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Transactions of the American Mathematical Society

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The uniform Martin's conjecture for many-one degrees


Authors: Takayuki Kihara and Antonio Montalbán
Journal: Trans. Amer. Math. Soc. 370 (2018), 9025-9044
MSC (2010): Primary 03D30; Secondary 03E15, 03E60
DOI: https://doi.org/10.1090/tran/7519
Published electronically: September 18, 2018
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Abstract: We study functions from reals to reals which are uniformly degree invariant from Turing equivalence to many-one equivalence, and we compare them ``on a cone''. We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing equivalence to Turing equivalence.

Our proof works in the general case of many-one degrees on $ \mathcal {Q}^{\omega }$ and Wadge degrees of functions $ {\omega }^{\omega }\to \mathcal {Q}$ for any better-quasi-ordering $ \mathcal {Q}$.


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Additional Information

Takayuki Kihara
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan
Email: kihara@i.nagoya-u.ac.jp

Antonio Montalbán
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: antonio@math.berkeley.edu

DOI: https://doi.org/10.1090/tran/7519
Received by editor(s): October 5, 2017
Received by editor(s) in revised form: January 23, 2018
Published electronically: September 18, 2018
Additional Notes: The first-named author was partially supported by JSPS KAKENHI grants 17H06738 and 15H03634, and the JSPS Core-to-Core Program (A. Advanced Research Networks).
The second-named author was partially supported by NSF grant DMS-0901169 and the Packard Fellowship.
Article copyright: © Copyright 2018 American Mathematical Society

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