Simultaneous representations of semilattices by lattices with permutable congruences
Résumé
The Congruence Lattice Problem (CLP), stated by R. P. Dilworth in the forties, asks whether every distributive {∨, 0}-semilatticeS 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 $|S|\leq\aleph_1$. Furthermore, one can then take L with permutable congruences. This contrasts with the case where $|S| \geq\aleph_2$, 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, $\ga\jj\gb=\ga\gb\uu\gb\ga$, for all congruences a and b. 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.