An Axiomatic Approach to Gr ¨obner Basis Theory by Examining Several Reduction Principles

: Several variants of the classical theory of Gr¨obner bases can be found in the literature. They come, depending on the structure they operate on, with their own speciﬁc peculiarity. Setting up an expedient reduction concept depends on the arithmetic equipment that is provided by the structure in question. Often it is necessary to introduce a term order that can be used for determining the orientation of the reduction, the choice of which might be a delicate task. But there are other situations where a different type of structure might give the appropriate basis for formulating adequate rewrite rules. In this paper we have tried to ﬁnd a uniﬁed concept for dealing with such situations. We develop a global theory of Gr ¨ obner bases for modules over a large class of rings. The method is axiomatic in that we demand properties that should be satisﬁed by a reduction process. Reduction concepts obeying the principles formulated in the axioms are then guaranteed to terminate. The class of rings we consider is large enough to subsume interesting candidates. Among others this class contains rings of differential operators, Ore-algebras and rings of difference-differential operators. The theory is general enough to embrace the well-known classical Gr ¨ obner basis concepts of commutative algebra as well as several modern approaches for modules over relevant noncommutative rings. We start with introducing the appropriate axioms step by step, derive consequences from them and end up with the Buchberger Algorithm, that makes it possible to compute a Gr¨obner basis. At the end of the paper we provide a few examples to illustrate the abstract concepts in concrete situations.


