Skip to main content
SpringerLink
Log in

Freely Generated Projective Planes with Finite Computable Dimension

  • Published:
Algebra and Logic Aims and scope

It is proved that for every natural n ≥ 1, there exists a computable freely generated projective plane with computable dimension n. It is stated that the class of freely generated projective planes is complete with respect to degree spectra of automorphically nontrivial structures, effective dimensions, expansions by constants, and degree spectra of relations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. T. Kogabaev, “The class of projective planes is noncomputable,” Algebra and Logic, 47, No. 4, 242-257 (2008).

  2. S. S. Goncharov, “Problem of number of nonautoequivalent constructivizations,” Algebra and Logic, 19, No. 6, 401-414 (1980).

  3. S. S. Goncharov, A. V. Molokov, and N. S. Romanovskii, “Nilpotent groups of finite algorithmic dimension,” Sib. Math. J., 30, No. 1, 63-68 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. B. Khoussainov and R. A. Shore, “Computable isomorphisms, degree spectra of relations, and Scott families,” Ann. Pure Appl. Log., 93, Nos. 1-3, 153-193 (1998).

  5. P. Cholak, S. Goncharov, B. Khoussainov, and R. A. Shore, “Computably categorical structures and expansions by constants,” J. Symb. Log., 64, No. 1, 13-37 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. M. Khoussainov and R. A. Shore, “Solution of the Goncharov–Ash problem and the spectrum problem in the theory of computable models,” Dokl. Akad. Nauk, 371, No. 1, 30/31 (2000).

  7. R. Miller, B. Poonen, H. Schoutens, and A. Shlapentokh, “A computable functor from graphs to fields,” submitted; http://arxiv.org/abs/1510.07322.

  8. L. Richter, “Degrees of structures,” J. Symb. Log., 46, 723-731 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. F. Knight, “Degrees coded in jumps of orderings,” J. Symb. Log., 51, No. 4, 1034-1042 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Slaman, “Relative to any nonrecursive set,” Proc. Am. Math. Soc., 126, No. 7, 2117-2122 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Wehner, “Enumeration, countable structure and Turing degrees,” Proc. Am. Math. Soc., 126, No. 7, 2131-2139 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Sh. Kalimullin, “Spectra of degrees of some structures,” Algebra and Logic, 46, No. 6, 399-408 (2007).

  13. A. Frolov, I. Kalimullin, V. Harizanov, O. Kudinov, and R. Miller, “Spectra of high n and non-low n degrees,” J. Log. Comput., 22, No. 4, 755-777 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. S. Harizanov, “Uncountable degree spectra,” Ann. Pure Appl. Log., 54, No. 3, 255-263 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. S. Harizanov, “Some effects of Ash–Nerode and other decidability conditions on degree spectra,” Ann. Pure Appl. Log., 55, No. 1, 51-65 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  16. 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).

    Article  MathSciNet  MATH  Google Scholar 

  17. D. R. Hirschfeldt, “Degree spectra of relations on structures of finite computable dimension,” Ann. Pure Appl. Log., 115, Nos. 1-3, 233-277 (2002).

  18. D. R. Hirschfeldt, B. Khoussainov, R. A. Shore, and A. M. Slinko, “Degree spectra and computable dimensions in algebraic structures,” Ann. Pure Appl. Log., 115, Nos. 1-3, 71-113 (2002).

  19. N. T. Kogabaev, “The theory of projective planes is complete with respect to degree spectra and effective dimensions,” Algebra and Logic, 54, No. 5, 387-407 (2015).

  20. A. I. Shirshov and A. A. Nikitin, “Theory of projective planes,” Algebra and Logic, 20, No. 3, 220-239 (1981).

  21. D. R. Hughes and F. C. Piper, Projective Planes, Grad. Texts Math., 6, Springer-Verlag, New York (1973).

  22. N. T. Kogabaev, “Undecidability of the theory of projective planes,” Algebra and Logic, 49, No. 1, 1-11 (2010).

  23. A. A. Nikitin, “Freely generated projective planes,” Algebra and Logic, 22, No. 1, 45-57 (1983).

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    N. T. Kogabaev

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    N. T. Kogabaev

Authors
  1. N. T. Kogabaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. T. Kogabaev.

Additional information

(N. T. Kogabaev) Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1) and by RFBR (projects No. 14-01-00376 and 13-01-91001-ANF_a).

Translated from Algebra i Logika, Vol. 55, No. 6, pp. 704-737, November-December, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kogabaev, N.T. Freely Generated Projective Planes with Finite Computable Dimension. Algebra Logic 55, 461–484 (2017). https://doi.org/10.1007/s10469-017-9418-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-017-9418-9

Keywords

Search

Navigation