Two generalizations of homogeneity in groups with applications to regular semigroups
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/10400.2/3811 |
Resumo: | Let X be a finite set such that |X| = n and let i 6 j 6 n. A group G 6 Sn is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.) A group G 6 Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.) In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1, k)-homogeneous groups (for 2 < k 6 ⌊n+12 ⌋). As a corollary of the classification we prove that a (k − 1, k homogeneous group is also (k − 2, k − 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k − 1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 6 k 6 ⌊n+1 2 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory. |
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Two generalizations of homogeneity in groups with applications to regular semigroupsTransformation semigroupsRegular semigroupsPermutation groupsPrimitive groupsHomogeneous groupsLet X be a finite set such that |X| = n and let i 6 j 6 n. A group G 6 Sn is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.) A group G 6 Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.) In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1, k)-homogeneous groups (for 2 < k 6 ⌊n+12 ⌋). As a corollary of the classification we prove that a (k − 1, k homogeneous group is also (k − 2, k − 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k − 1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 6 k 6 ⌊n+1 2 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.Repositório AbertoAraújo, JoãoCameron, Peter J.2015-03-24T15:37:05Z20142014-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/3811engAraújo João; Cameron, Peter J. - Two generalizations of homogeneity in groups with applications to regular semigroups. "Transactions of the American Mathematical Society" [Em linha]. ISSN 0002-9947 (Print) 1088-6850 (Online). (2014), p. 1-290002-9947info: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:RCAAP2023-11-16T15:19:09Zoai:repositorioaberto.uab.pt:10400.2/3811Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:44:59.318043Repositório 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 |
Two generalizations of homogeneity in groups with applications to regular semigroups |
title |
Two generalizations of homogeneity in groups with applications to regular semigroups |
spellingShingle |
Two generalizations of homogeneity in groups with applications to regular semigroups Araújo, João Transformation semigroups Regular semigroups Permutation groups Primitive groups Homogeneous groups |
title_short |
Two generalizations of homogeneity in groups with applications to regular semigroups |
title_full |
Two generalizations of homogeneity in groups with applications to regular semigroups |
title_fullStr |
Two generalizations of homogeneity in groups with applications to regular semigroups |
title_full_unstemmed |
Two generalizations of homogeneity in groups with applications to regular semigroups |
title_sort |
Two generalizations of homogeneity in groups with applications to regular semigroups |
author |
Araújo, João |
author_facet |
Araújo, João Cameron, Peter J. |
author_role |
author |
author2 |
Cameron, Peter J. |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Repositório Aberto |
dc.contributor.author.fl_str_mv |
Araújo, João Cameron, Peter J. |
dc.subject.por.fl_str_mv |
Transformation semigroups Regular semigroups Permutation groups Primitive groups Homogeneous groups |
topic |
Transformation semigroups Regular semigroups Permutation groups Primitive groups Homogeneous groups |
description |
Let X be a finite set such that |X| = n and let i 6 j 6 n. A group G 6 Sn is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.) A group G 6 Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.) In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1, k)-homogeneous groups (for 2 < k 6 ⌊n+12 ⌋). As a corollary of the classification we prove that a (k − 1, k homogeneous group is also (k − 2, k − 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k − 1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 6 k 6 ⌊n+1 2 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory. |
publishDate |
2014 |
dc.date.none.fl_str_mv |
2014 2014-01-01T00:00:00Z 2015-03-24T15:37:05Z |
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/10400.2/3811 |
url |
http://hdl.handle.net/10400.2/3811 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Araújo João; Cameron, Peter J. - Two generalizations of homogeneity in groups with applications to regular semigroups. "Transactions of the American Mathematical Society" [Em linha]. ISSN 0002-9947 (Print) 1088-6850 (Online). (2014), p. 1-29 0002-9947 |
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 |
repository.mail.fl_str_mv |
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1799135021415530496 |