Gröbner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras

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1 Gröbner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras arxiv: v2 [math.ra] 4 Mar 2009 L. A. Bokut School of Mathematical Sciences, South China Normal University Guangzhou , P. R. China Sobolev Institute of Mathematics, Russian Academy of Sciences Siberian Branch, Novosibirsk , Russia bokut@math.nsc.ru Yuqun Chen School of Mathematical Sciences, South China Normal University Guangzhou , P. R. China yqchen@scnu.edu.cn Jianjun Qiu Mathematics and Computational Science School Zhanjiang Normal University Zhanjiang , China jianjunqiu@126.com Abstract: 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, λ-differential algebra and 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 the recent results obtained to those obtained by K. Ebrahimi-Fard L. Guo, and L. Guo W. Keigher by using other methods. Key words: Rota-Baxter algebra; λ-differential algebra; Associative algebra with multiple operators; Gröbner-Shirshov basis. AMS 2000 Subject Classification: 16S15, 13P10, 16W99, 17A50 Dedicated to Professor Yu.G. Reshetnyak on the occasion of his 80th birthday. Supported by the NNSF of China (No ) and the NSF of Guangdong Province (No ). Supported by RFBR , LSS and SB RAS Integration grant No (Russia). Corresponding author. 1

2 1 Introduction Gröbner and Gröbner-Shirshov bases theory were initiated independently by A.I. Shirshov [59, 60], H. Hironaka [38] and B. Buchberger [22, 23]. In [59], Shirshov first defined Gröbner-Shirshov basis of a (anti-) commutative nonassociative algebra presentation with a fix monomial well ordering under the name reduced set of relations and he proved the Composition-Diamond Lemma (Lemma 2 of [59]) for any reduced set of a free (anti-) commutative algebra. Also Shirshov mentioned that his theory is equally valid for a non-associative algebra presentation and a free nonassociative algebra. Twelve years earlier, A.I. Zhukov [63] invented a weaker forms of Gröbner-Shirshov bases of non-associative algebras and Composition-Diamond lemma for a free non-associative algebra. The difference is that Zhukov [63], contrary to Shirshov [59], did not use any linear order of monomials. Instead, he defined the leading word for a non-associative polynomial as any of its monomial of maximal degree. In [60], Shirshov invented a much more involved Gröbner-Shirshov bases theory for Lie algebras explicitly and for associative algebras implicitly, and he proved the Composition- Diamond Lemma for a free Lie algebra again explicitly and for a free associative algebra implicitly (Lemma 3 of [60]). The main differences with [59] are as follows. For Lie algebra case, one needs to use a very special linear basis of a free Lie algebra, Lyndon Shirshov basis (see Shirshov ([58], 1958), Chen Fox Lyndon ([25], 1958)). Since that time, the Lyndon Shirshov basis and Lyndon Shirshov words (due to R. Lyndon ([53], 1954), and independently to Shirshov ([58], 1958)) play a significant role in algebra and combinatorics (see, for example, [2, 3, 20, 26, 52, 56]; they are often called Lyndon basis and Lyndon words, see, for example, [52, 56]). Also in Lie and associative algebra cases, one needs to use a notion of composition (of intersection) of two Lie (associative) polynomials (spolynomial under the later Buchberger s terminology [23]). Shirshov did it in two steps. First, he defined the composition of two Lie polynomials as associative (non-commutative) polynomials, and then he put in special bracketing using his the Special Bracketing Lemma 4 of [58]. Finally for Lie algebra case, one needs to use Shirshov s reduction algorithm to eliminate the leading (in degree-lexicographical ordering) monomial of one Lie polynomial in another Lie polynomial (again using Lemma 4 of [58]); for associative case, the reduction algorithm is just non-commutative Euclidean algorithm. An ingenious Shirshov s algorithm was defined for any reduced set S of Lie (associative) polynomials: to joint to S a composition of two polynomials from S and to apply the Shirshov s (Euclidean) reduction algorithm to the result set, and then by repeating this process again. A detailed description of this algorithm for associative polynomials (more pricesely, for associative binomials) can be found, for example, in the text [31] under the name Knuth-Bendix algorithm (cf. D. Knuth, P. Bendix ([43], 1970)). Shirshov algorithm is in general infinite as well as the Knuth Bendix algorithm (in contrary with Shirshov algorithm, the latter is applied to any universal algebra presentation (not only to Lie, associative, or associative-commutative algebra presentations, or a semigroup presentation), it does not use any specific normal forms (in free Lie, associative, associative commutative algebras, or free semigroups, free groups), and it does not use any effectively defined composition (s-polynomial); it means that Knuth-Bendix algorithm is too slowly to be effective for groups, semigroups, Lie algebras, associative algebras, associative-commutative algebras, and so on). Using transfinite induction, for any reduced set S of Lie (associative) polynomials, one may find a reduced set S c such that any composition of polynomials from 2

