# Compact Hausdorff MV-algebras: Structure, Duality and Projectivity

## TL;DR

This paper explores the structure and duality of compact Hausdorff MV-algebras, establishing categorical equivalences with extended multisets and complete distributive MV-algebras, and investigates key topological properties and projectivity within topological MV-algebras.

## Contribution

It establishes dualities and categorical equivalences for compact Hausdorff MV-algebras and investigates fundamental topological properties and projectivity in topological MV-algebras.

## Key findings

- Category of extended multisets is dually equivalent to compact Hausdorff MV-algebras.
- Compact Hausdorff MV-algebras are equivalent to complete, completely distributive MV-algebras.
- Urysohn-Strauss's Lemma, Gleason's Theorem, and projective objects are analyzed for topological MV-algebras.

## Abstract

It is proved that the category $\mathbb{EM}$ of extended multisets is dually equivalent to the category $\mathbb{CHMV}$ of compact Hausdorff MV-algebras with continuous homomorphisms, which is in turn equivalent to the category of complete and completely distributive MV-algebras with homomorphisms that reflect principal maximal ideals. Urysohn-Strauss's Lemma, Gleason's Theorem, and projective objects are also investigated for topological MV-algebras.