Composition–Diamond lemma for tensor product of free algebras

In this paper, we establish a Composition–Diamond lemma for the tensor product k〈X〉⊗k〈Y〉 of two free algebras over a field. As an application, we construct a Gröbner–Shirshov basis in k〈X〉⊗k〈Y〉 by lifting a given Gröbner–Shirshov basis in the tensor product k[X]⊗k〈Y〉 in which k[X] is the polynomial algebra.