Ordered Abelian Groups are Bi-Ordered Groups

Last updated Nov 1, 2022

Let $(G, \leq)$ be an Ordered Abelian Group. Then $G$ is a Bi-Ordered Group.

Let $g, h, h’ \in G$ so that $h \leq h’$. Then because $G$ is a Left Ordered Group, $$hg = gh \leq gh’ = h’g$$ so $G$ is also a Right Ordered Group. Thus $G$ is a Bi-Ordered Group. $\blacksquare$