I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice structure of $\mathbb{Z}^n$. So if I want noetherianness AND the lattice structure, a natural place to look is polycyclic $\ell$-groups. My issue is that I can't find any (other than $\mathbb{Z}^n$) that have nonlinear orders.
I am far from an expert in this area, but I have just spent a fair bit of time working with groups of the form $\mathbb{Z}\ltimes\mathbb{Z}^n$ using the map $z\mapsto A$ where $A\in SL(n,\mathbb{Z})$ with all positive eigenvalues. This group was suggested by a colleague as it is known to be linearly ordered in a natural way, but that natural way breaks down for any nonlinear ordering I have tried.
I have also looked into the possibility of finitely generated abelian groups, as they are polycyclic. This breaks down almost immediately, though, as $\ell$-groups that are finitely generated AS $\ell$-groups are rarely finitely generated as abelian groups, too.
Chapter 2, exercise 11 of Steinberg's book Lattice-Ordered Rings and Modules gives $G=(\mathbb{Z}\oplus\mathbb{Z})\ltimes_{\phi} \mathbb{Z}$ with $\phi(1)$ being the automorphism $\tau(x,y)=(y,x)$ as an example of an $\ell$-group. Since $G$ is the semidirect product of polycyclic groups, it is polycyclic itself. Thus, $G$ is an example of a polycyclic $\ell$-group that is not a linear order.