n-Permutability is not join-prime for n ≥ 5.

Let V-P be the variety generated by an order primal algebra of finite signature associated with a finite bounded poset P that admits a near-unanimity operation. Let. be a finite set of linear identities that does not interpret in V-P. Let V-Lambda be the variety defined by Lambda. We prove that V-P boolean OR V-Lambda is n-permutable for some n. This implies that there is an n such that n-permutability is not join-prime in the lattice of interpretability types of varieties. In fact, it follows that n-permutability where n runs through the integers greater than 1 is not prime in the lattice of interpretability types of varieties.We strengthen this result by making P and Lambda more special. We let P be the 6-element bounded poset that is not a lattice and V-m the variety defined by the set of majority identities for a ternary operational symbol m. We prove in this case that V-P boolean OR V-m is 5-permutable. This implies that n-permutability is not join-prime in the lattice of interpretability types of varieties whenever n >= 5. We also provide an example demonstrating that V-P boolean OR V-m is not 4-permutable.