On Some Compatible Operations on Heyting Algebras

Autores
Ertola Biraben, Rodolfo Cristian; San Martín, Hernán Javier
Año de publicación
2011
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.
Facultad de Ciencias Exactas
Facultad de Humanidades y Ciencias de la Educación
Materia
Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/137724

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spelling On Some Compatible Operations on Heyting AlgebrasErtola Biraben, Rodolfo CristianSan Martín, Hernán JavierMatemáticaIntuitionistic logicHeyting algebraCompatible operationWe study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.Facultad de Ciencias ExactasFacultad de Humanidades y Ciencias de la Educación2011info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf331-345http://sedici.unlp.edu.ar/handle/10915/137724enginfo:eu-repo/semantics/altIdentifier/issn/0039-3215info:eu-repo/semantics/altIdentifier/issn/1572-8730info:eu-repo/semantics/altIdentifier/doi/10.1007/s11225-011-9338-yinfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-09-03T11:04:23Zoai:sedici.unlp.edu.ar:10915/137724Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-09-03 11:04:23.627SEDICI (UNLP) - Universidad Nacional de La Platafalse
dc.title.none.fl_str_mv On Some Compatible Operations on Heyting Algebras
title On Some Compatible Operations on Heyting Algebras
spellingShingle On Some Compatible Operations on Heyting Algebras
Ertola Biraben, Rodolfo Cristian
Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
title_short On Some Compatible Operations on Heyting Algebras
title_full On Some Compatible Operations on Heyting Algebras
title_fullStr On Some Compatible Operations on Heyting Algebras
title_full_unstemmed On Some Compatible Operations on Heyting Algebras
title_sort On Some Compatible Operations on Heyting Algebras
dc.creator.none.fl_str_mv Ertola Biraben, Rodolfo Cristian
San Martín, Hernán Javier
author Ertola Biraben, Rodolfo Cristian
author_facet Ertola Biraben, Rodolfo Cristian
San Martín, Hernán Javier
author_role author
author2 San Martín, Hernán Javier
author2_role author
dc.subject.none.fl_str_mv Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
topic Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
dc.description.none.fl_txt_mv We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.
Facultad de Ciencias Exactas
Facultad de Humanidades y Ciencias de la Educación
description We study some operations that may be defined using the minimum operator in the context of a Heyting algebra. Our motivation comes from the fact that 1) already known compatible operations, such as the successor by Kuznetsov, the minimum dense by Smetanich and the operation G by Gabbay may be defined in this way, though almost never explicitly noted in the literature; 2) defining operations in this way is equivalent, from a logical point of view, to two clauses, one corresponding to an introduction rule and the other to an elimination rule, thus providing a manageable way to deal with these operations. Our main result is negative: all operations that arise turn out to be Heyting terms or the mentioned already known operations or operations interdefinable with them. However, it should be noted that some of the operations that arise may exist even if the known operations do not. We also study the extension of Priestley duality to Heyting algebras enriched with the new operations.
publishDate 2011
dc.date.none.fl_str_mv 2011
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format article
status_str publishedVersion
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dc.language.none.fl_str_mv eng
language eng
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info:eu-repo/semantics/altIdentifier/issn/1572-8730
info:eu-repo/semantics/altIdentifier/doi/10.1007/s11225-011-9338-y
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
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331-345
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