Locally Finite Reducts of Heyting Algebras and Canonical Formulas

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]. In this paper 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.