Distributive congruence lattices of congruence-permutable algebras

Abstract

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.