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Distributive Lattices with a Generalized Implication: Topological Duality

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Abstract

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.

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Authors and Affiliations

  1. Universidad Nacional de San Juan, San Juan, Argentina

    Jorge E. Castro

  2. CONICET and Departamento de Matemáticas, Universidad Nacional del Centro, Tandil, Argentina

    Sergio Arturo Celani

  3. Dept. Lògica, Història i Filsofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001, Barcleona, Spain

    Ramon Jansana

Authors
  1. Jorge E. Castro
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  2. Sergio Arturo Celani
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  3. Ramon Jansana
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Corresponding author

Correspondence to Sergio Arturo Celani.

Additional information

The research of the second author was supported by the research grant PIP-112-200801-02542 from the CONICET, Argentina. The research of the third author was done with the support of grants MTM2008-01139 from the Spanish government, including Feder funds, and 2005SGR-00083 and 2009SGR-1433 from the Catalan government.

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Castro, J.E., Celani, S.A. & Jansana, R. Distributive Lattices with a Generalized Implication: Topological Duality. Order 28, 227–249 (2011). https://doi.org/10.1007/s11083-010-9168-y

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  • DOI: https://doi.org/10.1007/s11083-010-9168-y

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