On Some Identifies Valid in Modular Congruence Varieties

Freese and J6nsson [8] showed that the congruence lattice of a (universal) algebra in a congruence modular variety is always arguesian. On the other hand J6nsson [16] constructed arguesian lattices which cannot be embedded into the normal subgroup lattice of a group. These lattices consist of two arguesian planes of different prime order glued together over a two element sublattice (cf. Dilworth and Hall [3]). In [11], Herrmann and Poguntke derived identities not valid in those lattices but valid in all lattices of normal subgroups. In the present paper we show that these (and more general) identities hold in the congruence lattice of any algebra in a congruence modular variety. This implies, in particular, that the class of arguesian tattices does not form a congruence variety in the sense of J6nsson [17]. (This result has been proved by the first author and announced in [7]). Moreover, one concludes as in [11] that a modular congruence variety cannot be defined by finitely many identities provided it contains the rational projective plane or two projective planes of distinct prime orders or a subgroup lattice of a group C~:. 1. Definitions and main results For subgroups the verification of the lattice identities to be constructed reduces to the trivial observation that isomorphic abelian quotients have the same exponent. Consequently, we introduce "projective" lattice relations which yield for certain quotients: I isomorphy II "coordinate systems" allowing one to speak about "exponents". Ad L Projective quotients. If a and b are elements of a modular lattice such that a>-b then we write a/b={xl a>-x>-b}. We write a/b/~c/d as well as