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A note on the join property

Author: Andrew E. M. Lewis
Journal: Proc. Amer. Math. Soc. 140 (2012), 707-714
MSC (2010): Primary 03D28; Secondary 03D10
Published electronically: June 6, 2011
MathSciNet review: 2846340
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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

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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.