Some results on semigroups of transformations with restricted range

Let XX be a non-empty set and YY a non-empty subset of XX. Denote the full transformation semigroup on XX by T(X)T\left(X) and write f(X)={f(x)∣x∈X}f\left(X)=\{f\left(x)| x\in X\} for each f∈T(X)f\in T\left(X). It is well known that T(X,Y)={f∈T(X)∣f(X)⊆Y}T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subse...

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Autores principales:	Yan Qingfu, Wang Shoufeng
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	green’s ∼-relation left (resp. right) ehresmann (resp. restriction) semigroup orthodox completely regular 20m20 20m10 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/409be9a1e905499f8ab9600ee5b52f9b
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spelling	oai:doaj.org-article:409be9a1e905499f8ab9600ee5b52f9b2021-12-05T14:10:52ZSome results on semigroups of transformations with restricted range2391-545510.1515/math-2021-0017https://doaj.org/article/409be9a1e905499f8ab9600ee5b52f9b2021-04-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0017https://doaj.org/toc/2391-5455Let XX be a non-empty set and YY a non-empty subset of XX. Denote the full transformation semigroup on XX by T(X)T\left(X) and write f(X)={f(x)∣x∈X}f\left(X)=\{f\left(x)\| x\in X\} for each f∈T(X)f\in T\left(X). It is well known that T(X,Y)={f∈T(X)∣f(X)⊆Y}T\left(X,Y)=\{f\in T\left(X)\| f\left(X)\subseteq Y\} is a subsemigroup of T(X)T\left(X) and RT(X,Y)RT\left(X,Y), the set of all regular elements of T(X,Y)T\left(X,Y), also forms a subsemigroup of T(X,Y)T\left(X,Y). Green’s ∗\ast -relations and Green’s ∼<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> \sim </mml:mpadded>\hspace{0.08em}-relations (with respect to a non-empty subset UU of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T(X,Y)T\left(X,Y). This investigation proves that the semigroup T(X,Y)T\left(X,Y) is always a left Ehresmann semigroup, and RT(X,Y)RT\left(X,Y) is orthodox (resp. completely regular) if and only if YY contains at most two elements.Yan QingfuWang ShoufengDe Gruyterarticlegreen’s ∼-relationleft (resp. right) ehresmann (resp. restriction) semigrouporthodoxcompletely regular20m2020m10MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 69-76 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	green’s ∼-relation left (resp. right) ehresmann (resp. restriction) semigroup orthodox completely regular 20m20 20m10 Mathematics QA1-939
spellingShingle	green’s ∼-relation left (resp. right) ehresmann (resp. restriction) semigroup orthodox completely regular 20m20 20m10 Mathematics QA1-939 Yan Qingfu Wang Shoufeng Some results on semigroups of transformations with restricted range
description	Let XX be a non-empty set and YY a non-empty subset of XX. Denote the full transformation semigroup on XX by T(X)T\left(X) and write f(X)={f(x)∣x∈X}f\left(X)=\{f\left(x)\| x\in X\} for each f∈T(X)f\in T\left(X). It is well known that T(X,Y)={f∈T(X)∣f(X)⊆Y}T\left(X,Y)=\{f\in T\left(X)\| f\left(X)\subseteq Y\} is a subsemigroup of T(X)T\left(X) and RT(X,Y)RT\left(X,Y), the set of all regular elements of T(X,Y)T\left(X,Y), also forms a subsemigroup of T(X,Y)T\left(X,Y). Green’s ∗\ast -relations and Green’s ∼<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> \sim </mml:mpadded>\hspace{0.08em}-relations (with respect to a non-empty subset UU of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T(X,Y)T\left(X,Y). This investigation proves that the semigroup T(X,Y)T\left(X,Y) is always a left Ehresmann semigroup, and RT(X,Y)RT\left(X,Y) is orthodox (resp. completely regular) if and only if YY contains at most two elements.
format	article
author	Yan Qingfu Wang Shoufeng
author_facet	Yan Qingfu Wang Shoufeng
author_sort	Yan Qingfu
title	Some results on semigroups of transformations with restricted range
title_short	Some results on semigroups of transformations with restricted range
title_full	Some results on semigroups of transformations with restricted range
title_fullStr	Some results on semigroups of transformations with restricted range
title_full_unstemmed	Some results on semigroups of transformations with restricted range
title_sort	some results on semigroups of transformations with restricted range
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/409be9a1e905499f8ab9600ee5b52f9b
work_keys_str_mv	AT yanqingfu someresultsonsemigroupsoftransformationswithrestrictedrange AT wangshoufeng someresultsonsemigroupsoftransformationswithrestrictedrange
_version_	1718371642597441536

Some results on semigroups of transformations with restricted range

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