Homotopy type of Frobenius complexes II

A finitely generated additive submonoid $\mathrm{\Lambda }$ of ${\mathbb{N}}^d$ has the partial order defined by $\lambda \le \lambda +\mu$ for $\lambda, \mu \in \Lambda$\mathrm{.}TheFrobeniuscomplexistheordercomplexofanopenintervalof$\Lambda$. In this paper, we express the homotopy type of the Frobenius complex of $\mathrm{\Lambda }[\rho \mathrm{/}r]$ in terms of those of $\mathrm{\Lambda }$, where $\Lambda[\rho / r]$istheadditivemonoidwhichisaddedthe$r$-thpartof$\rho$to$\Lambda$. As applications, we determine the homotopy type of the Frobenius complex of some submonoids of $\mathbb{N}$, for example, the submonoid generated by a finite geometric sequence. We also state an application to the multi-graded Poincar'e series.