3 S c is reduced to zero by using Shirshov (Euclidean) reduction algorithm, and S and S c generate the same ideal of the free Lie (associative) algebra. The set S c is now called a Gröbner-Shirshov basis for S or a Gröbner-Shirshov basis of the ideal Id(S) generated by S. Shirhsov proved the Composition-Diamond Lemma for any subset S of a free Lie algebra (Lemma 3 of [60]). It is clear that Shirshov algorithm as well as the definition of S c and the proof of Composition-Diamond Lemma for S c are equally valid for associative polynomials. In this case, one dose not need to use Lyndon Shirshov words, Special Bracketing Lemma, and Shirshov reduction algorithm (just Euclidean algorithm for non-commutative polynomials will be enough). An easy proof of the Composition-Diamond Lemma for associative polynomials is just following the proof of Lemma 3 of [60] (cf. [20]). Later the Composition-Diamond Lemma for associative polynomials was explicitly formulated and proved in a paper by L.A. Bokut, [6]) as a specialization of the Shirshov s Lemma and also by G. Bergman in [5]). In recent years there were quite a few results on Gröbner-Shirshov bases for associative algebras, semisimple Lie (super) algebras, irreducible modules, Kac Moode algebras, Coxeter groups, braid groups, conformal algebras, free inverse semigroups, Lodays dialgebras, Leibniz algebras and so on, see, for example, monographs [20, 55], and papers [7, 8, 9, 11, 12, 14, 16, 17, 21, 39, 40, 41, 42, 48, 49, 50, 51, 54, 61, 62], and also surveys [10, 13, 15, 18, 19]. Actually, conformal algebras and Loday s dialgebras are multioperator algebras (conformal algebras have infinitely many bilinear operations x(n)y, n = 0, 1, 2,... and a linear operation D(x), dialgebras have two bilinear operations, left and right products). The concept of multioperator algebra (Ω-algebra) was invented by A.G. Kurosh [46] under an influence of the concept of multioperator group of P.J. Higgins [37]. In [46], Kurosh generalized on a free Ω-algebra his result [45] that any subalgebra of a free nonassociative algebra is again free (the Nielsen Schreier property). According to Kurosh, free Ω-algebras would enjoy many combinatorial properties of free non-associative algebras. Indeed, for example, recently V. Drensky and R. Holtkamp [27] have proved an analogy of above Shirshov Zhukov s Composition-Diamond lemma for free Ω-algebras. It means that they extended Shirshov Zhukov s Gröbner-Shirshov bases theory from non-associative algebras to Ω-algebras. In 1969, Kurosh initiated to publish in the Uspehi Mat. Nauk (Russian Math. Surveys) a series of papers by V.A. Artamonov, M.S. Burgin, F.I. Kizner, S.V. Polin, and Yu.K. Rebane on the theory of Ω-algebras and wrote a special introduction paper on this series [47]. He predicted that the theory of Ω-algebras would have a role in future algebra as a subject between the theory of (non-associative) rings and the theory of universal algebras. Now after 40 years, his prediction is fulfilled. Multioperator Rota Baxter algebras were invented by G. Baxter [4] and was given the current name after an important paper by G.-C. Rota [57]. Let K be a commutative (associative) ring with unity and λ K. A Rota Baxter algebra of weight λ is an associative K-algebra R with a linear operator P(x) satisfying the Rota Baxter identity P(x)P(y) = P(xP(y)) + P(P(x)y) + λp(xy), x, y R. A differential algebra of weight λ, called λ-differential algebra, is an associative K- 3

