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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Defining totality in the enumeration degrees
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by Mingzhong Cai, Hristo A. Ganchev, Steffen Lempp, Joseph S. Miller and Mariya I. Soskova PDF
J. Amer. Math. Soc. 29 (2016), 1051-1067 Request permission

Abstract:

We show that if $A$ and $B$ form a nontrivial $\mathcal {K}$-pair, then there is a semi-computable set $C$ such that $A\leq _e C$ and $B\leq _e \overline {C}$. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal $\mathcal {K}$-pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in” relation for total degrees in the enumeration degrees.
References
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Additional Information
  • Mingzhong Cai
  • Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
  • MR Author ID: 816369
  • Email: Mingzhong.Cai@dartmouth.edu
  • Hristo A. Ganchev
  • Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
  • MR Author ID: 873534
  • Email: ganchev@fmi.uni-sofia.bg
  • Steffen Lempp
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
  • MR Author ID: 247988
  • Email: lempp@math.wisc.edu
  • Joseph S. Miller
  • Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
  • MR Author ID: 735102
  • Email: jmiller@math.wisc.edu
  • Mariya I. Soskova
  • Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
  • MR Author ID: 802392
  • Email: msoskova@fmi.uni-sofia.bg
  • Received by editor(s): January 27, 2014
  • Received by editor(s) in revised form: May 13, 2015, and August 26, 2015
  • Published electronically: November 2, 2015
  • Additional Notes: The first author was partially supported by NSF Grants DMS-1266214 and DMS-1458061.
    The second and fifth authors were also partially supported by BNSF Grant No. DMU 03/07/12.12.2011 and by NSF Binational Grant DMS-1101123 from the United States, Russia, Kazakhstan, and Bulgaria.
    The third author was partially supported by AMS-Simons Foundation Collaboration Grant 209087.
    The fourth author was partially supported by NSF Grant DMS-1001847.
    The fifth author was partially supported by a Marie Curie international outgoing fellowship STRIDE (298471) within the 7th European Community Framework Programme.
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 1051-1067
  • MSC (2010): Primary 03D30
  • DOI: https://doi.org/10.1090/jams/848
  • MathSciNet review: 3522609