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On automatic presentations of projective planes

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Abstract

We study automatic presentations of projective planes and prove that no freely generated projective plane has any automatic presentations. Every desarguesian (pappian) projective plane is shown to be automatically presentable if and only if it is finite.

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References

  1. Khoussainov B. and Nerode A., “Automatic presentations of structures,” in: Logic and Computational Complexity, Proc. LCC-1994, Springer-Verlag, Berlin, 1995, pp. 367–392 (Lecture Notes Comput. Sci.; V. 960).

    Google Scholar 

  2. Khoussainov B. and Nerode A., “Open questions in the theory of automatic structures,” Bull. Eur. Assoc. Theor. Comput. Sci., 94, 181–204 (2008).

    MATH  MathSciNet  Google Scholar 

  3. Rubin S., “Automata presenting structures: A survey of the finite string case,” Bull. Symbolic Logic, 14, No. 2, 169–209 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  4. Khoussainov B. and Minnes M., “Three lectures on automatic structures,” in: Logic Colloquium 2007, Cambridge Univ. Press, Cambridge, 2010, pp. 132–176 (Lecture Notes in Logic; V. 35).

    Chapter  Google Scholar 

  5. Tsankov T., “The additive group of the rationals does not have an automatic presentation,” J. Symbolic Logic, 76, No. 4, 1341–1351 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  6. Kuske D., Liu J., and Lohrey M., “The isomorphism problem on classes of automatic structures with transitive relations,” Trans. Amer. Math. Soc., 365, No. 10, 5103–5151 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  7. Shirshov A. I. and Nikitin A. A., “On the theory of projective planes,” Algebra i Logika, 20, No. 3, 330–356 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  8. Shirshov A. I. and Nikitin A. A., The Algebraic Theory of Projective Planes [in Russian], Novosibirsk Univ., Novosibirsk (1987).

    Google Scholar 

  9. Nikitin A. A., “On freely generated projective planes,” Algebra and Logic, 22, No. 1, 45–57 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  10. Nikitin A. A., “Homomorphisms of freely generated projective planes,” Algebra and Logic, 20, No. 4, 277–282 (1981).

    Article  MathSciNet  Google Scholar 

  11. Hughes D. R. and Piper F. C., Projective Planes, Springer-Verlag, New York, Heidelberg, and Berlin (1973).

    MATH  Google Scholar 

  12. Khoussainov B., Nies A., Rubin S., and Stephan F., “Automatic structures: richness and limitations,” Logical Methods in Computer Science, 3, No. 2, 1–18 (2007).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics Novosibirsk State University, Novosibirsk, Russia

    A. S. Denisenko & N. T. Kogabaev

Authors
  1. A. S. Denisenko
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  2. N. T. Kogabaev
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Corresponding author

Correspondence to A. S. Denisenko.

Additional information

Original Russian Text Copyright © 2014 Denisenko A.S. and Kogabaev N.T.

The authors were supported by the Russian Foundation for Basic Research (Grants 11-01-00236a and 13-01-91001-FWF_a) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-276.2012.1).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 66–78, January–February, 2014.

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Denisenko, A.S., Kogabaev, N.T. On automatic presentations of projective planes. Sib Math J 55, 53–62 (2014). https://doi.org/10.1134/S0037446614010078

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  • DOI: https://doi.org/10.1134/S0037446614010078

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