Gr\"obner-Shirshov bases for Coxeter groups I

A conjecture of Gr"obner-Shirshov basis of any Coxeter group has proposed by L.A. Bokut and L.-S. Shiao \cite{bs01}. In this paper, we give an example to show that the conjecture is not true in general. We list all possible nontrivial inclusion compositions when we deal with the general cases of the Coxeter groups. We give a Gr"obner-Shirshov basis of a Coxeter group which is without nontrivial inclusion compositions mentioned the above.