Properties of Ordered Group

Theorem
Let $\struct {G, \circ, \preceq}$ be an ordered group with identity $e$.

Let $x, y, z, w \in G$.

Then the following hold:

Corollary: Properties of Ordered Group/OG5
If $n \in \N_{>0}$ then:

Also see

 * Ordered Group Equivalences, which presents a subset of these properties in a very different fashion.
 * Properties of Relation Compatible with Group Operation
 * Properties of Ordered Ring