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 bi-Vietoris 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. The main results are bitopological and coalgebraic duality for Fitting's many-valued modal logic. We develop a bitopological duality for algebras of Fitting's Heyting-valued modal logic by extending known bitopological duality for Fitting's non-modal logic. To develop coalgebraic duality, we adapt Lauridsen's bi-Vietoris construction from the category of pairwise Stone spaces to the category $PBS_{\mathcal{L}}$ of $\mathcal{L}$-valued (with $\mathcal{L}$ a bounded finite distributive lattice, i.e., a Heyting algebra) pairwise Boolean spaces by incorporating a structure map, and from this obtain the $\mathcal{L}$-biVietoris functor. Finally, we establish dual equivalence between coalgebras for the $\mathcal{L}$-biVietoris functor and algebras of Fitting's $\mathcal{L}$-valued modal logic. As a result, we conclude that Fitting's Heyting-valued modal logic is sound and complete with respect to the coalgebras of the $\mathcal{L}$-biVietoris functor. We also apply this coalgebraic approach to the bitopological duality to show the existence of cofree and final coalgebras and to establish a Hennessy-Milner property.