Ordered vector space

A point x in R² and the set of all y such that x≤y (in red). The order here is x≤y 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 partial order ≤ on the set V, the pair (V, ≤) is called an ordered 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.

Notes

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.

If ≤ is only a preorder, (V, ≤) is called a preordered vector space.

Ordered vector spaces are ordered groups.

Positive cone

Given an ordered 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. V⁺ has the property that V⁺∩(−V⁺)={0}, so V⁺ is a proper cone. That it is convex can be seen by combining the above two axioms with the transitivity property of the (pre)order.

If V is a real vector space and C is a proper convex cone in V, there exists exactly one 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

• The real numbers with the usual order is an ordered vector space.
• R² is an ordered vector space with the ≤ relation defined in any of the following ways (in order of increasing strength, i.e., decreasing sets of pairs):
  • Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order. The positive cone is given by x > 0 or (x = 0 and y ≥ 0), i.e., in polar coordinates, the set of points with the angular coordinate satisfying -π/2 < θ ≤ π/2, together with the origin.
  • (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product order of two copies of R with "≤"). This is a partial order. The positive cone is given by x ≥ 0 and y ≥ 0, i.e., in polar coordinates 0 ≤ θ ≤ π/2, together with the origin.
  • (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the direct product of two copies of R with "<"). This is also a partial order. The positive cone is given by (x > 0 and y > 0) or (x = y = 0), i.e., in polar coordinates, 0 < θ < π/2, together with the origin.

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.

• Rⁿ is an ordered vector space with the ≤ relation defined similarly. For example, for the second order mentioned above:
  • x ≤ y if and only if x_i ≤ y_i for i = 1, … , n.
• A Riesz space is an ordered vector space where the order gives rise to a lattice.
• The space of continuous function on [0,1] where f ≤ g iff f(x) ≤ g(x) for all x in [0,1]

Remarks

• An interval in a partially ordered vector space is a convex set. If [a,b] = { x : a ≤ x ≤ b }, from axioms 1 and 2 above it follows that x,y in [a,b] and λ in (0,1) implies λx+(1-λ)y in [a,b].

References

• Bourbaki, Nicolas; Elements of Mathematics: Topological Vector Spaces; ISBN 0-387-13627-4.
• Schaefer, Helmut H; Wolff, M.P. (1999). Topological vector spaces, 2nd ed. New York: Springer. pp. 204–205. ISBN 0387987266. 
• Aliprantis, Charalambos D; Burkinshaw, Owen (2003). Locally solid Riesz spaces with applications to economics (Second ed.). Providence, R. I.: American Mathematical Society. ISBN 0821834088.