Example

— In the additive group $\mathbf{Z} \times \mathbf{Z}$, let $P$ be the set of elements $(x, y)$ satisfying two inequalities $ax + by \geq 0$ and $cx + dy \geq 0$, where $a, b, c, d$ are integers (* or real numbers *) such that $ad - bc \neq 0$; the « cone » $P$ satisfies the first two conditions of Prop. 3. In this way various orderings compatible with the group structure of $\mathbf{Z} \times \mathbf{Z}$ can be defined ; the group is not totally ordered under any of these orderings.