Linearly ordered group

Summary

A linearly ordered group is an ordered group whose order is a total order. Any two of its elements are comparable, and the order is preserved by multiplication on both sides. The additive groups of $\Z$, $\Q$ and $\R$ are the basic examples. Linearly ordered abelian groups are the value groups of valuations, and they are described by Hölder's theorem and the Hahn embedding theorem.