  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.
