Remarks on Priestley duality for distributive lattices

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category o...

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Autores principales: Cignoli, R., Lafalce, S., Petrovich, A.
Formato: JOUR
Materias:
AMS subject classification (1991): 06D05
Bounded distributive lattices
closure operators
filters
ideals
lattice homomorphisms
Priestley spaces
quantifiers
sublattices
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_01678094_v8_n3_p299_Cignoli
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_01678094_v8_n3_p299_Cignoli
record_format dspace
spelling todo:paper_01678094_v8_n3_p299_Cignoli2023-10-03T15:05:23Z Remarks on Priestley duality for distributive lattices Cignoli, R. Lafalce, S. Petrovich, A. AMS subject classification (1991): 06D05 Bounded distributive lattices closure operators filters ideals lattice homomorphisms Priestley spaces quantifiers sublattices The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix. © 1991 Kluwer Academic Publishers. Fil:Petrovich, A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_01678094_v8_n3_p299_Cignoli
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic AMS subject classification (1991): 06D05
Bounded distributive lattices
closure operators
filters
ideals
lattice homomorphisms
Priestley spaces
quantifiers
sublattices
spellingShingle AMS subject classification (1991): 06D05
Bounded distributive lattices
closure operators
filters
ideals
lattice homomorphisms
Priestley spaces
quantifiers
sublattices
Cignoli, R.
Lafalce, S.
Petrovich, A.
Remarks on Priestley duality for distributive lattices
topic_facet AMS subject classification (1991): 06D05
Bounded distributive lattices
closure operators
filters
ideals
lattice homomorphisms
Priestley spaces
quantifiers
sublattices
description The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix. © 1991 Kluwer Academic Publishers.
format JOUR
author Cignoli, R.
Lafalce, S.
Petrovich, A.
author_facet Cignoli, R.
Lafalce, S.
Petrovich, A.
author_sort Cignoli, R.
title Remarks on Priestley duality for distributive lattices
title_short Remarks on Priestley duality for distributive lattices
title_full Remarks on Priestley duality for distributive lattices
title_fullStr Remarks on Priestley duality for distributive lattices
title_full_unstemmed Remarks on Priestley duality for distributive lattices
title_sort remarks on priestley duality for distributive lattices
url http://hdl.handle.net/20.500.12110/paper_01678094_v8_n3_p299_Cignoli
work_keys_str_mv AT cignolir remarksonpriestleydualityfordistributivelattices
AT lafalces remarksonpriestleydualityfordistributivelattices
AT petrovicha remarksonpriestleydualityfordistributivelattices
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