Graph products and the absence of property (AR)

TL;DR

This paper investigates the structure of graph products of right LCM semigroups, demonstrating many examples, especially right-angled Artin monoids, that lack property (AR), a feature previously thought common.

Contribution

It provides the first examples of right LCM semigroups without property (AR), especially highlighting the case of right-angled Artin monoids.

Findings

Many graph product examples lack property (AR)

First known examples of right LCM semigroups without property (AR)

Results are particularly sharp for right-angled Artin monoids

Abstract

We discuss the internal structure of graph products of right LCM semigroups and prove that there is an abundance of examples without property (AR). Thereby we provide the first examples of right LCM semigroups lacking this seemingly common feature. The results are particularly sharp for right-angled Artin monoids.