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

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

Maximal Pre-Primal Clusters

A number of unsolved problems of primal algebra theory concern the existence of certain collections of dependent primal algebras. In [3] E. S. O'Keefe showed that any collection of pairwise non-isomorphic primal algebras of type {n} with n > 1 forms a primal cluster. Recently the author has discovered that if τ is any type containing at least two elements, one of which is > 1, then there are at least two non-isomorphic dependent primal algebras of type τ, except possibly in the case = {2, 0}; this result will appear later.