Congruence lattices of semilattices

The main result of this paper is that the class of con-gruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if 5^ * is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of 5^ " are groups with exponent dividing a fixed finite number. Given a variety (equational class) J ^ of algebras, among the inter-esting questions we can ask about the members of SίΓ is the following: does there exist a lattice identity δ such that for each algebra A e S?~, the congruence lattice Θ(A) satisfies S? In the case that 5ίΓ has dis-tributive congruences, many strong conclusions can be drawn about the algebras of J%Γ [1, 2, 7]. In the case that 3ίΓ has permutable con-gruences or modular congruences, there is reason to hope that som