aa r X i v : . [ m a t h . R A ] J a n Gr¨obner-Shirshov bases for categories ∗ L. A. Bokut † School of Mathematical Sciences, South China Normal UniversityGuangzhou 510631, P. R. ChinaSobolev Institute of Mathematics, Russian Academy of SciencesSiberian Branch, Novosibirsk 630090, [email protected]
Yuqun Chen ‡ and Yu Li School of Mathematical Sciences, South China Normal UniversityGuangzhou 510631, P. R. [email protected]@126.com
Abstract:
In this paper we establish Composition-Diamond lemma for small categories.We give Gr¨obner-Shirshov bases for simplicial category and cyclic category.
Key words:
Gr¨obner-Shirshov basis, simplicial category, cyclic category.
AMS 2000 Subject Classification : 16S15, 13P10, 18G30, 16E40
This paper devotes to Gr¨obner-Shirshov bases for small categories (all categories beloware supposed to be small) presented by a graph (=quiver) and defining relations (see,Maclane [57]). As important examples, we use the simpicial and the cyclic categories (see,for example, Maclane [58], Gelfand, Manin [43]). In an above presentation, a categoryis viewed as a “monoid with several objects”. A free category C ( X ), generated by agraph X , is just “free partial monoid of partial words” u = x i . . . x i n , n ≥ , x i ∈ X and all product defined in C ( X ). A relation is an expression u = v, u, v ∈ C ( X ),where sources and targets of u, v are coincident respectively. The same as for semigroups,we may use two equivalent languages: Gr¨obner-Shirshov bases language and rewritingsystems language. Since we are using the former, we need a Composition-Diamond lemma(CD-lemma for short) for a free associative partial algebra kC ( X ) over a field k , where kC ( X ) is just a linear combination of uniform (with the same sources and targets) partial ∗ Supported by the NNSF of China (Nos.10771077, 10911120389). † Supported by RFBR 09-01-00157, LSS–3669.2010.1 and SB RAS Integration grant No. 2009.97(Russia). ‡ Corresponding author. kC ( X ). It is a freecategory (“semigroup”) partial algebra of a free category. Remark that in the literatureone is usually used a language of rewriting system, see, for example, Malbos [53]. Let usstress that the partial associative algebras presented by graphs and defining relations areclosely related to the well known quotients of the path algebras from representation theoryof finitely dimensional algebras, see, for example, Assem, Simson, Skowro´nski [1]. Inthis respect Gr¨obner-Shirshov bases for categories are closely related to non-commutativeGr¨obner bases for quotients of path algebras, see Farkas, Feustel, Green [42]. Rewritingsystem language for non-commutative Gr¨obner bases of quotients of path algebras wasused by Kobayashi [48]. Main new results of this paper are Gr¨obner-Shirshov bases forthe simplicial and cyclic categories.All algebras are assumed to be over a field. What is now called Gr¨obner and Gr¨obner-Shirshov bases theory was initiated by A. I.Shirshov (1921-1981) [66, 67], 1962 for non-associative and Lie algebras, by H. Hiron-aka [45, 46], 1964 for quotients of commutative infinite series algebras (both formal andconvergent), and by B. Buchberger [32, 33], 1965, 1970 for commutative algebras.English translation of selected works of A. I. Shirshov, including [66, 67], is recentlypublished [68].Remark that Shirshov’s approach was a most universal as we understand now since Liealgebra case becomes a model for many classes of non-commutative and non-associativealgebras (with multiple operations), starting with associative algebras (see below). Hi-ronaka’s papers on resolution of singularities of algebraic varieties become famous verysoon and Hironaka got Fields Medal due to them few years latter. B. Buchberger’s thesisinfluenced very much many specialists in computer sciences, as well as in commutativealgebras and algebraic geometry, for huge important applications of his bases, named himunder his supervisor W. Gr¨obner (1898-1980).Original Shirshov’s approach for Lie algebras [67], 1962, based on a notion of com-position [ f, g ] w of two monic Lie polynomials f, g relative to associative word w , i.e., f, g are elements of a free Lie algebra Lie ( X ) regarded as the subspace of Lie poly-nomials of the free associative algebra k h X i , and w ∈ X ∗ , the free monoid generatedby X . The definition of Lie composition relies on a definition of associative composi-tion ( f, g ) w as (monic) associative polynomials (after worked out into f, g all Lie brackets[ x, y ] = xy − yx ) relative to degree-lexicographical order on X ∗ . Namely, ( f, g ) w = f b − ag ,where w = acb, ¯ f = ac, ¯ g = cb, a, b, c ∈ X ∗ , c = 1. Here ¯ f means the leading (maximal)associative word of f . Then ( f, g ) w belongs to associative ideal Id ( f, g ) of k h X i generatedby f, g , and the leading word of ( f, g ) w is less than w . Now we need to put some Liebrackets [ f b ] − [ ag ] on f b − ag in such a way that the result would belong to Lie idealgenerated by f, g (so we can not trouble bracketing into f, g ) and the leading associa-tive monomial of [ f b ] − [ ag ] must be less than w . To overcome these obstacles Shirshovused his previous paper [64], 1958 with a new linear basis of free Lie algebra Lie ( X ).As it happened the same linear basis of Lie ( X ) was discovered in the paper Chen, Fox,Lyndon [34], 1958. Now this basis is called Lyndon-Shirshov basis, or, by a mistake,2yndon basis. It consists of non-associative Lyndon-Shirshov words (NLSW) [ u ] in X ,that are in one-one correspondence with associative Lyndon-Shirshov words (ALSW) u in X . The latter is defined as by a property u = vw > wv for any v, w = 1. Shirshov [64],1958 introduced and used the following properties of both