Archimedean ordered vector space

Summary

In mathematics, specifically in order theory, a binary relation $\,\leq\,$ on a vector space $X$ over the real or complex numbers is called Archimedean if for all $x \in X,$ whenever there exists some $y \in X$ such that $n x \leq y$ for all positive integers $n,$ then necessarily $x \leq 0.$ An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space $X$ is called almost Archimedean if for all $x \in X,$ whenever there exists a $y \in X$ such that $-n^{-1} y \leq x \leq n^{-1} y$ for all positive integers $n,$ then $x = 0.$