Duality for Fitting's Multi-valued Modal logic via bitopology and biVietoris coalgebra

Abstract: Fitting's Heyting-valued logic and Heyting-valued modal logic have already been studied from an algebraic viewpoint. In addition to algebraic axiomatizations with the completeness of Fitting's Heyting-valued logic and Heyting-valued modal logic, both topological and coalgebraic dualities have also been developed for algebras of Fitting's Heyting-valued modal logic. Bitopological methods have recently been employed to investigate duality for Fitting's Heyting-valued logic. However, the concepts of bitopology and biVietoris coalgebras are conspicuously absent from the development of dualities for Fitting's many-valued modal logic. With this study, we try to bridge that gap. We develop a bitopological duality for algebras of Fitting's Heyting-valued modal logic. We construct a bi-Vietoris functor on the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued ($\mathcal{L}$ is a Heyting algebra) pairwise Boolean spaces. Finally, we obtain a dual equivalence between categories of biVietoris coalgebras and algebras of Fitting's Heyting-valued modal logic. As a result, we conclude that Fitting's many-valued modal logic is sound and complete with respect to the coalgebras of a biVietoris functor. We discuss the application of this coalgebraic approach to bitopological duality.