Rota-Baxter Type Operators, Rewriting Systems, and Gröbner-Shirshov Bases, Part II

Rota-Baxter Type Operators, Rewriting Systems, and Gröbner-Shirshov Bases, Part II William Sit 1 The City College of The City University of New York Kolchin Seminar in Differential Algebra December 9, 2016 (Graduate Center, CUNY) 1 Joint work with Shanghua Zheng, Xing Gao and Li Guo

Part II Outline (Review) TRS Defined by Certain Operated Polynomials Rota-Baxter Type Identities and Conjecture Main Theorems on RBT OPI Characterization Theorems of RBT OPI Monomial Orders on Operated Monomials TRS in Free Operated Algebras Defined by Subsets Compatibility of Monomial Order and Rewriting Systems Tools to Prove Termination Restriction and a Lemma on Confluence Gröbner-Shirshov Basis Composition Diamond Lemma Construction of Free RBT OPI-Algebras Construction of Monomial Order for a RBT OPI Equivalences for Compatibility with db,z Recap of Main Theorems and Further Discussions

TRS Defined by Certain Operated Polynomials Let ϕ(x, y) k x, y be an operated polynomial of the form x y B(x, y), where B(x, y) k x, y. The rewriting system Π ϕ (Z) defined by ϕ on k Z with basis M(Z) is the relation Π ϕ (Z) := { (q u v, q B(u,v) ) q M (Z), u, v M(Z) } We write ϕ instead of Πϕ(Z) when Z is fixed. Proposition: Π ϕ (Z) is simple. f ϕ g if for some u, v M(Z), g is obtained from f by replacing exactly once a subword u v in one monomial w Supp(f ) by B(u, v). A bracketed polynomial g k Z is said to be a normal ϕ-form for f if g is in RBNF and f ϕ g.

Totally Linear Expressions For ϕ = x y B(x, y) to be of a linear operator, we need to put some conditions on B(x, y). An expression B k X is totally linear in X if every variables x X, when counted with multiplicity in repeated multiplications, appears exactly once in every monomial w Supp(B). Examples: Let X = {x, y}. The expression x y + x y + xy is totally linear in X, but the monomials x (missing y), x 2 y and x y 2 are not.q

Rota-Baxter Type OPI, Operators, and Algebras An OPI ϕ k x, y is of Rota-Baxter type if ϕ has the form x y B(x, y) for some B(x, y) k x, y and if: (a) : B(x, y) is totally linear in x, y; (b) : B(x, y) is in RBNF; (c) : for every well-ordered set Z, the rewriting system Π ϕ (Z) defined by ϕ is terminating; (d) : for every well-ordered set Z, the expression B(B(u, v), w) B(u, B(v, w)) is ϕ-reducible to zero for all u, v, w M(Z). If ϕ := x y B(x, y) is of Rota-Baxter type, then we say the defining operator P = of a ϕ-algebra R, and (by abuse) the expression B(x, y), are of Rota-Baxter type. By a Rota-Baxter type algebra, we mean some ϕ-algebra R where ϕ is some OPI in k x, y of Rota-Baxter type.

Examples Let B(x, y) := x y. Then ϕ = 0 is the OPI defining the average operator and it is of Rota-Baxter type. The identity defining a Rota-Baxter operator and that defining a Nijenhuis operator are OPIs of Rota-Baxter type. The expression B(x, y) := y x is not of Rota-Baxter type. This is because in k u, v, w, the operated monomial B(B(u, v), w) = w B(u, v) = w v u is in RBNF, while B(u, B(v, w)) = B(v, w) u = w v u ϕ w u v is also in RBNF and the two operated monomials are not ϕ-joinable. By the Hierarchy Lemma, B(B(u, v), w) B(u, B(v, w)) ϕ 0. This also shows the base fork ( u v w ϕ B(u, v) w, u v w ϕ u B(v, w) ) is not ϕ-joinable, and the TRS Π ϕ (u, v, w) is not confluent.

Remarks on Definition of Rota-Baxter Type OPIs The well-order on Z need not be given if Z is denumerable or if Z is uncountable and we accept the Axiom of Choice. Total linearity on B(x, y) is imposed since we are considering linear operators. B(x, y) in RBNF and Π ϕ (Z) terminating are necessary to avoid obvious infinite rewriting under Π ϕ (Z). This rules out the Reynolds identity: x y = x y + x y + x y. ϕ-reduction to zero of B(B(u, v), w) B(u, B(v, w)) for all u, v, w M(Z) is to ensure Π ϕ (Z) is confluent. Note that in the definition for ϕ to be of Rota-Baxter type, there is no mention of monomial orders on M(Z) and hence no requirement of compatibility.

