It is given in [Regular left-order in groups][1] that Solvable Baum-Slag Solitar Group $BS(1,n)=\langle a, b\mid aba^{-1}=b^n\rangle $ is isomorphic to  $\mathbb{Z}[1/n]\rtimes \mathbb{Z}$ for $n\in \mathbb{Z}.$ Can we find an isomorphism $\phi:\mathbb{Z}[1/n]\rtimes \mathbb{Z}\rightarrow BS(1,n) $ such that if $a<b$ implies $\phi(a)<\phi(b),$ where $\mathbb{Z}[1/n]\rtimes \mathbb{Z}.$


  [1]: https://www.google.com/search?q=Regular%20left-orders%20on%20groups&rlz=1C1JJTC_enIN1108IN1108&oq=Regular%20left-orders%20on%20groups&gs_lcrp=EgZjaHJvbWUqBggAEEUYOzIGCAAQRRg7MgYIARBFGD3SAQgxODQ4ajBqN6gCALACAA&sourceid=chrome&ie=UTF-8