Lemma 1

For a commutative group $\Delta$ to be such that there exists on $\Delta$ a total ordering compatible with the group structure of $\Delta$, it is necessary and sufficient that $\Delta$ be torsion-free.

If there exists such an order structure on $\Delta$ and if $\lambda > 0$, then $\lambda + \mu > 0$ for all $\mu \geq 0$ and in particular, by induction on the integer $n > 0$, $n.\lambda > 0$, which proves that $\Delta$ is torsion-free (since every element $\neq 0$ of $\Delta$ is either $> 0$ or $< 0$). Conversely, if $\Delta$ is torsion-free, $\Delta$ is a sub-$\mathbf{Z}$-module of a vector $\mathbf{Q}$-space (§ 7, no. 10, Corollary 1 to Proposition 26) which may be assumed of the form $\mathbf{Q}^{(I)}$; if $I$ is given a well-ordering (Set Theory, III, § 2, no. 3, Theorem 1) and $\mathbf{Q}$ its usual ordering, the set $\mathbf{Q}^{(I)}$ with the lexicographical ordering is totally ordered (Set Theory, III, § 2, no. 6); it is immediate that this ordering is compatible with the additive group structure of $\mathbf{Q}^{(I)}$.