# Mv-algebras And Partially Cyclically Ordered Groups

## TL;DR

This paper establishes a functorial link between MV-algebras and partially cyclically ordered groups, enabling the transfer of properties and characterizations between these algebraic structures.

## Contribution

It introduces a new correspondence between MV-algebras and cyclically ordered groups, extending the understanding of their properties and classifications.

## Key findings

- Characterization of groups of unimodular complex numbers
- Description of finite cyclic groups via cyclic orders
- Classification of pseudofinite and pseudo-simple MV-chains

## Abstract

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.