Finitely Based Modular Congruence Varieties Are Distributive

R. Dedekind introduced the modular law, a lattice equation true in most of the lattices associated with classical algebraic systems, see [4]. Although this law is one of the most important tools for working with these lattices, it does not fully describe the equational properties of these lattices. This was made clear in [8] where B. Jónsson and the author showed that if any modular congruence variety actually satisfies the (stronger) Arguesian law. (A congruence variety is the variety generated by all the congruence lattices of the members of a variety of algebras.) In this note we show that no finite set of lattice equations is strong enough to describe the equational properties of the lattices associated with classical algebraic systems in the following strong sense: there is no modular, nondistributive congruence variety which has a finite basis for its equational theory. This question was posed by George McNulty in the problem session on lattice theory at the Jónsson symposium. Jónsson had asked a similar question in [17]. He asked, in Problem 9.12, whether there is a nontrivial variety whose congruence variety is neither the variety of all lattices nor the variety of all distributive lattices, but which is finitely based. Some partial results on this problem had been obtained previously. Let Vn denote the class of lattices embeddable into the lattice of subgroups of an abelian group of exponent n and let V0 be those lattices embeddable into the lattice of subspaces of a rational vector space. Thus HVn is the congruence variety associated with the variety of abelian groups of exponent n. In [7] it was shown that if L is a class of lattices contained in a modular congruence variety and containing Vn for some n not a prime, then L cannot be defined by finitely many first order axioms. It was also shown that every modular, nondistributive congruence variety contains HVn for some n. Mark Haiman, in his proof that there are Arguesian lattices which are not representable as lattices of permuting equivalence relations (solving an old problem of Jónsson), shows that HVp is not finitely based for any prime p, [11] and [12].