Limit elements in the configuration algebra for a cancellative monoid

We introduce two spaces $\mathrm{\Omega }(\mathrm{\Gamma },G)$ and $\mathrm{\Omega }(P_{\mathrm{\Gamma },G})$ of pre-partition functions and of opposite series, respectively, which are associated with a Cayley graph $(\mathrm{\Gamma },G)$ of a cancellative monoid $\mathrm{\Gamma }$ with a finite generating system $G$ and with its growth function $P_{\mathrm{\Gamma },G}(t)$. Under mild assumptions on $(\mathrm{\Gamma },G)$, we introduce a fibration ${\pi }_{\mathrm{\Omega }}:\mathrm{\Omega }(\mathrm{\Gamma },G)\to \mathrm{\Omega }(P_{\mathrm{\Gamma },G})$ equivariant with a ${\mathbb{Z}}_{\ge 0}$-action, which is transitive if it is of finite order. Then, the sum of pre-partition functions in a fiber is a linear combination of residues of the proportion of two growth functions $P_{\mathrm{\Gamma },G}(t)$ and $P_{\mathrm{\Gamma },G}\mathcal{M}(t)$ attached to $(\mathrm{\Gamma },G)$ at the places of poles on the circle of the convergent radius.