---
tag: RQ4X0EH6NH
kind: concept
name: Archimedean equivalence classes
msc: ["06F20"]
sources:
  - url: https://en.wikipedia.org/wiki/Archimedean_property
---

## Summary

Let $G$ be an abelian [[linearly ordered group|P2E8BNYC7B]].
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|FVPXXFNVQY]].

## Explanations

### L1 human

Compare the sizes of things up to any finite factor.
A millimetre and a kilometre are in the same class, since a million millimetres make a kilometre.
In a group with infinitely large elements, some element is larger than every multiple of another one, and then they lie in different classes.
In $\Z^2$ with the dictionary order, $(0, 1)$ and $(0, 5)$ are in the same class, but $(1, 0)$ is infinitely larger than both.