 If "X" is preordered then we may form an equivalence relation on "X" by defining "x" is equivalent to "y" if and only if "x" ≤ "y" and "y" ≤ "x"; 
if "N" is the equivalence class containing the origin then "N" is a vector subspace of "X" and "X"/"N" is an ordered vector space under the relation