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2017 Locally Finite Reducts of Heyting Algebras and Canonical Formulas
Guram Bezhanishvili, Nick Bezhanishvili
Notre Dame J. Formal Logic 58(1): 21-45 (2017). DOI: 10.1215/00294527-3691563

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 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 D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, 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.

Citation

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Guram Bezhanishvili. Nick Bezhanishvili. "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

Information

Received: 19 March 2013; Accepted: 9 March 2014; Published: 2017
First available in Project Euclid: 17 November 2016

zbMATH: 06686416
MathSciNet: MR3595340
Digital Object Identifier: 10.1215/00294527-3691563

Subjects:
Primary: 03B55, 06D20

Rights: Copyright © 2017 University of Notre Dame

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Vol.58 • No. 1 • 2017
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