Generalization of Stanley's monster reciprocity theorem

By studying the reciprocity property of linear Diophantine systems in light of Malcev–Neumann series, we present in this paper a new approach to and a generalization of Stanley's monster reciprocity theorem. A formula for the “error term” is given in the case when the system does not have the reciprocity property. We also give an inductive proof of Stanley's reciprocity theorem for linear homogeneous Diophantine systems.