2006/06/01 by Jiří Tůma, Friedrich Wehrung · 9 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #semigroups and automata theory #Rough Sets and Fuzzy Logic #Mathematics #Semilattice #Functor #Congruence lattice problem #Lattice (music) #Pure mathematics #Homomorphism #Congruence (geometry) #Congruence relation #Combinatorics #Discrete mathematics #Distributive lattice #Geometry
paper · doi:10.1142/s0218196706003049
published in International Journal of Algebra and Computation 16(03), 541-550 (World Scientific)
openalex publication_date 2006/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We construct a diagram [Formula: see text], indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 2 4 , such that for any variety V of algebras, if [Formula: see text] has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M 3 . In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.