Matrix theory for independence algebras

Detalhes bibliográficos
Autor(a) principal: Araújo, João
Data de Publicação: 2022
Outros Autores: Bentz, Wolfram, Cameron, Peter, Kinyon, Michael, Konieczny, Janusz
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10400.2/13251
Resumo: Preprint de J. Araújo, W. Bentz, P.J. Cameron, M. Kinyon, J. Konieczny, “Matrix Theory for Independence Algebras”, Linear Algebra and its Applications 642 (2022), 221-250.
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spelling Matrix theory for independence algebrasMatrix theorySemigroupsUniversal algebraGroupsFieldsModel theoryPreprint de J. Araújo, W. Bentz, P.J. Cameron, M. Kinyon, J. Konieczny, “Matrix Theory for Independence Algebras”, Linear Algebra and its Applications 642 (2022), 221-250.A universal algebra A with underlying set A is said to be a matroid algebra if (A, 〈·〉), where 〈·〉 denotes the operator subalgebra generated by, is a matroid. A matroid algebra is said to be an independence algebra if every mapping α : X → A defined on a minimal generating X of A can be extended to an endomorphism of A. These algebras are particularly well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics, such as model theory, group theory, and semigroup theory. It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n, with at least two elements. Denote by End(A) the monoid of endomorphisms of A. In the 1970s, Glazek proposed the problem of extending the matrix theory for vector spaces to a class of universal algebras which included independence algebras. In this paper, we answer that problem by developing a theory of matrices for (almost all) finite-dimensional independence algebras. In the process of solving this, we explain the relation between the classification of inde- pendence algebras obtained by Urbanik in the 1960s, and the classification of finite indepen- dence algebras up to endomorphism-equivalence obtained by Cameron and Szab ́o in 2000. (This answers another question by experts on independence algebras.) We also extend the classification of Cameron and Szab ́o to all independence algebras. The paper closes with a number of questions for experts on matrix theory, groups, semi- groups, universal algebra, set theory or model theory.This work was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020, UIDP/00297/2020 (Center for Mathematics and Applications) and PTDC/MAT/PUR/31174/2017.PTDC/MAT/PUR/31174/2017ElsevierRepositório AbertoAraújo, JoãoBentz, WolframCameron, PeterKinyon, MichaelKonieczny, Janusz2023-06-01T00:30:23Z2022-06-012022-06-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.2/13251engJ. Araújo, W. Bentz, P.J. Cameron, M. Kinyon, J. Konieczny, “Matrix Theory for Independence Algebras”, Linear Algebra and its Applications 642 (2022), 221-250.0024-379510.1016/j.laa.2022.02.0211873-1856info: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:RCAAP2023-12-03T01:47:57Zoai:repositorioaberto.uab.pt:10400.2/13251Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T22:52:17.573986Repositório 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 Matrix theory for independence algebras
title Matrix theory for independence algebras
spellingShingle Matrix theory for independence algebras
Araújo, João
Matrix theory
Semigroups
Universal algebra
Groups
Fields
Model theory
title_short Matrix theory for independence algebras
title_full Matrix theory for independence algebras
title_fullStr Matrix theory for independence algebras
title_full_unstemmed Matrix theory for independence algebras
title_sort Matrix theory for independence algebras
author Araújo, João
author_facet Araújo, João
Bentz, Wolfram
Cameron, Peter
Kinyon, Michael
Konieczny, Janusz
author_role author
author2 Bentz, Wolfram
Cameron, Peter
Kinyon, Michael
Konieczny, Janusz
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Repositório Aberto
dc.contributor.author.fl_str_mv Araújo, João
Bentz, Wolfram
Cameron, Peter
Kinyon, Michael
Konieczny, Janusz
dc.subject.por.fl_str_mv Matrix theory
Semigroups
Universal algebra
Groups
Fields
Model theory
topic Matrix theory
Semigroups
Universal algebra
Groups
Fields
Model theory
description Preprint de J. Araújo, W. Bentz, P.J. Cameron, M. Kinyon, J. Konieczny, “Matrix Theory for Independence Algebras”, Linear Algebra and its Applications 642 (2022), 221-250.
publishDate 2022
dc.date.none.fl_str_mv 2022-06-01
2022-06-01T00:00:00Z
2023-06-01T00:30:23Z
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/10400.2/13251
url http://hdl.handle.net/10400.2/13251
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv J. Araújo, W. Bentz, P.J. Cameron, M. Kinyon, J. Konieczny, “Matrix Theory for Independence Algebras”, Linear Algebra and its Applications 642 (2022), 221-250.
0024-3795
10.1016/j.laa.2022.02.021
1873-1856
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.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
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instacron:RCAAP
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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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