Introduction
The principal reason for studying Gröbner bases is their utility when dealing with computational questions regarding submodules and quotient modules. As a paradigma we may consider the membership problem for submodules: Given modules M, N with N ⊂ M . Decide whether a given element of M belongs to N .
The solution strategy provided by Gröbner basis theory for solving this problem is to set up a family of reduction steps u −→ v. The decision process consists in repeatedly applying these reduction steps to an element u ∈ M until an element v is reached which does not allow further reduction. The nature of this v gives the answer to the question.
In order to satisfy its demands the set ρ of all permitted reduction steps has to obey certain regulations.
1. Being a member of ρ must be effectively decidable. 2. ρ should allow no infinite sequences of reduction steps i.e., every such sequence has to have finite length. 3. The relation ρ has to be designed in such a way that admitted reduction steps do not leave the congruence classes mod N . 4. When m = u 0 −→ u 1 −→ · · · −→ u r = v is a maximally exhausted sequence then its terminal node v being irreducible (w.r.t. ρ) is called a normal form of u. The demand is now that we must know all possible normal forms contained within N . These requirements are enough to solve the membership problem for N : To decide whether a given element u is a member of N , simply reduce u until an irreducible v is reached; if v ∈ N then u is in N , otherwise not.
It is common and useful to request additional properties for ρ. The first is that normal forms be unique. This draft is then described by the two conditions 'Noetherian' and 'Church-Rosser'. Details on confluence of reduction processes are described in the fundamental paper [3]. The second property we want is additivity of irreducibles, i.e., the set I ρ of normal forms of all elements of M has to be an additive group. Necessarily, this group has to have a decidable membership problem.
The entire conception results in a direct sum decomposition M = N ⊕ I -at least as abelian groups. Obviously then the sequence 0 −→ N −→ M −→ M/N −→ 0 must split. This splitting capability of the module pair N ⊂ M is the ultimate limit beyond which the concept is meaningless. In addition to the amenity of invertible coefficients, this is the reason why we will focus on modules over rings R that contain a field K. All R-modules are then K-vector spaces and the mentioned limitation vanishes.
Starting in 1965 with his dissertation, Buchberger developed the theory of Gröbner bases into an indispensible tool for algorithmic algebra [5]. The scope of problems that can be solved by means of this tool is vast. Among others, the computation of Hilbert functions and their differential variants (Dimension polynomials) has been a central point of interest.
In 2006, Winkler and Zhou introduced Gröbner bases in difference-differential modules using a generalized term order that uses a cover of the group Z n by finitely many copies of N n (orthant decomposition) [35].
In 2008, Winkler and Zhou extended their 2006-approach to the notion of relative Gröbner bases and applied it to the computation of difference-differential dimension polynomials Splitting the set of derivations and the set of automorphisms, they provided algorithms for the univariate and the bivariate case [36,37].
In his 2013 paper, C. Dönch pointed out that the algorithm which generates a relative Gröbner basis out of a finite set of generators might not terminate [12]. See also [11]. In the meantime this has been fixed (cf. [18]).
Different viewpoints on the computation and applications of dimension polynomials are presented in [24]. For results on extending a presentation of a base algebra A to the free differential algebra on A see [38].
At the ISSAC 2015 conference, the authors have introduced the notion of Gröbner reduction, a general concept that covers the reduction part of several Gröbner basis techniques appearing in the literature. There, the principal intention has been to provide a scheme that makes it possible to compare such constructions. In particular, the Gröbner basis concepts developed in the papers of Levin, Pauer, Winkler, Zhou for rings of differential operators, Ore algebras etc. have been shown to be subordinate to this scheme. It has then be proved that reduction concepts which obey the axioms of Gröbner reduction allow the derivation of the dimension polynomial of finitely generated modules [15]. Later it has been shown that concepts like reduction relative to several term orders or relative reduction can be expressed in terms of Gröbner reductions [16].
Since its starting point with Buchbergers dissertation the theory has developed in several directions. Gröbner bases of particular types of ideals are treated in [33]. The evolution of the theory towards integro-differential algebras can be seen in [17]. Gröbner bases for operads is developed in [13]. Performing adjoint functor constructions for operated algebras is done in [39]. For Gröbner bases whose element have specific properties see e.g. [4].
In this paper we leave the tight environment that was dictated by the conception of [15]. The notion of reduction relation in a module is analyzed with regard to maximizing the scope of its models. This results in the formulation of four properties that are central for all such relations.
It turns out that the reduction is steered by two parameters: a binary predicate P , and a unary one, i.e., a set X. Accordingly, we are concerned with three items that have to be varied: P , X and the ground ring R. It is not surprising that the nature of R occupies central attentiveness.
Interestingly, the predicate P can be discussed in full generality, ignoring the particular nature of R. This is owed to the fact that the principal task of P is a restrictive one. The reduction has to terminate, and it does so because the predicate P forces this by importing some external structure to the ring. As a consequence, the choice of P has impact on the method of reduction rather than its actual power.
In contrast to the significance of the binary predicate, the unary predicate X is subordinate to the nature of the ring, and it is this nature that has greatest influence on the strength of reduction.
We are interested in rings R that contain a field K as a subring. R is then a free K-module and we assume a fixed Kbasis Λ ⊂ R that is considered as part of the structure -we call this a ring with basis. There is then a plethora of concepts that allow developing the facets of the theory, in particular when a well-order on the basis Λ is imposed. The nature of R is then discerned by the two basic structure formulae, answering the following questions: Given c ∈ K, λ, µ ∈ Λ, what is λ·c and λ·µ, both expressed in terms of the basis Λ? If λ · c = cλ, then K is central and consequently R is an algebra over K. If, in addition, 1 ∈ Λ and λ · µ ∈ Λ, then R is the monoid ring K[Λ]. These cases are discussed in the literature.
In this paper we develop the theory of reductions for rings with basis without assuming such restrictions. As the theory evolves, the unary predicate X, playing the active part in determining the strength of the reduction, is subject to increasing bondage. First, we assume that X is closed under multiplication by scalars. Later we will consider elements of X as being built from two components, one coming from the ground ring, the other from the module. It is this second component which then is called a Gröbner basis, provided that all conditions are satisfied. Under the validity of an appropriate ascending chain condition, by stepwise enlarging X, it is possible to increase the strength of the resulting reduction until all axioms of a Gröbner reduction are satisfied.
The structure of the paper is the following: First we present our four properties axiomatically and derive some elementary consequences. Then we take care of construction principles that eventually result in models of the axioms.
According to the logical conception of the paper, the setup is organized in decreasing generality. We start with the seemingly most comprehensive concepts, and subject them to a process of increasing specialization.
In the last section we consider applications of the concepts to particular rings. The first three examples discuss briefly cases of finite dimension. As Gröbner bases in these can be treated by linear methods, the concepts are of little computational value. Nethertheless we include them here because we feel that they reflect the ideas in a particularly simple way.
Concerning performance, the concepts are suited to turn into algorithmic procedures when specialized to particular situations. For aspects of complexity and computation see e.g. [32] or [23]. Further strengthening of computational power by combining Gröbner bases with characteristic pairs is performed in [31].
In any concrete instance the spezialized concepts will incorporate the computational advantages that come from the specific equipment.

