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On certain semigroups of contraction mappings of a finite chain

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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Main Authors: Umar, A., Zubairu, M.M.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2021
Series:Algebra and Discrete Mathematics
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/188755
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spelling irk-123456789-1887552023-03-15T01:27:29Z On certain semigroups of contraction mappings of a finite chain Umar, A. Zubairu, M.M. 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. 2021 Article 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. http://dspace.nbuv.gov.ua/handle/123456789/188755 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Umar, A.
Zubairu, M.M.
spellingShingle Umar, A.
Zubairu, M.M.
On certain semigroups of contraction mappings of a finite chain
Algebra and Discrete Mathematics
author_facet Umar, A.
Zubairu, M.M.
author_sort Umar, A.
title On certain semigroups of contraction mappings of a finite chain
title_short 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_sort on certain semigroups of contraction mappings of a finite chain
publisher Інститут прикладної математики і механіки НАН України
publishDate 2021
url http://dspace.nbuv.gov.ua/handle/123456789/188755
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.
series Algebra and Discrete Mathematics
work_keys_str_mv AT umara oncertainsemigroupsofcontractionmappingsofafinitechain
AT zubairumm oncertainsemigroupsofcontractionmappingsofafinitechain
first_indexed 2023-10-18T23:09:06Z
last_indexed 2023-10-18T23:09:06Z
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