# Limit Elements in the Conﬁguration Algebra for a Cancellative Monoid

### Kyoji Saito

Kyoto University, Japan

## Abstract

Associated with the Cayley graph $(Γ,G)$ of a cancellative monoid $Γ$ with a ﬁnite generating system $G$, we introduce two compact spaces: $Ω(Γ,G)$ consisting of pre-partition functions and $Ω(P_{Γ,G})$ consisting of series opposite to the growth function $P_{Γ,G}(t):=∑_{n=0}#Γ_{n}⋅t_{n}$ (where $Γ_{n}$ is the ball of radius $n$ centered at the unit element in the Cayley graph). Under mild assumptions on $(Γ,G)$, we introduce a ﬁbration $π:Ω(Γ,G)→Ω(P_{Γ,G})$, which is equivariant with respect to a $(τ~,τ)$-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 $π$ as a linear combination of the ratios of the residues of the two growth functions $P_{Γ,G}(t)$ and $P_{Γ,G}M(t):=∑_{∞}n=0M(Γ_{n})t_{n}$ (where $M(Γ_{n})/#Γ_{n}$ is the free energy of the ball $Γ_{n}$) at the poles on the circle of their convergence radius.

## Cite this article

Kyoji Saito, Limit Elements in the Conﬁguration Algebra for a Cancellative Monoid. Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, pp. 37–113

DOI 10.2977/PRIMS/2