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Algebra universalis

, Volume 78, Issue 2, pp 193–213 | Cite as

Spectral-like duality for distributive Hilbert algebras with infimum

  • Sergio A. CelaniEmail author
  • María Esteban
Article
  • 65 Downloads

Abstract

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 \({\wedge}\)-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.

Key words and phrases

Hilbert algebras topological representation spectral spaces distributive meet-semilattices 

2000 Mathematics Subject Classification

Primary: 03G25 Secondary: 06F35 

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CONICET and Departamento de Matemáticas, Facultad de Ciencias Exactas. Univ. Nac. del Centro.TandilArgentina
  2. 2.Departament de Lógica, História i Filosofia de la Ciéncia, Facultat de FilosofíaUniversitat de Barcelona (UB).BarcelonaSpain

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