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.

A new construction of a free inverse semigroup was obtained by Poliakova and
Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a
free inverse semigroup relative to the deg-lex order of words. In particular,
we give the (unique and shortest) Groebner-Shirshov normal forms in the classes
of equivalent words of a free inverse semigroup together with the
Groebner-Shirshov algorithm to transform any word to its normal form.

In this paper, we establish the Composition-Diamond lemma for
$\lambda$-differential associative algebras over a field $K $ with multiple
operators. As applications, we obtain Gr\"{o}bner-Shirshov bases of free
$\lambda$-differential Rota-Baxter algebras. In particular, linear bases of
free $\lambda$-differential Rota-Baxter algebras are obtained and consequently,
the free $\lambda$-differential Rota-Baxter algebras are constructed by words.