MV-algebras and Partially Cyclically Ordered Groups

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are the wound-rounds 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 generalize these results to some non-linearly ordered MV-algebras, for example hyperarchimedean MV-algebras.