  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 $W_{t}$. A simple
criterion is given to recognize when two such sequences classify isomorphic
unital $\ell$-groups. Many properties of the unital $\ell$-group $(G,u)$ can be
directly read off from its associated sequence: for instance, the properties of
being totally ordered, archimedean, finitely presented, simplicial, free.