List of Rota-Baxter Type Operators Conjecture: For any c, λ k, the operated polynomial ϕ := x y B(x, y), where B(x, y) is taken from the list below, is of Rota-Baxter type. Moreover, any OPI ϕ of Rota-Baxter type is necessarily defined as above by a B(x, y) from among this list. (a) : x y (average) (b) : x y (inverse average) (c) : x y + y x, (symmetric average) (d) : x y + y x, (symmetric inverse average) (e) : x y + x y xy (Nijenhuis) (f ) : x y + x y + λxy (Rota-Baxter) (g) : x y x 1 y + λxy, (average TD) (h) : x y x 1 y + λxy, (inverse average TD)

List continued (i) : x y + x y x 1 y + λxy (TD) (j) : x y + x y x 1 y xy 1 + λxy, (right TD) (k) : x y + x y x 1 y xy + λxy, (Nijenhuis TD) (l) : x y + x y x 1 y 1 xy + λxy, (left TD) (m) : cx 1 y + λxy (generalized endomorphism) (n) : cy 1 x + λyx (generalized antimorphism) A new type is underlined followed by a proposed name. When λ is present, of weight λ should be added.

Main Result on RBT OPI, Sufficiency Theorem. Let k be a field. Let ϕ := x y B(x, y) k x, y, where B(x, y) is in the list in the Conjecture. Then for every well-ordered set (Z, Z ), the following statements hold : 1. The operated polynomial ϕ is of Rota-Baxter type. 2. Z can be extended to the monomial order db,z on M(Z) with which ϕ compatible. 3. The rewriting system Π ϕ (Z) on k Z is convergent. 4. The set S ϕ (Z) is a Gröbner-Shirshov basis in k Z with respect to the monomial order db,z. 5. There is a (uniform) construction of the free ϕ-algebra over Z, which is (k R(Z),, P r ). Moreover, the constructions of Π ϕ (Z), db,z, S ϕ (Z) and (k R(Z),, P r ) are uniformly defined.

Main Result on RBT OPI: Necessity Theorem. Let k be a field. Let ϕ := x y B(x, y). Then ϕ is of Rota-Baxter type and is compatible with db.z on M(Z) for some well-ordered set Z (possibly the empty set) if and only if B(x, y) is among the list given in Conjecture. When Z is the empty set, M(Z) is the free operated monoid M(1) on 1 where 1 is the monoid identity element and M(1) contains (not consists of) monomials of the form M = n i=1 Pe i (1) and P(P i (1)) = P i+1 (1). It also contains P(M). Remark. Recall that in the definition for ϕ to be of Rota-Baxter type, there is no mention of monomial orders on M(Z) and hence no requirement of compatibility. So, there might be some ϕ of Rota-Baxter type (as we define it) that is not on the list.

A Local Characterization Theorem Theorem: Let k be a field. Let Z be a set, let Z be a monomial order on M(Z), let B(x, y) be in RBNF and totally linear in x, y, and let ϕ(x, y) := x y B(x, y) k x, y be compatible with Z on M(Z). TFAE: (a): For all u, v, w M(Z), B(B(u, v), w) B(u, B(v, w)) ϕ 0. (b): For all u, v, w M(Z), B(B(u, v), w) ϕ B(u, B(v, w)). (c): Π ϕ (Z) is confluent. (d): Π ϕ (Z) is convergent. Moreover, these hold if the following holds, and the converse is true if Z has at least 3 elements. (e): B(B(µ, ν), ω) Πϕ(µ,ν,ω) B(µ, B(ν, ω)) under the rewriting system Π ϕ (µ, ν, ω) for k µ, ν, ω.

A Global Characterization for Rota-Baxter Type Let k be a field. Let B(x, y) k x, y be in RBNF and totally linear in x, y. Let ϕ(x, y) := x y B(x, y). Let Z = { µ, ν, ω }. Suppose for every well-ordered set (Z, Z ), there is a monomial order with which ϕ is compatible (also denoted by Z ) extending Z from Z to M(Z). Then the following four statements are equivalent. (a) ϕ is a Rota-Baxter type OPI. (b) The two expressions B(B(µ, ν), ω) and B(µ, B(ν, ω)) k Z are joinable under Π ϕ (Z ). (c) Π ϕ (Z ) is convergent. (d) Π ϕ (Z) is convergent for every well-ordered set Z.

