The modularity conjecture holds for linear idempotent varieties

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning $n$-permutability for some $n$, and the satisfaction of nontrivial congruence identities. Our theorems require a technical result about the equational theory of linear varieties, which might be of independent interest.