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 element if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and it is called the positive cone of G. So we have a ≤ b if and only if -a+b ∈ G+.

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

For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially ordered 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 natural number 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 a 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 x_i ≤ y_j, then there exists z ∈ G such that x_i ≤ z ≤ y_j.

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.