  Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl
group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on
the set of all subsets of $I$ using unipotent $\chi$-linear bicrystals. In this
paper, we will generalize this result to all types of finite Coxeter groups
(including non-crystallographic types). Our approach is more elementary, based
on some combinatorics of Coxeter groups. Moreover, we will calculate this
monoid structure explicitly for each type.
