Seminar of Algebra

A classical result of Ore says that, if $M$ is a cancellative monoid where any two elements admit a least common multiple, that every element of the enveloping group $U(M)$ of $M$ can be represented uniquely as an irreducible fraction on $M$. We shall show how to weaken the hypothesis about the existence of common multiples at the expense of considering what we call multifractions. When $M$ admits a finite Garside family, this leads to an algorithm of a new type (but one that is reminiscent of Dehn's algorithm for hyperbolic groups) for solving the Word Problem of the group $U(M)$.

We shall see that this algorithm may fail for certain monoids, but massive computer experiments suggest that it might work for every Artin-Tits monoid, at last giving a clue for a question that remains open for decades.