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# 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 x1y1 and x2y2.

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. xy implies x + zy + z
2. yx implies λ y ≤ λ x.

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