Order unit

Definition

For the ordering cone K ⊆ X in the vector space X , the element e ∈ K is an order unit (more precisely an K -order unit) if for every x ∈ X there exists a λ x > 0 such that λ x e − x ∈ K (i.e. x ≤ K λ x e ).

Equivalent definition

The order units of an ordering cone K ⊆ X are those elements in the algebraic interior of K , i.e. given by core ⁡ ( K ) .

Examples

Let X = R be the real numbers and K = R + = { x ∈ R : x ≥ 0 } , then the unit element 1 is an order unit.

Let X = R n and K = R + n = { x ∈ R : ∀ i = 1 , … , n : x i ≥ 0 } , then the unit element 1 → = ( 1 , … , 1 ) is an order unit.