The nondeterministic advice complexity of the P-selective sets is known to be exactly linear.Regarding the deterministic advice complexity of the P-selective sets-i.e., the amount of KarpLipton advice needed for polynomial-time machines to recognize them in general-the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96].We prove that every associatively P-selective set is commutatively, associatively P-selective. Using this, we establish an algebraic sufficient condition for the P-selective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all P-selective sets are associatively P-selective then the deterministic advice complexity of the P-selective sets is linear. The weakest previously known sufficient condition was P = NP.We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.

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Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k > 2, if PWe extend this to obtain a more general downward collapse result.