Semilattices of finitely generated ideals of exchange rings with finite stable rank

We find a distributive -semilattice of size 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:

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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  .

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There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to .

These results are established by constructing an infinitary statement, denoted here by , 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  .