TITLE: Lattice ordered group
QUESTION [5 upvotes]: It seems the basic fact but I did not get the idea how to prove it. Let $G$ be a partially ordered group. Then $G$ is an $\ell$-group if and only if every pair of positive elements has least upper bound.

REPLY [4 votes]: This follows from the assertion:
A partially ordered group $G$ is an $l$-group if and only if for any $a\in G$ there is the least upper bound of $a$ and $1$.
[Fuchs, L. Partially ordered algebraic systems. Pergamon Press, 1963, chapt.5, sec.1]