 Furthermore, there exist vector a vector sublattice "N" of this space "X" such that "N" ∩ "C" has empty interior in "X" but no positive linear functional on "N" can be extended to a positive linear functional on "X".Let "M" be a vector subspace of an ordered vector space "X" having positive cone "C", let formula_65 be the canonical projection, and let formula_66.Then formula_67 is a cone in "X"/"M" that induces a canonical preordering on the quotient space "X"/"M".