Distributive congruence lattices of congruence-permutable algebras

Abstract

We prove that every distributive algebraic lattice with at most $\aleph\_1$ compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The $\aleph\_1$ bound is optimal, as we find a distributive algebraic lattice $D$ with $\aleph\_2$ 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 $\aleph\_2$ 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. Jonsson 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 join-semilattice with zero is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.