4 algebra with linear λ-differential operator D(x) such that D(xy) = D(x)y + xd(y) + λd(x)d(y) for any x, y R. L. Guo and W. Keigher [35] introduced a notion of λ-differential Rota- Baxter algebra which is a Rota-Baxter K-algebra of weight λ with λ-differential operator D such that DP = Id R. There have been some constructions of free (commutative) Rota-Baxter algebras. In this aspect, G.-C. Rota [57] and P. Cartier [24] gave the explicit constructions of free commutative Rota-Baxter algebras of weight λ = 1, which they called shuffle Baxter and standard Baxter algebras, respectively. Recently L. Guo and W. Keigher [33, 34] constructed the free commutative Rota-Baxter algebras (with unit or without unit) for any λ K using the mixable shuffle product. These are called the mixable shuffle product algebras, which generalize the classical construction of shuffle product algebras (see, for example, [52]). K. Ebrahimi-Fard and L. Guo [29] further constructed the free associative Rota-Baxter algebras by using the Rota-Baxter words. E. Kolchin [44] considered the differential algebras and constructed a free differential algebra. L. Guo and W. Keigher [35] dealt with a λ-generalization of this algebras. In [35], 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 [30] use rooted trees and forests to give explicit construction of free noncommutative Rota-Baxter algebras on modules and sets. K. Ebrahimi- Fard, J. M. Gracia-Bondia, and F. Patras [28] gave solution of the word problem for free non-commutative Rota-Baxter algebra. A free Rota-Baxter algebra is constructed on decorated trees by M. Aguiar and M. Moreira [1]. In this paper, we deal with associative algebras with multiple linear operators. We construct free associative algebras with multiple linear operators and establish the Composition- Diamond lemma for such algebras. As applications, we give the Gröbner-Shirshov bases of free Rota-Baxter algebra, λ-differential algebra and λ-differential Rota-Baxter algebra, respectively. Thus the linear bases of these three free algebras can be hence obtained, respectively, which are essentially the same or similar to those obtained by using other methods in the papers by K. Ebrahimi-Fard and L. Guo [29] and L. Guo and W. Keigher [35]. Let X be a set. Then we denote by X and S(X) the free monoid and free semigroup generated by X, respectively. Let X be a well ordered set. By the deg-lex ordering on X, we mean the ordering which compares two words first by their degrees and then compares lexicographically. 2 Free associative algebra with multiple operators A free associative algebra with unary operators are constructed by L. Guo [32]. A free associative algebra with one operator is constructed by K. Ebrahimi-Fard and L. Guo [29]. In this section, we construct the free associative algebra with multiple linear operators. Let K be a commutative ring with unit. An associative algebra with multiple linear operators is an associative K-algebra R with a set Ω of multilinear operators (operations). 4

5 Let X be a set and Ω = Ω n, where Ω n is the set of n-ary operations, for example, ary (δ) = n if δ Ω n. Define S 0 = S(X 0 ), X 0 = X, where For n > 1, define where Let Then we have Ω(S 0 ) = Ω(S n 1 ) = n=1 S 1 = S(X 1 ), X 1 = X Ω(S 0 ) {δ(u 1, u 2,...,u t ) δ Ω t, u i S 0, i = 1, 2,..., t}. t=1 S n = S(X n ), X n = X Ω(S n 1 ) {δ(u 1, u 2,...,u t ) δ Ω t, u i S n 1, i = 1, 2,..., t}. t=1 S 0 S 1 S n. S(X) = n 0S n. Then, it is easy to see that S(X) is a semigroup such that Ω(S(X)) S(X). Now, we describe an element in S(X) by a labeled reduced planar rooted forest, see [27] and [30]. For x X, we now describe x by a labeled reduced planar rooted tree x. Thus, for any element u = x 1 x 2... x n S(X), u can be described by x1 x2 xn which is called a labeled reduced planar rooted forest. For any u Ω(S 0 ), say u = θ n (u 1,...,u n ) where each u i S 0, u can be described by u1 u2 θn For example, if u = θ 3 (u 1, u 2, u 3 ) Ω(S 0 ) where u 1 = x 1 x 2, u 2 = x 3, u 3 = x 4 x 5, then u1 u2 u3 x1 x2 x3 x4 x5 θ3 = un θ3 Now, for any v = v 1 v 2 v n S 1 = S(X 1 ) (v i X 1, i = 1,...,n), u can be described by labeled reduced planar rooted forest: v1 v2 vn 5

