Congruence lattices of free lattices in non-distributive varieties

Miroslav Ploscica (1) , Jiri Tuma (2) , Friedrich Wehrung (3)

1 Institute of Mathematics
2 MFF-UK - Department of Algebra
3 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We prove that for any free lattice F with at least $\aleph_2$ 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 Grätzer and Schmidt in 1962. This yields in turn 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.

Mots clés: Congruence lattice; congruence splitting lattice; Uniform Refinement Property; Kuratowski's Theorem; diamond; pentagon

Domaines: Mathématiques générales [math.GM]

Miroslav Ploscica, Jiri Tuma, Friedrich Wehrung. Congruence lattices of free lattices in non-distributive varieties. Colloquium Mathematicum, 1998, 76, no. 2, pp.269-278. ⟨hal-00004064⟩