Ishibe Tadashi | Searcharxiv

arXiv2013

Infinite examples of cancellative monoids that do not always have least common multiple

We will study the presentations of fundamental groups of the complement of complexified real affine line arrangements that do not contain two parallel lines. By Yoshinaga's minimal presentation, we can give positive homogeneous presentations of the fundamental groups. We consider the associated monoids defined by the presentations. It turns out that, in some cases, left (resp. right) \emph{least common multiple} does not always exist. Hence, the monoids are neither \emph{Garside} nor \emph{Artin}. Nevertheless, we will show that they carry certain particular elements similar to the \emph{fundamental elements} in Artin monoids, and that, by improving the classical method in combinatorial group theory, they are \emph{cancellative monoids}. As a result, we will show that the word problem can be solved and the center of them are determined.

math.GR↗

arXiv2013

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

Let $M$ be a positive homogeneously presented cancellative monoid ${< L \mid R >}_{mo}$ equipped with the degree map $°:M \to \Z_{\ge0}$ 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 $Δ_{J}$ 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 $Δ_{J}$ 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 $\mathrm{B_{ii}}$ and others whose towers do not stop on the first stage $J_1$.

math.CO↗