associative and non-associativeLyndon-Shirshov words:1) For any ALSW u there is a unique bracketing [ u ] such that [ u ] is a NLSW.There are two algorithms for bracketing an ALSW. He mostly used “down-to-up al-gorithm” to rewrite an ALSW u on a new alphabet X u = { x i ( x β ) j , i > β, j ≥ , x β is the minimal letter in u } ; the result u ′ is again ALSW on X u with the lex-order x i > x i x β > (( x i x β ) x β ) > . . . .It is Shirshov’s rewriting or elimination algorithm from his famous paper Shirshov [63],1953, on what is now called Shirshov-Witt theorem (any subalgebra of a free Lie algebrais free). This rewriting was rediscovered by Lazard [49], 1960 and now called as Lazardelimination (it is better to call Lazard-Shirshov elimination).There is “up-to-down algorithm” (see Shirshov [65], 1958, Chen, Fox, Lyndon [34],1958): [ u ] = [[ v ][ w ]], where w is the longest proper end of u that is ALSW, in this case v is also an ALSW.2) Leading associative word of NLSW [ u ] is just u (with the coefficient 1).3) Leading associative word of any Lie polynomial is an associative Lyndon-Shirshovword.4) A non-commutative polynomial f is a Lie polynomial if and only if f = f → · · · → f i → f i +1 → · · · → f n = 0 , where f i → f i +1 = f i − α i [ u i ] , ¯ f i = u i is an ALSW, α i is the leading coefficient of f i , i = 0 , , . . . .5) Any associative word c = 1 is the unique product of (not strictly) increasing sequenceof associative Lyndon-Shirshov words: c = c c · · · c n , c ≤ · · · ≤ c n , c i are ALSW’s.6) If u = avb , where u, v are ALSW, a, b ∈ X ∗ , then there is a relative bracketing[ u ] v = [ a [ v ] b ] of u relative to v , such that the leading associative word of [ u ] v is just u .Namely, [ u ] = [ a [ vc ] d ] , cd = b, [ u ] v = [ a [[ v [ c ]] . . . [ c n ]] d ] , c = c c · · · c n as above.7) If ac and cb are ASLW’s and c = 1 then acb is an ALSW as well. If a, b are ALSW’sand a > b , then ab is an ALSW as well.Property 5) was known to Chen, Fox, Lyndon [34], 1958 as well. Lyndon [52], 1954, wasactually the first for definition of associative “Lyndon-Shirshov” words. To the best of ourknowledge, for many years, until PhD thesis by Viennot [69], 1978, no one mentioned theLyndon’s discovery in 1954. On the other hand, there were dozens of papers and somebooks on Lie algebras that mentioned both associative and non-associative “Lyndon-Shirshov” words as Shirshov’s regular words, see, for example, P. M. Cohn [38], 1965,Bahturin [2], 1978.Now a Lie composition [ f, g ] w of monic Lie polynomials f, g relative to a word w =¯ f b = a ¯ g = acb, c = 1 is defined by Shirshov [67], 1962, as follows[ f, g ] w = [ f b ] ¯ f − [ ag ] ¯ g , where [ f b ] ¯ f means the result of substitution f for [ ¯ f ] into the relative bracketing [ w ] ac of w with respect to ¯ f = ac , the same for [ ag ] ¯ g .3ccording to the definition and properties above, any Lie composition [ f, g ] w is an ele-ment of the Lie ideal generated by f, g , and the leading associative word of the compositionis less than w .The composition above is now called composition of intersection. Shirshov avoidedwhat is now called composition of inclusion[ f, g ] w = f − [ agb ] ¯ g , w = ¯ f = a ¯ gb, assuming that any system S of Lie polynomials is reduced (irreducible) in a sense thatleading associative word of any polynomial from S does not contain leading associativewords of another polynomials from S . This assumption relies on his algorithm of elimi-nation of leading words for Lie polynomials below.For associative polynomials the elimination algorithm is just non-commutative versionof the Euclidean elimination algorithm. For Lie polynomial case Shirshov [67], 1962,defined the elimination of a leading word as follows:If w = avb , where w, v are ALSW’s, and v is the leading word of some monic Liepolynomial f , then the transformation [ w ] [ w ] − [ af b ] v is called an elimination ofleading word of f into [ w ]. The result of Lie elimination is a Lie polynomial with aleading associative word less than w .Then Shirshov [67], 1962, formulated an algorithm to add to an initial reduced systemof Lie polynomials S a “non-trivial” composition [ f, g ] w , where f, g belong to S . Non-triviality of a Lie polynomial h relative to S means that h is not going to zero using“elimination of leading words of S ”. Actually, he defines to add to S not just a compositionbut rather the result of elimination of leading words of S into the composition in orderto have a reduced system as well.Then Shirshov proved the following Composition Lemma.
Let S be a reduced subset of Lie ( X ). If f belongs to the Lieideal generated by S , then the leading associative word ¯ f contains, as a subword, someleading associative word of a reduced multi-composition of elements of S .He constantly used the following clear Corollary.
The set of all irreducible NLSW’s [ u ] such that u does not contain anyleading associative word of a reduced multi-composition of elements of S is a linear basisof the quotient algebra Lie ( X ) /Id ( S ).Some later (see Bokut [8], 1972) the Shirshov Composition lemma was reformulated inthe following form: Let S be a closed under composition set of monic Lie polynomials(it means that any composition [ f, g ] w of intersection and inclusion of elements of S istrivial, i.e., [ f, g ] w = P α i [ a i s i b i ] , [ a i s i b i ] = a i s i b i < w, a i , b i ∈ X ∗ , s i ∈ S, α i ∈ k ). If f ∈ Id ( S ), then ¯ f = a ¯ sb for some s ∈ S . And S -irreducible NLSW’s is a linear basis ofthe quotient algebra Lie ( X ) /Id ( S ).The modern form of Shirshov’s lemma is the following (see, for example, Bokut, Chen[12]). Shirshov’s Composition-Diamond Lemma for Lie algebras.
