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Abstract. The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras while the variety of implicative semilattices by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short).The ∨-free reducts of Heyting algebras give rise to the (∧, →)-canonical formulas that were studied in [3]. Here we introduce the (∧, ∨)-canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by (∧, ∨)-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D = A 2 , we show that the (∧, ∨)-canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D = ∅, the (∧, ∨)-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.

Abstract. We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object.

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.

We introduce modal compact Hausdorff spaces as generalizations of modal spaces, and show these are coalgebras for the Vietoris functor on compact Hausdorff spaces. Modal compact regular frames and modal de Vries algebras are introduced as algebraic counterparts of modal compact Hausdorff spaces, and dualities are given for the categories involved. These extend the familiar Isbell and de Vries dualities for compact Hausdorff spaces, as well as the duality between modal spaces and modal algebras. As the first step in the logical treatment of modal compact Hausdorff spaces, a version of Sahlqvist correspondence is given for the positive modal language.