Ordered vector space

A point $x$ in $\mathbb {R} ^{2}$ and the set of all $y$ such that $x\leq y$ (in red). The order here is $x\leq y$ if and only if $x_{1}\leq y_{1}$ and $x_{2}\leq y_{2}.$

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.

Definition

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

1. $x\leq y$ implies $x+z\leq y+z,$
2. $y\leq x$ implies $ry\leq rx.$

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

Positive cones and their equivalence to orderings

A subset $C$ of a vector space $X$ is called a cone if for all real $r>0,$ $rC\subseteq C.$ A cone is called pointed if it contains the origin. A cone $C$ is convex if and only if $C+C\subseteq C.$ 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 $C$ in a vector space $X$ is said to be generating if $X=C-C.$ A positive cone is generating if and only if it is a directed set under $\,\leq .$

Given a preordered vector space $X,$ the subset $X^{+}$ of all elements $x$ in $(X,\leq )$ satisfying $x\geq 0$ is a pointed convex cone with vertex $0$ (that is, it contains $0$) called the positive cone of $X$ and denoted by $\operatorname {PosCone} X.$ The elements of the positive cone are called positive. If $x$ and $y$ are elements of a preordered vector space $(X,\leq ),$ then $x\leq y$ if and only if $y-x\in X^{+}.$ Given any pointed convex cone $C$ with vertex $0,$ one may define a preorder $\,\leq \,$ on $X$ that is compatible with the vector space structure of $X$ by declaring for all $x,y\in X,$ that $x\leq y$ if and only if $y-x\in C;$ the positive cone of this resulting preordered vector space is $C.$ There is thus a one-to-one correspondence between pointed convex cones with vertex $$