Reference for a proof of cancellation property of braid monoids

Question

Let $M$ be a monoid. If $ab=ac$ implies that $b=c$, $a,b,c \in M$, then $M$ is said to have the left cancellation property. Similarly, the right cancellation property is $ba=ca$ implies that $b=c$.

A braid monoid is a monoid generated by $T_1, \ldots, T_n$ subject to the relations: $T_i T_j = T_j T_i$, $|i-j|>1$, and $T_i T_j T_i = T_j T_i T_j$, $|i-j|=1$.

Are there some references about a proof of cancellation property of braid monoids? Thank you very much.

Answer 1

In the book Braids and Self-Distributivity by Patrick Dehornoy, Chapter 2, Proposition 4.5 states that the positive braid monoid $B_{n}^{+}$ is left-cancellative. Dehornoy actually proves left-cancellativity for monoids constructed from a coherent complement (Prop. 2.14). Later Dehornoy shows that the braid monoid complement is coherent (Lem. 4.1) and therefore the braid monoid is left-cancellative (and also right-cancellative by symmetry).