Archimedean equivalence classes

Summary

Let $G$ be an abelian linearly ordered group. Two nonzero elements of $G$ are Archimedean equivalent when each is bounded by a multiple of the other, in absolute value. The equivalence classes measure the size of elements up to finite factors, in the way orders of magnitude do. They form a totally ordered set, and the group is Archimedean exactly when it has at most one class of nonzero elements. These classes index the coordinates in the Hahn embedding theorem.