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

## Abstract

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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Official URL: | http://www.ams.org/publications/journals/journalsf... |

Additional Information: | © 2012 American Mathematical Society |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 06 Aug 2013 11:31 |

Last Modified: | 20 Jan 2020 04:46 |

URI: | http://eprints.lse.ac.uk/id/eprint/51445 |

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