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:
“Then he rang the bell and ordered a ham sandwich. When the maid placed the plate on the table, he deliberately looked away but as soon as the door had shut, he grabbed the sandwich with both hands, immediately soiled his fingers and chin with the hanging margin of fat and, grunting greedily, began to much.”
—Vladimir Nabokov (1899–1977)
“Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.”
—Woodrow Wilson (1856–1924)