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 ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
Read more about Congruence Lattice Problem: Preliminaries, Semilattice Formulation of CLP, Schmidt's Approach Via Distributive Join-homomorphisms, Pudlák's Approach; Lifting Diagrams of (∨,0)-semilattices, Congruence Lattices of Lattices and Nonstable K-theory of Von Neumann Regular Rings, A First Application of Kuratowski's Free Set Theorem, Solving CLP: The Erosion Lemma, A Positive Representation Result For Distributive Semilattices
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