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Extensions of embeddings below computably enumerable degrees


Authors: Rod Downey, Noam Greenberg, Andrew Lewis and Antonio Montalbán
Journal: Trans. Amer. Math. Soc. 365 (2013), 2977-3018
MSC (2010): Primary 03D28; Secondary 03D25
DOI: https://doi.org/10.1090/S0002-9947-2012-05660-1
Published electronically: December 13, 2012
MathSciNet review: 3034456
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Abstract: Toward establishing the decidability of the two-quantifier theory of the $ \Delta ^0_2$ Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular, the degree of the halting set $ \boldsymbol {0}'$. We obtain a good deal of sufficient and necessary conditions.


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Additional Information

Rod Downey
Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University, P.O. Box 600, Wellington, New Zealand
Email: Rod.Downey@msor.vuw.ac.nz

Noam Greenberg
Affiliation: School of Mathematics, Statistics and Computer Science, Victoria University. P.O. Box 600, Wellington, New Zealand
Email: greenberg@msor.vuw.ac.nz

Andrew Lewis
Affiliation: School of Mathematics, University of Leeds, Leeds, United Kingdom
Email: andy@aemlewis.co.uk

Antonio Montalbán
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637
Email: antonio@math.uchicago.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05660-1
Keywords: Turing degrees, decidability, extensions of embeddings
Received by editor(s): December 9, 2009
Received by editor(s) in revised form: June 30, 2011
Published electronically: December 13, 2012
Additional Notes: The first two authors were supported by the Marsden Fund of New Zealand.
The third author was supported by a Royal Society University Research Fellowship.
The fourth author was partially supported by NSF Grant DMS-0600824, and by the Marsden Fund of New Zealand via a postdoctoral fellowship.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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