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CEA Operators and the Ershov Hierarchy

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Abstract

We examine the relationship between the CEA hierarchy and the Ershov hierarchy within \(\Delta_2^0\) Turing degrees. We study the long-standing problem raised in [1] about the existence of a low computably enumerable (c.e.) degree \(\bf a\) for which the class of all non-c.e. \(CEA(\bf a)\) degrees does not contain 2-c.e. degrees. We solve the problem by proving a stronger result: there exists a noncomputable low c.e. degree \(\textbf{a}\) such that any \(CEA(\bf a)\) \(\omega\)-c.e. degree is c.e. Also we discuss related questions and possible generalizations of this result.

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ACKNOWLEDGMENTS

Authors would like to thank Faizrakhmanov Marat for useful tips and comments during the initial stage of this work. M.M. Yamaleev also would like to thank Wu Guohua for helpful discussions on the works [1] and [5].

Funding

M.M. Arslanov and M.M. Yamaleev are partially supported by the Program of development of Scientific and Educational Mathematical Center of Volga Region (project no. 075-02-2021-1393). I.I. Batyrshin is partially supported by the Russian Science Foundation (project no. 18-11-00028).

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Authors and Affiliations

  1. Kazan Federal University, 18 Kremlyovskaya str., 420008, Kazan, Russia

    M. M. Arslanov, I. I. Batyrshin & M. M. Yamaleev

Authors
  1. M. M. Arslanov
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  2. I. I. Batyrshin
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  3. M. M. Yamaleev
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Correspondence to M. M. Arslanov, I. I. Batyrshin or M. M. Yamaleev.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 8, pp. 72–79.

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Arslanov, M.M., Batyrshin, I.I. & Yamaleev, M.M. CEA Operators and the Ershov Hierarchy. Russ Math. 65, 63–69 (2021). https://doi.org/10.3103/S1066369X21080089

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  • DOI: https://doi.org/10.3103/S1066369X21080089

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