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

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 MValgebras 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. * This work was carried out within the IRSES project MaToMUVI, funded by the EU 7th Framework Programme.

Locally compact Stone duality

“…Also, the idea of representing continuous functions by certain relations, as in §4, already appears in formal topology (see [CMS13]), where constructions similar to those in §5 also appear. There have also been various other extensions of Stone duality, often in the context of categorical or continuous rather than classical logic and/or based on the ring structure of C(X, C), the lattice structure of C(X, R) or the MV-algebra structure of C(X, [0, 1]) (see [KR16], [MR15], [Rus16] and the references therein).…”