🤖 AI Summary
This paper addresses the absence of a unified bitopological–coalgebraic duality theory for Fitting’s many-valued modal logic. To resolve this, we introduce bitopologies and bi-Vietoris coalgebras into its duality analysis for the first time. Specifically, we construct a bi-Vietoris coalgebra functor on the category of PBSₗ-algebras and integrate Heyting algebra semantics with categorical methods to establish a strict dual equivalence between the algebraic semantics of the logic and bi-Vietoris coalgebras over bitopological spaces. This duality yields a soundness and completeness theorem for the logic and fills a foundational gap in the bitopological–coalgebraic treatment of many-valued modal logics. Moreover, it provides a novel categorical framework for higher-order modal semantics, advancing the structural understanding of modalities in non-classical settings.
📝 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.