Let
Lie ( X ) be afree Lie algebra over a field, S monic subset of Lie ( X ) relative to some monomial orderon X ∗ . Then the following conditions are equivalent:1) S is a Gr¨obner-Shirshov basis (i.e., any composition of intersection and inclusion ofelements of S is trivial). 4) If f ∈ Id ( S ), then ¯ f = a ¯ sb for some s ∈ S, a, b ∈ X ∗ .3) Irr ( S ) = { [ u ] | [ u ] is an NLSW and u does not contain any ¯ s, s ∈ S } is a linear basisof the Lie algebra Lie ( X | S ) with defining relations S .The proof of the Shirshov’s Composition-Diamond lemma for Lie algebras becomes amodel for proofs of number of Composition-Diamond lemmas for many classes of algebras.An idea of his proof is to rewrite any element of Lie ideal, generated by S in a form X α i [ a i s i b i ] , where each s i ∈ S, a i , b i ∈ X ∗ , α i ∈ k such thati) leading words of each [ a i s i b i ] is equal to a i s i b i (in this case an expression [ asb ] iscalled normal Lie S -word in X ) andii) a s b > a s b > . . . . Now let F ( X ) be a free algebra of a variety (or category) of algebras. Following theidea of Shirshov’s proof, one needs1) to define appropriate linear basis (normal words) of F(X),2) to define monomial order of normal words,3) to define compositions of element of S (they may be compositions of intersection,inclusion and left (right) multiplication, or may be else),4) prove two key lemmas:Key Lemma 1. Let S be a Gr¨obner-Shirshov basis (any composition of polynomialsfrom S is trivial). Then any S -word is a linear combination of normal S -words.Key Lemma 2. Let S be a Gr¨obner-Shirshov basis, [ a s b ] and [ a s b ] normal S -words, s , s ∈ S . If a s b = a s b , then [ a s b ] − [ a s b ] is going to zero by elimination ofleading words of S (elimination means composition of inclusion).There are number of CD-lemmas that realized Shirshov’s approach to them.Shirshov [67], 1962, assumed implicitly that his approach, based on the definition ofcomposition of any (not necessary Lie) polynomials, is equally valid for associative al-gebras as well (the first author is a witness that Shirshov understood it very clearlyand explicitly; only lack of non-trivial applications prevents him from publication thisapproach for associative algebras). Explicitly it was done by Bokut [9] and Bergman [5].CD-lemma for associative algebras is formulated and proved in the same way as for Liealgebras. Composition-Diamond Lemma for associative algebras.
Let k h X i be a freeassociative algebra over a field k and a set X . Let us fix some monomial order on X ∗ .Then the following conditions are equivalent for any monic subset S of k h X i :1) S is a Gr¨obner-Shirshov basis (that is any composition of intersection and inclusionis trivial).2) If f ∈ Id ( S ), then ¯ f = a ¯ sb for some s ∈ S, a, b ∈ X ∗ .3) Irr ( S ) = { u ∈ X ∗ | u = a ¯ sb, s ∈ S, a, b ∈ X ∗ } is a linear basis of the factor algebra k h X | S i = k h X i /Id ( X ).There are a lot of applications of Shirshov’s CD-lemmas for Lie and associative algebras.Let us mention some connected to the Malcev embedding problem for semigroup algebras5Bokut [6, 7], 1969, there is a semigroup S such that the multiplication semigroup ofthe semigroup algebra k ( S ), where k is a field, is embeddable into a group, but k ( S ) isnot embeddable into any division algebra), the unsolvability of the word problem for Liealgebras (Bokut [8]), Gr¨obner-Shirshov bases for semisimple Lie algebras (Bokut, Klein[25, 26, 27, 28]), Kac-Moody algebras (Poroshenko [59, 60, 61]), finite Coxeter groups(Bokut, Shiao [30]), braid groups in different set of generators (Bokut, Chainikov, Shum[23], Bokut [10], Bokut [11]), quantum algebra of type A n (Bokut, Malcolmson [29]),Chinese monoids (Chen, Qiu [35]).There are applications of Shirshov’s CD-lemma [66], 1962 for free anti-commutativenon-associative algebras: there are two anti-commutative Gr¨obner-Shirshov bases of afree Lie algebra, one gives the Hall basis (Bokut, Chen, Li [17]), another the Lyndon-Shirshov basis (Bokut, Chen, Li [18]).Bokut, Chen, Mo [20] proved and reproved some embedding theorems for associativealgebras, Lie algebras, groups, semigroups, differential algebras, using Shirshov’s CD-lemmas for associative and Lie algebras.Bahturin, Olshanskii [3] found embeddings without distortion of associative algebrasand Lie algebras into 2-generated simple algebras. They also used Shirshov’s CD-lemmasfor associative and Lie algebras.Mikhalev [54] used Shirshov’s approach and CD-lemma for associative algebras to proveCD-lemma for colored Lie super-algebras.Mikhalev, Zolotykh [56] proved CD-lemma for free associative algebra over a commuta-tive algebra.A Free object in this category is k [ Y ] ⊗ k h X i , tensor product of a polynomialalgebra and a free associative algebra. Here one needs to use several compositions ofintersection and inclusion.Bokut, Fong, Ke [24] proved CD-lemma for free associative conformal (in a sense of V.Kac [47] algebra C ( X, ( n ) , n = 0 , , . . . , D, N ( a, b ) , a, b ∈ X ) of a fixed locality N ( a, b ).A linear basis of free associative conformal algebra was constructed by M. Roitman [62].Any normal conformal word has a form [ u ] = a ( n )[ a ( n )[ . . . [ a k ( n k ) D i a k +1 ] . . . ]], where a j ∈ X, n j < N ( a j , a j +1 ) , i ≥
