Archimedean ordered vector space

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