Notation
A ring is always an associative ring with a unit element 1 which is preserved by ring homomorphisms. Modules are assumed to be left and unitary. N, Z and Z + denote the sets of non-negative integers, integers and integers > 0. Most often we consider the additive monoid N p equipped with the product partial order ≤ π given by Throughout, R will be a ring, M an arbitrary module over R and K a field. If X is a subset of M then ZX and RX stand for the abelian group resp. the R-submodule of M generated by X. If S ⊂ R is a set without additive structure then SX = {sx | s ∈ S, x ∈ X}. This applies in particular to the group of units K × of K in case that K ⊆ R.
If not locally introduced otherwise, F denotes a free module over R. The letters X, Y and W are chosen to designate an unspecified set, most often W will be equipped with some kind of (partial) order. P(X) denotes the set of all subsets of the set X and P fin (X) is the subset of all finite subsets. X + Y = X \ Y ∪ Y \ X denotes symmetric difference of sets. A binary relation ρ ⊆ M × M is considered as a reduction and we write f ρ (r ≥ 0) and f ρ + G G h claims that this chain has positive length. Thus, ρ + = • ρ + G G • is the transitive closure whereas ρ = • ρ G G • denotes the reflexive and transitive closure of ρ. We write I ρ = {f ∈ M | ∃h such that f ρ G G h } for the set of ρ-irreducible elements and Z ρ = (ρ ) −1 (0) for the set of elements f with the property f ρ G G 0 . Using these notions we will omit reference to ρ when the situation under discussion is unambiguous.
The relation ρ is Noetherian if it does not allow an infinite sequence f 1 −→ f 2 −→ · · · . The equivalence relation generated by ρ is written ρ or, on occasion, o o

The General Theory
Let N ⊆ M be an additive subgroup and ρ ⊆ M × M an arbitrary relation. Consider the following requirements on ρ: Axiom 1: ρ is Noetherian; Axiom 2: ρ is congruence preserving, i.e., f −→ h ⇒ f ≡ h mod N ; Axiom 3: I is an additive group; Axiom 4: I ∩ N = 0 Definition 3.1. 1. ρ is a reduction for N iff it satisfies Axioms 1 and 2. 2. ρ is additive iff it satisfies Axiom 3. 3. ρ is a Gröbner reduction for N iff it satisfies Axioms 1,2,3,4. The following example demonstrates that the system of Axioms 1 to 4 is trivially consistent. Then: If, in addition, ρ is a Gröbner reduction for N i.e., n ∈ Z. If, conversely, N = Z then an i ∈ I ∩ N must eventually reduce to 0; as i is irreducible, this reduction is improper, that is, i = 0. It remains to show that the RHS implies irreducibility of 0. But if 0 would reduce to some h, this h must be in N and, by assumption, h G G 0 . This would produce a chain of reductions 0 −→ h G G 0 which contradicts the Noetherian property. 4. is a consequence of Proposition 3.1. Assume that h 1 ←− f −→ h 2 . Reduce both h 1 , h 2 till irreducibles i 1 , i 2 are reached. Then i 1 ≡ i 2 mod N and therefore i 1 − i 2 ∈ I ∩ N = 0 which proves 5. Point 6. is obvious. Thus, in case of a Gröbner reduction, given an additive section s : M/N −→ M to π, the endomorphism s•π provides the normal form NF(f ) = (s • π)(f ) of elements f ∈ M . Moreover im(s • π) = I and each f ∈ M has a unique representation Corollary 3.1. Let ρ be a Gröbner reduction for N ⊆ M . Then ρ equals the congruence modulo N . Proof Then ϕ ρ \ 1 M is a Gröbner reduction for M . In fact, ϕ ρ is the projection M −→ M onto a direct complement of M .
In the next sections we shall focus on accomplishing the environment for constructing relations that satisfy the Axioms 1 to 4.

