Distributive Lattices with a Generalized Implication: Topological Duality

JORGE CASTRO, SERGIO ARTURO CELANI AND RAMÓN JANSANA

ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS

SPRINGER

Año: 2011 vol. 28 p. 227 - 249

0167-8094

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 ageneralized implication ⇒ defined in A satisfying certain conditions, and we studythe class of bounded distributive lattices A endowed with a generalized implicationas a common abstraction of the notions of annihilator (Mandelker, Duke Math J37:377–386, 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 styletopological duality for the bounded distributive lattices with a generalized implication.This duality generalizes the duality given in Celani and Jansana (Math Log Quartely 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (MathBohem 126:721–736, 2001) for Quasi-modal algebras.