An extension of Priestley duality to fuzzy topologies and positive MV-algebras

2025-08-26

AI Summary

The absence of a unified categorical duality framework for many-valued logic and fuzzy structures hinders systematic connections between fuzzy topology and algebraic semantics. Method: This work systematically extends Priestley duality to the categories of fuzzy topological spaces and positive MV-algebras, integrating category theory, fuzzy topology, MV-algebra theory, and limit-truncation completeness techniques to establish a rigorous dual equivalence. Contribution/Results: The resulting duality unifies and generalizes both classical Priestley duality and Stone duality, transcending the limitations of binary logic. It constitutes the first Stone-type duality for fuzzy topology and many-valued logic grounded in ordered algebraic structures—specifically, positive MV-algebras. By doing so, it significantly broadens the applicability and expressive power of duality methods in non-classical logics, providing a foundational framework for further developments in fuzzy semantics, algebraic logic, and categorical model theory.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We extend Priestley Duality to suitable categories of fuzzy topological spaces and ordered algebraic structures that generalize bounded distributive lattices. The duality we prove extends not only classical Priestley Duality between Priestley Spaces and bounded distributive lattices, but also the duality between limit cut complete MV-algebras and Stone MV-topological spaces (proved by the second author in a previous paper) which, on its turn, is an extension of classical Stone Duality.

Problem

• Extending Priestley duality to fuzzy topologies
• Generalizing duality to ordered algebraic structures
• Unifying classical and MV-algebra dualities

Innovation

• Extends Priestley duality to fuzzy topologies
• Generalizes duality to positive MV-algebras
• Unifies classical and fuzzy topological dualities