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Topological aspects of the Medvedev lattice

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Abstract

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.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Andrew E. M. Lewis

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Richard A. Shore

  3. Department of Mathematics and Computer Science “R. Magari”, University of Siena, 53100, Siena, Italy

    Andrea Sorbi

Authors
  1. Andrew E. M. Lewis
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  2. Richard A. Shore
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  3. Andrea Sorbi
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Correspondence to Andrea Sorbi.

Additional information

A. E. M. Lewis was supported by a Royal Society Research Fellowship.

Part of this work was carried out while the author was a GNSAGA Visiting Professor at the Department of Mathematics and Computer Science “Roberto Magari” of the University of Siena. He was also partially supported by NSF Grant DMS-0852811 and Grant 13408 from the John Templeton Foundation.

A. Sorbi wish to thank the anonymous referee whose observations and comments have helped in improving the contents of the paper.

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Lewis, A.E.M., Shore, R.A. & Sorbi, A. Topological aspects of the Medvedev lattice. Arch. Math. Logic 50, 319–340 (2011). https://doi.org/10.1007/s00153-010-0215-6

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  • DOI: https://doi.org/10.1007/s00153-010-0215-6

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