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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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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