Archive for Mathematical Logic

, Volume 45, Issue 6, pp 769–781 | Cite as

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

  • Bakhadyr Khoussainov
  • Theodore Slaman
  • Pavel Semukhin
Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Computability theory and its applications: current trends and open problems. In: Cholak P., Lempp S., Lerman M., Shore R. (eds.) Proceedings of a AMS-IMS-SIAM joint summer research conference (1999)Google Scholar
  2. 2.
    Feiner L. (1970) Hierarchies of Boolean algebras. J. Symbo. Log. 35, 365–374CrossRefMathSciNetGoogle Scholar
  3. 3.
    In: Griffor E. (ed.) Handbook of computability theory, Elsevier Amsterdam (1999)Google Scholar
  4. 4.
    In: Marek V., Remmel J., Nerode A., Goncharov S., Ershov Yu. (eds.) Handbook of recursive mathematics, vol. 1, 2, Elsevier Amsterdam (1998)Google Scholar
  5. 5.
    Kasymov N. (1987) Algebras with finitely approximable positively representable enrichments. Algebra Log. 26(6): 715–730MATHMathSciNetGoogle Scholar
  6. 6.
    Kasymov N. (1991) Positive algebras with congruences of finite index. Algebra Log. 30(3): 293–305MATHMathSciNetGoogle Scholar
  7. 7.
    Kasymov N., Khoussainov B. (1986) Finitely generated enumerable and absolutely locally finite algebras. Vychisl. Systemy. 116, 3–15MATHMathSciNetGoogle Scholar
  8. 8.
    Khoussainov B., Lempp S. Slaman T. (2006) Computably enumerable algebras, their expansions and isomorphisms. The Int. J. Algebra Comput. (accepted)Google Scholar
  9. 9.
    Love J. (1993) Stability among r.e. quotient algebras. Ann. Pure Appl. Log. 59, 55–63MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Soare R. (1987) Recursively enumerable sets and degrees. Springer Berlin Heidelberg, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Bakhadyr Khoussainov
    • 1
  • Theodore Slaman
    • 2
  • Pavel Semukhin
    • 1
    • 3
  1. 1.Department of Computer ScienceThe University of AucklandAucklandNew Zealand
  2. 2.Department of MathematicsThe University of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsNovosibirsk State UniversityNovosibiriskRussia

Personalised recommendations