A cancellativity criterion for presented monoids

We establish a new, fairly general cancellativity criterion for a presented monoid that properly extends the previously known related criteria. It is based on a new version of the word transformation called factor reversing, and its specificity is to avoid any restriction on the number of relations in the presentation. As an application, we deduce the cancellativity of some natural extension of Artin's braid monoid in which crossings are colored.