# Interval orders, semiorders and ordered groups

## TL;DR

This paper characterizes when the order of an ordered group is an interval order or semiorder, and explores the structure of semiorders in ordered groups, including examples like free groups and Thompson's group.

## Contribution

It establishes the equivalence between interval orders and semiorders in ordered groups and characterizes semiorders via intervals in totally ordered abelian groups.

## Key findings

- Order of an ordered group is an interval order iff it is a semiorder.
- Every semiorder corresponds to intervals in a totally ordered abelian group.
- Certain groups like free groups and Thompson's group admit compatible semiorders, unlike Clifford's group.

## Abstract

We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ \alpha[$ for some positive $\alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.