The Commutation with Codes and Ternary Sets of Words

  • Juhani Karhumäki
  • Michel Latteux
  • Ion Petre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2607)


We prove several results on the commutation of languages. First, we prove that the largest set commuting with a given code X, i.e., its centralizer C(X), is always ρ(X)., where ρ(X) is the primitive rootof X. Using this result, we characterize the commutation with codes similarly as for words, polynomials, and formal power series: a language commutes with X if and only if it is a union of powers of ρ(X). This solves a conjecture of Ratoandromanana, 1989, and also gives an affermative answer to a special case of an intriguing problem raised by Conway in 1971. Second, we prove that for any nonperiodic ternary set of words F ⊆∑+,C(F) = F*., and moreover, a language commutes with F if and only if it is a union of powers of F, results previously known only for ternary codes. A boundary point is thus established, as these results do not hold for all languages with at least four words.


Regular Languages Combinatories on Words 


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  1. 1.
    Autebert, J.M., Boasson, L., Latteux, M.: Motifs et bases de langages, RAIRO Inform. Theor., 23(4) (1989) 379–393.MathSciNetMATHGoogle Scholar
  2. 2.
    Bergman, G.: Centralizers in free associative algebras, Transactions of the American Mathematical Society 137 (1969) 327–344.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berstel, J., Perrin, D.: Theory of Codes, Academic Press, New York (1985).MATHGoogle Scholar
  4. 4.
    Choffrut, C., Karhumäki, J.: Combinatorics of Words. In Rozenberg, G., Salomaa, A. (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag (1997) 329–438.Google Scholar
  5. 5.
    Choffrut, C., Karhumäki, J.: On Fatou properties of rational languages, in Martin-Vide, C., Mitrana, V. (eds.), Where mathematics, Computer Science, Linguistics and Biology Meet, Kluwer, Dordrecht (2000).Google Scholar
  6. 6.
    Choffrut, C., Karhumäki, J., Ollinger, N.: The commutation of finite sets: a challenging problem, Theoret. Comput. Sci., 273 (1–2) (2002) 69–79.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cohn, P.M.: Factorization in noncommuting power series rings, Proc. Cambridge Philos. Soc. 58 (1962) 452–464.Google Scholar
  8. 8.
    Cohn, P.M.: Centralisateurs dans les corps libres, in Berstel, J. (ed.), Sℰies formelles, Paris, (1978) 45–54.Google Scholar
  9. 9.
    Conway, J.H.: Regular Algebra and Finite Machines, Chapman Hall (1971).Google Scholar
  10. 10.
    Devolder, J., Latteux, M., Litovsky, I., Staiger, L.: Codes and infinite words, Acta Cybernetica 11 (1994) 241–256.MATHMathSciNetGoogle Scholar
  11. 11.
    Harju, T., Petre, I.: On commutation and primitive roots of codes, submitted. A preliminary version of this paper has been presented at WORDS 2001, Palermo, Italy.Google Scholar
  12. 12.
    Karhumäki, J.: Challenges of commutation: an advertisement, in Proc. of FCT 2001, LNCS 2138, Springer (2001) 15–23.Google Scholar
  13. 13.
    Karhumäki, J., Petre, I.: On the centralizer of a finite set, in Proc. of ICALP 2000, LNCS 1853, Springer (2000) 536–546.Google Scholar
  14. 14.
    Karhumäki, J., Petre, I.: Conway’s Problem for three-word sets, Theoret. Comput. Sci., 289/1 (2002) 705–725.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Karhumäki, J., Petre, I.: Conway’s problem and the commutation of languages, Bulletin of EATCS 74 (2001) 171–177.MATHGoogle Scholar
  16. 16.
    Karhumäki, J., Petre, I.: The branching point approach to Conway’s problem, LNCS 2300, Springer (2002) 69–76.Google Scholar
  17. 17.
    Lothaire, M.: Combinatorics on Words (Addison-Wesley, Reading, MA., (1983).Google Scholar
  18. 18.
    Lothaire, M.: Algebraic Combinatorics on Words (Cambridge University Press), (2002).Google Scholar
  19. 19.
    Mateescu, A., Salomaa, A., Yu, S.: On the decomposition of finite languages, TUCS Technical Report 222, http://www.tucs../ (1998).
  20. 20.
    Petre, I.: Commutation Problems on Sets of Words and Formal Power Series, PhD Thesis, University of Turku (2002).Google Scholar
  21. 21.
    Ratoandromanana, B.: Codes et motifs, RAIRO Inform. Theor., 23(4) (1989) 425–444.MathSciNetMATHGoogle Scholar
  22. 22.
    Restivo, A.: Some decision results for recognizable sets in arbitrary monoids, in Proc. of ICALP 1978, LNCS 62 Springer (1978) 363–371.Google Scholar
  23. 23.
    Shyr, H.J.: Free monoids and languages, Hon Min Book Company, (1991).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Juhani Karhumäki
    • 1
  • Michel Latteux
    • 2
  • Ion Petre
    • 3
  1. 1.Department of MathematicsUniversity of Turku andTurku Centre for Computer Science (TUCS)TurkuFinland
  2. 2.LIFL, URA CNRS 369Université des Sciences et Technologie de LilleVilleneuve d’AscqFrance
  3. 3.Department of MathematicsUniversity of Turku andTurku Centre for Computer Science (TUCS)TurkuFinland

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