Characterizing the Join-Irreducible Medvedev Degrees

Abstract
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
Keywords Medvedev degrees   lattices   Brouwer algebras   intermediate logics
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Reprint years 2011
DOI 10.1215/00294527-2010-034
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2-Minimality, Jump Classes and a Note on Natural Definability.Mingzhong Cai - 2014 - Annals of Pure and Applied Logic 165 (2):724-741.
Coding True Arithmetic in the Medvedev Degrees of Classes.Paul Shafer - 2012 - Annals of Pure and Applied Logic 163 (3):321-337.

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