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
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/137724
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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 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Articulo 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://sedici.unlp.edu.ar/handle/10915/137724 |
url |
http://sedici.unlp.edu.ar/handle/10915/137724 |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/issn/0039-3215 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) |
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openAccess |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) |
dc.format.none.fl_str_mv |
application/pdf 331-345 |
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