6 For example, if u = θ 3 (u 1, u 2, u 3 )θ 2 (u 4, u 5 ) where u 1 = x 1 x 2, u 2 = x 3, u 3 = x 4 x 5, u 4 = x 6, u 5 = x 7 x 8, then u can be described by that is by u1 u2 θ3 u3 u4 θ2 u5 x1 x2 x3 θ3 x4 x5 x6 θ2 x7 x8 Continue this step, we can describe any element in S(X) by labeled reduced planar rooted forest. Therefore, with the above way, each element in S(X) corresponds uniquely a labeled reduced planar rooted forest. For any u S(X), dep(u) = min{n u S n } is called the depth of u. Let K X; Ω be the K-algebra spanned by S(X). Then, the element in S(X) (resp. K X; Ω ) is called a Ω-word (resp. Ω-polynomial). If u X Ω(S(X)), we call u a prime Ω-word and define bre(u) = 1 (the breadth of u). If u = u 1 u 2 u n S(X), where u i is prime Ω-word for all i, then we define bre(u) = n. Extend linearly each δ Ω n, δ : S(X) n S(X), (x 1, x 2,, x n ) δ(x 1, x 2,, x n ) to K X; Ω. Then, it is easy to see that K X; Ω is a free associative algebra with multiple linear operators Ω on set X. 3 Composition-Diamond lemma for associative algebras with multiple operators In this section, we introduce the notions of Gröbner-Shirshov bases for associative algebras with multiple linear operators and establish the Composition-Diamond lemma for such algebras. Let K X; Ω be free associative algebra with multiple linear operators Ω on X and / X. By a -Ω-word we mean any expression in S(X { }) with only one occurrence of. The set of all -Ω-words on Xwe define by S (X). Let u be a -Ω-word and s K X; Ω. Then we call u s = u s an s-ω-word. In other words, an s-ω-word u s means that we have replaced the leaf of u by s. For example, 6

7 u s = x1 x2 s x4 x5 δ2 δ1 δ3 where in u we has instead of s. Similarly, we can define ( 1, 2 )-Ω-words as expressions in S(X { 1, 2 }) with only one occurrence of 1 and only one occurrence of 2. Let us denote by S 1, 2 (X) the set of all ( 1, 2 )-Ω-words. Let u S 1, 2 (X). Then we call an s 1 -s 2 -Ω-word. u s1, s 2 = u 1 s 1, 2 s 2, Now, we assume that S(X) is equipped with a monomial ordering >. This means that > is a well ordering on S(X) such that for any v, w S(X), u S (X), w > v u w > u v. Note that such an order on S(X) exists, see order (1) in the next section. For every Ω-polynomial f K X; Ω, let f be the leading Ω-word of f. If the coefficient of f is 1, then we call f monic. Let f, g be two monic Ω-polynomials. Then, there are two kinds of compositions. (I) If there exists a Ω-word w = fa = bḡ for some a, b S(X) such that bre(w) < bre( f) + bre(ḡ), then we call (f, g) w = fa bg the intersection composition of f and g with respect to w. (II) If there exists a Ω-word w = f = u ḡ for some u S (X), then we call (f, g) w = f u g the including composition of f and g with respect to w. In this case transformation f (f, g) w is called the Elimination of the Leading Word (ELW) of g in f. In the above definition, w is called the ambiguity of the composition. Clearly, (f, g) w Id(f, g) and (f, g) w < w, where Id(f, g) is the Ω-ideal of K X; Ω generated by f, g. Let S be a set of monic Ω-polynomials. Then the composition (f, g) w is called trivial modulo (S, w) if (f, g) w = α i u i si, where each α i K, u i S (X), s i S and u i si < w. If this is the case, we write (f, g) w 0 mod(s, w). 7

8 In general, for any two Ω-polynomials p and q, p q mod(s, w) means that p q = αi u i si, where each α i K, u i S (X), s i S and u i si < w. S is called a Gröbner-Shirshov basis in K X; Ω if any composition (f, g) w of f, g S is trivial modulo (S, w). Lemma 3.1 Let S be a Gröbner-Shirshov basis in K X; Ω, u 1, u 2 S (X) and s 1, s 2 S. If w = u 1 s1 = u 2 s2, then u 1 s1 u 2 s2 mod(s, w). Proof: There are three cases to consider. (I) The Ω-words s 1 and s 2 are disjoint in w. Then there exits a ( 1, 2 )-Ω-word Π such that Π s1, s 2 = u 1 s1 = u 2 s2. Then u 2 s2 u 1 s1 = Π s1, s 2 Π s1, s 2 = ( Π s1 s 1, s 2 + Π s1, s 2 ) + (Π s1, s 2 s 2 Π s1, s 2 ) = Π s1 s 1, s 2 + Π s1, s 2 s 2. Let Π s1 s 1, s 2 = α 2t u 2t s2 and Π s1, s 2 s 2 = α 1l u 1l s1. Since s 1 s 1 < s 1 and s 2 s 2 < s 2, we have u 2t s2, u 1l s1 < w. Therefore with each u 2t s2, u 1l s1 < w. It follows that u 2 s2 u 1 s1 = α 2t u 2t s2 + α 1l u 1l s1 u 1 s1 u 2 s2 mod(s, w). (II) The Ω-words s 1 and s 2 have nonempty intersection in w but do not include each other. Without lost of generality we can assume that s 1 a = bs 2 is a Ω-subword of w for some Ω-words a, b. Then there exists a -Ω-word Π such that Π s1 a = u 1 s1 = u 2 s2 = Π bs2. Thus, we have u 2 s2 u 1 s1 = Π bs2 Π s1 a = Π s1 a bs 2. Since S is a Gröbner-Shirshov basis in K X; Ω, we have s 1 a bs 2 = α j v j sj, where each α j K, v j S (X), s j S and v j sj < s 1 a. Let Π vj sj = Π j sj. Then u 2 s2 u 1 s1 = α j Π j sj with Π j sj < w. It follows that u 1 s1 u 2 s2 mod(s, w). 8

