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

TL;DR

This paper investigates the skew growth functions of certain cancellative monoids, providing explicit calculations for the monoid of type B_{ii} and others with complex tower structures, extending previous inversion formulas.

Contribution

It introduces a method to compute skew growth functions for monoids with complex tower structures, generalizing existing inversion formulas to new classes of monoids.

Findings

Explicit formulas for skew growth functions of type B_{ii} monoids.

Examples of monoids with towers that do not stop at the first stage.

Extension of inversion formulas to broader classes of monoids.

Abstract

Let $M$ be a positive homogeneously presented cancellative monoid ${< L \mid R >}_{mo}$ equipped with the degree map $\deg:M \to \Z_{\ge0}$ defined by assigning to each equivalence class of words the length of the words, and let $P_{M, \deg}(t):= \sum_{u \in\ M}t^{\deg(u)}$ 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 $P_{M, \deg}(t)^{-1}$ is given by the polynomial $\sum_{J \subset I_{0}}(-1)^{#J} t^{\deg(\Delta_{J})}$, 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 \[ P_{M,\deg}(t). N_{M,\deg}(t)=1, \] where the second factor in LHS is a suitably signed generating series \[ N_{M,\deg}(t):= 1 + \sum_{T\in \mathrm{Tmcm}(M)}(-1)^{#J_1+...+#J_{n}-n+1}\sum_{\Delta\in \mathrm{mcm}(J_n)} t^{\deg(\Delta)}, \] 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 $\mathrm{B_{ii}}$ and others whose towers do not stop on the first stage $J_1$.