Kerodon

Remark 2.4.7.8. Let $(W,S)$ be a Coxeter system. Then an element $\overrightarrow {w} = (w_1, w_2, \ldots , w_ n )$ of $M_0(W)$ is minimal (with respect to the refinement ordering $\preceq $) if and only if each $w_ i$ belongs to $S$. Moreover, every element $\overrightarrow {w} \in M_0(W)$ admits a refinement $\overrightarrow {s} = (s_1, s_2, \ldots , s_ m )$ which is minimal in $M_0(W)$ (given by choosing a decomposition of each $w_ i$ as a product of elements of $S$). In particular, every connected component of the simplicial set $M_{\bullet }(W)$ contains a vertex $\overrightarrow {s} = (s_1, \ldots , s_ m)$, where each $s_{i}$ belongs to $S$.