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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Wadge-like degrees of Borel bqo-valued functions
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by Takayuki Kihara and Victor Selivanov PDF
Proc. Amer. Math. Soc. 150 (2022), 3989-4003 Request permission


We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., $\{0,1\}$-valued functions) to $Q$-valued functions, for a better quasiorder $Q$. In this article, we consider more general reducibilities on the $Q$-valued functions and generalise some results of L. Motto Ros [J. Symbolic Logic 74 (2009), pp. 27–49] in the first direction and of T. Kihara and A. Montalbán [Trans. Amer. Math. Soc. 370 (2018), pp. 9025–9044] in the second direction: Our main result states that the structure of the $\mathbf {\Delta }^0_\alpha$-degrees of $\mathbf {\Delta }^0_{\alpha +\gamma }$-measurable $Q$-valued functions is isomorphic to the $\mathbf {\Delta }^0_\beta$-degrees of $\mathbf {\Delta }^0_{\beta +\gamma }$-measurable $Q$-valued functions, and these are isomorphic to the generalized homomorphism order on the $\gamma$-th iterated $Q$-labeled forests.
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Additional Information
  • Takayuki Kihara
  • Affiliation: Department of Mathematical Informatics, Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan
  • MR Author ID: 892476
  • ORCID: 0000-0002-1611-952X
  • Email:
  • Victor Selivanov
  • Affiliation: A. P. Ershov Institute of Informatics Systems, SB Russian Academy of Sciences, Lavrentyev av. 6, 630090 Novosibirsk, Russia
  • MR Author ID: 195695
  • Email:
  • Received by editor(s): July 14, 2020
  • Received by editor(s) in revised form: November 28, 2021
  • Published electronically: April 7, 2022
  • Additional Notes: The first author was partially supported by JSPS KAKENHI Grant 19K03602, 21H03392, the JSPS Core-to-Core Program (A. Advanced Research Networks), and the JSPS-RFBR Bilateral Program (Grant JPJSBP120204809). The research of the second author was supported by RFBR-JSPS Grant 20-51-50001.
  • Communicated by: Vera Fischer
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 150 (2022), 3989-4003
  • MSC (2020): Primary 03E15; Secondary 03D55
  • DOI:
  • MathSciNet review: 4446246