On the use of majority for investigating primeness of 3-permutability

We have recently published a result that [Formula: see text]-permutability is not join-prime in the lattice of interpretability types of varieties whenever [Formula: see text]. In the proof, we showed that if [Formula: see text], then the join of a properly chosen finitely generated non-[Formula: see text]-permutable variety and the variety [Formula: see text] defined by the majority identities is [Formula: see text]-permutable. In the present note, we prove that the join of any locally finite non-3-permutable variety with [Formula: see text] is non-3-permutable. We also prove that the join of any non-2-permutable variety with [Formula: see text] is non-2-permutable. Our non-3-permutable result gives that one has to use a nonlocally finite non-3-permutable variety [Formula: see text] if they want to prove that 3-permutability is not join-prime by arguing that [Formula: see text] is 3-permutable.