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Fitting's Heyting-valued logic and Heyting-valued modal logic have alreadybeen studied from an algebraic viewpoint. In addition to algebraicaxiomatizations with the completeness of Fitting's Heyting-valued logic andHeyting-valued modal logic, both topological and coalgebraic dualities havealso been developed for algebras of Fitting's Heyting-valued modal logic.Bitopological methods have recently been employed to investigate duality forFitting's Heyting-valued logic. However, the concepts of bitopology andbiVietoris coalgebras are conspicuously absent from the development ofdualities for Fitting's many-valued modal logic. With this study, we try tobridge that gap. We develop a bitopological duality for algebras of Fitting'sHeyting-valued modal logic. We construct a bi-Vietoris functor on the category$PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued ($\mathcal{L}$ is a Heytingalgebra) pairwise Boolean spaces. Finally, we obtain a dual equivalence betweencategories of biVietoris coalgebras and algebras of Fitting's Heyting-valuedmodal logic. As a result, we conclude that Fitting's many-valued modal logic issound and complete with respect to the coalgebras of a biVietoris functor. Wediscuss the application of this coalgebraic approach to bitopological duality.

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