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

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