Theorem: Translation and dilation produce a cone order

Suppose a partial order on a real vector space $V$ satisfies

Then $K=\{v:0\le v\}$ is a pointed convex cone and

If the order graph is closed, $K$ is closed. Conversely, a pointed convex cone defines such an order.

Proof. Reflexivity gives $0\in K$. If $u,v\in K$, translation gives $u\le u+v$, and transitivity gives $u+v\in K$. Positive scaling and addition imply convexity. If $v,-v\in K$, antisymmetry gives $v=0$. Translation by $-x$ gives the difference formula. Closedness follows by taking the inverse image of the order graph under $v\mapsto (0,v)$.

Conversely, membership of zero, addition, and pointedness establish reflexivity, transitivity, and antisymmetry; the difference formula establishes both invariances.