Partially ordered group


In abstract algebra, a partially ordered group is a group 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 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.
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 ≤ if and only if ai ≤ bi 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 is a partially ordered abelian group.

    Properties

Archimedean

The Archimedean property of the real numbers can be generalized to partially ordered groups.

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.

Note