A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions
- Autores
- Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.
- Año de publicación
- 2022
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbertstyle axiomatization of a new logic called “Dually hemimorphic semi-Heyting logic” (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil’s logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the “Deduction Theorem” holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semiHeyting algebras, as well as a new axiomatization for the 3-valued Lukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan G¨odel logics and dually pseudocomplemented G¨odel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.
Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentina
Fil: Sankappanavar, Hanamantagouda P.. State University of New York; Estados Unidos - Materia
-
Dually Hemimorphic Semi-Heyting Algebras
SEMI HEYTING ALGEBRAS
HEYTING ALGEBRAS - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/231843
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A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic ExtensionsCornejo, Juan ManuelSankappanavar, Hanamantagouda P.Dually Hemimorphic Semi-Heyting AlgebrasSEMI HEYTING ALGEBRASHEYTING ALGEBRAShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbertstyle axiomatization of a new logic called “Dually hemimorphic semi-Heyting logic” (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil’s logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the “Deduction Theorem” holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semiHeyting algebras, as well as a new axiomatization for the 3-valued Lukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan G¨odel logics and dually pseudocomplemented G¨odel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras.Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Sankappanavar, Hanamantagouda P.. State University of New York; Estados UnidosLodz University Press2022-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/231843Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.; A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions; Lodz University Press; Bulletin Of The Section Of Logic; 51; 4; 12-2022; 555-6450138-0680CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://czasopisma.uni.lodz.pl/bulletin/article/view/12505info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:58:50Zoai:ri.conicet.gov.ar:11336/231843instacron: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:34982025-09-29 09:58:51.079CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
title |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
spellingShingle |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions Cornejo, Juan Manuel Dually Hemimorphic Semi-Heyting Algebras SEMI HEYTING ALGEBRAS HEYTING ALGEBRAS |
title_short |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
title_full |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
title_fullStr |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
title_full_unstemmed |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
title_sort |
A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions |
dc.creator.none.fl_str_mv |
Cornejo, Juan Manuel Sankappanavar, Hanamantagouda P. |
author |
Cornejo, Juan Manuel |
author_facet |
Cornejo, Juan Manuel Sankappanavar, Hanamantagouda P. |
author_role |
author |
author2 |
Sankappanavar, Hanamantagouda P. |
author2_role |
author |
dc.subject.none.fl_str_mv |
Dually Hemimorphic Semi-Heyting Algebras SEMI HEYTING ALGEBRAS HEYTING ALGEBRAS |
topic |
Dually Hemimorphic Semi-Heyting Algebras SEMI HEYTING ALGEBRAS HEYTING ALGEBRAS |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
dc.description.none.fl_txt_mv |
The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbertstyle axiomatization of a new logic called “Dually hemimorphic semi-Heyting logic” (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil’s logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the “Deduction Theorem” holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semiHeyting algebras, as well as a new axiomatization for the 3-valued Lukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan G¨odel logics and dually pseudocomplemented G¨odel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. Fil: Cornejo, Juan Manuel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentina Fil: Sankappanavar, Hanamantagouda P.. State University of New York; Estados Unidos |
description |
The variety DHMSH of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety DHMSH from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a Hilbertstyle axiomatization of a new logic called “Dually hemimorphic semi-Heyting logic” (DHMSH, for short), as an expansion of semi-intuitionistic logic SI (also called SH) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety DHMSH. It is deduced that the logic DHMSH is algebraizable in the sense of Blok and Pigozzi, with the variety DHMSH as its equivalent algebraic semantics and that the lattice of axiomatic extensions of DHMSH is dually isomorphic to the lattice of subvarieties of DHMSH. A new axiomatization for Moisil’s logic is also obtained. Secondly, we characterize the axiomatic extensions of DHMSH in which the “Deduction Theorem” holds. Thirdly, we present several new logics, extending the logic DHMSH, corresponding to several important subvarieties of the variety DHMSH. These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semiHeyting algebras, as well as a new axiomatization for the 3-valued Lukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan G¨odel logics and dually pseudocomplemented G¨odel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. |
publishDate |
2022 |
dc.date.none.fl_str_mv |
2022-12 |
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/11336/231843 Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.; A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions; Lodz University Press; Bulletin Of The Section Of Logic; 51; 4; 12-2022; 555-645 0138-0680 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/231843 |
identifier_str_mv |
Cornejo, Juan Manuel; Sankappanavar, Hanamantagouda P.; A Logic for Dually Hemimorphic Semi-Heyting Algebras and Axiomatic Extensions; Lodz University Press; Bulletin Of The Section Of Logic; 51; 4; 12-2022; 555-645 0138-0680 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://czasopisma.uni.lodz.pl/bulletin/article/view/12505 |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-nd/2.5/ar/ |
eu_rights_str_mv |
openAccess |
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https://creativecommons.org/licenses/by-nc-nd/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
Lodz University Press |
publisher.none.fl_str_mv |
Lodz University Press |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) |
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CONICET Digital (CONICET) |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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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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13.070432 |