SIMULTANEOUS REPRESENTATIONS OF SEMILATTICES BY LATTICES WITH PERMUTABLE CONGRUENCES

Abstract

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 Con c 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 Lwith permutable congruences. This contrasts with the case where |S|≥ℵ 2 , where there are counterexamples S for which Lcannot 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 Con c functor. The strongest known positive results, which originate in earlier work by the first author, imply that many diagrams of semilattices indexed by the square2 2 can be lifted with respect to the Con c 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 Con c 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 Con c functor on finite Boolean semilattices.