Ordered group - Definition and Overview

In abstract algebra, an ordered group is a group G equipped with a partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb. Note that sometimes the term ordered group is used for linearly/totally ordered group, and what we describe here is called partially ordered group.

By the definition we can reduce the partial order to a monadic property: a ≤ b iff 1 ≤ a^-1 b. The set of elements x ≥ 1 of G are often denoted with G⁺. So we have a ≤ b iff a^-1b ∈ G⁺. That G is an ordered group can be expressed only in terms of G⁺: A group is an ordered group iff there exists a subset H (which is G⁺) of G such that:

• 1 ∈ H
• a ∈ H and b ∈ H then ab ∈ H
• if a ∈ H then x^-1ax ∈ H for each x of G

A typical example of an ordered group is Zⁿ, where the group operation is componentwise addition, and we write (a₁,...,a_n) ≤ (b₁,...,b_n) iff a_i ≤ b_i (in the usual order of integers) for all i=1,...,n. More generally, if G is an ordered group and X is some set, then the set of all functions from X to G is again an ordered group: all operations are performed componentwise. Furthermore, every subgroup of G is an ordered group: it inherits the order from G.

If the order on the group is a linear order, we speak of a linearly ordered group.

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

Ordered groups are used in the definition of valuations of fields.