Archimedean equivalence classes

Remarks

Exactly one of $a \sim b$, "$a$ is infinitely larger than $b$" and "$b$ is infinitely larger than $a$" holds, so the classes of nonzero elements are totally ordered by size. Example: in $\Z^2$ with the lexicographic order, $(0, 2) \sim (0, 3)$, since $\abs{(0, 2)} \le 1 \cdot (0, 3)$ and $(0, 3) \le 2 \cdot (0, 2)$. Non-example: $(1, 0) \not\sim (0, 1)$, since $n \cdot (0, 1) = (0, n) < (1, 0)$ for all $n$, so no multiple of $(0, 1)$ bounds $(1, 0)$. So $\Z^2$ with this order has exactly two classes of nonzero elements, those with first coordinate $0$ and the others. The group $G$ is Archimedean if and only if all its nonzero elements are equivalent. Conventions on the direction of the order of classes differ between texts.