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On Brzozowski’s Conjecture for the Free Burnside Semigroup Satisfying x2 = x3

  • Andrey N. Plyushchenko
  • Arseny M. Shur
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

In this paper we examine Brzozowski’s conjecture for the two-generated free Burnside semigroup satisfying x 2 = x 3. The elements of this semigroup are classes of equivalent words, and the conjecture claims that all elements are regular languages. The case of the identity x 2 = x 3 is the only one, for which Brzozowski’s conjecture is neither proved nor disproved. We prove the conjecture for all the elements containing an overlap-free or an “almost” overlap-free word. In addition, we show that all but finitely many of these elements are “big” languages in terms of growth rate.

Keywords

Word Problem Regular Language Congruence Class Free Semigroup Primary Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bakirov, M.F., Sukhanov, F.V.: Thue-Morse words and \(\mathcal D\)-structure of the free Burnside semigroup. Proc. Ural State Univ. Ser. Mathematics and Mechanics 18(3), 5–19 (2000) (Russian)Google Scholar
  2. 2.
    Brzozowski, J., Culik, K., Gabrielian, A.: Classification of non-counting events. J. Computer and System Sci. 5, 41–53 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brzozowski, J.: Open problems about regular languages. In: Formal language theory: perspectives and open problems, pp. 23–47. Academic Press, New York (1980)Google Scholar
  4. 4.
    Green, J. A., Rees, D.: On semigroups in which x r = x. Proc. Cambridge Phil. Soc. 48, 35–40 (1952)Google Scholar
  5. 5.
    Guba, V.S.: The word problem for the relatively free semigroups satisfying t m = t m + n with m ≥ 4 or m ≥ 3, n = 1. Internat. J. Algebra Comput. 3(2), 125–140 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Guba, V.S.: The word problem for the relatively free semigroups satisfying t m = t m + n with m ≥ 3. Internat. J. Algebra Comput. 3(3), 335–348 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kad’ourek, L., Polák, J.: On free semigroups satisfying x r ≃ x. Simon Stevin 64(1), 3–19 (1990)MathSciNetMATHGoogle Scholar
  8. 8.
    do Lago, A.P.: On the Burnside semigroups x n = x n + m. Internat. J. Algebra Comput. 6(2), 179–227 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    do Lago, A.P.: Maximal groups in free Burnside semigroups. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 70–81. Springer, Heidelberg (1998)Google Scholar
  10. 10.
    do Lago, A.P., Simon, I.: Free Burnside semigroups. RAIRO Theoret. Inform. Appl. 35(6), 579–595 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    de Luca, A., Varricchio, S.: On non-counting regular classes. In: Proc. ICALP 1990. LNCS, vol. 443, pp. 74–87. Springer, Berlin (1990)Google Scholar
  12. 12.
    McCammond, J.: The solution to the word problem for the relatively free semigroups satisfying t a = t a + b with a ≥ 6. Internat. J. Algebra Comput. 1(1), 1–32 (1991)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Plyushchenko, A.N.: Overlap-free words and the two-generated free Burnside semigroup satisfying x 2 = x 3. Siberian Electron. Math. Rep. 6, 166–181 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Plyushchenko, A.N.: On the word problem for the free Burnside semigroups satisfying x 2 = x 3. Russian Math, Iz. Vuz. (2011) (submitted)Google Scholar
  15. 15.
    Plyushchenko, A.N., Shur, A.M.: Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x 2 = x 3. In: Proceedings of WORDS 2007, Marseille, France, pp. 245–253 (2007)Google Scholar
  16. 16.
    Plyushchenko, A.N., Shur, A.M.: Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x 2 = x 3. Internat. J. Algebra Comput. (submitted), http://arxiv.org/abs/1102.4315
  17. 17.
    Shur, A.M.: Overlap-free words and Thue-Morse sequences. Internat. J. Algebra Comput. 6(3), 353–367 (1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Thue, A.: Uber die gegenseitige Lage gleicher Teile gewisser Zeichentreihen. Norske Videnskabssellskabets Skrifter I Mat. Nat. Kl. 1, 1–67 (1912)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrey N. Plyushchenko
    • 1
  • Arseny M. Shur
    • 1
  1. 1.Ural State UniversityEkaterinburgRussia

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