Frontal operators in distributive lattices with a generalized implication

Abstract

We introduce a family of extensions of bounded distributive lattices. These extensions are obtained by adding two operations: an internal unary operation, and a function (called generalized implication) that maps pair of elements to ideals of the lattice. A bounded distributive lattice with a generalized implication is called gi-lattice in [4]. The main goal of this paper is to introduce and study the category of frontal gi-lattices (and some subcategories of it). This category can be seen as a generalization of the category of frontal weak Heyting algebras ([9]). In particular, we study the case of frontal gi-lattices where the generalized implication is defined as the annihilator ([11], [15]). We give a Priestley’s style duality for each one of the new classes of structures considered.