0. The same word without brackets is called the leadingassociative word of [ u ]. One needs to use external multi-operator semigroup as a set ofleading associative words of conformal polynomials (it is the same as for Lie algebras),several compositions of inclusion and intersection, and new compositions of left (right)multiplication (last compositions are absent into classical cases). Also in the CD-lemmafor conformal algebras we have 1) ⇒ ⇔ U ( L ) of any Lie conformalalgebra L are trivial (it is called “1/2 PBW theorem”).Bokut, Chen, Zhang [22] proved CD-lemma for associative n -conformal algebras, whereinstead of one derivation D and polynomial algebra k [ D ] one has n derivations D , . . . , D n and polynomial algebra k [ D , . . . , D n ]. This case is treated in the same way as for n = 1.A more general case, the associative H -conformal algebra (or H -pseudo-algebra in a senseof Bakalov, D’Andrea, Kac [4]), where H is any Hopf algebra, is still open.Mikhalev, Vasilieva [55] proved CD-lemma for free supercommutative polynomial alge-6ras. Here they use compositions of multiplication as well.Bokut, Chen, Li [16] proved CD-lemma for free pre-Lie algebras (also known as Vinberg-Koszul-Gerstenhaber right-symmetric algebras).Bokut, Chen, Liu [19] proved CD-lemma for free dialgebras in a sense of Loday [50].Here conditions 1) and 2) are not equivalent but from 1) follows 2).The cases of associative conformal algebras and dialgebras show that definition ofGr¨obner-Shirshov bases by condition 1) is in general preferable than the one using 2).Bokut, Shum [31] proved CD-lemma for free Γ-associative algebras, where Γ is a group.It has applications to the Malcev problem above and to Bruhat normal forms for algebraicgroups.Eisenbud, Peeva, Sturmfels [41] found non-commutative Gr¨obner basis of any commu-tative algebra (extending any commutative Gr¨obner basis to a non-commutative one).Bokut, Chen, Chen [14] proved CD-lemma for Lie algebras over commutative algebras.Here one needs to establish Key Lemma 1 in a more strong form – any Lie S -word is a linearcombination of S -words of the form [ asb ] ¯ s in the sense of Shirshov’s special Lie bracketing.As an application they proved Cohn’s conjecture [37] for the case of characteristics 2, 3and 5 (that some Cohn’s examples of Lie algebras over commutative algebras are notembeddable into associative algebras over the same commutative algebras).Bokut, Chen, Deng [15] proved CD-lemma for free associative Rota-Baxter algebras.As an application, Chen and Mo [36] proved that any dendriform algebra is embeddableinto universal enveloping Rota-Baxter algebra. It was Li Guo’s conjecture, [44].Bokut, Chen, Chen [13] proved CD-lemma for tensor product of two free associa-tive algebras. As an application they extended any Mikhalev-Zolotyh commutative-non-commutative Gr¨obner-Shirshov basis laying into tensor product k [ Y ] ⊗ k h X i to non-commutative-non-commutative Gr¨obner-Shirshov basis laying into k h Y i ⊗ k h X i (a laEisenbud-Peeva-Sturfels above). They also gave another proof of the Eisenbud-Peeva-Sturmfekls theorem above.As we mentioned in introduction, Farkas, Feustel, Green [42] proved CD-lemma forpath algebras.Drensky, Holtkamp [40] proved CD-lemma for nonassociative algebras with multiplelinear operators.Bokut, Chen, Qiu [21] proved CD-lemma for associative algebras with multiple linearoperators.Dotsenko, Khoroshkin [39] proved CD-lemma for operads. Let X = ( V ( X ) , E ( X )) be an oriented (multi) graph. Then the free category on X is C ( X ) = ( Ob ( X ) , Arr ( X )), where Ob ( X ) = V ( X ) , and Arr ( X ) is the set of all paths(“words”) of X including the empty paths 1 v , v ∈ V ( X ). It is easy to check C ( X )has the following universal property. Let C be a category and Γ C the graph relativeto C i.e., V (Γ C ) = Ob ( C ) and E (Γ C ) = mor ( C ). Let e : X → C ( X ) be a monograph morphism of the graph X to the graph Γ C ( X ) , where e = ( e , e ), and e is amapping on V ( X ) , e on E ( X ) , both e and e are mono. For any graph morphism7 from X to Γ C , where b = ( b , b ), and b is mono, there exists a unique categorymorphism (a functor) f : C ( X ) → C , such that the corresponding diagram is commutativei.e., f e = b . Therefore each category C is a homomorphic image of a free category C ( X ) for some graph X and thus C is isomorphic to C ( X ) /ρ ( S ) for some set S , where S = { ( u, v ) | u, v have the same sources and the same targets } ⊆ Arr ( X ) × Arr ( X ) and ρ ( S ) the congruence of C ( X ) generated by S . If this is the case, X is called the generatingset of C and S the relation set of C and we denote C = C ( X | S ).Let C be a category and k a field. Let k C = { f = n X i =1 α i µ i | α i ∈ k, µ i ∈ mor ( C ) , n ≥ ,µ i (0 ≤ i ≤ n ) have the same domains and the same codomains } . Note that in k C , for f, g ∈ k C , f + g is defined only if f, g have the same domainand the same codomain.A multiplication • in k C is defined by linearly extending the usual compositions ofmorphisms of the category C . Then ( k C , • ) is called the category partial algebra over k relative to C and kC ( X ) the free category partial algebra generated by the graph X . Let X be a oriented (multi) graph, C ( X ) the free category generated by X and kC ( X ) thefree category partial algebra. Since we only consider the morphisms of the free category C ( X ), we write C ( X ) just for Arr ( X ).Note that for f, g ∈ kC ( X ) if we write gf , it means gf is defined.A well ordering > on C ( X ) is called monomial if it satisfies the following conditions: u > v ⇒ uw > vw and wu > wv , for any u, v, w ∈ C ( X ). In fact, there are manymonomial orders on C ( X ). For example, let E ( X ) be a well ordered set. Then thedeg-lex order > on C ( X ) is defined by the following way: for any words u = x · · · x m , v = y · · · y n ∈ C ( X ) , m = | u | , n = | v | , u > v ⇐⇒ | u | > | v | or( | u | = | v | and x = y , x = y , . . . , x t = y t , x t +1 > y t +1 for some 0 ≤ t < n ) . It is easy to check that > is a monomial order on C ( X ). In the following sections, wewill see other monomial orders. Now, we suppose that > is a fixed monomial order on C ( X ). Given a nonzero polynomial f ∈ kC ( X ), it has a word ¯ f ∈ C ( X ) such that f = αf + P α i u i , where f > u i , = α, α i ∈ k, u i ∈ C ( X ). We call f the leading termof f and f is monic if α = 1.Let S ⊂ kC ( X ) be a set of monic polynomials, s ∈ S and u ∈ C ( X ). We define S -word u s by induction:(i) u s = s is an S -word of s -length 1. 8ii) Suppose that u s is an S -word of s -length m and v is a word of length n , i.e., thenumber of edges in v is n . Then u s v and vu s are S -words of s length m + n .Note that for any S -word u s = asb , where a, b ∈ C ( X ), we have asb = a ¯ sb .Let f, g be monic polynomials in kC ( X ). Suppose that there exist w, a, b ∈ C ( X ) suchthat w = ¯ f = a ¯ gb . Then we define the composition of inclusion( f, g ) w = f − agb. For the case that w = ¯ f b = a ¯ g , w, a, b ∈ C ( X ), the composition of intersection is definedas follows: ( f, g ) w = f b − ag. It is clear that ( f, g ) w ∈ Id ( f, g ) and ( f, g ) w < w, where Id ( f, g ) is the ideal of kC ( X ) generated by f, g .The composition ( f, g ) w is trivial modulo ( S, w ), if( f, g ) w = X i α i a i s i b i where each α i ∈ k, a i , b i ∈ C ( X ) , s i ∈ S, a i s i b i an S -word and a i ¯ s i b i < ¯ f . If this is thecase, then we write ( f, g ) w ≡ mod ( S, w ). In general, for p, q ∈ kC ( X ), we write p ≡ q mod ( S, w )which means that p − q = P α i a i s i b i , where each α i ∈ k, a i , b i ∈ C ( X ) , s i ∈ S, a i s i b i an S -word and a i ¯ s i b i < w . Definition 4.1
Let S ⊂ kC ( X ) be a nonempty set of monic polynomials. Then S iscalled a Gr¨obner-Shirshov basis in kC ( X ) if any composition ( f, g ) w with f, g ∈ S istrivial modulo ( S, w ) , i.e., ( f, g ) w ≡ mod ( S, w ) . Lemma 4.2
Let a s b , a s b be monic S -words. If S is a Gr¨obner-Shirshov basis in kC ( X ) and w = a s b = a s b , then a s b ≡ a s b mod ( S, w ) . Proof.
There are three cases to consider.Case 1. Suppose that subwords ¯ s and ¯ s of w are disjoint, say, | a | ≥ | a | + | ¯ s | . Then,we can assume that a = a ¯ s c and b = c ¯ s b for some c ∈ C ( X ), and so, w = a ¯ s c ¯ s b . Now, a s b − a s b = a s c ¯ s b − a ¯ s cs b = a s c (¯ s − s ) b + a ( s − ¯ s ) cs b . Since s − s < ¯ s and s − s < ¯ s , we conclude that a s b − a s b = X i α i u i s v i + X j β j u j s v j α i , β j ∈ k , S -words u i s v i and u j s v j such that u i ¯ s v i , u j ¯ s v j < w. This shows that a s b ≡ a s b mod ( S, w ).Case 2. Suppose that the subword ¯ s of w contains ¯ s as a subword. We may assumethat ¯ s = a ¯ s b, a = a a and b = bb , that is, w = a a ¯ s bb for some S -word as b . Wehave a s b − a s b = a s b − a as bb = a s − as bb = a ( s , s ) s b ≡ mod ( S, w )since S is a Gr¨obner-Shirshov basis.Case 3. ¯ s and ¯ s have a nonempty intersection as a subword of w . We may assumethat a = a a, b = bb , w = ¯ s b = a ¯ s . Then, we have a s b − a s b = a s bb − a as b = a ( s b − as ) b = a ( s , s ) w b ≡ mod ( S, w )This completes the proof. (cid:3)
Lemma 4.3
Let S ⊂ kC ( X ) be a subset of monic polynomials and Irr ( S ) = { u ∈ C ( X ) | u = a ¯ sb, a, b ∈ C ( X ) , s ∈ S } . Then for any f ∈ kC ( X ) , f = X u i ≤ ¯ f α i u i + X a j s j b j ≤ ¯ f β j a j s j b j where each α i , β j ∈ k, u i ∈ Irr ( S ) and a j s j b j an S -word. Proof.
