A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees
S. K. Thomason · Canadian Journal of Mathematics · 1969
In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem. THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees. For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j: 1 2 Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.