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Transactions of the American Mathematical Society

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

 

 

A join theorem for the computably enumerable degrees


Authors: Carl G. Jockusch Jr., Angsheng Li and Yue Yang
Translated by:
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
Published electronically: February 27, 2004
MathSciNet review: 2052189
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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.


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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

DOI: https://doi.org/10.1090/S0002-9947-04-03585-8
Keywords: Join theorem, computably enumerable degree, definable ideals, Turing jump
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)
Article copyright: © Copyright 2004 American Mathematical Society