Limit elements in the configuration algebra for a cancellative monoid

We introduce two spaces $\Omega(\Gamma,G)$ and $\Omega(P_{\Gamma,G})$ of pre-partition functions and of opposite series, respectively, which are associated with a Cayley graph $(\Gamma,G)$ of a cancellative monoid $\Gamma$ with a finite generating system $G$ and with its growth function $P_{\Gamma,G}(t)$. Under mild assumptions on $(\Gamma,G)$, we introduce a fibration $\pi_\Omega:\Omega(\Gamma,G)\to \Omega(P_{\Gamma,G})$ equivariant with a $\Z_{\ge0}$-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 $P_{\Gamma,G}(t)$ and $P_{\Gamma,G}\mathcal{M}(t)$ attached to $(\Gamma,G)$ at the places of poles on the circle of the convergent radius.