Integrally closed ordered group
In algebra, an ordered group G is called integrally closed if for all elements a and b of G, if an ≤ b for all natural n then a ≤ 1.
This property is somewhat stronger than the fact that an 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.
- A. M. W. Glass, Partially Ordered Groups, World Scientific, 1999