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 satisfies the *join* property if, for every non-zero , there exists with . It was observed by Downey, Greenberg, Lewis and Montalbán that all degrees which are non-GL 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 satisfies the cupping property if, for every , there exists with . 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 fail to satisfy join and, moreover, that the non-zero degree below 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

Email:
andy@aemlewis.co.uk

DOI:
https://doi.org/10.1090/S0002-9939-2011-10908-0

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.