Monomial Order on M(Z) A monomial order on M(Z) is a well order on M(Z) such that: u < v = q u < q v, for all u, v M(Z) and all q M (Z). Given a monomial order, and f k Z, f / k, the notions of leading bracketed monomial f and leading coefficient c(f ) are clear. The remainder R(f ) is defined as f c(f )f. If f k (including f = 0), we define the leading monomial of f to be f = 1 M(Z), the leading coefficient to be c(f ) = f, and the remainder is R(f ) = 0. We say f is monic with respect to if c(f ) = 1. A subset S k Z is monic if every s S is.

TRS on Free Operated Algebras For Subsets Let be a monomial order on M(Z). Let S k Z be monic with respect to. We associate a term-rewriting system Π S, (Z) on k Z with basis M(Z): Π S, (Z) := {(q s, q R(s) ) s S, q M (Z) } Z k Z. We will often abbreviate Π S, (Z) to Π S, and denote the relation by S and its reflexive transitive closure by S. If a rule (q s, q R(s) ) Π S is used to reduce f k Z to q,s g k Z in one step, we shall write f S g, in which case, f g = q s, and f Id(S) if and only if g Id(S), where Id(S) is the operated ideal of k Z generated by S.

Compatible Rewriting Systems Let Π be a rewriting system on V = k M(Z) with basis W = M(Z). We say Π is compatible with the monomial order on M(Z) if v < t for all (t, v) Π. Every rewriting system Π,S (Z) is by definition compatible with respect to. Let ϕ = x y B(x, y) k x, y. We say ϕ is compatible with on M(Z) if the rewriting system Π ϕ (Z) is compatible with on M(Z), or equivalently, if the rewriting subsystem Π ϕ(z) is, or equivalently, if B(u, v) < u v for every u, v M(Z). If ϕ is compatible with, then Π ϕ (Z) = Π,S (Z), with S = { ϕ(u, v) q,u,v u, v M(Z) }. If s = ϕ(u, v), then f ϕ g q,s if and only if f S g. Hence f k Z is Π ϕ -reducible if and only if f is Π S -reducible. The set Irr(S) of Π S -irreducible monomials is precisely the set R(Z) of monomials in RBNF.

The Rewriting System Π S is Terminating Let be a monomial order on M(Z) for a set Z. Let S k Z be monic and let f k Z. For the rewriting system Π S, we define the leading S-reducible monomial of f to be the monomial L(f ) maximal with respect to among monomials m appearing in f that are not in Irr(S), that is, L(f ) := max{m m Supp(f ), m / Irr(S)}. Lemma: Suppose g, g k Z are both S-reducible and for some q M (Z) and s S, g (q,s) S g. Then L(g ) L(g), where equality holds if and only if L(g) q s. Theorem: The rewriting system Π S is terminating. In particular, the rewriting system Π ϕ (Z) is terminating if ϕ := x y B(x, y) k x, y is compatible with.

Descent of Leading Term After Rewriting The following are more refined versions of the previous Lemma. Lemma: Let g k Z be Π S -reducible and for some s S, q M (Z), suppose g (q,s) S g k Z. Then g g, where equality holds if and only if q s < g. Corollary: Let ϕ = x y B(x, y) k Z be compatible with on M(Z). Let g k Z be ϕ-reducible and let u, v M(Z), q M (Z) and g k Z be such that g q,u,v ϕ g. Then g g, where equality holds if and only if q u v < g.

Restriction to a Rewritng Subsystem Let W be a subset of M(Z) and let V be the free k-submodule of k Z with basis W. We say the rewriting system Π ϕ (Z) on k Z restricts to a rewriting system Π on V with basis W if for all f V and all triples (q, u, v) for f, we have f q,u,v ϕ g implies g V. A necessary and sufficient condition that Π ϕ (Z) restricts to Π is that for all f V and all triples (q, u, v) of f, q u v W implies q B(u,v) V. A similar definition can be given for the more general TRS Π S (Z) for a subset S k Z.

A Crucial Lemma on Confluence Let Z be a set, let be a monomial order on M(Z), let Y M(Z) be monic and let ϕ(x, y) := x y B(x, y) k x, y be compatible with on M(Z). Suppose the rewriting system Π ϕ (Z) restricts to a rewriting system Π ϕ,y on k Y with basis Y and suppose Π ϕ,y is confluent. For 1 i n, let c i k, q i M (Z) and s i S ϕ (Z) be such that q i si Y and n i=1 c iq i si s i = 0. Lemma: Then n i=1 c iq i si ϕ 0.

