An Algebraic Approach to Subframe Logics. Intuitionistic Case

We develop duality between nuclei on Heyting algebras and certain binary relations on Heyting spaces. We show that those binary relations are in 1-1 correspondence with subframes of Heyting spaces. We introduce the notions of nuclear and dense nuclear varieties of Heyting algebras, and prove that a variety of Heyting algebras is nuclear iff it is a subframe variety, and that it is dense nuclear iff it is a cofinal subframe variety. We give an alternative proof that every (cofinal) subframe variety of Heyting algebras is generated by its finite members.