Congruence lattices of free lattices in non-distributive varieties

Abstract

We prove that for any free lattice F with at least ℵ 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.

A survey of recent results on congruence lattices of lattices

“…It is proved in M. Ploščica, J. Tůma, and F. Wehrung [38] that for any nondistributive variety V of lattices and any set X with at least ℵ 2 elements, the congruence lattice of the free lattice F V (X) satisfies many negative properties with respect to CLP, see Theorem 2.6; in particular, its semilattice of compact elements is not representable via Schmidt's Lemma (see E. T. Schmidt [41], Proposition 2.5, and Theorem 6.6) and it is not isomorphic to Con c L, for any sectionally complemented lattice L. A variety of lattices is nondistributive iff it contains as an element either the diamond M 3 or the pentagon N 5 . As a surprising consequence, even the very "simple" finitely generated lattice varieties M 3 and N 5 have complicated congruence classes (see Definition 8.1), not completely understood yet.…”

Regular congruence-preserving extensions of lattices

“…It is natural to ask: which lattices have a congruence-preserving embedding into a relatively complemented lattice? M. Ploščica, J. Tůma, and F. Wehrung [6] proved that not every lattice admits such an embedding. G. Grätzer, H. Lakser, and F. Wehrung [2] proved that if the congruence lattice is finite, then there is such an embedding.…”

Gamma invariants for dense lattices

“…So A$ i B m M p i (K i ) and B $ j B n M q j (L j ) as F-algebras, where all m, n, p i , q j are finite, and K i , L j are finite fields containing F. Then It has been shown recently ([17, [11]) that there are elements of L which are not realized as L 2 (R) for any regular ring R. For lattices in L with Ò 1 many compact elements the question remains open whether they are always realized as L 2 (R).…”