A Duality for Algebras of Lattice-Valued Modal Logic

Abstract

In this paper, we consider some versions of Fitting’s L-valued logic and L-valued modal logic for a finite distributive lattice L. Using the theory of natural dualities, we first obtain a natural duality for algebras of L-valued logic (i.e., L -VL-algebras), which extends Stone duality for Boolean algebras to the L-valued case. Then, based on this duality, we develop a Jónsson-Tarski-style duality for algebras of L-valued modal logic (i.e., L -ML-algebras), which extends Jónsson-Tarski duality for modal algebras to the L-valued case. By applying these dualities, we obtain compactness theorems for L-valued logic and for L-valued modal logic, and the classification of equivalence classes of categories of L -VL-algebras for finite distributive lattices L.