9 (III) One of Ω-words s 1, s 2 is contained in the other. For example, let s 1 = u s2 for some -Ω-word u. Then w = u 2 s2 = u 1 u s2 and u 2 s2 u 1 s1 = u 1 u s2 u 1 s1 = u 1 s1 u s2. By using similar method as in (II), we obtain the result. The following theorem is an analogy of Shirshov s Composition lemma for Lie algebras [60], which was specialized to associative algebras by Bokut [6], see also Bergman [5]. Theorem 3.2 (Composition-Diamond lemma) Let S be a set of monic Ω-polynomials in K X; Ω, > a monomial ordering on S(X) and Id(S) the Ω-ideal of K X; Ω generated by S. Then the following statements are equivalent: (I) S is a Gröbner-Shirshov basis in K X; Ω. (II) f Id(S) f = u s for some u S (X) and s S. (II ) f Id(S) f = α 1 u 1 s1 + α 2 u 2 s2 + + α n u n sn where each α i K, s i S, u i S (X) and u 1 s1 > u 2 s2 > > u n sn. (III) Irr(S) = {w S(X) w u s for any u S (X) and s S} is a K-basis of K X; Ω S = K X; Ω /Id(S). Proof: (I)= (II) Let 0 f Id(S). Then f = n α i u i si where each α i K, u i S (X) and s i S. Let w i = u i si. We arrange these leading Ω-words in non-increasing order by i=1 w 1 = w 2 = = w m > w m+1 w n. Now we prove the result by induction on m. If m = 1, then f = u 1 s1. Now we assume that m 2. Then u 1 s1 = w 1 = w 2 = u 2 s2. We prove the result by induction on w 1. It is clear that w 1 f. If w 1 = f then the result is clear. Let w 1 > f. Since S is a Gröbner-Shirshov basis in K X; Ω, by Lemma 3.1, we have u 2 s2 u 1 s1 = β j v j sj where β j K, s j S, v j S (X) and v j sj < w 1. Therefore, since α 1 u 1 s1 + α 2 u 2 s2 = (α 1 + α 2 )u 1 s1 + α 2 (u 2 s2 u 1 s1 ), we have f = (α 1 + α 2 )u 1 s1 + α 2 β j v j sj + 9 n α i u i si. i=3

10 If either m > 2 or α 1 + α 2 0, then the result follows from induction on m. If m = 2 and α 1 + α 2 = 0, then the result follows from induction on w 1. From the above proof, it follows also (II) (II ). (II)= (III) For any f K X; Ω, by induction on f, using the ELW s of S, we have f = α i u i + β j v j sj. u i Irr(S),u i h s j S,v j sj h It follows that any f K X; Ω can be expressed modulo Id(S) by a linear combination of elements of Irr(S). Now suppose that α 1 u 1 + α 2 u 2 + α n u n = 0 in K X; Ω S with each u i Irr(S). Then, in K X; Ω, g = α 1 u 1 + α 2 u α n u n Id(S). By (II), we have ḡ / Irr(S), a contradiction. This shows that Irr(S) is a K-basis of K X; Ω S. (III)= (I) In particular, for any composition (f, g) w in S, since (f, g) w Id(S) and (III), we have (f, g) w = β j v j sj, where each β j K, v j S (X), s j S and v j sj (f, g) w < w. This shows (I). Remark: If Ω is empty, then Theorem 3.2 is the Composition-Diamond Lemma for associative algebras (cf. [6]). 4 Gröbner-Shirshov bases for free Rota-Baxter algebras In this section, we give a Gröbner-Shirshov basis and a linear basis for a free Rota- Baxter algebra. In fact, we construct the free Rota-Baxter algebra on a set X by using the Composition-Diamond Lemma above for associative algebras with multiple linear operators (Theorem 3.2). First of all, we define an order on S(X), which will be used in the sequel. Let X and Ω be well ordered sets. We order X by the deg-lex ordering. For any u S(X), u can be uniquely expressed without brackets as u = u 0 δ i1 xi1 u 1 δ it xit u t, where each u i X, δ ik Ω ik, x ik = (x k1, x k2,, x kik ) S(X) i k. It is reminded that for each i k, dep(u) > dep( x ik ). Denote by wt(u) = (t, δ i1, x i1,, δ it, x it, u 0, u 1,, u t ). Let v = v 0 µ j1 yj1 v 1 µ js yjs v s S(X). Then, we order S(X) as follows: u > v wt(u) > wt(v) lexicographically (1) 10

