An Algebraic Approach to Canonical Formulas: intuitionistic Case

We introduce partial Esakia morphisms, well partial Esakia morphisms, and strong partial Esakia morphisms between Esakia spaces and show that they provide the dual description of (∧,→) homomorphisms, (∧,→, 0) homomorphisms, and (∧,→,∨) homomorphisms between Heyting algebras, thus establishing a generalization of Esakia duality. This yields an algebraic characterization of Zakharyaschev’s subreductions, cofinal subreductions, dense subreductions, and the closed domain condition. As a consequence, we obtain a new simplified proof (which is algebraic in nature) of Zakharyaschev’s theorem that each intermediate logic can be axiomatized by canonical formulas. §