Graph products of right cancellative monoids

TL;DR

This paper investigates the properties of graph products of right cancellative monoids, establishing their cancellativity, describing their inverse hulls, and introducing polygraph monoids with specific inverse properties.

Contribution

It proves that graph products of right cancellative monoids are right cancellative and characterizes their inverse hulls, introducing polygraph monoids as a new class with F*-inverse properties.

Findings

Graph products of right cancellative monoids are right cancellative.

The inverse hull of such graph products can be explicitly described.

Polygraph monoids are F*-inverse monoids.

Abstract

Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the case of the inverse hulls of graph monoids, obtaining what we call polygraph monoids. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterisation of those right cancellative monoids with inverse hulls that are F*-inverse.