有序交换群系指一对 ，其中 为交换群， 为其上的一个二元关系，且满足如下条件：

另一种等价的描述是：给定一个子集 ，使得 对加法封闭，且 。

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.