An extension of Stone duality to fuzzy topologies and MV-algebras

TL;DR

This paper extends Stone duality to fuzzy topologies and MV-algebras by introducing MV-topologies and establishing a duality between certain MV-algebras and MV-spaces, enriching the categorical framework.

Contribution

It introduces MV-topologies and proves a duality between limit cut complete MV-algebras and Stone MV-spaces, expanding the classical Stone duality to fuzzy settings.

Findings

Established a duality between limit cut complete MV-algebras and Stone MV-spaces.

Showed every semisimple MV-algebra has a minimal limit cut complete extension.

Connected new dualities with existing categorical frameworks.

Abstract

In this paper we introduce the concept of MV-topology, a special class of fuzzy topological spaces, and prove a proper extension of Stone Duality to the categories of limit cut complete MV-algebras and Stone MV-spaces, namely, zero-dimensional compact Hausdorff MV-topological spaces. Then we describe the object class of limit cut complete MV-algebras, and show that any semisimple MV-algebra has a limit cut completion, namely, a minimum limit cut complete extension. Last, we compose our duality with other known ones, thus obtaining new categorical equivalences and dualities involving categories of MV-algebras.