# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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 propertyHTML articles powered by AMS MathViewer

by Andrew E. M. Lewis
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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• 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