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

TL;DR

This paper constructs infinite examples of cancellative monoids derived from fundamental groups of complexified line arrangements, which lack least common multiples and are neither Garside nor Artin, yet are cancellative and solvable.

Contribution

It introduces new cancellative monoids from line arrangement groups that do not have least common multiples, expanding understanding of monoid structures beyond Garside and Artin types.

Findings

Monoids lack least common multiples in some cases.

They are cancellative and have solvable word problems.

Centers of these monoids are explicitly determined.

Abstract

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.