We define the Möbius function for a language S being closed under factors as the formal inverse of the characteristic series over the set S. We derive identities in commuting and non-commutating variables characterizing this function as a quotient of polynomials which can be expressed as certain determinants. These determinants in turn are obtained by some matrix related to the minimal automaton recognizing S. Our contribution extends some recent work of Choffrut and Goldwurm.