Construction of Reduction Relations
Because the general complexion of a reduction ρ ⊂ M × M for a subgroup N ⊆ M is determined by it, we start discussing Axiom 2.

Axiom 2
Let d denote subtraction in the module M . That a relation ρ ⊆ M × M satisfies Axiom 2 can be expressed by the commutativity of the diagram where d = d|ρ and the upwardly directed arrows denote inclusion. If we let P denote the relation ρ written as a binary predicate and set X = d(ρ) then we obtain If, conversely, X ⊆ N and P (f, h) is an arbitrary predicate then (1) read as a definition for its LHS results in a relation that satisfies Axiom 2. We will therefore constrain reduction relations to this formula. So (1) is the primary scheme subsuming all relations that will emerge in this paper. It is then clear that Axiom 2 is always fulfilled. Moreover As X and P are the involved parameters we will denote the relation (1) by the symbol ρ (X,P ) .
Proposition 4.1. Consider a subgroup N ⊆ M , a set X ⊆ N and a binary predicate P . If ρ (X,P ) is a Gröbner reduction for N then N = ZX.
Proof. ZX ⊆ N and I + ZX ⊆ M . Take f ∈ M and reduce it to an irreducible Proposition 4.2. Let X, Y be sets and let P, Q denote predicates. Assume that Y ⊆ X and Q ⊆ P . If ρ (X,P ) is Noetherian then ρ (Y,Q) is a reduction for ZY .

Axiom 1
We will modulate the parameters present in Formula (1) in order to come along with the remaining axioms. This depends on the situation, that is, on the equipment provided by the actual candidates for R, M and N .
In order to warrant Axiom 1 we need to import a Noetherian structure from some well-founded set.
Definition 4.1. Let (W, ≤) be a partially ordered set, v : M −→ W and V : W −→ P(M ) functions and X ⊆ M a set. We consider the following relations.
Evidently ρ (X,v) and ρ (X,V ) are of type (1) for the obvious predicates P . We will refer to a function v : M −→ W as a rank for M with values in W .
The function V : W −→ P(M ) is called monotone and exhaustive when Moreover, V is stable w.r.t. ρ (X,V ) . Proof. Both relations are congruence preserving w.r.t. ZX. Since W is well-founded ρ (X,v) satisfies Axiom 1. As to the second relation, consider a sequence of ρ (X,V ) -steps Choose w ∈ W with f ∈ V (w). Then there exists z 1 < w with f 1 ∈ V (z 1 ). To the same effect there is an z 2 < z 1 with f 2 ∈ V (z 2 ). Iteration produces a sequence in W , descending w.r.t. <. Since this sequence must terminate after finitely many steps so does (3), thus ρ (X,V ) is Noetherian. From its definition it is clear that the family (V (w)) w∈W is ρ (X,V )stable.

Axiom 3
Fixing the predicate P , the only tool for influencing the behavior of the irreducibles is scaling the generating set X. This will be elaborated in the sequel by adjusting X due to the configuration available in the type of ring under consideration.

Axiom 4
In general when constructing a reduction for N Axiom 4 will not hold. In this case we will try to achieve it by extending the set X. This is what the classical Buchberger algorithm does. Also here a general instruction is beyond our control.
Example 4.1. Consider M = R 2 as a vector space over R, N = 0 × R, X ⊆ N . Let the rank function rk : M −→ N, be given as rk(f 1 , f 2 ) = |f 2 | , and consider the reduction This example shows that, in this generality, we cannot expect to construct a Gröbner reduction by simply enlarging X. The reason is, that the group N contains elements f = 0 of minimal rank. To achieve Axiom 4 this has to be prevented.
Proposition 4.4. Let N ⊆ M be a subgroup and assume that the rank function rk : M −→ W satisfies rk(0) < rk(f ) ∀f = 0. Then any X ⊆ N can be extended to a set Y ⊇ X such that ρ (Y,rk) satisfies Axiom 4.
Proof. As a witness choose Y = N and take f ∈ N \ 0.

