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 is generated 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 we studied in an earlier work. 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 of pairs of elements of a finite subdirectly irreducible Heyting algebra . When , we show that the -canonical formula of is equivalent to the Jankov formula of . On the other hand, when , 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.
"Locally Finite Reducts of Heyting Algebras and Canonical Formulas." Notre Dame J. Formal Logic 58 (1) 21 - 45, 2017. https://doi.org/10.1215/00294527-3691563