The skew growth functions $N_{M,\mathrm{deg}}(t)$ for the monoid of type ${\mathrm{B}}_{\mathrm{ii}}$ and others

Abstract

Let $M$ be a positive homogeneously presented cancellative monoid ${<L\mid R>}_{mo}$ equipped with the degree map $°:M\to {\mathbb{Z}}_{\ge 0}$ defined by assigning to each equivalence class of words the length of the words, and let $P_{M,°}(t):={\sum }_{u\in M}{t}^{°(u)}$ be its generating series, called the growth function. If $M$ satisfies the condition that any subset $J$ of $I_0$ ($:=$ the image of the set $L$ in $M$) admits either the least right common multiple ${\Delta }_J$ or no common multiple in $M$, then the inversion function $P_{M,°}(t{)}^{-1}$ is given by the polynomial $\sum_{J \subset I_{0}}(-1)^{#J} t^{°(Δ_{J})}$, where the summation index $J$ runs over all subsets of $I_0$ whose least right common multiple exists. Since a monoid $M$ generally may not admit the least right common multiple ${\Delta }_J$ for a given subset $J$ of it, if we attempt to generalize the formula, the consideration to obtain the above formula is invalid. To resolve this obstruction, we will examine the set $\mathrm{mcm}(J)$ of minimal common right multiples of $J$. Then, we need to introduce a concept of a tower of minimal common multiples of elements of $M$ and denote the set of all the towers in $M$ by $\mathrm{Tmcm}(M)$. Considering the structure of the set $\mathrm{Tmcm}(M)$, K. Saito has proved the inversion formula \[ P_{M,°}(t). N_{M,°}(t)=1, \] where the second factor in LHS is a suitably signed generating series \[ N_{M,°}(t):= 1 + \sum_{T\in \mathrm{Tmcm}(M)}(-1)^{#J_1+...+#J_{n}-n+1}\sum_{Δ\in \mathrm{mcm}(J_n)} t^{°(Δ)}, \] called the skew growth function. In this article, we present several explicit calculations of examples of the skew growth functions for the monoid of type ${\mathrm{B}}_{\mathrm{ii}}$ and others whose towers do not stop on the first stage $J_1$.

References in corpus (1)

• Inversion formula for the growth function of a cancellative monoid