Embedding $FD(ω)$ into $\mathcal{P}_s$ densely
Joshua A. Cole · arXiv (Cornell University) · 2007
Let $\mathcal{P}_s$ be the lattice of degrees of non-empty $Π_1^0$ subsets of $2^ω$ 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}