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 ${\displaystyle X}$ over the real numbers ${\displaystyle \mathbb {R} }$ and a preorder ${\displaystyle \,\leq \,}$ on the set ${\displaystyle X,}$ the pair ${\displaystyle (X,\leq )}$ is called a preordered vector space and we say that the preorder ${\displaystyle \,\leq \,}$ is compatible with the vector space structure of ${\displaystyle X}$ and call ${\displaystyle \,\leq \,}$ a vector preorder on ${\displaystyle X}$ if for all ${\displaystyle x,y,z\in X}$ and ${\displaystyle r\in \mathbb {R} }$ with ${\displaystyle r\geq 0}$ the following two axioms are satisfied

If ${\displaystyle \,\leq \,}$ is a partial order compatible with the vector space structure of ${\displaystyle X}$ then ${\displaystyle (X,\leq )}$ is called an ordered vector space and ${\displaystyle \,\leq \,}$ is called a vector partial order on ${\displaystyle X.}$ The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping ${\displaystyle x\mapsto -x}$ is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation. Note that ${\displaystyle x\leq y}$ if and only if ${\displaystyle -y\leq -x.}$

A subset ${\displaystyle C}$ of a vector space ${\displaystyle X}$ is called a cone if for all real ${\displaystyle r>0,}$ ${\displaystyle rC\subseteq C}$. A cone is called pointed if it contains the origin. A cone ${\displaystyle C}$ is convex if and only if ${\displaystyle C+C\subseteq C.}$ The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones). A cone ${\displaystyle C}$ in a vector space ${\displaystyle X}$ is said to be generating if ${\displaystyle X=C-C.}$

Given a preordered vector space ${\displaystyle X,}$ the subset ${\displaystyle X^{+}}$ of all elements ${\displaystyle x}$ in ${\displaystyle (X,\leq )}$ satisfying ${\displaystyle x\geq 0}$ is a pointed convex cone (that is, a convex cone containing ${\displaystyle 0}$) called the positive cone of ${\displaystyle X}$ and denoted by ${\displaystyle \operatorname {PosCone} X.}$ The elements of the positive cone are called positive. If ${\displaystyle x}$ and ${\displaystyle y}$ are elements of a preordered vector space ${\displaystyle (X,\leq ),}$ then ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in X^{+}.}$ The positive cone is generating if and only if ${\displaystyle X}$ is a directed set under ${\displaystyle \,\leq .}$ Given any pointed convex cone ${\displaystyle C}$ one may define a preorder ${\displaystyle \,\leq \,}$ on ${\displaystyle X}$ that is compatible with the vector space structure of ${\displaystyle X}$ by declaring for all ${\displaystyle x,y\in X,}$ that ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in C;}$ the positive cone of this resulting preordered vector space is ${\displaystyle C.}$ There is thus a one-to-one correspondence between pointed convex cones and vector preorders on ${\displaystyle X.}$ If ${\displaystyle X}$ is preordered then we may form an equivalence relation on ${\displaystyle X}$ by defining ${\displaystyle x}$ is equivalent to ${\displaystyle y}$ if and only if ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x;}$ if ${\displaystyle N}$ is the equivalence class containing the origin then ${\displaystyle N}$ is a vector subspace of ${\displaystyle X}$ and ${\displaystyle X/N}$ is an ordered vector space under the relation: ${\displaystyle A\leq B}$ if and only there exist ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$ such that ${\displaystyle a\leq b.}$

A subset of ${\displaystyle C}$ of a vector space ${\displaystyle X}$ is called a proper cone if it is a convex cone satisfying ${\displaystyle C\cap (-C)=\{0\}.}$ Explicitly, ${\displaystyle C}$ is a proper cone if (1) ${\displaystyle C+C\subseteq C,}$ (2) ${\displaystyle rC\subseteq C}$ for all ${\displaystyle r>0,}$ and (3) ${\displaystyle C\cap (-C)=\{0\}.}$ The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone ${\displaystyle C}$ in a real vector space induces an order on the vector space by defining ${\displaystyle x\leq y}$ if and only if ${\displaystyle y-x\in C,}$ and furthermore, the positive cone of this ordered vector space will be ${\displaystyle C.}$ Therefore, there exists a one-to-one correspondence between the proper convex cones of ${\displaystyle X}$ and the vector partial orders on ${\displaystyle X.}$

By a total vector ordering on ${\displaystyle X}$ we mean a total order on ${\displaystyle X}$ that is compatible with the vector space structure of ${\displaystyle X.}$ The family of total vector orderings on a vector space ${\displaystyle X}$ is in one-to-one correspondence with the family of all proper cones that are maximal under set inclusion. A total vector ordering cannot be Archimedean if its dimension, when considered as a vector space over the reals, is greater than 1.

If ${\displaystyle R}$ and ${\displaystyle S}$ are two orderings of a vector space with positive cones ${\displaystyle P}$ and ${\displaystyle Q,}$ respectively, then we say that ${\displaystyle R}$ is finer than ${\displaystyle S}$ if ${\displaystyle P\subseteq Q.}$