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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A join theorem for the computably enumerable degrees
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by Carl G. Jockusch Jr., Angsheng Li and Yue Yang PDF
Trans. Amer. Math. Soc. 356 (2004), 2557-2568 Request permission

Abstract:

It is shown that for any computably enumerable (c.e.) degree $\mathbf {w}$, if $\mathbf {w\not =0}$, then there is a c.e. degree $\mathbf {a}$ such that $\mathbf {(a\lor w)’} = \mathbf {a''}= \mathbf {0''}$ (so $\mathbf {a}$ is low$_2$ and $\mathbf {a\lor w}$ is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low$_2$ c.e. degrees are not elementarily equivalent as partial orderings.
References
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Additional Information
  • Carl G. Jockusch Jr.
  • Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, Illinois 61801
  • Email: jockusch@math.uiuc.edu
  • Angsheng Li
  • Affiliation: Institute of Software, Chinese Academy of Sciences, P. O. Box 8718, Beijing, 100080, People’s Republic of China
  • Email: angsheng@gcl.iscas.ac.cn
  • Yue Yang
  • Affiliation: Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 119260
  • Email: matyangy@leonis.nus.edu.sg
  • Received by editor(s): June 11, 2002
  • Published electronically: February 27, 2004
  • Additional Notes: The first author was partially supported by NSF Grant DMS-98-03073. The second author was supported by EPSRC Research Grant no. GR/M 91419, “Turing Definability” (UK), by NSF Grant No. 69973048, by NSF Major Grant No. 19931020 (P. R. China), and by National Distinguished Young Investigator Award no. 60325206 (China). The third author was partially supported by the NSTB OAP programme and NUS Grant No. R-146-000-028-112 (Singapore). All three authors were partially supported by NSFC grant No. 60310213 “New Directions in Theory and Applications of Models of Computation" (China)
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2557-2568
  • MSC (2000): Primary 03D25, 03D30; Secondary 03D35
  • DOI: https://doi.org/10.1090/S0002-9947-04-03585-8
  • MathSciNet review: 2052189