The commuting graph of the symmetric inverse semigroup
Autor(a) principal: | |
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Data de Publicação: | 2014 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.2/3813 |
Resumo: | The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dol˘zan and Oblak claimed that this upper bound is in fact the exact value.The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory. |
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The commuting graph of the symmetric inverse semigroupCommuting graphs of semigroupsSymmetric inverse semigroupCommutative semigroupsInverse semigroupsNilpotent semigroupsClique numberDiameterThe commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dol˘zan and Oblak claimed that this upper bound is in fact the exact value.The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.Repositório AbertoAraújo, JoãoBentz, WolframKonieczny, Janusz2015-03-24T17:21:14Z20142014-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/3813engAraújo, João; Bentz, Wolfram; Konieczny, Janusz - The commuting graph of the symmetric inverse semigroup. "Israel Journal of Mathematics" [Em linha]. ISSN 0021-2172 (Print) 1565-8511 (Online). (2014), p. 1-290021-2172info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-16T15:19:10Zoai:repositorioaberto.uab.pt:10400.2/3813Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:44:59.611045Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
The commuting graph of the symmetric inverse semigroup |
title |
The commuting graph of the symmetric inverse semigroup |
spellingShingle |
The commuting graph of the symmetric inverse semigroup Araújo, João Commuting graphs of semigroups Symmetric inverse semigroup Commutative semigroups Inverse semigroups Nilpotent semigroups Clique number Diameter |
title_short |
The commuting graph of the symmetric inverse semigroup |
title_full |
The commuting graph of the symmetric inverse semigroup |
title_fullStr |
The commuting graph of the symmetric inverse semigroup |
title_full_unstemmed |
The commuting graph of the symmetric inverse semigroup |
title_sort |
The commuting graph of the symmetric inverse semigroup |
author |
Araújo, João |
author_facet |
Araújo, João Bentz, Wolfram Konieczny, Janusz |
author_role |
author |
author2 |
Bentz, Wolfram Konieczny, Janusz |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
Repositório Aberto |
dc.contributor.author.fl_str_mv |
Araújo, João Bentz, Wolfram Konieczny, Janusz |
dc.subject.por.fl_str_mv |
Commuting graphs of semigroups Symmetric inverse semigroup Commutative semigroups Inverse semigroups Nilpotent semigroups Clique number Diameter |
topic |
Commuting graphs of semigroups Symmetric inverse semigroup Commutative semigroups Inverse semigroups Nilpotent semigroups Clique number Diameter |
description |
The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse semigroup of partial injective transformations on a finite set X. The semigroup I(X) has the symmetric group Sym(X) of permutations on X as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of Sym(X). In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of G(Sym(X)), and in 2011, Dol˘zan and Oblak claimed that this upper bound is in fact the exact value.The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup I(X). We calculate the clique number of G(I(X)), the diameters of the commuting graphs of the proper ideals of I(X), and the diameter of G(I(X)) when |X| is even or a power of an odd prime. We show that when |X| is odd and divisible by at least two primes, then the diameter of G(I(X)) is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of I(X) of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of I(X). The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory. |
publishDate |
2014 |
dc.date.none.fl_str_mv |
2014 2014-01-01T00:00:00Z 2015-03-24T17:21:14Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10400.2/3813 |
url |
http://hdl.handle.net/10400.2/3813 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Araújo, João; Bentz, Wolfram; Konieczny, Janusz - The commuting graph of the symmetric inverse semigroup. "Israel Journal of Mathematics" [Em linha]. ISSN 0021-2172 (Print) 1565-8511 (Online). (2014), p. 1-29 0021-2172 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799135021426016256 |