu Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain Scientific African Partial contraction mappings Regularity Green’s relations |
| title | Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain |
| title_full | Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain |
| title_fullStr | Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain |
| title_full_unstemmed | Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain |
| title_short | Regularity and Green’s relations for the semigroups of partial and full contractions of a finite chain |
| title_sort | regularity and green s relations for the semigroups of partial and full contractions of a finite chain |
| topic | Partial contraction mappings Regularity Green’s relations |
| url | http://www.sciencedirect.com/science/article/pii/S2468227623003459 |
| work_keys_str_mv | AT bashirali regularityandgreensrelationsforthesemigroupsofpartialandfullcontractionsofafinitechain AT abdullahiumar regularityandgreensrelationsforthesemigroupsofpartialandfullcontractionsofafinitechain AT muhammadmansurzubairu regularityandgreensrelationsforthesemigroupsofpartialandfullcontractionsofafinitechain |