A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a
translation-invariant lattice-order and a distinguished element $u$, called
order-unit, whose positive integer multiples eventually dominate each element
of $G$. We classify finitely generated unital $\ell$-groups by sequences
$\mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes,
where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander
binary stellar operation, or by deleting a maximal simplex of $W_{t}$.