Varieties of Algebras and their Congruence Varieties

1. Congruence lattices. G. Birkhoff and O. Frink noted that the congruence lattice Con04) of an algebra A (with operations of finite rank) is algebraic or compactly generated. The celebrated Grätzer-Schmidt theorem states that, conversely, every algebraic lattice is isomorphic to Con(A) for some algebra A. The importance of this is obvious, for it shows that unless something more is known about the algebra A9 nothing can be said about Con(A) that does not follow from the fact that it is algebraic. This, however, leaves open the question of what happens when additional conditions are imposed on A. The most obvious question is for what similarity types %9 if any, it is true that every algebraic lattice is isomorphic to Con(>4) for some T-algebra A. It is easy to see that no preassigned number of unary operations will suffice, but other than that the problem is completely open. In particular, as several people have observed, it is not even known whether every algebraic lattice is isomorphic to the congruence lattice of some groupoid, Little progress has also been made on the problem of characterizing the congruence lattices of the members of various familiar varieties, or equational classes, of algebras, e.g., it seems unlikely that every algebraic lattice is isomorphic to the congruence lattice of some semigroup, but no counterexample is known. (However, an unpublished example by R. N. McKenzie shows commutative semigroups will not suffice.) The congruence lattices of groups or, equivalently, the lattices of normal subgroups, are known to be modular and to satisfy certain even stronger identities, but there is little hope of characterizing these lattices. The congruence lattices of lattices are distributive (N. Funayama and T. Nakayama), but it is an open question whether every distributive algebraic lattice is isomorphic to the congruence lattice of some lattice, The situation is no better concerning most other varieties, e.g., various subvarieties of the above three. Among the exceptions are