  We find a distributive (v, 0, 1)-semilattice S of size $ aleph\_1$ that is
not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed
with an order-unit of finite stable rank. We thus obtain solutions to various
open problems in ring theory and in lattice theory. In particular: - There is
no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real
rank zero) with finite stable rank whose semilattice of finitely generated,
idempotent-generated two-sided ideals is isomorphic to S. - There is no locally
finite, modular lattice whose semilattice of finitely generated congruences is
isomorphic to S. These results are established by constructing an infinitary
statement, denoted here by URPsr, that holds in the maximal semilattice
quotient of every Riesz monoid endowed with an order-unit of finite stable
rank, but not in the semilattice S.
