caa a22 4500 44581389X CHVBK 20180317145220.0 cr unu---uuuuu 170323e20110701xx s 000 0 eng 10.1007/s11083-010-9168-y doi (NATIONALLICENCE)springer-10.1007/s11083-010-9168-y Distributive Lattices with a Generalized Implication: Topological Duality [Elektronische Daten] [Jorge Castro, Sergio Celani, Ramon Jansana] 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. Springer Science+Business Media B.V., 2010 Generalized implication nationallicence Quasi-modal operator nationallicence Annihilator nationallicence Weakly Heyting algebras nationallicence Priestley's duality nationallicence Castro Jorge Universidad Nacional de San Juan, San Juan, Argentina aut Celani Sergio CONICET and Departamento de Matemáticas, Universidad Nacional del Centro, Tandil, Argentina aut Jansana Ramon Dept. Lògica, Història i Filsofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001, Barcleona, Spain aut Order Springer Netherlands 28/2(2011-07-01), 227-249 0167-8094 28:2<227 2011 28 11083 https://doi.org/10.1007/s11083-010-9168-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11083-010-9168-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Castro Jorge Universidad Nacional de San Juan, San Juan, Argentina aut NATIONALLICENCE 700 1- Celani Sergio CONICET and Departamento de Matemáticas, Universidad Nacional del Centro, Tandil, Argentina aut NATIONALLICENCE 700 1- Jansana Ramon Dept. Lògica, Història i Filsofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001, Barcleona, Spain aut NATIONALLICENCE 773 0- Order Springer Netherlands 28/2(2011-07-01), 227-249 0167-8094 28:2<227 2011 28 11083 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer