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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A note on the join property
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by Andrew E. M. Lewis PDF
Proc. Amer. Math. Soc. 140 (2012), 707-714 Request permission

Abstract:

A Turing degree $\boldsymbol {a}$ satisfies the join property if, for every non-zero $\boldsymbol {b}<\boldsymbol {a}$, there exists $\boldsymbol {c}<\boldsymbol {a}$ with $\boldsymbol {b} \vee \boldsymbol {c}= \boldsymbol {a}$. It was observed by Downey, Greenberg, Lewis and Montalbán that all degrees which are non-GL$_2$ satisfy the join property. This, however, leaves open many questions. Do all a.n.r. degrees satisfy the join property? What about the PA degrees or the Martin-Löf random degrees? A degree $\boldsymbol {b}$ satisfies the cupping property if, for every $\boldsymbol {a}>\boldsymbol {b}$, there exists $\boldsymbol {c}<\boldsymbol {a}$ with $\boldsymbol {b}\vee \boldsymbol {c}=\boldsymbol {a}$. Is satisfying the cupping property equivalent to all degrees above satisfying join? We answer all of these questions by showing that above every low degree there is a low degree which does not satisfy join. We show, in fact, that all low fixed point free degrees $\boldsymbol {a}$ fail to satisfy join and, moreover, that the non-zero degree below $\boldsymbol {a}$ without any joining partner can be chosen to be a c.e. degree.
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Additional Information
  • Andrew E. M. Lewis
  • Affiliation: Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
  • MR Author ID: 748032
  • Email: andy@aemlewis.co.uk
  • Received by editor(s): June 28, 2009
  • Received by editor(s) in revised form: July 15, 2009, August 12, 2010, and November 21, 2010
  • Published electronically: June 6, 2011
  • Additional Notes: The author was supported by a Royal Society University Research Fellowship
  • Communicated by: Julia Knight
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 707-714
  • MSC (2010): Primary 03D28; Secondary 03D10
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10908-0
  • MathSciNet review: 2846340