Basics on Gröbner-Shirshov Basis Let f, g k Z be two distinct bracketed polynomials, each monic with respect to. Let f be the leading monomial of f, and let f be the breadth of f. If there exist µ, ν, b M(Z) such that b = f µ = νg with max{ f, g } < b < f + g, we call the operated polynomial (f, g) µ,ν b := f µ νg the intersection composition of f and g with respect to (µ, ν). If there exist q M (Z) and b M(Z) such that b = f = q g, we call the operated polynomial (f, g) q b := f q g the including composition of f and g with respect to q.

Gröbner-Shirshov Basis Let b M(Z), let S be a set of monic bracketed polynomials in k Z, and let Id(S) be the bracketed ideal generated by S. An operated polynomial f k Z is called trivial modulo S with bound b or, in short, trivial modulo (S, b) if f Id(S) and can be expressed as a sum i c iq i si where 0 c i k, q i M (Z), s i S and q i si < b. The polynomial 0, expressed as the empty sum, is trivial modulo (S, b) for any S and b. A set S k Z of monic bracketed polynomials is called a Gröbner-Shirshov basis with respect to if, for all pairs f, g S with f g, every intersection composition of the form (f, g) µ,ν b is trivial modulo S with bound b, and every including composition of the form (f, g) q b is trivial modulo S with bound b.

The Composition-Diamond Lemma Let η : k Z k Z /Id(S) be the canonical homomorphism of k-modules. Then the following statements are equivalent. (a) : S is a Gröbner-Shirshov basis in k Z. (b) : For every non-zero f Id(S), f = q s for some q M (Z) and some s S. (c) : For every non-zero f Id(S), f can be expressed in triangular form: f = c 1 q 1 s1 + c 2 q 2 s2 + + c n q n sn, (1) where c i k (c i 0), s i S, q i M (Z) for 1 i n, and q 1 s1 > q 2 s2 > > q n sn. (d) : As k-modules, k Z = k Irr(S) Id(S) and η(irr(s)) is a k-basis of k Z /Id(S).

Second Local Characterization Theorem Let Z be a set, and let be a monomial order on M(Z). Let ϕ(x, y) := x y B(x, y) k x, y with B(x, y) in RBNF and totally linear in x, y be compatible with. Then the following conditions are equivalent. (a) : The rewriting system Π ϕ (Z) is convergent. (b) : With respect to, the set S := S ϕ (Z) is a Gröbner-Shirshov basis in k Z.

Construction of Free ϕ-algebras Let Z be a set, let be a monomial order on M(Z), and let ϕ(x, y) := x y B(x, y) k x, y be of Rota-Baxter type and compatible with. Let S = S ϕ (Z). Then the following holds: Recall that R(Z) M(Z) is the set of monomials in RBNF and η : k Z k Z /Id(S) =: k ϕ Z is the canonical homomorphism of k-modules. The composition η ι: k R(Z) ι k M(Z) k Z η k ϕ Z is an isomorphism of operated k-modules, taking the k-basis R(Z) of k R(Z) to a k-basis of k ϕ Z and P r the operator induced by P, that is, P r (f + Id(S)) = P(f ) + Id(S). Let α : k ϕ Z k R(Z) be the inverse of η ι and let ρ := ρ ϕ be the composition k Z η k ϕ Z α k R(Z). For u, v R(Z), let u = u 1 u s and v = v 1 v t be standard decompositions.

Construction of Diamond Product Let be the multiplication on k R(Z) that is uniquely determined by bilinearity and the products u v, where 1. 1 v = v and u 1 := u, if either u = 1 or v = 1 (or both) where 1 is the empty word in M(Z); 2. u v := uv if either u S(Z) or v S(Z) (or both); 3. u v := ρ(b(u, v )) if u = u and v = v are both in R(Z) ; 4. u v := u 1 u s 1 (u s v 1 )v 2 v t if either s > 1, or t > 1 (or both). Here the multiplications are concatenations, except for u s v 1, where is defined recursively by Step 2 or 3. With the above construction, (k R(Z),, P r ) is a free ϕ-algebra on the set Z.

Monomial order on M(Z) We construct a monomial order on M(Z) inductively in several steps. Recall that M(Z) is the direct limit of ι n : M n (Z) M n+1 (Z). The degree lexicographical order dlex on M(Z). A well-order n on M n (Z) for every non-negative integer n which is induced by Z on M n (Z) and the preorders induced by degree and breadth. The degree-breadth lexicographic order db,z is the direct limit of n. Lemma. db,z is a well order and the restriction of Z to M n (Z) is n. Theorem. db,z is a monomial order on M(Z).

