Order unit

An order unit is an element of an ordered vector space which can be used to bound all elements from above.^[1] In this way (as seen in the first example below) the order unit generalizes the unit element in the reals.

Definition

For the ordering cone $K\subseteq X$ in the vector space $X$, the element ${\displaystyle e\in K}$ is an order unit (more precisely an $K$-order unit) if for every $x\in X$ there exists a ${\displaystyle \lambda _{x}>0}$ such that ${\displaystyle \lambda _{x}e-x\in K}$ (i.e. ${\displaystyle x\leq _{K}\lambda _{x}e}$).^[2]

Equivalent definition

The order units of an ordering cone $K\subseteq X$ are those elements in the algebraic interior of $K$, i.e. given by ${\displaystyle \operatorname {core} (K)}$.^[2]

Examples

Let $X=\mathbb {R}$ be the real numbers and ${\displaystyle K=\mathbb {R} _{+}=\{x\in \mathbb {R} :x\geq 0\}}$, then the unit element $1$ is an order unit.

Let $X=\mathbb {R} ^{n}$ and ${\displaystyle K=\mathbb {R} _{+}^{n}=\{x\in \mathbb {R} :\forall i=1,\ldots ,n:x_{i}\geq 0\}}$, then the unit element ${\displaystyle {\vec {1}}=(1,\ldots ,1)}$ is an order unit.

References

1. Fuchssteiner, Benno; Lusky, Wolfgang (1981). Convex Cones. Elsevier. ISBN 9780444862907.
2. Charalambos D. Aliprantis; Rabee Tourky (2007). Cones and Duality. American Mathematical Society. ISBN 9780821841464.