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).

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).