Varieties of Algebras whose Congruence Lattices Satisfy Lattice Identities

Given a variety K of algebras, among the interesting questions we can ask about the members of K is the following: does there exist a lattice identity S such that for each algebra A ε K, the congruence lattice ϴ(A) satisfies S ? This thesis deals with questions of this type. First, the thesis shows that the congruence lattices of relatively free unary algebras satisfy no nontrivial lattice identities. It is also shown that the class of congruence lattices of semi-lattices satisfies no nontrivial lattice identities. As a consequence it is shown that if K is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of K are groups with exponent dividing a fixed finite number. In particular, the congruence lattices of members of K are modular Finally, it is shown that the varieties whose congruence lattices satisfy one of a class of lattice identities of a fairly general form are in fact congruence modular.