I am reading a book entitled An introduction to the classification of amenable C*-algebras. It reads,
Definition 3.3.1 An ordered group $(G,G_+)$ is an abelian group $G$ with a distinguished subsemigroup $G_+$ containing zero, called the positive cone of $G$ satisfying the properties
(1) $G_+-G_+=G$ and
(2) $G_+\cap (-G_+)=\{0\}$
$G_+$ induces a partial ordering on $G$ by $x\leq y$ if $y-x\in G_+$. Also, we write $g<f$, if $g\leq f$ and $g\neq f$.
An order ideal $I$ of an ordered group $(G,G_+)$ is a subgroup of $G$ such that $g\leq f$ for some $f\in I$ implies that $g\in I$.
Since $I$ is a subgroup, $0\in I$. So for every $g\in G_+$, $0-(-g)=g\in G_+$ implies $-g\in I$ and $g\in I$. Eventually, $G=G_+-G_+\subset I$ and $G=I$.
I wonder if I misunderstood something or if the book is wrong.