Abstract
Dickert (1988) settled a well-known conjecture on unambiguous liftings to words of Möbius functions for free partially commutative monoids. He established thereby a new bridge from the theory of formal power series to the theory of complete semi-Thue systems. As an open question, it was asked whether this approach has a generalization to a relative situation where we do not lift to the free (word-) case but only to some level of less commutation. This question has been reconsidered by König (1992) and a conjecture has been formulated there. Our results solve the open problem and settle this conjecture (Remark 4.8). We show that there is a canonical one-to-one correspondence between unambiguous Möbius functions and confluent semi-commutation systems (Theorem 5.1). This identification is due to some graph-theoretic characterizationobtained by Diekert (1991) and which allows one to apply some interesting complexity results (Section 6). Our results can be viewed as a contribution to the combinatorial theory of Möbius functions and to the theory of rewriting on traces.
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