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.
We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results.
The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilattice S is isomorphic to the semilattice Conc L of compact congruences of a lattice L.While this problem is still open, many partial solutions have been obtained, positive and negative as well. The solution to CLP is known to be positive for all S such that |S| ≤ ℵ 1 . Furthermore, one can then take L with permutable congruences. This contrasts with the case where |S| ≥ ℵ 2 , where there are counterexamples S for which L cannot be, for example, sectionally complemented. We prove in this paper that the lattices of these counterexamples cannot have permutable congruences as well.We also isolate finite, combinatorial analogues of these results. All the "finite" statements that we obtain are amalgamation properties of the Conc functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square 2 2 can be lifted with respect to the Conc functor.We prove that the latter results cannot be extended to the cube, 2 3 . In particular, we give an example of a cube diagram of finite Boolean semilattices and semilattice embeddings that cannot be lifted, with respect to the Conc functor, by lattices with permutable congruences.We also extend many of our results to lattices with almost permutable congruences, that is, α ∨ β = αβ ∪ βα, for all congruences α and β.We conclude the paper with a very short proof that no functor from finite Boolean semilattices to lattices can lift the Conc functor on finite Boolean semilattices.
We prove that every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ 1 bound is optimal, as we find a distributive algebraic lattice D with ℵ 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E.T. Schmidt from 1969. Furthermore, D may be taken as the congruence lattice of the free bounded lattice on ℵ 2 generators in any non-distributive lattice variety.Some of our results are obtained via a functorial approach of the semilattice-valued 'distances' used by B. Jónsson in his proof of Whitman's Embedding Theorem. In particular, the semilattice of compact elements of D is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive ∨, 0 -semilattice is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.
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