Partially ordered group

In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a + g ≤ b + g and g + a ≤ g + b.

An element x of G is called positive if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the positive cone of G.

By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b. So we can reduce the partial order to a monadic property: if and only if

For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially orderable group if and only if there exists a subset H (which is G+) of G such that:
 * 0 ∈ H
 * if a ∈ H and b ∈ H then a + b ∈ H
 * if a ∈ H then -x + a + x ∈ H for each x of G
 * if a ∈ H and -a ∈ H then a = 0

A partially ordered group G with positive cone G+ is said to be unperforated if n · g ∈ G+ for some positive integer n implies g ∈ G+. Being unperforated means there is no "gap" in the positive cone G+.

If the order on the group is a linear order, then it is said to be a linearly ordered group. If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group (shortly l-group, though usually typeset with a script l: ℓ-group).

A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the Riesz interpolation property: if x1, x2, y1, y2 are elements of G and xi ≤ yj, then there exists z ∈ G such that xi ≤ z ≤ yj.

If G and H are two partially ordered groups, a map from G to H is a morphism of partially ordered groups if it is both a group homomorphism and a monotonic function. The partially ordered groups, together with this notion of morphism, form a category.

Partially ordered groups are used in the definition of valuations of fields.

Examples

 * 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)

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 $$b < a^n$$.

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.