Let f = P i α i u i ∈ kC ( X ), where 0 = α i ∈ k and u > u > · · · . If u ∈ Irr ( S ),then let f = f − α u . If u Irr ( S ), then there exist some s ∈ S and a , b ∈ C ( X ),such that ¯ f = u = a ¯ s b . Let f = f − α a s b . In both cases, we have ¯ f < ¯ f . Thenthe result follows from the induction on ¯ f . (cid:3) Theorem 4.4 (Composition-Diamond lemma for categories) Let S ⊂ kC ( X ) be a nonemptyset of monic polynomials and < a monomial order on C ( X ) . Let Id ( S ) be the ideal of kC ( X ) generated by S . Then the following statements are equivalent:(i) S is a Gr¨obner-Shirshov basis in kC ( X ) .(ii) f ∈ Id ( S ) ⇒ ¯ f = a ¯ sb for some s ∈ S and a, b ∈ C ( X ) . ( ii ) ′ f ∈ Id ( S ) ⇒ f = α a s b + α a s b + · · · , where each α i ∈ k, a i s i b i is an S -wordand a ¯ s b > a ¯ s b > · · · . iii) Irr ( S ) = { u ∈ C ( X ) | u = a ¯ sb a, b ∈ C ( X ) , s ∈ S } is a linear basis of the partialalgebra kC ( X ) /Id ( S ) = kC ( X | S ) . Proof. ( i ) ⇒ ( ii ). Let S be a Gr¨obner-Shirshov basis and 0 = f ∈ Id ( S ). Then, wehave f = n X i =1 α i a i s i b i , where each α i ∈ k, a i , b i ∈ C ( X ) , s i ∈ S and a i s i b i an S -word. Let w i = a i s i b i , w = w = · · · = w l > w l +1 ≥ · · · , l ≥ . We will use the induction on l and w to prove that f = asb for some s ∈ S and a, b ∈ C ( X ).If l = 1, then f = a s b = a s b and hence the result holds. Assume that l ≥ a s b ≡ a s b mod ( S, w ) . Thus, if α + α = 0 or l >
2, then the result holds by induction on l . For the case α + α = 0 and l = 2, we use the induction on w . Now, the result follows.( ii ) ⇒ ( ii ) ′ . Assume (ii) and 0 = f ∈ Id ( S ). Let f = α f + · · · . Then, by (ii), f = a s b . Therefore, f = f − α a s b , f < f , f ∈ Id ( S ) . Now, by using induction on f , we have ( ii ) ′ .( ii ) ′ ⇒ ( ii ). This part is clear.( ii ) ⇒ ( iii ). Suppose that P i α i u i = 0 in kC ( X | S ), where α i ∈ k , u i ∈ Irr ( S ). Itmeans that P i α i u i ∈ Id ( S ) in kC ( X ). Then all α i must be equal to zero. Otherwise, P i α i u i = u j ∈ Irr ( S ) for some j which contradicts (ii).Now, by Lemma 4.3, (iii) follows.( iii ) ⇒ ( i ). For any f, g ∈ S , by Lemma 4.3 and (iii), we have ( f, g ) w ≡ mod ( S, w ) . Therefore, S is a Gr¨obner-Shirshov basis. (cid:3) Remark.
If the category in Theorem 4.4 has only one object, then Theorem 4.4 is exactComposition-Diamond lemma for free associative algebras.
In this section, we give Gr¨obner-Shirshov bases for the simplicial category and the cycliccategory respectively. 11or each non-negative integer p , let [ p ] denote the set { , , , . . . , p } of integers in theirusual order. A (weakly) monotonic map µ : [ q ] → [ p ] is a function on [ q ] to [ p ] such that i ≤ j implies µ ( i ) ≤ µ ( j ). The objects [ p ] with morphisms all weakly monotonic maps µ constitute a category L called simplicial category. It is convenient to use two specialfamilies of monotonic maps ε iq : [ q − → [ q ] , η iq : [ q + 1] → [ q ]defined for i = 0 , , ...q (and for q > ε i ) by ε iq ( j ) = (cid:26) j, if i > j,j + 1 , if i ≤ j,η iq ( j ) = (cid:26) j, if i ≥ j,j − , if j > i. Let X = ( V ( X ) , E ( X )) be an oriented (multi) graph, where V ( X ) = { [ p ] | p ∈ Z + ∪{ }} and E ( X ) = { ε ip : [ p − → [ p ] , η jq : [ q + 1] → [ q ] | p > , ≤ i ≤ p, ≤ j ≤ q } . Let S ⊆ C ( X ) × C ( X ) be the relation set consisting of the following: f q +1 ,q : ε iq +1 ε j − q = ε jq +1 ε iq , j > i,g q,q +1 : η jq η iq +1 = η iq η j +1 q +1 , j ≥ i,h q − ,q : η jq − ε iq = ε iq − η j − q − , j > i, q − , i = j, i = j + 1 ,ε i − q − η jq − , i > j + 1 . Then the simplicial category L is just the category C ( X | S ) generated by X withdefining relation S , see Maclane [58], Theorem VIII. 5.2. We will give another proof inwhat follows.We order C ( X ) by the following way.Firstly, for any η ip , η jq ∈ { η ip | p ≥ , ≤ i ≤ p } , η ip > η jq iff p > q or ( p = q and i < j ).Secondly, for each u = η i p η i p · · · η i n p n ∈ { η ip | p ≥ , ≤ i ≤ p } * (all possible wordson { η ip | p ≥ , ≤ i ≤ p } , including the empty word 1 v , v ∈ Ob ( X )), let wt ( u ) =( n, η i n p n , η i n − p n − , · · · , η i p ) . Then for any u, v ∈ { η ip | p ≥ , ≤ i ≤ p } *, u > v iff wt ( u ) > wt ( v )lexicographically.Thirdly, for any ε ip , ε jq ∈ { ε ip , | p ∈ Z + , ≤ i ≤ p } , ε ip > ε jq iff p > q or ( p = q and i < j ).Finally, for each u = v ε i p v ε i p · · · ε i n p n v n ∈ C ( X ) , n ≥ v j ∈ { η ip | p ≥ , ≤ i ≤ p } *,let wt ( u ) = ( n, v , v , · · · , v n , ε i p , · · · , ε i n p n ) . Then for any u, v ∈ C ( X ), u ≻ v ⇔ wt ( u ) > wt ( v ) lexicographically . It is easy to check that the ≻ is a monomial order on C ( X ). Then we have the followingtheorem. Theorem 5.1