11 by induction on dep(u) + dep(v). It is clear that the order (1) is a monomial ordering on S(X). Let K be a commutative ring with unit and λ K. A Rota-Baxter K-algebra of weight λ ([4, 57]) is an associative algebra R with a K-linear operation P : R R satisfying the Rota-Baxter relation: P(x)P(y) = P(P(x)y + xp(y) + λxy), x, y R. Thus, any Rota-Baxter algebra is a special case of associative algebra with multiple operators when Ω = {P }. Now, let Ω = {P } and S(X) be as before. Let K X; P be the free associative algebra with one operator Ω = {P } on a set X. Theorem 4.1 With the order (1) on S(X), S = {P(x)P(y) P(P(x)y) P(xP(y)) λp(xy) x, y S(X)} is a Gröbner-Shirshov basis in K X; P. Proof: The ambiguities of all possible compositions of Ω-polynomials in S are: (i) P(x)P(y)P(z) (ii) P(u P(x)P(y) )P(z) (iii) P(x)P(u P(y)P(z) ) where x, y, z S(X), u S (X). It is easy to check that all these compositions are trivial. Here, for example, we just check (ii). Others are similarly proved. Let Then f(x, y) = P(x)P(y) P(P(x)y) P(xP(y)) λp(xy). (f(u P(x)P(y), z), f(x, y)) P(u P(x)P(y) )P(z) = P(P(u P(x)P(y) )z) P(u P(x)P(y) P(z)) λp(u P(x)P(y) z) +P(u P(P(x)y) )P(z) + P(u P(xP(y)) )P(z) + λp(u P(xy) )P(z) P(P(u P(P(x)y) )z) P(P(u P(xP(y)) )z) λp(p(u P(xy) )z) P(u P(P(x)y) P(z)) P(u P(xP(y)) P(z)) λp(u P(xy) P(z)) λp(u P(P(x)y) z) λp(u P(xP(y)) z) λ 2 P(u p(xy) z) +P(P(u P(P(x)y) )z) + P(u P(P(x)y) P(z)) + λp(u P(P(x)y) z) +P(P(u P(xP(y)) )z) + P(u P(xP(y)) P(z)) + λp(u P(xP(y)) z) +λp(p(u P(xy) )z) + λp(u P(xy) P(z)) + λ 2 P(u P(xy) z) 0 mod(s, P(u P(x)P(y) )P(z)). Now, we describe the set Irr(S). Let Y and Z be two subsets of S(X). Define the alternating product of X and Y with P (see also [29]): Λ P X (Y, Z) = ( r 1(Y P(Z)) r ) ( r 0 (Y P(Z)) r Y ) ( r 1 (P(Z)Y ) r ) ( r 0 (P(Z)Y ) r P(Z)). 11

12 Clearly, Λ P X (Y, Z) S(X). Now let Φ 0 = S(X). For n > 0, define Let Φ n = Λ P X(Φ 0, Φ n 1 ). Φ(X) = n 0Φ n. Then, we call the elements in Φ(X) the Rota-Baxter words. It is easy to see that Φ 0 Φ 1 Φ n Theorem 4.2 ([29]) Irr(S) = Φ(X) is a linear basis of K X; P S. Proof: The result follows from Theorem 3.2 and Theorem 4.1. Thus, we have the following theorem. Theorem 4.3 ([29]) K X; P S is a free Rota-Baxter algebra on set X with a K-basis Φ(X). By using ELWs, we have the following algorithm. In fact, it is an algorithm to compute the product of two Rota-Baxter words in the free Rota-Baxter algebra K X; P S. It coincides with the product defined in [29]. Algorithm Let u, v Φ(X). We define u v by induction on n = dep(u) + dep(v). (a) If n = 0, then u, v S(X) and u v = uv. (b) If n 1, there are two cases to consider: (i) If bre(u) = bre(v) = 1, i.e., u, v X P(Φ(X)), then { uv if u X or v X u v = P(P(u ) v ) + P(u P(v )) + λp(u v ) if u = P(u ) and v = P(v ) (ii) If bre(u) > 1 or bre(v) > 1, say, u = u 1 u 2 u t or v = v 1 v 2 v l, where u i and v j are prime, then u v = u 1 u 2 u t 1 (u t v 1 )v 2 v l. 5 Gröbner-Shirshov bases for free λ-differential algebras In this section, we give two Gröbner-Shirshov bases and two linear bases for a free λ- differential algebra. We construct the free λ-differential algebra on a set X by using the Composition-Diamond Lemma (Theorem 3.2). 12

