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=0_M(Γ_n)tn (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