 Translating back to the lattice world by using the theorem above and using a lattice-theoretical analogue of the "V(R)" construction, called the "dimension monoid", introduced by Wehrung in 1998, yields the following result.Theorem (Wehrung 2004).There exists a distributive (∨,0,1)-semilattice of cardinality ℵ1 that is not isomorphic to Conc "L", for any modular lattice "L" every finitely generated sublattice of which has finite length.