Well-founded Orders
Throughout this chapter we assumme that (W, ≤) is a well-founded partially ordered set. Regarding Axiom 1., the importance of such a set is evident. Later we will consider the case where (W, ≤) is even a well-order.
We consider the following requirements on the function V : .
For the second equation take a surjection v : The proposition says that surjective rank-functions are oneone associated with certain monotone exhaustive functions, we call them the corresponding monotone exhaustive families. The next proposition says that their associated reduction relations coincide.
Plainly we may always assume that a rank function is surjective. By violating some of the properties 1. 2., Proposition 5.1 shows that the set of monotonous and exhaustive functions W −→ M is richer than the set E(M, W ).
A particularly important class of monotone and exhaustive families of subsets of M is the class of N p -indexed filtrations.
Definition 5.2. By a p-fold filtration on R we mean a family of additive subgroups R k ⊆ R, indexed by k ∈ N p , such that M together with a filtration is called a filtered module over the filtered ring R.
From its axioms a filtration is a monotone and exhaustive function on the well-founded set (N p , ≤ π ). Proposition 4.3 yields Corollary 5.2. Let F be a filtration on M and X ⊆ M . The relation ρ (X,F ) is a reduction for ZX and the filtration F is stable w.r.t. ρ (X,F ) .
The characterization of the irreducibles is obvious.
Plainly, well-ordered sets are optimal for the values of a rank function. Often we will use them in an extended version.
Definition 5.4. Let W be a linearly ordered set. For A, B ∈ P fin (W ) we set Proposition 5.3. Let W be linearly ordered, A, B ∈ P fin (W ) and x, y ∈ W .
1. (4) is a linear order on P fin (W ). (4) is a well-order on P fin (W ); Thus, the well-order (4) extends both, inclusion and a given well-order on W .

Rings and Modules with Basis
Assume that the field K is a subring of R. We fix a Kbasis Λ ⊂ R, so that R = K (Λ) . Let F = R (E) be the free R-module on the set E. Then F = K (ΛE) and each element f ∈ F has a unique representation The K-bases Λ and ΛE will be considered as part of the structure and R (F ) is called a ring (module) with K-basis. We write π t (f ) = f t for the projection function π t : F −→ K (t ∈ ΛE). For a term s = λe ∈ ΛE (λ ∈ Λ, e ∈ E) we set π 1 (s) = λ, π 2 (s) = e. This gives the projection functions π 1 : ΛE −→ Λ, π 2 : ΛE −→ E. We say that the term s involves the basis element e (cf. [14]). This notation will stay in force for the remainder of the paper.
For the product of a, b ∈ R we obtain The two expressions λ · b u and ν · µ are responsible for the behaviour of multiplication in R and scalar operation in R-modules. When explicitly exposed these two expressions provide the basic structure formulae of R.
In concrete instances the monomials λ ∈ R often carry additional structure. The basic structure formulae simplify then by expressing them in terms of the structural components of the monomials.
We call a submodule V ⊆ F monomial when it is generated as an R-module by a subset of ΛE. The submodule V is called homogeneous when it is generated over K by such a subset.

Term Orders
Now assume that < ⊂ ΛE × ΛE is a well-order on the set ΛE. Each non-zero f ∈ F has then a leading term LT(f ) = max T(f ). The leading coefficient of f is the coefficient LC(f ) = f LT(f) ∈ K. If X ⊆ F , we write LT(X) and LC(X) for the sets {LT(x) | x ∈ X} and {LC(x) | x ∈ X} respectively.

Reduction in a Module with Basis
We proceed assuming presence of a well-order < on ΛE. Let X ⊆ F . We consider three types of reduction relations We call ρ LT (X) leading term reduction, ρ T (X) T-reduction and ρ CR (X) classical reduction.
ρ LT (X) uses the leading term LT : F −→ ΛE ∪ {0} as a rank. The extended well-order on P fin (ΛE) provides the map T : F −→ P fin (ΛE) as a rank function for ρ T (X). So these are relations of the type considered in Definition 4.1 and therefore both are reductions for ZX. That also ρ CR (X) is a reduction for ZX is a consequence of the next proposition.
To reduce clumsy notation we shall on occasion suppress the letter ρ writing CR(X) instead of ρ CR(X) . We even may omit the 'X' when it is obvious from context. A similar convention will be used for the other reduction relations. From (2) we derive immediately It is plain that these inclusions may be strict.
In particular,

