We affirm a conjecture of Sacks  by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, 𝒟h. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of 𝒟h. Corollaries include the decidability of the two quantifier theory of 𝒟h and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of ω₁CK. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of 𝒟h.
"Lattice initial segments of the hyperdegrees." J. Symbolic Logic 75 (1) 103 - 130, March 2010. https://doi.org/10.2178/jsl/1264433911