On certain semigroups of contraction mappings of a finite chain
Let [n] = {1, 2, . . . , n} be a finite chain and let Pn (resp. , Tn) be the semigroup of partial transformations on [n] (resp. , full transformations on [n]). Let CPn = {α ∈ Pn : (for all x, y ∈ Dom α) |xα−yα| ≤ |x−y|} (resp. , CT n = {α ∈ Tn : (for all x, y ∈ [n]) |xα−yα| ≤ |x−y|} ) be the subsemi...
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2021 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/188755 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On certain semigroups of contraction mappings of a finite chain / A. Umar, M.M. Zubairu // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 299-320. — Бібліогр.: 37 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862605845671444480 |
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| author | Umar, A. Zubairu, M.M. |
| author_facet | Umar, A. Zubairu, M.M. |
| citation_txt | On certain semigroups of contraction mappings of a finite chain / A. Umar, M.M. Zubairu // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 299-320. — Бібліогр.: 37 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let [n] = {1, 2, . . . , n} be a finite chain and let Pn (resp. , Tn) be the semigroup of partial transformations on [n] (resp. , full transformations on [n]). Let CPn = {α ∈ Pn : (for all x, y ∈ Dom α) |xα−yα| ≤ |x−y|} (resp. , CT n = {α ∈ Tn : (for all x, y ∈ [n]) |xα−yα| ≤ |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 CPn 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 CPn and CT n, and some of their subsemigroups are left abundant semigroups for all n but not right abundant for n ≥ 4. We further show that the set of regular elements of the semigroup 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.
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| first_indexed | 2025-11-28T11:21:53Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-188755 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-28T11:21:53Z |
| publishDate | 2021 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Umar, A. Zubairu, M.M. 2023-03-14T17:16:37Z 2023-03-14T17:16:37Z 2021 On certain semigroups of contraction mappings of a finite chain / A. Umar, M.M. Zubairu // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 2. — С. 299-320. — Бібліогр.: 37 назв. — англ. 1726-3255 DOI:10.12958/adm1816 2020 MSC: 20M20. https://nasplib.isofts.kiev.ua/handle/123456789/188755 Let [n] = {1, 2, . . . , n} be a finite chain and let Pn (resp. , Tn) be the semigroup of partial transformations on [n] (resp. , full transformations on [n]). Let CPn = {α ∈ Pn : (for all x, y ∈ Dom α) |xα−yα| ≤ |x−y|} (resp. , CT n = {α ∈ Tn : (for all x, y ∈ [n]) |xα−yα| ≤ |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 CPn 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 CPn and CT n, and some of their subsemigroups are left abundant semigroups for all n but not right abundant for n ≥ 4. We further show that the set of regular elements of the semigroup 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. The second author would like to thank Bayero University and TET Fund for financial support. He would also like to thank The Petroleum Institute, Khalifa University of Science and Technology for hospitality during his 3-months research visit (November 2017 to February 2018) to the institution. The authors thank the referee for providing useful suggestions. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On certain semigroups of contraction mappings of a finite chain Article published earlier |
| spellingShingle | On certain semigroups of contraction mappings of a finite chain Umar, A. Zubairu, M.M. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188755 |
| work_keys_str_mv | AT umara oncertainsemigroupsofcontractionmappingsofafinitechain AT zubairumm oncertainsemigroupsofcontractionmappingsofafinitechain |