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Computably enumerable Turing degrees and the meet property


Authors: Benedict Durrant, Andy Lewis-Pye, Keng Meng Ng and James Riley
Journal: Proc. Amer. Math. Soc. 144 (2016), 1735-1744
MSC (2010): Primary 03D25
Published electronically: August 12, 2015
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Abstract: Working in the Turing degree structure, we show that those degrees which contain computably enumerable sets all satisfy the meet property, i.e. if $ \boldsymbol {a}$ is c.e. and $ \boldsymbol {b}<\boldsymbol {a}$, then there exists non-zero $ \boldsymbol {m}<\boldsymbol {a}$ with $ \boldsymbol {b} \wedge \boldsymbol {m}= \boldsymbol {0}$. In fact, more than this is true: $ \boldsymbol {m}$ may always be chosen to be a minimal degree. This settles a conjecture of Cooper and Epstein from the 1980s.


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Additional Information

Benedict Durrant
Affiliation: School of Mathematics, University of Leeds, Leeds, United Kingdom

Andy Lewis-Pye
Affiliation: Department of Mathematics, London School of Economics, London, United Kingdom
Email: andy@aemlewis.co.uk

Keng Meng Ng
Affiliation: School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore
Email: kmng@ntu.edu.sg

James Riley
Affiliation: School of Mathematics, University of Leeds, Leeds, United Kingdom

DOI: https://doi.org/10.1090/proc/12808
Received by editor(s): November 7, 2014
Received by editor(s) in revised form: March 24, 2015, and April 2, 2015
Published electronically: August 12, 2015
Additional Notes: The second author was supported by a Royal Society University Research Fellowship
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society