A solution to Dilworth's Congruence Lattice Problem

Abstract

We construct a distributive algebraic lattice D that is not isomorphic to the congruence lattice of any lattice. This solves a long-standing open problem, traditionally attributed to R. P. Dilworth, from the forties. The lattice D has compact top element and aleph omega+1 compact elements. Our results extend to all algebras possessing a polynomially definable structure of a join-semilattice with a largest element.

Version 1 presents a longer and slightly more general proof, based on so-called "uniform refinement properties". Version 2 presents a shorter proof. Versions 3 an 4 add a few minor improvements. Version 5 fixes a minor oversight in the proof of Theorem 7.1

References in corpus (1)

• Non-extendability of semilattice-valued measures on partially ordered sets

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• Categories of partial algebras for critical points between varieties of algebras
• Infinite combinatorial issues raised by lifting problems in universal algebra
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