On the structure of groups endowed with a compatible C-relation

TL;DR

This paper explores the structure of C-groups using quasi-orders, providing a new framework to understand their properties and reformulating existing results in the context of quasi-ordered groups.

Contribution

It introduces a novel approach by associating quasi-orders to compatible C-relations, offering a detailed structural description of C-groups and reinterpreting prior results.

Findings

Established a correspondence between C-relations and quasi-orders in groups

Provided a structural classification of quasi-ordered C-groups

Reformulated results on C-minimal groups using quasi-order theory

Abstract

We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given by Macpherson and Steinhorn.