Algebra and Logic

, Volume 42, Issue 2, pp 105–111 | Cite as

Degree Spectra of Relations on Boolean Algebras

  • S. S. Goncharov
  • R. G. Downey
  • D. R. Hirschfeldt
Article

Abstract

We show that every computable relation on a computable Boolean algebra \(\mathfrak{B}\) is either definable by a quantifier-free formula with constants from \(\mathfrak{B}\) (in which case it is obviously intrinsically computable) or has infinite degree spectrum.

computable Boolean algebra computable relation intrinsically computable relation 

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REFERENCES

  1. 1.
    Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Handbook of Recursive Mathematics, Stud. Log. Found. Math., Vols. 138/139, Elsevier, Amsterdam (1998).Google Scholar
  2. 2.
    R. I. Soare, Recursively Enumerable Sets and Degrees, Persp. Math. Log., Springer, Heidelberg (1987).Google Scholar
  3. 3.
    W. Hodges, Model Theory, Enc. Math. Appl., Vol. 42, Cambridge Univ. Press, Cambridge (1993).Google Scholar
  4. 4.
    S. S. Goncharov, Countable Boolean Algebras and Decidability, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1996).Google Scholar
  5. 5.
    C. J. Ash and A. Nerode, “Intrinsically recursive relations,” in Aspects of Effective Algebra, Proc. Conf. Monash Univ., Australia (1981), pp. 26–41.Google Scholar
  6. 6.
    M. Moses, “Relations intrinsically recursive in linear orders,”Z.Math. Log. Grund. Math., 32, No. 5, 467–472 (1986).Google Scholar
  7. 7.
    V. S. Harizanov, “Degree spectrum of a recursive relation on a recursive structure,” Ph. D Thesis, University of Wisconsin, Madison, WI (1987).Google Scholar
  8. 8.
    J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Log., 46, No. 3, 572–594 (1981).Google Scholar
  9. 9.
    D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimension in algebraic structures,” Ann. Pure Appl. Log., 115, Nos. 1–3, 71–113 (2002).Google Scholar
  10. 10.
    V. S. Harizanov, “The possible Turing degree of the nonzero member in a two element degree spectrum,” Ann. Pure Appl. Log., 60, No. 1, 1–30 (1993).Google Scholar
  11. 11.
    D. R. Hirschfeldt, “Degree spectra of relations on computable structures in the presence of Δ0 2 isomorphisms,” J. Symb. Log., 67, No. 2, 697–720 (2002).Google Scholar
  12. 12.
    M. Moses, “Recursive linear orderings with recursive successivities,” Ann. Pure Appl. Log., 27, No. 3, 253–264 (1984).Google Scholar
  13. 13.
    S. B. Cooper, L. Harrington, A. H. Lachlan, et al., “The d.r.e. degrees are not dense,” Ann. Pure Appl. Log., 55, No. 2, 125–151 (1991).Google Scholar
  14. 14.
    R. G. Downey and M. F. Moses, “Recursive linear orders with incomplete successivities,” Trans. Am. Math. Soc., 326, 653–668 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • S. S. Goncharov
    • 1
  • R. G. Downey
    • 2
  • D. R. Hirschfeldt
    • 3
  1. 1.Akademika Koptyuga Prospekt, 4Institute of Mathematics SB RASNovosibirsk
  2. 2.School of Mathematical and Computing SciencesVictoria UniversityWellingtonNew Zealand
  3. 3.Department of MathematicsUniversity of ChicagoUSA

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