13 Let K be a commutative ring with unit and λ K. A λ-differential algebra over K ([35]) is an associative K-algebra R together with a K-linear operator D : R R such that D(xy) = D(x)y + xd(y) + λd(x)d(y), x, y R. Any λ-differential algebra is also an associative algebra with one operator Ω = {D}. Let X be well ordered and K X; D the free associative algebra with one operator Ω = {D} defined as before. For any u S(X), u has a unique expression u = u 1 u 2 u n, where each u i X D(S(X)). Denote by deg X (u) the number of x X in u. For example, if u = D(x 1 x 2 )D(D(x 1 ))x 3 S(X), then deg X (u) = 4. Let Let v = v 1 v 2 v m S(X). Define wt(u) = (deg X (u), u 1, u 2,, u n ). u > v wt(u) > wt(v) lexicographically, (2) where for each t, u t > v t if one of the following holds: (a) u t, v t X and u t > v t ; (b) u t = D(u t), v t X; (c) u t = D(u t ), v t = D(v t ) and u t > v t. It is easy to see the order (2) is a monomial ordering on S(X). Theorem 5.1 With the order (2) on S(X), S = {D(xy) D(x)y xd(y) λd(x)d(y) x, y S(X)} is a Gröbner-Shirshov basis in K X; D. Proof: The ambiguities of all possible compositions of the Ω-polynomials in S are: (a) D(v(u D(xy) )) (b) D(u D(xy) v) where x, y, v S(X), u S (X). It is easy to check that all these compositions are trivial. Here, for example, we just check (b). Others are similarly proved. Let g(x, y) = D(xy) D(x)y xd(y) λd(x)d(y). 13

14 Then (g(u D(xy), v), g(x, y)) D((u D(xy) )v) = D(u D(xy) )v u D(xy) D(v) λd(u D(xy) )D(v) +D((u D(x)y )v) + D((u xd(y) )v) + λd((u D(x)D(y) )v) D(u D(x)y )v D(u xd(y) )v λd(u D(x)D(y) )v u D(x)y D(v) u xd(y) D(v) λu D(x)D(y) D(v) λd(u D(x)y )D(v) λd(u xd(y) )D(v) λ 2 D(u D(x)D(y) )D(v) +D(u D(x)y )v + u D(x)y D(v) + λd(u D(x)y )D(v) +D(u xd(y) )v + u xd(y) D(v) + λd(u xd(y) )D(v) +λd(u D(x)D(y) )v + λu D(x)D(y) D(v) + λ 2 D(u D(x)D(y) )D(v) 0 mod(s, D((u D(xy) )v)). Let D ω (X) = {D i (x) i 0, x X}, where D 0 (x) = x and S(D ω (X)) the free semigroup generated by D ω (X). Theorem 5.2 ([35]) Irr(S) = S(D ω (X)) is a K-basis of K X; D S. Proof: The result follows from Theorem 3.2 and Theorem 5.1. Thus, we have the following theorem. Theorem 5.3 ([35]) K X; D S is a free λ-differential algebra on set X with a linear basis S(D ω (X)). By using ELWs, we have the following algorithm. Algorithm ([35]) Let u = u 1 u 2 u t S(D ω (X)), where u i D ω (X). Define D(u) by induction on t. (a) If t = 1, i.e., u = D i (x) for some i 0, x X, then D(u) = D (i+1) (x). (b) If t 1, then D(u) = D(u 1 u 2 u t ) = D(u 1 )(u 2 u t ) + u 1 D(u 2 u t ) + λd(u 1 )D(u 2 u t ). The following theorem gives another Gröbner-Shirshov basis in K X; D when λ K is invertible. Theorem 5.4 Let λ K be invertible and S(X) with the order (1). Set T = {D(x)D(y) + λ 1 D(x)y + λ 1 xd(y) λ 1 D(xy) x, y S(X)}. Then K X; D S = K X; D T and T is a Gröbner-Shirshov basis in K X; D. 14

