Invariant ideals in Leavitt path algebras

It is known that the ideals of a Leavitt path algebra LK(E) generated by Pl(E), by Pc(E), or by Pec(E) are invariant under isomorphism. Though the ideal generated by Pb∞(E) is not invariant we find its "natural" replacement (which is indeed invariant): the one generated by the vertices of...

Descripción completa

Detalles Bibliográficos
Autores: Gil Canto, Cristobal|||0000-0002-4975-1935, Martín Barquero, Dolores|||0000-0002-7210-1578, Martín González, Cándido|||0000-0003-2796-7417
Tipo de recurso: artículo
Fecha de publicación:2022
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:264508
Acceso en línea:https://ddd.uab.cat/record/264508
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622203
Access Level:acceso abierto
Palabra clave:Leavitt path algebra
Annihilator
Socle
Invariant ideal
Dcc topology
Hereditary and saturated point functors
id ES_870deef2f55efd1986c8ec47715bbace
oai_identifier_str oai:ddd.uab.cat:264508
network_acronym_str ES
network_name_str España
repository_id_str
spelling Invariant ideals in Leavitt path algebrasGil Canto, Cristobal|||0000-0002-4975-1935Martín Barquero, Dolores|||0000-0002-7210-1578Martín González, Cándido|||0000-0003-2796-7417Leavitt path algebraAnnihilatorSocleInvariant idealDcc topologyHereditary and saturated point functorsIt is known that the ideals of a Leavitt path algebra LK(E) generated by Pl(E), by Pc(E), or by Pec(E) are invariant under isomorphism. Though the ideal generated by Pb∞(E) is not invariant we find its "natural" replacement (which is indeed invariant): the one generated by the vertices of Pb∞p (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the DCC topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if H is a hereditary saturated subset of vertices providing an invariant ideal, its exterior ext(H) in the DCC topology of E0 generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance Pl, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain Soc(1)( ) ⊆ Soc(2)( ) ⊆ · · · , all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative. 22022-01-0120222022-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/264508https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622203reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104236GB-I00open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2645082026-06-06T12:50:31Z
dc.title.none.fl_str_mv Invariant ideals in Leavitt path algebras
title Invariant ideals in Leavitt path algebras
spellingShingle Invariant ideals in Leavitt path algebras
Gil Canto, Cristobal|||0000-0002-4975-1935
Leavitt path algebra
Annihilator
Socle
Invariant ideal
Dcc topology
Hereditary and saturated point functors
title_short Invariant ideals in Leavitt path algebras
title_full Invariant ideals in Leavitt path algebras
title_fullStr Invariant ideals in Leavitt path algebras
title_full_unstemmed Invariant ideals in Leavitt path algebras
title_sort Invariant ideals in Leavitt path algebras
dc.creator.none.fl_str_mv Gil Canto, Cristobal|||0000-0002-4975-1935
Martín Barquero, Dolores|||0000-0002-7210-1578
Martín González, Cándido|||0000-0003-2796-7417
author Gil Canto, Cristobal|||0000-0002-4975-1935
author_facet Gil Canto, Cristobal|||0000-0002-4975-1935
Martín Barquero, Dolores|||0000-0002-7210-1578
Martín González, Cándido|||0000-0003-2796-7417
author_role author
author2 Martín Barquero, Dolores|||0000-0002-7210-1578
Martín González, Cándido|||0000-0003-2796-7417
author2_role author
author
dc.subject.none.fl_str_mv Leavitt path algebra
Annihilator
Socle
Invariant ideal
Dcc topology
Hereditary and saturated point functors
topic Leavitt path algebra
Annihilator
Socle
Invariant ideal
Dcc topology
Hereditary and saturated point functors
description It is known that the ideals of a Leavitt path algebra LK(E) generated by Pl(E), by Pc(E), or by Pec(E) are invariant under isomorphism. Though the ideal generated by Pb∞(E) is not invariant we find its "natural" replacement (which is indeed invariant): the one generated by the vertices of Pb∞p (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the DCC topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if H is a hereditary saturated subset of vertices providing an invariant ideal, its exterior ext(H) in the DCC topology of E0 generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance Pl, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain Soc(1)( ) ⊆ Soc(2)( ) ⊆ · · · , all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.
publishDate 2022
dc.date.none.fl_str_mv 2
2022-01-01
2022
2022-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/264508
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622203
url https://ddd.uab.cat/record/264508
https://dx.doi.org/urn:doi:10.5565/PUBLMAT6622203
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2019-104236GB-I00
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869412418419949568
score 15,301603

Similares en LA Referencia