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

Litan Kumar Das, Kumar Sankar Ray, Prakash Chandra Mali

TL;DR

The paper develops a unified bitopological and coalgebraic semantic framework for Fitting's $ L$-valued multi-modal logic. By extending Lauridsen's bi-Vietoris construction to $ L$-valued pairwise Boolean spaces, it defines a coalgebra endofunctor $V_{ L}^{bi}$ and proves a dual equivalence between algebras of Fitting's $ L$-valued modal logic and coalgebras for $V_{ L}^{bi}$. This yields soundness and completeness, a Hennessy–Milner property, and the existence of cofree and final coalgebras, all within a bitopological setting. The work generalizes Jónsson–Tarski-like dualities to the multi-valued modal context and provides a robust semantic bridge between algebraic and coalgebraic perspectives. It also outlines potential future directions, including extensions to bitopological Esakia spaces and broader $ L$-valued frameworks.

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.

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

TL;DR

The paper develops a unified bitopological and coalgebraic semantic framework for Fitting's -valued multi-modal logic. By extending Lauridsen's bi-Vietoris construction to -valued pairwise Boolean spaces, it defines a coalgebra endofunctor and proves a dual equivalence between algebras of Fitting's -valued modal logic and coalgebras for . This yields soundness and completeness, a Hennessy–Milner property, and the existence of cofree and final coalgebras, all within a bitopological setting. The work generalizes Jónsson–Tarski-like dualities to the multi-valued modal context and provides a robust semantic bridge between algebraic and coalgebraic perspectives. It also outlines potential future directions, including extensions to bitopological Esakia spaces and broader -valued frameworks.

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 of -valued (with a bounded finite distributive lattice, i.e., a Heyting algebra) pairwise Boolean spaces by incorporating a structure map, and from this obtain the -biVietoris functor. Finally, we establish dual equivalence between coalgebras for the -biVietoris functor and algebras of Fitting's -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 -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.
Paper Structure (13 sections, 33 theorems, 54 equations, 1 figure)

This paper contains 13 sections, 33 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bitopological spaces
  4. Fitting’s Heyting-valued logics
  5. Bitopological duality for Fitting’s Heyting-valued logic
  6. Bitopological duality for Fitting’s many-valued modal logic
  7. Category
  8. Functors
  9. Bitopological Duality for $\mathcal{L}$-$\mathcal{ML}$ Algebras
  10. Coalgebraic Duality Theorem for Fitting's multi-valued logic
  11. The structure of the endofunctor $V_{\mathcal{L}}^{bi}$
  12. Applications
  13. Conclusion

Key Result

Proposition 1

If $T_1$ and $T_2$ are subbasis for the topologies $\tau_1$ and $\tau_2$, respectively, then $T_1\cup T_2$ is a subbasis for the topology $\tau_1\vee\tau_2$.

Figures (1)

  • Figure :

Theorems & Definitions (81)

  • Definition 1: salbany1974bitopological
  • Proposition 1: lauridsen2015bitopological
  • Proposition 2: lauridsen2015bitopological
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 2: maruyama2009algebraic
  • Definition 3
  • Definition 4: maruyama2009algebraic
  • Definition 5: maruyama2009algebraic
  • ...and 71 more