A Study on Some Classes of Distributive Lattices with a Generalized Implication

A generalized implication on a distributive lattice \(\varvec{A}\) is a function between \(\varvec{A} \times \varvec{A}\) to ideals of \(\varvec{A}\) satisfying similar conditions to strict implication of weak Heyting algebras. Relative anihilators and quasi-modal operators are examples of generalized implication in distributive lattices. The aim of this paper is to study some classes of distributive lattices with a generalized implication. In particular, we prove that the class of Boolean algebras endowed with a quasi-modal operator is equivalent to the class of Boolean algebras with a generalized implication. This equivalence allow us to give another presentation of the class of quasi-monadic algebras and the class of compingent algebras defined by H. De Vries. We also introduce the notion of gi-sublattice and we characterize the simple and subdirectly irreducible distributive lattices with a generalized implication through topological duality.