Abstract. We find a distributive (∨, 0, 1)-semilattice Sω 1 of size ℵ 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 Sω 1 . -There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to Sω 1 . 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 Sω 1 .