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Semilattices of finitely generated ideals of exchange rings with finite stable rank. (English) Zbl 1034.06007
The author gives an example of a distributive \((\vee, 0,1)\)-semilattice \(S_{w_1}\) 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 (the origin of this work is in the representation problem of distributive \((\vee,0)\)-semilattices as semilattices of compact ideals of von Neumann regular rings). So the author obtains solutions to various open problems in ring theory and lattice theory, such as:
– 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 \(S_{w_1}\).
– There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to \(S_{w_1}\).
These results are established by constructing an infinitary statement 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 \(S_{w_1}\).

MSC:
06A12 Semilattices
20M14 Commutative semigroups
06B10 Lattice ideals, congruence relations
19K14 \(K_0\) as an ordered group, traces
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