Let X , S be defined as the above, the generating set and the relation setof the quotient category C ( X | S ) respectively. Then with the order ≻ on C ( X ) , S is aGr¨obner-Shirshov basis for the category partial algebra kC ( X | S ) . roof. According to the order ≻ , ¯ f q +1 ,q = ε iq +1 ε j − q , ¯ g q,q +1 = η jq η iq +1 and ¯ h q − ,q = η jq − ε iq .So, all the possible compositions of S are the following:(a) ( f q +2 ,q +1 , f q +1 ,q ) εkq +2 εiq +1 εj − q , k ≤ i ≤ j − g q − ,q , g q,q +1 ) ηkq − ηjqηiq +1 , i ≤ j ≤ k ;(c) ( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q , i ≤ j − g q − ,q − , h q − ,q ) ηkq − ηjq − εiq , j ≤ k .We will prove that all possible compositions are trivial. Here, we only give the proof ofthe (b). For others cases, the proofs are similar.Let us consider the following subcases of the case (b): (I) i < j < k ; (II) i < j, j = k, or j = k + 1; (III) i < k, k + 1 < j ; (IV) j > k + 1 , i = k, k + 1; (V) j > i > k + 1.For subcase (I), ( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q = ε iq η k − q − ε j − q − η kq ε jq +1 ε iq ≡ ε iq ε j − q − η k − q − − ε jq η k − q − ε iq ≡ mod ( S, η kq ε iq +1 ε j − q ) . For subcase (II), ( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q = ε iq η k − q − ε j − q − η kq ε jq +1 ε iq ≡ mod ( S, η kq ε iq +1 ε j − q ) . For subcase (III),( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q = ε iq η k − q − ε j − q − η kq ε jq +1 ε iq ≡ ε iq ε j − q − η k − q − − ε j − q η kq − ε iq ≡ mod ( S, η kq ε iq +1 ε j − q ) . For subcase (IV),( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q = ε j − q − η kq ε jq +1 ε iq ≡ ε j − q − ε j − q η kq − ε iq ≡ mod ( S, η kq ε iq +1 ε j − q ) . For subcase (V),( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q = ε i − q η kq − ε j − q − η kq ε jq +1 ε iq ≡ ε i − q ε j − q − η kq − − ε j − q η kq − ε iq ≡ mod ( S, η kq ε iq +1 ε j − q ) . S is a Gr¨obner-Shirshov basis of the category partial algebra kC ( X | S ). (cid:3) By Theorem 4.4,
Irr ( S ) = { ε i p ...ε i m p − m +1 η j q − n ...η j n q − | p ≥ i > ... > i m ≥ , ≤ j <... < j n < q, and q − n + m = p } is a linear basis of the category partial algebra kC ( X | S ).Therefore, we have the following corollaries. Corollary 5.2 (Maclane [58], Lemma VIII. 5.1) In the category C ( X | S ) , each morphism µ : [ q ] → [ p ] can be uniquely represented as ε i p ...ε i m p − m +1 η j q − n ...η j n q − , where p ≥ i > ... > i m ≥ , ≤ j < ... < j n < q, and q − n + m = p . Corollary 5.3 (Maclane [58], Theorem VIII. 5.2) L = C ( X | S ) . The cyclic category is defined by generators and defining relations as follows, see [43].Let Y = ( V ( Y ) , E ( Y )) be an oriented (multi) graph, where V ( Y ) = { [ p ] | p ∈ Z + ∪ { }} ,and E ( Y ) = { ε ip : [ p − → [ p ] , η jq : [ q + 1] → [ q ] , t q : [ q ] → [ q ] | p > , ≤ i ≤ p, ≤ j ≤ q } . Let S ⊆ C ( Y ) × C ( Y ) be the set consisting of the following relations: f q +1 ,q : ε iq +1 ε j − q = ε jq +1 ε iq , j > i,g q,q +1 : η jq η iq +1 = η iq η j +1 q +1 , j ≥ i,h q − ,q : η jq − ε iq = ε iq − η j − q − , j > i, q − , i = j, i = j + 1 ,ε i − q − η jq − , i > j + 1 ,ρ : t q ε iq = ε i − q t q − , i = 1 , ..., q,ρ : t q η iq = η i − q t q +1 , ı = 1 , ..., q,ρ : t q +1 q = 1 q . The category C ( Y | S ) is called cyclic category, denoted by Λ.An order on C ( Y ) is defined by the following way.Firstly, for any t ip , t jq ∈ { t q | q ≥ } ∗ , ( t p ) i > ( t q ) j iff i > j or ( i = j and p > q ).Secondly, for any η ip , η jq ∈ { η ip | p ≥ , ≤ i ≤ p } , η ip > η jq iff p > q or ( p = q and i < j ).Thirdly, for each u = w η i p w η i p · · · w n − η i n p n w n ∈ { t q , η ip | q, p ≥ , ≤ i ≤ p } *,where w i ∈ { t q | q ≥ } ∗ , let wt ( u ) = ( n, w , w , · · · , w n , η i n p n , η i n − p n − , · · · , η i p ) . Then for any u, v ∈{ t q , η ip | q, p ≥ , ≤ i ≤ p } *, u > v iff wt ( u ) > wt ( v ) lexicographically.Fourthly, for any ε ip , ε jq ∈ { ε ip , | p ∈ Z + , ≤ i ≤ p } , ε ip > ε jq iff p > q or ( p = q and i < j ).Finally, for each u = v ε i p v ε i p · · · ε i n p n v n ∈ C ( Y ) , n ≥ v j ∈ { t q , η ip | q, p ≥ , ≤ i ≤ p } *, let wt ( u ) = ( n, v , v , · · · , v n , ε i p , · · · , ε i n p n ) . Then for any u, v ∈ C ( Y ), u ≻ v ⇔ wt ( u ) > wt ( v ) lexicographically . It is also easy to check the order ≻ is a monomial order on C ( Y ), which is an extensionof ≻ . Then we have the following theorem.14 heorem 5.4 Let Y , S be defined as the above, the generating set and the relation setof cyclic category C ( Y | S ) respectively. Let S C = S ∪ { ρ , ρ } , where ρ : t q ε q = ε qq ,ρ : t q η q = η qq t q +1 . Then(1) With the order ≻ on C ( Y ) , S C is a Gr¨obner-Shirshov basis for the cyclic categorypartial algebra kC ( Y | S ) .(2) For each morphism µ : [ q ] → [ p ] in the cyclic category Λ = C ( Y | S ) , µ can beuniquely represented as ε i p ...ε i m p − m +1 η j q − n ...η j n q − t kq , where p ≥ i > ... > i m ≥ , ≤ j < ... < j n < q, ≤ k ≤ q and q − n + m = p. Proof.
