2007/08/30 by Joshua A. Cole, Cole, Joshua A.
Computer Science · Mathematics · #03D30 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03D30 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.0708.4215
14 pages
arxiv created 2007/08/30 · openalex publication_date 2007/08/30 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Let Ps be the lattice of degrees of non-empty Π10 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 Ps. Cenzer and Hinman proved that Ps 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 U <s V, we can lattice embed FD(ω) into Ps strictly between degs(U) and degs(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.