On Some Compatible Operations on Heyting Algebras

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...

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Detalles Bibliográficos
Autores principales: Ertola Biraben, Rodolfo Cristian, San Martín, Hernán Javier
Formato: Articulo
Lenguaje:Inglés
Publicado: 2011
Materias:
Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/137724
Aporte de:
Aportado por : SEDICI (UNLP) de Universidad Nacional de La Plata .
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spelling I19-R120-10915-1377242022-06-14T04:04:50Z http://sedici.unlp.edu.ar/handle/10915/137724 issn:0039-3215 issn:1572-8730 On Some Compatible Operations on Heyting Algebras Ertola Biraben, Rodolfo Cristian San Martín, Hernán Javier 2011 2022-06-13T13:20:26Z en Matemática Intuitionistic logic Heyting algebra Compatible operation 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 Articulo Articulo http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) application/pdf 331-345
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
spellingShingle Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
Ertola Biraben, Rodolfo Cristian
San Martín, Hernán Javier
On Some Compatible Operations on Heyting Algebras
topic_facet Matemática
Intuitionistic logic
Heyting algebra
Compatible operation
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.
format Articulo
Articulo
author Ertola Biraben, Rodolfo Cristian
San Martín, Hernán Javier
author_facet Ertola Biraben, Rodolfo Cristian
San Martín, Hernán Javier
author_sort Ertola Biraben, Rodolfo Cristian
title On Some Compatible Operations on Heyting Algebras
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
publishDate 2011
url http://sedici.unlp.edu.ar/handle/10915/137724
work_keys_str_mv AT ertolabirabenrodolfocristian onsomecompatibleoperationsonheytingalgebras
AT sanmartinhernanjavier onsomecompatibleoperationsonheytingalgebras
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