An abelian totally ordered group or an ordered abelian group is an abelian group (written multiplicatively) $G$ endowed with a total order, such that $x\leq y$ implies $xz \leq yz$ for all $z\in G$. Since $1<x$ implies $1 < x < x^{2} < x^{3} < \cdot\cdot\cdot< x^{n}< \cdot\cdot\cdot$ (note that $1<x$ would imply $x^{-1}<1$), the abelian ordered group is torsion-free, i.e. no elements in it except the identity have finite order.