CONGRUENCE LIFTING OF DIAGRAMS OF FINITE BOOLEAN SEMILATTICES REQUIRES LARGE CONGRUENCE VARIETIES

We construct a diagram [Formula: see text], indexed by a finite partially ordered set, of finite Boolean 〈∨, 0, 1〉-semilattices and 〈∨, 0, 1〉-embeddings, with top semilattice 24, such that for any variety V of algebras, if [Formula: see text] has a lifting, with respect to the congruence lattice functor, by algebras and homomorphisms in V, then there exists an algebra U in V such that the congruence lattice of U contains, as a 0,1-sublattice, the five-element modular nondistributive lattice M3. In particular, V has an algebra whose congruence lattice is neither join- nor meet-semidistributive Using earlier work of K. A. Kearnes and Á. Szendrei, we also deduce that V has no nontrivial congruence lattice identity. In particular, there is no functor Φ from finite Boolean semilattices and 〈∨, 0, 1〉-embeddings to lattices and lattice embeddings such that the composition Con Φ is equivalent to the identity (where Con denotes the congruence lattice functor), thus solving negatively a problem raised by P. Pudlák in 1985 about the existence of a functorial solution of the Congruence Lattice Problem.

In Search of a Pappian Lattice Identity

In [8] and subsequent papers, Jônsson (et al) developed a lattice identity which reflects precisely Desargues Law in projective geometry in that a projective geometry satisfies Desargues Law if and only if the geometry, qua lattice, satisfies this identity. This identity, appropriately called the Arguesian law, has become exceedingly important in recent investigations in the variety of modular lattices (see for example [2], [3], [9], and [12]). In this note, we supply two possible lattice identities for the Pappus' Law of projective geometry.

A Characterization of Identities Implying Congruence Modularity I

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .