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→Congruence lattices of lattices and nonstable K-theory of von Neumann regular rings: clean up, typos fixed: well-known → well known using AWB (8097) |
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In [[mathematics]], the '''congruence lattice problem''' asks whether every algebraic [[distributive lattice]] is [[isomorphic]] to the [[congruence lattice]] of some other lattice. The problem was posed by [[Robert P. Dilworth]], and for many years it was one of the most famous and long-standing open problems in [[lattice theory]]; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most [[Aleph number|ℵ<sub>1</sub>]] [[compact element]]s, but F. Wehrung provided a counterexample for distributive lattices with ℵ<sub>2</sub> compact elements using a construction based on [[Kuratowski's free set theorem]].
==Preliminaries==
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