On the Centralizer of a Finite Set

  • Juhani Karhumäki
  • Ion Petre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1853)


We prove two results on commutation of languages. First, we show that the maximal language commuting with a three element language, i.e. its centralizer, is rational, thus giving an affirmative answer to a special case of a problem proposed by Conway in 1971. Second, we characterize all languages commuting with a three element code. The characterization is similar to the one proved by Bergman for polynomials over noncommuting variables, cf. Bergman, 1969 and Lothaire, 2000: A language commutes with a three element code X if and only if it is a union of powers of X.


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  1. 1.
    G. Bergman, Centralizers in free associative algebras, Transactions of the American Mathematical Society 137: 327–344, 1969.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    C. Choffrut, J. Karhumäki, Combinatorics on Words. In G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages, vol 1: 329–438, Springer-Verlag, 1997.Google Scholar
  3. 3.
    C. Choffrut, J. Karhumäki, On Fatou properties of rational languages, TUCS Technical Report 322,, 1999.
  4. 4.
    C. Choffrut, J. Karhumäki, N. Ollinger, The commutation of finite sets: a challenging problem, TUCS Technical Report 303,, 1999, submitted to a special issue of Theoret. Comput. Sci. on Words.
  5. 5.
    P. M. Cohn, Centralisateurs dans les corps libres, in J. Berstel (Ed.), S’eries formelles: 45–54, Paris, 1978.Google Scholar
  6. 6.
    J. H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971.Google Scholar
  7. 7.
    S. Eilenberg, Automata, Languages and Machines, Academic Press, 1974.Google Scholar
  8. 8.
    L. Kari, On insertion and deletion in formal languages, Ph.D. Thesis, University of Turku, 1991.Google Scholar
  9. 9.
    J. Karhumäki, I. Petre, On the centralizer of a finite set, Technical Report 342, TUCS,, 2000.
  10. 10.
    E. Leiss, Language Equations, Springer, 1998.Google Scholar
  11. 11.
    M. Lothaire, Combinatorics on Words (Addison-Wesley, Reading, MA., 1983).MATHGoogle Scholar
  12. 12.
    M. Lothaire, Combinatorics on Words II, to appear.Google Scholar
  13. 13.
    A. Mateescu, A. Salomaa, S. Yu, On the decomposition of finite languages, Technical Report 222, TUCS,, 1998.
  14. 14.
    B. Ratoandromanana, Codes et motifs, RAIRO Inform, Theor., 23(4): 425–444, 1989.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Juhani Karhumäki
    • 1
  • Ion Petre
    • 1
  1. 1.Department of MathematicsUniversity of Turku andTurku Centre for Computer Science (TUCS)TurkuFinland

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