Spectral-like duality for distributive Hilbert algebras with infimum
| Autor: | María Paz Sandín Esteban, Sergio Arturo Celani |
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| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Subcategory Algebra and Number Theory Matemáticas 010102 general mathematics Duality (mathematics) Structure (category theory) 0102 computer and information sciences Topological space 01 natural sciences Dual (category theory) Hilbert algebras Matemática Pura Distributive property 010201 computation theory & mathematics Lattice (order) distributive semilattices 0101 mathematics Birkhoff's representation theorem topological representation CIENCIAS NATURALES Y EXACTAS Mathematics |
| Popis: | Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and ∧ -semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters. Fil: Celani, Sergio Arturo. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Núcleo Consolidado de Matemática Pura y Aplicada; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Esteban, María. Universidad Central de Barcelona; España |
| Databáze: | OpenAIRE |
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