Ordered Groups
An ordered group G is called integrally closed if and only 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. We have the surprising 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. Furthermore, every archimedean lattice-ordered group is abelian.
Read more about this topic: Integrally Closed
Famous quotes containing the words ordered and/or groups:
“I am aware that I have been on many a man’s premises, and might have been legally ordered off, but I am not aware that I have been in many men’s houses.”
—Henry David Thoreau (1817–1862)
“Writers and politicians are natural rivals. Both groups try to make the world in their own images; they fight for the same territory.”
—Salman Rushdie (b. 1947)