c-Regular cyclically ordered groups

TL;DR

This paper characterizes regular and c-regular cyclically ordered abelian groups, showing their elementary equivalences to well-known structures like complex roots of unity, finite cyclic groups, and integers.

Contribution

It introduces the concepts of regular and c-regular cyclically ordered abelian groups and establishes their elementary equivalences to classical algebraic structures.

Findings

Dense c-regular groups are elementarily equivalent to complex roots of unity.

Discrete c-regular groups are elementarily equivalent to ultraproducts of finite cyclic groups.

Regular non-c-regular groups are elementarily equivalent to the integers.

Abstract

We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex numbers, that every discrete c-regular cyclically ordered abelian group is elementarily equivalent to some ultraproduct of finite cyclic groups, and that the discrete regular non-c-regular cyclically ordered abelian groups are elementarily equivalent to the linearly cyclically ordered group of integers.