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

We have recently published a result that n-permutability is not joinprime in the lattice of interpretability types of varieties whenever n ≥ 5. In the proof we showed that a properly chosen finitely generated non-5-permutable variety does not interpret in any non-5-permutable majority variety. In the present note we prove that every locally finite non-3-permutable variety interprets in a non-3-permutable majority variety. We also prove that every non-2-permutable variety interprets in a non-2-permutable majority variety. Our non-3-permutable result gives that one has to use a non-locally finite non-3-permutable variety if they want to prove by the use of the majority identities that 3-permutability is not join-prime in the lattice of interpretability types of varieties.