Derivatives and the "Modularity Conjecture"

In 1984, Garcia and Taylor formulated the "Modularity Conjecture", which states that the join of two non-congurence modular varieties is non-congruence modular. We show that the conjecture holds for the special case of linear idempotent varieties. Our proofs rely on Dent, Kearnes, and Szendrei´s recently introduced concept of the derivative of an equational theory. Similar consequences have been obtained using another version of the derivative due to Freese. We will give an overview of the notions and of our results. This is joint work with Luis Sequeira.