Limit Elements in the Configuration Algebra for a Cancellative Monoid

Abstract

Associated with the Cayley graph $(\Gamma ,G)$ of a cancellative monoid $\Gamma$ with a finite generating system $G$, we introduce two compact spaces: $\Omega (\Gamma ,G)$ consisting of pre-partition functions and $\Omega (P_{\Gamma ,G})$ consisting of series opposite to the growth function $P_{\Gamma ,G}(t):={\sum }_{n=0}^{\infty }\#{\Gamma }_n\cdot t^n$ (where ${\Gamma }_n$ is the ball of radius $n$ centered at the unit element in the Cayley graph). Under mild assumptions on $(\Gamma ,G)$, we introduce a fibration $\pi :\Omega (\Gamma ,G)\to \Omega (P_{\Gamma ,G})$, which is equivariant with respect to a $(\tilde{\tau },\tau )$-action. The action is transitive if it is of finite order. Then we express the finite sum of the pre-partition functions in each fiber of $\pi$ as a linear combination of the ratios of the residues of the two growth functions $P_{\Gamma ,G}(t)$ and $P_{\Gamma ,G}\mathcal{M}(t):={\sum }^{\infty }n=0\mathcal{M}({\Gamma }_n)t^n$ (where $\mathcal{M}({\Gamma }_n)/\#{\Gamma }_n$ is the free energy of the ball ${\Gamma }_n$) at the poles on the circle of their convergence radius.