Archive for Mathematical Logic

, Volume 50, Issue 3–4, pp 319–340 | Cite as

Topological aspects of the Medvedev lattice

  • Andrew E. M. Lewis
  • Richard A. Shore
  • Andrea SorbiEmail author
Original Article


We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.


Medvedev reducibility Baire space Cantor space 

Mathematics Subject Classification (2000)



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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Andrew E. M. Lewis
    • 1
  • Richard A. Shore
    • 2
  • Andrea Sorbi
    • 3
    Email author
  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of Mathematics and Computer Science “R. Magari”University of SienaSienaItaly

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