Definition 1

Given an ordered vector space E, a linear form L on E is said to be positive if $L(x) \geq 0$ for every $x \geq 0$ in E.

Since $L(y) - L(x) = L(y - x)$, it is the same to say that the relation $x \leq y$ implies $L(x) \leq L(y)$, or again that $L$ is an increasing function on E.