Ordered groups - examples Let $G=BS(m,n)$ denote the Baumslag–Solitar groups defined by 
the presentation $\langle a,b: b^m a=a b^n\rangle$.

Question: Find $m,n$ such that $G$ is an ordered group, i.e. $G$ is a group  on which a partial order relation $\le $ is given such that for any elements $x,y,z \in G$, from $x \le y$ it follows that $xz \le  yz$ and $zx \le  zy$.


Let $\alpha:BS(m,n)\rightarrow\mathbb{Z}$ be defined by $\alpha(a)=1,\alpha(b)=0$. Define a partial order on $BS(m,n)$ by $x\leq y$ if either $x=y$ or $\alpha(x)<\alpha(y)$. This partial order is bi-invariant. 
The question of existence of a total ordering of $BS(m,n)$, either bi-invariant or left invariant, seems interesting.
