Properties of Ordered Group

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

Let $\prec$ be the reflexive reduction of $\preccurlyeq$.

The following properties hold:

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