On the equivalence of Z-automata

We prove that two automata with multiplicity in Z are equivalent, i.e. define the same rational series, if and only if there is a sequence of Z-coverings, co-Z-coverings, and circulations of –1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective. This is obtained by combining two results: the first one relates coverings to conjugacy of automata, and is modeled after a theorem from symbolic dynamics; the second one is an adaptation of Schützenberger's reduction algorithm of representations in a field to representations in an Euclidean domain (and thus in Z)