 Ordered vector space

In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.Given a vector space "X" over the real numbers R and a preorder ≤ on the set "X", the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of "X" and call ≤ a vector preorder on "X" if for all "x", "y", "z" in "X" and in formula_1 the following two axioms are satisfied


If ≤ is a partial order compatible with the vector space structure of "X" then is called an ordered vector space and ≤ is called a vector partial order on "X".