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 $zx \le  zy$.


From Wikipedia:
A group $G$ is a partially ordered group if and only if there exists a subset $H\subset G$ such that:
i) $1 \in H$
ii) If $a ∈ H$ and $b ∈ H$ then $ab ∈ H$
iii) If $a ∈ H$ then $x^{-1}ax ∈ H$ for each $x\in G$
iv) If $a ∈ H$ and $a^{-1} ∈ H$ then $a=0$

For every $n,m$ can you find such a subset? Here's one thought: let $n=m=1$. Let $H=\{a^n:n\in\mathbb N\}$, where we include $a^0=1$. I believe this satisfies all the conditions and thus $G=BS(1,1)$ (which is the free abelian group on two generators) has a partial ordering.
