We characterize the join-irreducible Medvedev degrees as the degrees of complements of Turing ideals, thereby solving a problem posed by Sorbi. We use this characterization to prove that there are Medvedev degrees above the second-least degree that do not bound any join-irreducible degrees above this second-least degree. This solves a problem posed by Sorbi and Terwijn. Finally, we prove that the filter generated by the degrees of closed sets is not prime. This solves a problem posed by Bianchini and Sorbi.
"Characterizing the Join-Irreducible Medvedev Degrees." Notre Dame J. Formal Logic 52 (1) 21 - 38, 2011. https://doi.org/10.1215/00294527-2010-034