Remark

— By virtue of the condition $P + P \subset P$, the relation $x \geq 0$ in an ordered group implies that $nx \geq 0$ for every natural number $n$. If in addition the positive element $x$ of $G$ has finite order $n$, then $-x = (n-1)x$ is positive ; since

this implies that $x = 0$. In particular, if every element of $G$ has finite order, then $P = \{0\}$; the relation $x \leq y$ is then equivalent to $x = y$ (the discrete ordering).