Proof.
For r = 1 we obtain that Assuming that the assertion holds for r ≥ 1, let f G G f r+1 be a reduction chain of length r + 1. Then remaining inclusions are consequences of Proposition 6.2 and Lemma 6.2.
In the sequel we need to impose conditions on the set X ⊆ F . This is necessary in order to derive more specific properties of the relation ρ (X,P ) . A quite strong condition is to request that X = N .
Proof. Specializing X = N in Corollary 6.1 yields N ⊆ Z ρ ⊆ ZN = N . Proposition 3.1 gives the result.
Of course, the statement of this corollary is of little value. More moderate is it to require that K × X = X, a condition that we will meet on several occasions. Later we will presume that X = AG for certain sets A ⊆ R and G ⊆ F . Note that the condition K × X = X implies that ZX = KX; Axiom 2 is then talking about the congruence modulo a vector space. Proposition 6.3. Let ρ be one of ρ LT (X), ρ CR (X), ρ T (X)

Leading Term Reduction
The function LT : F −→ ΛE ∪ {0} is a surjection. The corresponding monotone exhaustive family is Corollary 5.1 provides that ρ LT (X) = ρ (X,F ) for arbitrary X ⊆ F .

Classical Reduction
, v such that the below diagram can be augmented by the dotted arrows.
Proof. Proposition 6.4 yields that Proof. From Proposition 6.2 and the fact that ρ T (X) is

T-Reduction
The rank-function T : F −→ P fin (ΛE) is a surjection. According to Corollary 5.1 we obtain ρ T (X) = ρ (X,τ ) (X ⊆ F arbitrary), where is the corresponding monotone exhaustive family.
Lemma 6.4. Let X ⊆ F be an arbitrary set, f ∈ F , x ∈ X.
Lemma 6.4 guarantees then that LT(x) ∈ T(f ). Therefore The second point of the same Lemma yields that Corollary 6.7. Assume that K × X = X and let ρ be one of ρ CR (X), ρ T (X). Then I ρ is an additive homogeneous subgroup of F .
Proof. This follows immediately from Corollary 6.5. Note that Corollary 6.7 does not hold for leading-term reduction; that is, in general the irreducibles of ρ LT (X) neither form a group nor is a term of an irreducible element necessarily irreducible.
We realize that under the assumtions of Corollary 6.7, which is a requirement on the set X, relations ρ CR(X) and ρ T(X) both are models of Axioms 1,2,3. Then Corollary 6.2 claims that at least for the case X = N , where N is a submodule of F , they provide Gröbner reductions.
These are of course not very useful. We need to specify a set X which is recursively accessible.
Proposition 6.5. Consider two sets X, Y ⊆ N ⊆ F with property K × X = X, K × Y = Y . Let ρ X stand for one of ρ CR (X), ρ T (X), and similar for ρ Y . Assume that ρ X and ρ Y are both Gröbner reductions for N . If X ⊆ Y then I X = I Y .
Proof. From X ⊆ Y we obtain I Y ⊆ I X and Corollary 6.7 yields that I X and I Y are both groups. Moreover Consequently I X = I Y .

Reduction for Finitely Generated Submodules
Assume that N ⊆ F is generated by a finite set G = {g 1 , . . . , g q } ⊂ N . Let X ⊆ F with K × X = X and ρ one of CR(X), T(X). The kernel of the corresponding presentation Φ : R q −→ N is ker(Φ) = t∈ΛE ker(π t • Φ) and we obtain that ker(π s • Φ).
Definition 6.1. Consider an R-submodule N ⊆ F . Let A ⊆ R and G = {g 1 , . . . , g q } ⊆ F be sets, X = AG and ρ one of ρ LT (X), ρ CR (X), ρ T (X). Then (A, G) is a ρ-Gröbner basis for N iff ρ is a Gröbner reduction for N .
Under ideal conditions statement (8) can be used to compute a Gröbner basis: Consider r = (r 1 , . . . , r q ) ∈ R q and write Assume we can solve the system of equations q j=1 λ∈Λ ξ∈Λ t∈ΛE r j λ (λ · g j t ) ξ (ξ · t) s = 0 (s ∈ LT(X)) (10) for the indeterminants r j λ and select a solution r with Φ(r) = 0. Then we may use Φ(r) to enlarge the set X, e.g., by setting G = G ∪ {Φ(r)} and defining X accordingly (such that K × X = X , ρ is defined by X and LT(X ) ⊃ LT(X)). If iterated replication of this process terminates, it will eventually result in a Gröbner reduction for N . Simple examples will follow in Section 7.
In general the system (10) will not be satisfactorily accessible. Then we need more sophisticated concepts which will defined next.

