2017/06/10 by Pouzet, Maurice, Zaguia, Imed
#06A05 #06A06 #06F15 #06F20 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.03276
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+ α[ for some positive α. 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.