Transitive orientations, möbius functions, and complete semi-thue systems for free partially commutative monoids

  • Volker Diekert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 317)


Let I\(\subseteq\)X×X be an independence relation over a finite alphabet X and M=X*/{(ab, ba)|(a, b)teI} the associated free partially commutative monoid. The Möbius function of M is a polynomial in the ring of formal power series ZM》. Taking representatives we may view it as a polynomial in ZX*》. We call it unambiguous if its formal inverse in ZX*》 is the characteristic series over a set of representatives of M. The main result states that there is an unambiguous Möbius function of M in ZX*》 if and only if there is a transitive orientation of I. It is known that transitive orientations correspond exactly to finite complete semi-Thue systems S\(\subseteq\)XX* which define M. We obtain a one-to-one correspondence between unambiguous Möbius functions, transitive orientations and finite (normalized) complete semi-Thue systems.


Characteristic Function Formal Power Series Independence Relation Finite Alphabet Commutative Monoid 


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  1. [CaFa69]
    P. Cartier, D. Faota: Problèmes combinatoires de commutation et réarrengements; Lect. Not. in Math., No. 85 (1969)Google Scholar
  2. [Chof86]
    C. Choffrut: Free partially commutative monoids; LITP-report 86/20, Université de Paris 7 (1986)Google Scholar
  3. [ClLa85]
    M. Clerbout, M. Latteux: Partial Commutations and faithful Rational Transductions; Theoret. Comp. Sci. 35 (1985), 241–254Google Scholar
  4. [CoPe85]
    R. Cori, D. Perrin: Automates et Commutations Partielles; R.A.I.R.O., Informatique theoriques 19, No. 1, (1985), 21–32Google Scholar
  5. [Golum80]
    M.C. Golumbic: Algorithmic graph theory and perfect graphs; Academic Press, New York 1986Google Scholar
  6. [KuSa86]
    W. Kuich, A. Salomaa: Semirings, Automata, Languages; EATCS monographs, Vol 5, Springer, Berlin 1986Google Scholar
  7. [MeOch87]
    Y. Métivier, E. Ochmanski: On lexicographic semi-commutations; Inform. Proc. Letters 26 (1987/88) 55–59Google Scholar
  8. [Otto87]
    F. Otto: Finite canonical rewriting systems for congruences generated by concurrency relations; to appear in: Math. Syst. TheoryGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Volker Diekert
    • 1
  1. 1.Institut für Informatik der Technischen Universität MünchenMünchen 2

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