15 Proof: The proof is similar to the Theorem 4.1. Now let For n > 0, define Let Φ 0 = S(X). Φ n = Λ D X (Φ 0, Φ n 1 ). Φ(X) = n 0Φ n. Theorem 5.5 Let λ K be invertible. Then Irr(T) = Φ(X) is a K-basis of K X; D T. Proof: By Theorem 5.4 and Theorem 3.2, we can obtain the result. By Theorem 5.5 and Theorem 5.3, we have the following theorem. Theorem 5.6 Let λ K be invertible. Then K X; D T is a free λ-differential algebra on set X with a linear basis Φ(X) as above. 6 Gröbner-Shirshov bases for free λ-differential Rota- Baxter algebras In this section, we obtain a Gröbner-Shirshov basis and a linear basis for a free λ- differential Rota-Baxter algebra. Let K be a commutative ring with unit and λ K. A differential Rota-Baxter algebra of weight λ ([35]), called also λ-differential Rota-Baxter algebra, is an associative K- algebra R with two K-linear operators P, D : R R such that for any x, y R, (I) (Rota-Baxter relation) P(x)P(y) = P(xP(y)) + P(P(x)y) + λp(xy); (II) (λ-differential relation) D(xy) = D(x)y + xd(y) + λd(x)d(y); (III) D(P(x)) = x. Hence, any λ-differential Rota-Baxter algebra is an associative algebra with two linear operators Ω = {P, D}. Let K X; Ω be the free associative algebra with multiple linear operators Ω on X, where Ω = {P, D}. For any u S(X), u has a unique expression u = u 0 P(b 1 )u 1 P(b 2 )u 2 P(b n )u n, where each u i (X D(S(X))) and b i S(X). Denote by wt(u) = (deg X (u), deg P (u), n, b 1,, b n, u 0,, u n ), where deg P (u) is the number of P in u. Also, for any u t (X D(S(X))), u t has a unique expression u t = u t1 u tk where each u tj X D(S(X)). 15

16 Let X be well ordered and v = v 0 P(c 1 )v 1 P(c 2 )v 2 P(c m )v m. Order S(X) as follows: where for each t, u t > v t if u > v wt(u) > wt(v) lexicographically (3) (deg X (u t ), deg P (u t ), u t1,, u tk ) > (deg X (v t ), deg P (v t ), v t1,, v tl ) lexicographically where for each j, u tj > v tj if one of the following holds: (a) u tj, v tj X and u tj > v tj ; (b) u tj = D(u t j ), v tj X; (c) u tj = D(u t j ), v tj = D(v t j ) and u t j > v t j. Clearly, the order (3) is a monomial ordering on S(X). Let S be the set consisting of the following Ω-polynomials: 1. P(x)P(y) P(xP(y)) P(P(x)y) λp(xy), 2. D(xy) D(x)y xd(y) λd(x)d(y), 3. D(P(x)) x, where x, y S(X). Theorem 6.1 With the order (3) on S(X), S is a Gröbner-Shirshov basis in K X; Ω. Proof. Denote by i j the composition of the Ω-polynomials of type i and type j. The ambiguities of all possible compositions of the Ω-polynomials in S are only as below: 3 3 D(P(u D(P(x)) )) 3 2 D(P(u D(xy) )) 3 1 D(P(u P(x)P(y) )) 2 3 D(x(u D(P(y)) )), D((u D(P(x)) )y) 2 2 D(x(u D(yz) )), D((u D(xy) )z) 2 1 D((u P(x)P(y) )z), D(x(u P(y)P(z) )) 1 3 P(u D(P(x)) )P(z), P(x)P(u D(P(y)) ) 1 2 P(u D(xy) )P(z), P(x)P(u D(yz) ) 1 1 P(x)P(y)P(z), P(u P(x)P(y) )P(z), P(x)P(u P(y)P(z) ) where x, y, z S(X), u S (X). Similar to the proofs of Theorem 4.1 and Theorem 5.1, we have 1 1 and 2 2 are trivial. Others are also easily checked. Here we just check 3 2: 3 2 = u D(xy) + D(P(u D(x)y )) + D(P(u xd(y) )) + λd(p(u D(x)D(y) )) u D(x)y u xd(y) λu D(x)D(y) + u D(x)y + u xd(y) + λu D(x)D(y) 0 mod(s, D(P(u D(xy) ))). 16

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