Formalization of Universal Algebra in Agda

Autores
Gunther, Emmanuel; Gadea, Alejandro Emilio; Pagano, Miguel Maria
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.
Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
Materia
EQUATIONAL LOGIC
FORMALIZATION OF MATHEMATICS
UNIVERSAL ALGEBRA
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/100027
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Formalization of Universal Algebra in AgdaGunther, EmmanuelGadea, Alejandro EmilioPagano, Miguel MariaEQUATIONAL LOGICFORMALIZATION OF MATHEMATICSUNIVERSAL ALGEBRAhttps://purl.org/becyt/ford/1.2https://purl.org/becyt/ford/1In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; ArgentinaElsevier2018-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/100027Gunther, Emmanuel; Gadea, Alejandro Emilio; Pagano, Miguel Maria; Formalization of Universal Algebra in Agda; Elsevier; Electronic Notes in Theoretical Computer Science; 338; 10-2018; 147-1661571-0661CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.entcs.2018.10.010info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S1571066118300768info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:54:27Zoai:ri.conicet.gov.ar:11336/100027instacron: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 09:54:28.074CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Formalization of Universal Algebra in Agda
title Formalization of Universal Algebra in Agda
spellingShingle Formalization of Universal Algebra in Agda
Gunther, Emmanuel
EQUATIONAL LOGIC
FORMALIZATION OF MATHEMATICS
UNIVERSAL ALGEBRA
title_short Formalization of Universal Algebra in Agda
title_full Formalization of Universal Algebra in Agda
title_fullStr Formalization of Universal Algebra in Agda
title_full_unstemmed Formalization of Universal Algebra in Agda
title_sort Formalization of Universal Algebra in Agda
dc.creator.none.fl_str_mv Gunther, Emmanuel
Gadea, Alejandro Emilio
Pagano, Miguel Maria
author Gunther, Emmanuel
author_facet Gunther, Emmanuel
Gadea, Alejandro Emilio
Pagano, Miguel Maria
author_role author
author2 Gadea, Alejandro Emilio
Pagano, Miguel Maria
author2_role author
author
dc.subject.none.fl_str_mv EQUATIONAL LOGIC
FORMALIZATION OF MATHEMATICS
UNIVERSAL ALGEBRA
topic EQUATIONAL LOGIC
FORMALIZATION OF MATHEMATICS
UNIVERSAL ALGEBRA
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.2
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.
Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina
description In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.
publishDate 2018
dc.date.none.fl_str_mv 2018-10
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/11336/100027
Gunther, Emmanuel; Gadea, Alejandro Emilio; Pagano, Miguel Maria; Formalization of Universal Algebra in Agda; Elsevier; Electronic Notes in Theoretical Computer Science; 338; 10-2018; 147-166
1571-0661
CONICET Digital
CONICET
url http://hdl.handle.net/11336/100027
identifier_str_mv Gunther, Emmanuel; Gadea, Alejandro Emilio; Pagano, Miguel Maria; Formalization of Universal Algebra in Agda; Elsevier; Electronic Notes in Theoretical Computer Science; 338; 10-2018; 147-166
1571-0661
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.entcs.2018.10.010
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S1571066118300768
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
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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