2009 | OriginalPaper | Chapter
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 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
.
