A Characterization of Identities Implying Congruence Modularity I
Alan Day, Ralph S. Freese · Canadian Journal of Mathematics · 1980
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 .