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