A Duality for Algebras of Lattice-Valued Modal Logic

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 Jonsson-Tarski-style duality for algebras of L -valued modal logic (i.e., L -ML-algebras), which extends Jonsson-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 .

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