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CUPPING AND JUMP CLASSES IN THE COMPUTABLY ENUMERABLE DEGREES

Part of: Computability and recursion theory

Published online by Cambridge University Press:  30 October 2020

NOAM GREENBERG ,
KENG MENG NG  and
GUOHUA WU
NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALAND E-mail: greenberg@msor.vuw.ac.nz
KENG MENG NG
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE 637371, SINGAPORE E-mail: kmng@ntu.edu.sg E-mail: guohua@ntu.edu.sg
GUOHUA WU
Affiliation:
DIVISION OF MATHEMATICAL SCIENCES SCHOOL OF PHYSICAL AND MATHEMATICAL SCIENCES NANYANG TECHNOLOGICAL UNIVERSITY SINGAPORE 637371, SINGAPORE E-mail: kmng@ntu.edu.sg E-mail: guohua@ntu.edu.sg
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Abstract

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are ${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed ${\operatorname {\mathrm {low}}}_n$ cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the ${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.

Keywords

computably enumerable degreesjoinjump classesarray computable

MSC classification

Primary: 03D25: Recursively (computably) enumerable sets and degrees

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1499 - 1545
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Copyright
© The Association for Symbolic Logic 2020

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References

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