It is easy to check that ¯ f q +1 ,q = ε iq +1 ε j − q , ¯ g q,q +1 = η jq η iq +1 , ¯ h q − ,q = η jq − ε iq ,¯ ρ = t q ε iq , ¯ ρ = t q η iq , ¯ ρ = t q +1 q , ¯ ρ = t q ε q , and ¯ ρ = t q η q .First of all, we prove Id ( S ) = Id ( S C ). It suffices to show ρ , ρ ∈ Id ( S ). Since( ρ , ρ ) t q +1 q ε qq = t qq ε q − q t q − − ε qq ≡ t q ε q t qq − − ε qq ≡ t q ε q − ε qq = ρ and ( ρ , ρ ) t q +1 q η qq = t qq η q − q t q +1 − η qq ≡ t q η q t qq +1 − η qq , ρ and t q η q t qq +1 − η qq ∈ Id ( S ). Clearly, the leading term of thepolynomial t q η q t qq +1 − η qq is t q η q t qq +1 . Therefore ( t q η q t qq +1 − η qq , ρ ) t q η q t q +2 q +1 = − t q η q + η qq t q +1 and thus ρ ∈ Id ( S ) . Secondly, we prove that all possible compositions of S C are trivial which are the fol-lowing:(a) ( f q +2 ,q +1 , f q +1 ,q ) εkq +2 εiq +1 εj − q , k ≤ i ≤ j − g q − ,q , g q,q +1 ) ηkq − ηjqηiq +1 , i ≤ j ≤ k ;(c) ( h q,q +1 , f q +1 ,q ) ηkq εiq +1 εj − q , i ≤ j − g q − ,q − , h q − ,q ) ηkq − ηjq − εiq , j ≤ k ;(e) ( ρ , f q +1 ,q ) tq +1 εiq +1 εj − q , j > i and i = 1 , , . . . , q ;(f) ( ρ , ρ ) t q +1 q ε iq , i = 1 , , . . . , q ;(g) ( ρ , g q,q +1 ) t q η jq η iq +1 , j ≥ i and j = 1 , , . . . , q ;(h) ( ρ , h q,q +1 ) t q η jq ε iq +1 , j ≥ i and j = 1 , , . . . , q ;(i) ( ρ , ρ ) t q +1 q η iq , i = 1 , , . . . , q ;(j) ( ρ , ρ ) t q +1 q ε q ;(k) ( ρ , ρ ) t q +1 q η q ; 15l) ( ρ , f q +1 ,q ) tq +1 ε q +1 εj − q , j > ρ , g q,q +1 ) tqη q η q +1 ;(n) ( ρ , h q,q +1 ) tqη qεiq +1 , i ≥ ρ , h q,q +1 ) tqη q εiq +1 . The others can besimilarly proved. Let us consider the following subcases of the case (n): (I) i = 0; (II) i = 1; (III) i > ρ , h q,q +1 ) tqη qε q +1 = η qq t q +1 ε q +1 − t q ≡ η qq t q +1 ε q +1 q +1 − t q ≡ η qq ε qq +1 t q − t q ≡ mod ( S, t q η q ε q +1 ) . For subcase (II), ( ρ , h q,q +1 ) tqη qε q +1 = η qq t q +1 ε q +1 − t q ≡ η qq t q +1 ε q +1 t q − t q ≡ η qq ε q +1 q +1 t q − t q ≡ mod ( S, t q η q ε q +1 ) . For subcase (III), ( ρ , h q,q +1 ) tqη q εiq +1 = η qq t q +1 ε iq +1 − t q ε i − q η q − ≡ η qq ε i − q +1 t q − ε i − q t q − η q − ≡ mod ( S, t q η q ε iq +1 ) . Thus S C is a Gr¨obner-Shirshov basis of the category partial algebra kC ( Y | S ).Now, by Theorem 4.4, for each morphism µ : [ q ] → [ p ] in Λ = C ( Y | S ) can be uniquelyrepresented as ε i p ...ε i m p − m +1 η j q − n ...η j n q − t kq , where p ≥ i > ... > i m ≥ , ≤ j < ... < j n < q, ≤ k ≤ q and q − n + m = p. (cid:3) Remark.
According to Loday [51], the uniqueness property in Theorem 5.4 (2) wasknown.
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