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\(\Pi^0_1\)-Presentations of Algebras

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Abstract

In this paper we study the question as to which computable algebras are isomorphic to non-computable \(\Pi_{1}^{0}\)-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\(\Pi_{1}^{0}\)-presentations. On the other hand, many of this structures fail to have non-computable \(\Sigma_{1}^{0}\)-presentation.

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Authors and Affiliations

  1. Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand

    Bakhadyr Khoussainov & Pavel Semukhin

  2. Department of Mathematics, The University of California, 719 Evans Hall #3840, Berkeley, CA, 94720-3840, USA

    Theodore Slaman

  3. Department of Mathematics, Novosibirsk State University, Novosibirisk, Russia

    Pavel Semukhin

Authors
  1. Bakhadyr Khoussainov
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  2. Theodore Slaman
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  3. Pavel Semukhin
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Corresponding author

Correspondence to Bakhadyr Khoussainov.

Additional information

This research was partially supported by the Marsden Fund of New Zealand. The third author’s research was partially supported by RFFR grant No. 02-01-00593 and Council for Grants under RF President, project NSh-2112.2003.1.

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Khoussainov, B., Slaman, T. & Semukhin, P. \(\Pi^0_1\)-Presentations of Algebras. Arch. Math. Logic 45, 769–781 (2006). https://doi.org/10.1007/s00153-006-0013-3

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  • DOI: https://doi.org/10.1007/s00153-006-0013-3

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