Garside monoids vs divisibility monoids

TL;DR

This paper explores the relationship between divisibility monoids and Garside monoids, analyzing their quasi-centers and identifying their intersection, thus advancing the understanding of their algebraic structures.

Contribution

It provides a comparative study of divisibility and Garside monoids, focusing on their quasi-centers and the intersection of these classes.

Findings

Quasi-centers can be studied similarly in both monoid classes.

The intersection between divisibility and Garside monoids is characterized.

Structural properties of these monoids are clarified.

Abstract

Divisibility monoids (resp. Garside monoids) are a natural algebraic generalization of Mazurkiewicz trace monoids (resp. spherical Artin monoids), namely monoids in which the distributivity of the underlying lattices (resp. the existence of common multiples) is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be commutations (resp. be of Coxeter type). Here, we show that the quasi-center of these monoids can be studied and described similarly, and then we exhibit the intersection between the two classes of monoids.