Integrally closed ordered group

In algebra, a partially ordered group "G" is called integrally closed if for all elements "a" and "b" of "G", if "a""n" ≤ "b" for all natural "n" then "a" ≤ 1.
This property is somewhat stronger than the fact that a partially ordered group is Archimedean, though for a lattice-ordered group to be integrally closed and to be Archimedean is equivalent.
There is a theorem that every integrally closed directed group is already abelian. This has to do with the fact that a directed group is embeddable into a complete lattice-ordered group if and only if it is integrally closed.