Generating primitive positive clones
Suppose \( \cal F \) is a set of operations on a finite set A. Define PPC(\(\cal F \)) to be the smallest primitive positive clone on A containing \( \cal F \). For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo n A). S. Burris and R. Willard  conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)2 so that these results are not vacuous.
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