S-polynomials
To construct Gröbner bases in a finitary way we need an appropriate concept describing S-polynomials.
In particular there is a quotient function for F when Λ · Λ ⊆ Λ ∼ = N n . Definition 6.
3. An S-polynomial is a function S ∈ (f,g)∈F ×F R · {f, g} subject to the following properties.

TO-pairs and Syzygies
In most sitations the well-order on ΛE is accompanied by a well-order on Λ.
Since ΛΛ ⊆ Λ, this sum must collapse: where we express λg j s as the sum µ≤λ c µ µ justified by the additional property. Each summand is smaller than t, because s ≺ g j and µ ≤ λ has as a consequence that µs < µ · LT(g j ) ≤ λ · LT(g j ) = t.
Therefore T(h) < t, and this means that t G G h which is impossible.
In order to use reduction relations for performing computations we have to impose finiteness conditions. Let N ⊆ F be finitely generated by the set G = {g 1 , . . . , g q }. We consider the associated representation Φ : R q −→ N , φ(r) = q j=1 r j g j together with the map For t ∈ ΛE we set R <t = δ −1 [0, t), i.e., R <t = {r ∈ R q | δ(r) < t}.
Then φ(r) δ(r) ∀r ∈ R q and R <t is a module over the ring K q i.e., r, s ∈ R <t ∧c ∈ K q ⇒ r+s ∈ R <t ∧(c 1 r 1 , . . . , c q r q ) ∈ R <t .

Moreover we have that
We proceed assuming available a TO-pair. Theorem 6.1. Assume that ∀ λ∈Λ ∀ f ∈F0 λf ∈ F 0 and that F admits an S-polynomial S : F × F −→ F . Let A ⊆ R be such that Λ ⊆ A = K × A, G = {g 1 , . . . , g q } ⊆ F 0 , X = AG and ρ one of ρ CR (X), ρ T (X). If for all 1 ≤ i = j ≤ q: S(g i , g j ) ρ G G 0 then (A, G) is a ρ-Gröbner basis for RG.
Proof. Because ZX ⊆ RG, ρ is a reduction for RG. Since K × X = K × AG = AG = X we obtain from Corollary 6.7 that ρ satisfies also Axiom 3. It remains to show that RG ∩ I ρ = 0.
Therefore k ≥ 2. Because λ i g i ∈ F 0 (1 ≤ i ≤ k), we derive from Lemma 6.5 that From the axioms on S-polynomials we obtain

Collecting like terms gives
Because a i j g j S(g i , g k ) = q j=1 a i j g j ∀ i ∀ j using Definition 6.4 we get Because λ i g i and λ k g k are in F 0 and LT(λ i g i ) = LT(λ k g k ) = δ(r) we obtain that S(λ i g i , λ k g k ) = λ i g i − λ k g k ≺ δ(r) ∀ i=1,...,k−1 . Thus e i η i · a i q ∈ R <δ(r) and thus ). Consequently It follows that ∃s ∈ R <δ(r) with φ(s) = f and therefore This proof when spezializing the Ring R to K[x 1 , . . . , x n ] corresponds to the one found in [1]. For other proofs of the polynomial case see [2], [7], [34] and the literature listed in [8].

The Buchberger Algorithm
As in the classical theory Theorem 6.1 allows the construction of Gröbner bases provided that the groundring R has a Noetherian structure.
Thus, supposing that an S-polynomial S : F × F −→ F is available, we fix a set A ⊆ R such that Λ ⊆ A = K × A.

Specialization to Particular Rings
As mentioned in Section 6.6, when dim K R as well as rank F are both finite, it may happen that a Gröbner basis for N ⊆ F can be computed without invoking the machinery of S-polynomials.
Assume we can compute such a solution (r l λ ), set G = G ∪ {n} and X = K × ΛG . If then LT(X ) is strictly containing LT(X), iteration of this procedure must terminate due to the finiteness of ΛE.

