Ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations. == Definition == Given a vector space X {\displaystyle X} over the real numbers R {\displaystyle \mathbb {R} } and a preorder ≤ {\displaystyle \,\leq \,} on the set X , {\displaystyle X,} the pair ( X , ≤ ) {\displaystyle (X,\leq )} is called a preordered vector space and we say that the preorder ≤ {\displaystyle \,\leq \,} is compatible with the vector space structure of X {\displaystyle X} and call ≤ {\displaystyle \,\leq \,} a vector preorder on X {\displaystyle X} if for all x , y , z ∈ X {\displaystyle x,y,z\in X} and r ∈ R {\displaystyle r\in \mathbb {R} } with r ≥ 0 {\displaystyle r\geq 0} the following two axioms are satisfied x ≤ y {\displaystyle x\leq y} implies x + z ≤ y + z , {\displaystyle x+z\leq y+z,} y ≤ x {\displaystyle y\leq x} implies r y ≤ r x .