Interval orders, semiorders and ordered groups
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.
 Publication:

arXiv eprints
 Pub Date:
 June 2017
 DOI:
 10.48550/arXiv.1706.03276
 arXiv:
 arXiv:1706.03276
 Bibcode:
 2017arXiv170603276P
 Keywords:

 Mathematics  Combinatorics;
 06A05;
 06A06;
 06F15;
 06F20
 EPrint:
 32 pages, 2 figures