Geodesic
A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 147-148

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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, a 0, a 1,...}, 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 A 0, A 1, ... 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. 
Thomason, S. K. A Note on Non-Distributive Sublattices of Degrees and Hyperdegrees. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 147-148. doi: 10.4153/CJM-1969-013-2
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[1] 1. Thomason, S. K., The forcing method and the upper semi-lattice of hyperdegrees, Trans. Amer. Math. Soc. 129 (1967), 38–57. Google Scholar

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