# 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,*)/#Γ_n_ is the free energy of the ball Γ_n_) at the poles on the circle of their convergence radius.

*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,*(Γ_n*G__M*(*t*) := ∑∞*n*=0_M*tn*(where*M*(Γ_n## 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