 If "x" and "y" are elements of a preordered vector space("X", ≤), then "x" ≤ "y" if and only if "y" - "x" ∈ "X"+.Given any pointed convex cone "C" with vertex 0, one may define a preorder ≤ on "X" that is compatible with the vector space structure of "X" by declaring for all "x" and "y" in "X", that "x" ≤ "y" if and only if "y" - "x" ∈ "C"; 
the positive cone of this resulting preordered vector space is "C".There is thus a one-to-one correspondence between pointed convex cones with vertex 0 and vector preorders on "X".