Determinants and Möbius functions in trace monoids

We show that the independence relation deening a trace monoid M admits a transitive orientation if and only if the characteristic series of a lexicographic cross section of M satisses an equation of the form Det(Id ? X) = 1, where X is a matrix representing the minimum nite automaton recognizing and Id is the identity matrix. This implies that, if the independence relation of a trace monoid M admits a transitive orientation, then any unambiguous lifting of the MM obius function of M is the determinant of a matrix deened by the smallest acceptor of the corresponding cross section.