Characterizing the Join-Irreducible Medvedev Degrees

Notre Dame Journal of Formal Logic 52 (1):21-38 (2011)
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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

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Citations of this work

Topological aspects of the Medvedev lattice.Andrew Em Lewis, Richard A. Shore & Andrea Sorbi - 2011 - Archive for Mathematical Logic 50 (3-4):319-340.
Coding true arithmetic in the Medvedev and Muchnik degrees.Paul Shafer - 2011 - Journal of Symbolic Logic 76 (1):267 - 288.
Natural factors of the Medvedev lattice capturing IPC.Rutger Kuyper - 2014 - Archive for Mathematical Logic 53 (7-8):865-879.
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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References found in this work

Some remarks on the algebraic structure of the Medvedev lattice.Andrea Sorbi - 1990 - Journal of Symbolic Logic 55 (2):831-853.
Embedding Brouwer algebras in the Medvedev lattice.Andrea Sorbi - 1991 - Notre Dame Journal of Formal Logic 32 (2):266-275.
Intermediate logics and factors of the Medvedev lattice.Andrea Sorbi & Sebastiaan A. Terwijn - 2008 - Annals of Pure and Applied Logic 155 (2):69-85.
Topological aspects of the Medvedev lattice.Andrew Em Lewis, Richard A. Shore & Andrea Sorbi - 2011 - Archive for Mathematical Logic 50 (3-4):319-340.
Some Quotient Lattices of the Medvedev Lattice.Andrea Sorbi - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (9-12):167-182.

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