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

Abstract

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 Z 《M》. Taking representatives we may view it as a polynomial in Z《X*》. We call it unambiguous if its formal inverse in Z《X*》 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 Z 《X*》 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 $\subseteq$ X*×X* which define M. We obtain a one-to-one correspondence between unambiguous Möbius functions, transitive orientations and finite (normalized) complete semi-Thue systems.