Positive cone

Remarks

A positive cone $P$ defines a partial order on $V$ by $x \le y \iff y - x \in P$. This order satisfies $x \le y \implies x + z \le y + z$ and $x \le y \implies \lambda x \le \lambda y$ for $\lambda \ge 0$, and $P = \set{x : 0 \le x}$. Conversely, for each partial order with these two properties, the set $\set{x : 0 \le x}$ is a positive cone. Example: $P = \set{x \in \R^n : x_i \ge 0 \text{ for all } i}$ is a positive cone, and it gives the componentwise order. Example: the positive semidefinite matrices form a positive cone in the space of real symmetric $n \times n$ matrices. Non-example: the half-plane $\set{(x, y) \in \R^2 : y \ge 0}$ satisfies clauses 1 and 2, but contains $(1, 0)$ and $(-1, 0)$, so it fails clause 3. Some authors drop clause 3 and call $P$ a wedge or a convex cone; then the relation $\le$ is only a preorder.