Lexicographically ordered group
= Lexicographically ordered group
For ordered <Abelian groups>, compare tuples at their first differing coordinate. This gives a two-sided invariant <total order> on their direct product. The additive group $\mathbb Z^2$ with its first coordinate dominant is not an <Archimedean ordered group>, since $n(0,1)<(1,0)$ for every positive integer $n$.