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Limit elements in the configuration algebra for a cancellative monoid

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