Composition-Diamond Lemma for Tensor Product of Free Algebras

In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k<X>\otimes k<Y>$ by lifting a Groebner-Shirshov basis in $k[X]\otimes k<Y>$, where $k[X]$ is a commutative algebra.