  Despite being a vast generalization of Garside groups, right $\ell$-groups
with noetherian lattice structure and strong order unit share a lot of the
properties of Garside groups. In the present work, we prove that every modular
noetherian right $\ell$-group with strong order unit decomposes as a direct
product of beams, which are sublattices that correspond to the directly
indecomposable factors of the strong order interval. Furthermore, we show that
the beams of dimension $\delta \geq 4$ can be coordinatized by the $R$-lattices
in $Q^{\delta}$, where $Q$ is a noncommutative discrete valuation field with
valuation ring $R$. In particular, this gives a precise description of a very
big family of modular Garside groups.
