On the free implicative semilattice extension of a Hilbert algebra

Autores
Celani, Sergio Arturo; Jansana, Ramon
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.
Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad Central de Barcelona; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; Argentina
Fil: Jansana, Ramon. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad de Barcelona; España
Materia
FREE EXTENSIONS
HILBERT ALGEBRAS
IMPLICATIVE 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:11336/199316
Acceder
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network_name_str CONICET Digital (CONICET)
spelling On the free implicative semilattice extension of a Hilbert algebraCelani, Sergio ArturoJansana, RamonFREE EXTENSIONSHILBERT ALGEBRASIMPLICATIVE SEMILATTICEShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad Central de Barcelona; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; ArgentinaFil: Jansana, Ramon. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad de Barcelona; EspañaWiley VCH Verlag2012-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/199316Celani, Sergio Arturo; Jansana, Ramon; On the free implicative semilattice extension of a Hilbert algebra; Wiley VCH Verlag; Mathematical Logic Quarterly; 58; 3; 5-2012; 188-2070942-5616CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.201020098info:eu-repo/semantics/altIdentifier/doi/10.1002/malq.201020098info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:52:23Zoai:ri.conicet.gov.ar:11336/199316instacron: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 09:52:23.967CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the free implicative semilattice extension of a Hilbert algebra
title On the free implicative semilattice extension of a Hilbert algebra
spellingShingle On the free implicative semilattice extension of a Hilbert algebra
Celani, Sergio Arturo
FREE EXTENSIONS
HILBERT ALGEBRAS
IMPLICATIVE SEMILATTICES
title_short On the free implicative semilattice extension of a Hilbert algebra
title_full On the free implicative semilattice extension of a Hilbert algebra
title_fullStr On the free implicative semilattice extension of a Hilbert algebra
title_full_unstemmed On the free implicative semilattice extension of a Hilbert algebra
title_sort On the free implicative semilattice extension of a Hilbert algebra
dc.creator.none.fl_str_mv Celani, Sergio Arturo
Jansana, Ramon
author Celani, Sergio Arturo
author_facet Celani, Sergio Arturo
Jansana, Ramon
author_role author
author2 Jansana, Ramon
author2_role author
dc.subject.none.fl_str_mv FREE EXTENSIONS
HILBERT ALGEBRAS
IMPLICATIVE SEMILATTICES
topic FREE EXTENSIONS
HILBERT ALGEBRAS
IMPLICATIVE 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 Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.
Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad Central de Barcelona; España. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tandil; Argentina
Fil: Jansana, Ramon. Universidad Nacional del Centro de la Provincia de Buenos Aires; Argentina. Universidad de Barcelona; España
description Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.
publishDate 2012
dc.date.none.fl_str_mv 2012-05
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/11336/199316
Celani, Sergio Arturo; Jansana, Ramon; On the free implicative semilattice extension of a Hilbert algebra; Wiley VCH Verlag; Mathematical Logic Quarterly; 58; 3; 5-2012; 188-207
0942-5616
CONICET Digital
CONICET
url http://hdl.handle.net/11336/199316
identifier_str_mv Celani, Sergio Arturo; Jansana, Ramon; On the free implicative semilattice extension of a Hilbert algebra; Wiley VCH Verlag; Mathematical Logic Quarterly; 58; 3; 5-2012; 188-207
0942-5616
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.201020098
info:eu-repo/semantics/altIdentifier/doi/10.1002/malq.201020098
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
dc.publisher.none.fl_str_mv Wiley VCH Verlag
publisher.none.fl_str_mv Wiley VCH Verlag
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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score 13.13397

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