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A note on the join property

Lewis-Pye, Andrew (2012) A note on the join property. Proceedings of the American Mathematical Society, 140 (2). pp. 707-714. ISSN 0002-9939

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Identification Number: 10.1090/S0002-9939-2011-10908-0


A Turing degree a satisfies the join property if, for every non-zero b<a, there exists c<a with b V c = a. It was observed by Downey, Greenberg, Lewis and Montalbán that all degrees which are non-GL2 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 b satisfies the cupping property if, for every a>b, there exists c<a with b V c = 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 a fail to satisfy join and, moreover, that the non-zero degree below a without any joining partner can be chosen to be a c.e. degree.

Item Type: Article
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Additional Information: © 2012 American Mathematical Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 06 Aug 2013 11:31
Last Modified: 20 Jul 2021 00:14

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