Ordered vector space

A point x in R2 and the set of all y such that xy (in red). The order here is xy if and only if x1 y1 and x2 y2.

In mathematics an ordered vector space or partially ordered vector space is a vector space equipped with a partial order which is compatible with the vector space operations.

Definition

Given a vector space V over the real numbers R and a preorder on the set V, the pair (V, ) is called an preordered vector space if for all x,y,z in V and 0 λ in R the following two axioms are satisfied

  1. x y implies x + z y + z
  2. y x implies λ y λ x.

If is a partial order, (V, ) is called an ordered vector space. The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping f(x) = x is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.

Positive cone

Given a preordered vector space V, the subset V+ of all elements x in V satisfying x  0 is a convex cone, called the positive cone of V. If V is an ordered vector space, then V+  (V+) = {0}, and hence V+ is a proper cone.

If V is a real vector space and C is a proper convex cone in V, there exists a unique partial order on V that makes V into an ordered vector space such V+ = C. This partial order is given by

x  y if and only if y  x is in C.

Therefore, there exists a one-to-one correspondence between the partial orders on a vector space V that are compatible with the vector space structure and the proper convex cones of V.

Examples

Only the second order is, as a subset of R4, closed, see partial orders in topological spaces.
For the third order the two-dimensional "intervals" p < x < q are open sets which generate the topology.

Remarks

See also

References

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