Construction 2.4.7.5 (The Braid Group). Let $(W,S)$ be a Coxeter system. We let $\mathrm{Br}(W)$ denote the quotient of the free group generated by $S$ by the relations $(st)^{m_{s,t}} = 1$, where $s$ and $t$ range over distinct elements of $S$ satisfying $m_{s,t} < \infty $; here $m_{s,t}$ denotes the order of the product $st$ in the group $W$. We will refer to $\mathrm{Br}(W)$ as the braid group of the Coxeter system $(W,S)$. By construction, the braid group $\mathrm{Br}(W)$ is equipped with a surjective group homomorphism $\mathrm{Br}(W) \twoheadrightarrow W$, which exhibits $W$ as the quotient of $\mathrm{Br}(W)$ by the relations $s^2 = 1$ for $s \in S$.
Let $\mathrm{Br}^{+}(W)$ denote the submonoid of $\mathrm{Br}(W)$ generated by the elements of $S$. We will refer to $\mathrm{Br}^{+}(W)$ as the braid monoid of the Coxeter system $(W,S)$.