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The Structure of Computably Enumerable Preorder Relations

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Algebra and Logic Aims and scope

We study the structure Ceprs induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤c. It is proved that the structure of computably enumerable equivalence relations is definable in Ceprs. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure Ceprs is computably isomorphic to first-order arithmetic. It is shown that a Σ1-fragment of the theory is decidable, while its Π3-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in Ceprs do not have a least upper bound, and that among minimal degrees in Ceprs, exactly two are c-degrees of computably enumerable linear preorders.

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Authors and Affiliations

  1. Kazakh-British Technical University, Alma-Ata, Kazakhstan

    S. A. Badaev

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    N. A. Bazhenov

  3. Al-Farabi Kazakh National University. Kazakh-British Technical University, Alma-Ata, Kazakhstan

    B. S. Kalmurzaev

Authors
  1. S. A. Badaev
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  2. N. A. Bazhenov
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  3. B. S. Kalmurzaev
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Corresponding author

Correspondence to S. A. Badaev.

Additional information

Translated from Algebra i Logika, Vol. 59, No. 3, pp. 293-314, May-June, 2020. Russian https://doi.org/10.33048/alglog.2020.59.301.

S. A. Badaev and B. S. Kalmurzaev are supported by SC MES RK, project No. AP05131579.

N. A. Bazhenov supported by Mathematical Center in Akademgorodok, Agreement with RF Ministry of Science and Higher Education No. 075-15-2019-1613.

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Badaev, S.A., Bazhenov, N.A. & Kalmurzaev, B.S. The Structure of Computably Enumerable Preorder Relations. Algebra Logic 59, 201–215 (2020). https://doi.org/10.1007/s10469-020-09592-x

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  • DOI: https://doi.org/10.1007/s10469-020-09592-x

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