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Concept
Ordered vector space
Summary
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 over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied
implies
implies
If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on
The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.
Note that if and only if
A subset of a vector space is called a cone if for all real A cone is called pointed if it contains the origin. A cone is convex if and only if 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 in a vector space is said to be generating if
Given a preordered vector space the subset of all elements in satisfying is a pointed convex cone with vertex (that is, it contains ) called the positive cone of and denoted by
The elements of the positive cone are called positive.
If and are elements of a preordered vector space then if and only if The positive cone is generating if and only if is a directed set under
Given any pointed convex cone with vertex one may define a preorder on that is compatible with the vector space structure of by declaring for all that if and only if
the positive cone of this resulting preordered vector space is
There is thus a one-to-one correspondence between pointed convex cones with vertex and vector preorders on
If is preordered then we may form an equivalence relation on by defining is equivalent to if and only if and
if is the equivalence class containing the origin then is a vector subspace of and is an ordered vector space under the relation: if and only there exist and such that
A subset of of a vector space is called a proper cone if it is a convex cone of vertex satisfying
Explicitly, is a proper cone if (1) (2) for all and (3)
The intersection of any non-empty family of proper cones is again a proper cone.
Official source
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