Congruence lattices of free lattices in non-distributive varieties

1998/01/01 by Miroslav Ploščica, Jiří Tůma, Friedrich Wehrung · 24 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #Semilattice #Distributive property #Congruence lattice problem #Mathematics #Distributive lattice #Lattice (music) #Congruence (geometry) #Pure mathematics #Von Neumann architecture #Complete lattice #Combinatorics #Universality (dynamical systems) #Physics #Geometry

paper · pdf · doi:10.4064/cm-76-2-269-278

published in Colloquium Mathematicum 76(2), 269-278 (Polish Academy of Sciences)

openalex publication_date 1998/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We prove that for any free lattice F with at least 2 generators in any non-distributive variety of lattices, there exists no sectionally complemented lattice L with congruence lattice isomorphic to the one of F . This solves a question formulated by Grtzer and Schmidt in 1962. This in turn yields further examples of simply constructed distributive semilattices that are not isomorphic to the semilattice of finitely generated two-sided ideals in any von Neumann regular ring.

Citations

• Sur une caractérisation des alephs
• Non-measurability properties of interpolation vector spaces
• On congruence lattices of lattices

Cited by

• Free trees and the optimal bound in Wehrung's theorem
• Congruence lattice problem [wikipedia]