Proposition 13

— (i) The set P, of elements $\geqslant 0$, of a preordered vector space E, is a pointed convex cone.

(ii) Conversely, if P is a pointed convex cone in E, then the relation $y - x \in P$ is a preorder relation on E, and the preorder structure that it defines is the only one that is compatible with the vector space structure of E and for which P is the set of elements $\geqslant 0$.

(iii) The relation $y - x \in P$, with P a pointed convex cone, is an order relation on E if and only if P is a proper cone.

(i) Axioms (EO_I) and (EO_{II}) imply $P + P \subset P$ and $\lambda P \subset P$ for all $\lambda > 0$. As $0 \in P$, it follows that P is a pointed convex cone (II, p. 11, prop. 10).

(ii) Conversely, if P is a pointed convex cone, the relation $P + P \subset P$ implies that the relation $y - x \in P$ is a preorder compatible with the additive group structure of E (A, VI, p. 3, prop. 3); clearly writing it $x \leqslant y$, the set P is identical with the set of $x \geqslant 0$; further the relation $\lambda P \subset P$ for all $\lambda \geqslant 0$ shows that axiom (EO_{II}) is satisfied.

(iii) To say that P is proper means that $P \cap (-P) = \{0\}$ (II, p. 11, cor. 2), hence that $y - x \in P$ is an order relation.