a b s t r a c tIn this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free RotaBaxter algebra, free λ-differential algebra and free λ-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.

In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov Ω-words on a set X and prove that the set of all non-associative Lyndon-Shirshov Ω-words forms a linear basis of the free Lie Ω-algebra on the set X. From this, we establish Gröbner-Shirshov bases theory for Lie Ω-algebras. As applications, we give Gröbner-Shirshov bases for free λ-Rota-Baxter Lie algebras, free modified λ-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.

In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Gröbner-Shirshov bases of free λ-differential Rota-Baxter algebras. In particular, linear bases of free λ-differential Rota-Baxter algebras are obtained and consequently, the free λ-differential Rota-Baxter algebras are constructed by words.Similar to the relation of integral and differential operators, L. Guo and W. Keigher [15] introduced the notion of λ-differential Rota-Baxter algebra which is a K-algebra R with a λ-differential operator D and a Rota-Baxter operator P such that DP = Id R .There have been some constructions of free Rota-Baxter algebras (commutative and associative). We note that G.-C. Rota [19] and P. Cartier [8] gave the explicit constructions of the free commutative Rota-Baxter algebras on a set when λ = 1, namely, the shuffle Baxter and standard Baxter algebras, respectively. Recently, L. Guo and W. Keigher [13,14] constructed the free commutative Rota-Baxter algebras (with identity or without identity) for any λ ∈ K by the mixable shuffle product. These algebras are now called the mixable shuffle product algebras. In fact, these algebras generalize the classical construction of shuffle product algebras. K. Ebrahimi-Fard and L. Guo [11] constructed recently the free associative Rota-Baxter algebras by Rota-Baxter words.E. Kolchin [16] considered the differential algebra and constructed a free differential algebra. L. Guo and W. Keigher [15] dealt with a generalization of this algebra. Also in [15], the free λ-differential Rota-Baxter algebra was obtained by using the free Rota-Baxter algebra on planar decorated rooted trees.K. Ebrahimi-Fard and L. Guo [12] used rooted trees and forests to give an explicit construction of free noncommutative Rota-Baxter algebras on modules and sets. K. Ebrahimi-Fard, J. M. Gracia-Bondia and F. Patras [10] gave the solution of the word problem for free non-commutative Rota-Baxter algebra. A free Rota-Baxter algebra was constructed on decorated trees by M. Aguiar and M. Moreira [1].The concept of multi-operators algebras (Ω-algebras) was first introduced by A. G. Kurosh in [17,18]. Also, Kurosh noticed that free Ω-algebras share many of the combinatorial properties of free non-associative algebras. On the other hand, the Gröbner-Shirshov bases theory for Lie algebras was first considered by A. I. Shirshov [20]. In fact, Shirshov [20] defined the composition of two Lie polynomials and established the Composition lemma for the Lie algebras. Later on, L. A. Bokut [4] specialized the approach of Shirshov to associative algebras, see also G. M. Bergman [3]. For commutative polynomials, this lemma is known as the Buchberger's Theorem in [6,7].Gröbner-Shirshov bases for Ω-algebras were given in the paper of V. Drensky and R. Holtkamp [9]. In a recent paper of L. B. Bokut, Y. Chen and J. Qiu [5], the Composition-Diamond lemma is established for associative Ω-algebras.In this paper, we construct free λ-...

In this paper, a Gröbner–Shirshov basis for the Chinese monoid is obtained and an algorithm for the normal form of the Chinese monoid is given.

In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gröbner-Shirshov bases method and then we obtain a left counital Hopf algebra structure on a free Rota-Baxter system.