Decomposition and Structure theorems for Garside-like groups with modular lattice structure

Despite being a vast generalization of Garside groups, right $\mathrm{\ell }$-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 $\mathrm{\ell }$-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 $\delta \ge 4$ 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.