Summary: The boolean hierarchy of \(k\)-partitions over NP for \(k\geq 2\) was introduced as a generalization of the well-known boolean hierarchy of sets. The classes of this hierarchy are exactly those classes of NP-partitions which are generated by finite labeled lattices. We refine the boolean hierarchy of NP-partitions by considering partition classes which are generated by finite labeled posets. Since we cannot prove it absolutely, we collect evidence for this refined boolean hierarchy to be strict. We give an exhaustive answer to the question which relativizable inclusions between partition classes can occur depending on the relation between their defining posets. The study of the refined boolean hierarchy is closely related to the issue of whether one can reduce the number of solutions of NP problems. For finite cardinality types, assuming the extended boolean hierarchy of \(k\)-partitions over NP is strict, we get a complete characterization when such solution reductions are possible.
For the entire collection see [Zbl 0944.00070].