Abstract
Let \({\mathcal{P}_s}\) be the lattice of degrees of non-empty \({\Pi_1^0}\) subsets of 2ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in \({\mathcal{P}_s}\) . Cenzer and Hinman proved that \({\mathcal{P}_s}\) is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any \({\mathcal{U} < _s \mathcal{V}}\) , we can lattice embed FD(ω) into \({\mathcal{P}_s}\) strictly between \({deg_s(\mathcal{U})}\) and \({deg_s({\mathcal V})}\) . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made.
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Thanks to my adviser Peter Cholak for his guidance in my research. I also wish to thank the anonymous referee for helpful comments and suggestions. My research was partially supported by NSF grants DMS-0245167 and RTG-0353748 and a Schmitt Fellowship at the University of Notre Dame.
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Cole, J.A. Embedding FD(ω) into \({\mathcal{P}_s}\) densely. Arch. Math. Logic 46, 649–664 (2008). https://doi.org/10.1007/s00153-007-0062-2
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DOI: https://doi.org/10.1007/s00153-007-0062-2