Distributive Congruence Lattices of Congruence-permutable Algebras

We prove that every distributive algebraic lattice with at most א1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The א1 bound is optimal, as we find a distributive algebraic lattice D with א2 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 א2 generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilatticevalued ‘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.