In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 an ordered vector space or partially ordered vector space is a vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 equipped with a partial order which is compatible with the vector space operations.

Definition

Given a vector space V over the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s 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.

Positive cone

Given an ordered vector space V, the subset V⁺ of all elements x in V satisfying x≥0 is a convex cone

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

, 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 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 number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s 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
    Lexicographical order
    In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
    
    : (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d). This is a total order
    Total order
    In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
    
    . The positive cone is given by x > 0 or (x = 0 and y ≥ 0), i.e., in polar coordinates
    Polar coordinate system
    In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....
    
    , 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
    Product order
    In mathematics, given two ordered sets A and B, one can induce a partial ordering on the Cartesian product A × B. Giventwo pairs and in A × B, one sets ≤...
    
     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
  Riesz space
  In mathematics a Riesz space, lattice-ordered vector space or vector lattice is an ordered vector space where the order structure is a lattice....
  
   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
  Convex set
  In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
  
  . 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].