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