Mathematical problems | Lattice theory

Congruence lattice problem

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. (Wikipedia).

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Related pages

Ideal (order theory) | Compact element | Module (mathematics) | Resolution (logic) | Vector space | Semilattice | Partially ordered set | Complemented lattice | Ideal (ring theory) | Robert P. Dilworth | Lattice (order) | Modular lattice | Group (mathematics) | Distributivity (order theory) | Distributive homomorphism | Congruence relation | Universal algebra | Distributive lattice | Direct limit | Monoid | Congruence-permutable algebra | Diagram (category theory) | Mathematics | Locally finite group | Von Neumann regular ring | Field (mathematics) | Equivalence of categories | Refinement monoid | Normal subgroup | Ring (mathematics) | Aleph number | Subset | Limit (category theory) | Signature (logic) | Magma (algebra) | Cardinality | Maximal semilattice quotient | Kuratowski's free set theorem | Boolean algebra (structure)