 sup {"x", "y"} exists if and only if inf {"x", "y"} exists, in which case for all "z" ∈ "X",
"X" is a vector lattice if and only if sup { 0, "x" } exists for all "x" ∈ "X".A cone "C" is said to be generating if "C" − "C" is equal to the whole vector space.If "X" and "W" are two non-trivial ordered vector spaces with respective positive cones "P" and "Q", then "P" is generating in "X" if and only if the set formula_8 is a proper cone in L("X"; "W"), which is the space of all linear maps from "X" into "W".