Spectral-like duality for distributive Hilbert algebras with infimum

Autores
Celani, Sergio Arturo; Esteban, María
Año de publicación
2017
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and ∧ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.
Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Núcleo Consolidado de Matemática Pura y Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Esteban, María. Universidad Central de Barcelona; España
Materia
Hilbert Algebras
Topological Representation
Distributive Semilattices
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/58915

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spelling Spectral-like duality for distributive Hilbert algebras with infimumCelani, Sergio ArturoEsteban, MaríaHilbert AlgebrasTopological RepresentationDistributive Semilatticeshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and ∧ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Núcleo Consolidado de Matemática Pura y Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Esteban, María. Universidad Central de Barcelona; EspañaSpringer2017-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/58915Celani, Sergio Arturo; Esteban, María; Spectral-like duality for distributive Hilbert algebras with infimum; Springer; Algebra Universalis; 78; 2; 10-2017; 193-2130002-52401420-8911CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1007/s00012-017-0451-2info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00012-017-0451-2info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:43:26Zoai:ri.conicet.gov.ar:11336/58915instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-03 09:43:26.466CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Spectral-like duality for distributive Hilbert algebras with infimum
title Spectral-like duality for distributive Hilbert algebras with infimum
spellingShingle Spectral-like duality for distributive Hilbert algebras with infimum
Celani, Sergio Arturo
Hilbert Algebras
Topological Representation
Distributive Semilattices
title_short Spectral-like duality for distributive Hilbert algebras with infimum
title_full Spectral-like duality for distributive Hilbert algebras with infimum
title_fullStr Spectral-like duality for distributive Hilbert algebras with infimum
title_full_unstemmed Spectral-like duality for distributive Hilbert algebras with infimum
title_sort Spectral-like duality for distributive Hilbert algebras with infimum
dc.creator.none.fl_str_mv Celani, Sergio Arturo
Esteban, María
author Celani, Sergio Arturo
author_facet Celani, Sergio Arturo
Esteban, María
author_role author
author2 Esteban, María
author2_role author
dc.subject.none.fl_str_mv Hilbert Algebras
Topological Representation
Distributive Semilattices
topic Hilbert Algebras
Topological Representation
Distributive Semilattices
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and ∧ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.
Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Núcleo Consolidado de Matemática Pura y Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Esteban, María. Universidad Central de Barcelona; España
description Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and ∧ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.
publishDate 2017
dc.date.none.fl_str_mv 2017-10
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/58915
Celani, Sergio Arturo; Esteban, María; Spectral-like duality for distributive Hilbert algebras with infimum; Springer; Algebra Universalis; 78; 2; 10-2017; 193-213
0002-5240
1420-8911
CONICET Digital
CONICET
url http://hdl.handle.net/11336/58915
identifier_str_mv Celani, Sergio Arturo; Esteban, María; Spectral-like duality for distributive Hilbert algebras with infimum; Springer; Algebra Universalis; 78; 2; 10-2017; 193-213
0002-5240
1420-8911
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1007/s00012-017-0451-2
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00012-017-0451-2
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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