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Congruence lattices of free lattices in non-distributive varieties

Miroslav Ploscica·Jiri Tuma·Friedrich Wehrung
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Abstract

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.

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Miroslav Ploscica, Jiri Tuma, Friedrich Wehrung. 2005-01-25. Congruence lattices of free lattices in non-distributive varieties. https://arxiv.org/abs/math/0501459

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