Proposition 3

— If $P$ is a subset of an abelian group $G$, containing $0$ and such that $P + P \subset P$, then the relation $y - x \in P$ is a pre-order relation compatible with the group structure of $G$. For this relation to make $G$ an ordered group it is necessary and sufficient to have $P \cap (-P) = \{0\}$; for $G$ to be a totally ordered group under this ordering it is necessary and sufficient to have in addition $P \cup (-P) = G$.

It is immediate that the relation $y - x \in P$ is reflexive and transitive, and (if it is written $x \leq y$) satisfies the axiom POM. To prove the second assertion it is enough to remark that $P \cap (-P)$ is the subgroup $G'$ consisting of elements $x$ such that $0 \leq x$ and $x \leq 0$. Finally, to say that $G$ is totally ordered means that, for each pair of elements $x, y$ of $G$, at least one of the elements $x - y, y - x$ belongs to $P$, which completes the proof.