It is given in [Regular left-orders on groups][1] that the solvable Baumslag-Solitar group $\mathrm{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 \mathrm{BS}(1,n) $ such that if $a<b$ implies $\phi(a)<\phi(b),$ where $\mathbb{Z}[1/n]\rtimes \mathbb{Z}$.


  [1]: https://arxiv.org/abs/2104.04475