Bounded distributive lattices with strict implication

Ramon Jansana

We will introduce the variety of weakly Heyting algebras. They are bounded distributive lattices with an implication that has the properties of the strict implication of the normal modal logic $K$. This variety has as subvarieties the varieties of Heyting algebras, Basic algebras of Visser's logic, and all the varieties that correspond to the strict implication fragments of the normal modal logics. We will present a Priestly style duality for weakly Heyting algebras and several results that show that the study of the lattice of varieties of weakly Heyting algebras encompasses both the stydy of the lattice of varieties of normal modal algebras and the study of the lattice of varieties of Heyting algebras.