Partially ordered group

• The integers with their usual order • An ordered vector space is a partially ordered group • A Riesz space is a lattice-ordered group • A typical example of a partially ordered group is Zn, where the group operation is componentwise addition, and we write (a1,...,an) ≤ (b1,...,bn) if and only if ai ≤ bi (in the usual order of integers) for all i = 1,..., n. • More generally, if G is a partially ordered group and X is some set, then the set of all functions from X to G is again a partially ordered group: all operations are performed componentwise. Furthermore, every subgroup of G is a partially ordered group: it inherits the order from G. • If A is an approximately finite-dimensional C*-algebra, or more generally, if A is a stably finite unital C*-algebra, then K0(A) is a partially ordered abelian group. (Elliott, 1976) ==Properties==

Archimedean The Archimedean property of the real numbers can be generalized to partially ordered groups. :Property: A partially ordered group G is called Archimedean when for any a, b \in G, if e \le a \le b and a^n \le b for all n \ge 1 then a=e. Equivalently, when a \neq e, then for any b \in G, there is some n\in \mathbb{Z} such that b . Integrally closed A partially 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 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. == See also ==