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 . 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.