Distributive Lattices with a Generalized Implication: Topological Duality |
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Authors: | Jorge E Castro Sergio Arturo Celani Ramon Jansana |
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Institution: | 1.Universidad Nacional de San Juan,San Juan,Argentina;2.CONICET and Departamento de Matemáticas,Universidad Nacional del Centro,Tandil,Argentina;3.Dept. Lògica, Història i Filsofia de la Ciència,Universitat de Barcelona,Barcleona,Spain |
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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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