Degree Lexicographic Order dlex on M(Z) Let Z be a set with a well order Z and M(Z) the free monoid on Z. For u = u 1 u r M(Z) with u 1,, u r Z, define deg Z (u) = r if u 1 and deg Z (1) = 0. Define the degree lexicographical order dlex on M(Z) by taking, for any u = u 1 u r, v = v 1 v s M(Z)\{1}, where u 1,, u r, v 1,, v s Z, { degz (u) < deg Z (v), u dlex v or deg Z (u) = deg Z (v)(= r) and u 1 u r lex v 1 where lex is the lexicographical order on M(Z), with the convention that the empty word 1 dlex u for all u M(Z). Lemma. If Z is a well order on Z, then dlex is a well order on M(Z).

Preorders and Pre-linear Orders Let Y be a nonempty set. A preorder or quasiorder Y on Y is a binary relation that is reflexive and transitive, that is, for all x, y, z Y, we have 1. x Y x; and 2. if x Y y, y Y z, then x Y z. We denote x = Y y if x Y y and y Y x. If x Y y but x Y y, we write x < Y y or y > Y x. A pre-linear order Y on Y is a preorder Y such that either x Y y or y Y x (or both) for all x, y Y.

P-degree and P-breadth: Examples of Preorders Let u = u 0 u1 u 1 u2 ur u r M(Z), where u 0, u 1,, u r M(Z) and u1, u2,, ur, M(Z). Same for v = v 0 v1 v 1 v2 vs v s M(Z). Define u dgp v deg P (u) deg P (v), (2) where the P-degree deg P (u) of u is the number of occurrences of P = in u. Define u brp v r s (that is u P v P ), (3) where u P := r is the P-breadth of u. Lemma. dgp and brp are pre-linear orders satisfying the descending chain condition on M(Z).

Isomorphisms between Ordered Free Monoids Proposition. For every n 0, we can construct a well order n on M n (Z) depending only on Z such that the natural embeddings ι n induces an isomorphism of ordered free monoids between (M n (Z), n ) and (Im(ι n ), n+1), where n+1 is the restriction of n+1 to Im(ι n ). For natural numbers m 0, let M m (Z) := {u M(Z) u P = m} and let M m n (Z) := M n (Z) M m (Z). Then M n (Z) is the infinite disjoint union m=0 Mm n (Z).

Linear Order lexn on M n (Z) Define a relation lex m n on M m n (Z) by u lex m n v (u 1, u 2, u m, u 0,, u m ) clexn,m (v 1, v 2,, v m, v 0,, v m ). Then lex m n is a well order on M m n (Z) for all m 0. Now M n (Z) = m=0 M m=0, hence we may define a relation on M n (Z) by u lexn v lexn { u <brp v, or u = brp v and u lex m n v, where m = u P = v P. lexn a linear order on M n (Z), and in fact, it is a well order.

Well Order n on M n (Z) and db,z on M(Z) Finally, define the relation n on M n (Z) by: { u <dgp v, u n v (4) or u = dgp v and u lexn v, where we have restricted dgp to M n (Z). We prove by induction on n that u n v ι n (u) n+1 ι n (v) for all n 0, and so the induced map M n (Z) Im(ι n ) is an isomorphism between ordered free monoids. {(M n (Z), n )} n=0 as a filtration of ordered free monoids for (M(Z), db ), where M(Z) = n=0 M n(z) and db = lim n. More specifically, for u, v M(Z), u db,z v := u n v for any n such that u, v M n (Z). (5)

db,z is a Monomial Order A well order on M(Z) is called bracket compatible if u v u v for all u, v, w M(Z). A well order on M(Z) is called left (multiplicatively) compatible if u v wu wv for all u, v, w M(Z). A well order on M(Z) is called right (multiplicatively) compatible if u v uw vw for all u, v, w M(Z). Lemma. A well order is a monomial order on M(Z) if and only if is bracket, left, and right compatible. Theorem. Let (Z, Z ) be a well-ordered set. The order db,z is a monomial order on M(Z), extending Z.

Equivalences for Compatibility with db,z Proposition. Let ϕ(x, y) = x y B(x, y) k x, y and suppose B(x, y) is totally linear and in RBNF. TFAE: (a) For every well-ordered set (Z, Z ), the rewriting system Π ϕ (Z) is compatible with db,z on M(Z). (b) For some well-ordered set (Z, Z ) (possibly the empty set), the rewriting system Π ϕ (Z ) is compatible with db,z. (c) The total P-degree of B(x, y) is 1. (d) B(x, y) has the form: a 1 y x + a 2 x y + a 3 y x + a 4 x y + a 5 yx + a 6 xy +a 7 y 1 x+a 8 x 1 y+a 9 yx 1 +a 10 xy 1 +a 11 1 yx+a 12 1 xy where a j k (1 j 14). +a 13 yx + a 14 xy