Ordered abelian group

Definition

An ordered abelian group is an abelian group $(\Gamma,+,0)$ equipped with a total order $\leq$ that is translation-invariant:

a\leq b\quad\Longrightarrow\quad a+c\leq b+c

for all $a,b,c\in\Gamma$ . Equivalently, its positive cone $\Gamma_{\geq0}=\{a:0\leq a\}$ is closed under addition, satisfies $\Gamma_{\geq0}\cap(-\Gamma_{\geq0})=\{0\}$ , and obeys $\Gamma_{\geq0}\cup(-\Gamma_{\geq0})=\Gamma$ .

Examples

The additive groups $\mathbb Z$ , $\mathbb Q$ , and $\mathbb R$ with their usual orders are ordered abelian groups. The lexicographic order makes $\mathbb Z^n$ an ordered abelian group that is generally non-Archimedean.

Value groups

The codomain of a valuation is commonly an ordered abelian group, often enlarged by an element $\infty$ . The group operation records multiplication of valued elements, while the order compares their sizes.

Terminology

Some sources use “ordered abelian group” for a translation-invariant partial order and say linearly ordered or totally ordered when every pair is comparable. Here the term uses the total-order convention standard for valuation value groups.

References

K. R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, 1986. DOI record. Relevant: Chapter 1.