1999/01/01 by Friedrich Wehrung · 40 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #Rings, Modules, and Algebras #semigroups and automata theory #Congruence lattice problem #Lattice (music) #Distributive property #Mathematics #Distributive lattice #Congruence (geometry) #Pure mathematics #Algebraic number #Combinatorics #Discrete mathematics #Mathematical analysis #Physics #Geometry
paper · pdf · doi:10.1090/s0002-9939-99-04558-x
published in Proceedings of the American Mathematical Society 127(2), 363-370 (American Mathematical Society)
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
The Congruence Lattice Problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of a lattice. It was hoped that a positive solution would follow from E. T. Schmidtâs construction or from the approach of P. Pudlák, M. Tischendorf, and J. Tůma. In a previous paper, we constructed a distributive algebraic lattice A with ℵ 2 compact elements that cannot be obtained by Schmidtâs construction. In this paper, we show that the same lattice A cannot be obtained using the Pudlák, Tischendorf, Tůma approach. The basic idea is that every congruence lattice arising from either method satisfies the Uniform Refinement Property, that is not satisfied by our example. This yields, in turn, corresponding negative results about congruence lattices of sectionally complemented lattices and two-sided ideals of von Neumann regular rings.