Partition functions associated with cancellative monoids

Inspired by the classical study of the partition function in Ising model, we introduce the space of partition functions associated with any cancellative monoid. Precisely, we replace the square lattice in the Ising model by the Cayley graph of a monoid, and replace the sum over states on the square lattice by the sum over configurations in the Cayley graph. Then, the space of partion functions (and free energies) forms a compact set in a certain complete Hopf algebra generated by configurations. Our goal is to give a presentation of these partition functions as proportions of the residues of the generating growth (Poincare) series of the monoid (if they are meromorphic functions). We calculate some examples of the partion functions, including the cases of Artin and braid monoids and the monoids of integral square matrices.