Gröbner–Shirshov bases for associative algebras with multiple operators and free Rota–Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner–Shirshov bases of free Rota–Baxter algebra, free λλ-differential algebra and free λλ-differential Rota–Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard–L. Guo, and L. Guo–W. Keigher by using other methods.