Vector Spaces
The most simple situation occurs when K = R. Then Λ = {1}, and if E = {e 1 , . . . , e n }, the module F = R (E) is just a finite dimensional vector space over the field K. ΛE = E and the structure formulae are 1 · c = c (c ∈ K) and 1 · 1 = 1.
Of course this is solvable, and we can produce a Gröbner reduction as described in Section 6.6.
In particular we obtain a Gröbner basis when X is defined as X = K × G and we proceed by extending it to X = K × (G ∪ {Φ(r)}) with a solution r ∈ ker(Φ).
Alternatively, we may take S(f, g) = f − g as an Spolynomial and compute a Gröbner basis using the Buchberger algorithm.

Monoid Rings
Assume that Λ is a finite monoid, R = K[Λ] the associated monoid ring, and F = R E with finite set E. We order ΛE arbitrarily. The linear system (16) associated to G = {g 1 , . . . , g q } ⊆ F amounts to q l=1 λs=t g l s r l λ = 0 (t ∈ LT(X)).
Having determined a solution matrix (r l λ ) ∈ K q×Λ such that the corresponding element n = t∈ΛE λs=t q l=1 r l λ g l s t = 0 we can set G = G∪{n}, X = K × ΛG. Since 1 ∈ Λ it is clear that n ∈ X and LT(n) ∈ T(n) ⊆ ΛE \LT(X). Therefore the set X properly contains X, and since ΛE is finite, iteration of this process must terminate and yields a Gröbner basis for RG.

Matrixrings
Let R = K n×n be the ring of square n × n-matrices over K. This ring has a natural K-basis Λ consisting of all matrices λ j i (1 ≤ i, j ≤ n) where λ j i is the matrix with 1 in position (j, i) and 0 else. Thus The basic structure formulae in this setting are then Let F = R E with finite set E be a free module and order ΛE arbitrarily. Consider a submodule N ⊆ F . Since R is simple Artinian, it is clear that F is Noetherian and therefore N is generated by a finite set G = {g 1 , . . . , g q } ⊆ F . Set X = K × ΛG. With notation as before we get So we have to compute the general solution of the system q l=1 n i=1 (g e l ) i u (r l ) j i = 0 (λ j u e ∈ LT(X)) and check if it is possible to single out a particular instance such that the corresponding element n is not zero. If this is the case we can proceed in the usual way: set X = K × Λ · (G ∪ {n}). Although n ∈ X it is obvious that LT(X ) ⊃ LT(X) whence the process terminates.

The Ring of Ore Polynomials
Given an endomorphism σ : K −→ K, a σ-skew derivation is an additive map δ : An Ore-variable over K is a pair ∂ = (σ, δ) where σ is an endomorphism and δ is a σ-skew derivation.
Let ∂ i = (σ i , δ i ) be Ore-variables (1 ≤ i ≤ n) such that all mappings σ i , δ j commute with each other. Then the Ore algebra O defined by X = (∂ 1 , . . . , ∂ n ) is the free K-module on the set of formal expressions ∂ k = ∂ k1 1 · · · ∂ kn n (k ∈ N n ) with multiplication determined by the rules For the set of monomials we use Λ = {∂ k : k ∈ N n }. With the notation and the binomial coefficient the product in O may be written explicitly where x, y ∈ K and k, l ∈ N n . Inparticular the two basic structure formulae are and a product that reflects the properties of derivations and automorphisms. We consider the elements of the set Λ = {δ k σ l | (k, l) ∈ N m × Z n } as the distinguished monomials. Consequently elements of D are finite K-linear combinations (k,l)∈N m ×Z n a k,l δ k σ l , (a k,l ∈ K) and the product is driven by the rules δ i · c = c δ i + δ i (c) σ j · c = σ j (c)σ j , (c ∈ K).
We call D a ∆Σ-ring over K.
A left module over D is also called a difference-differential module, or ∆Σ-module over K. In the literature the tuples δ and σ are denoted informally as the sets ∆ and Σ, whence the name. Note though, that the mappings δ i need not be distinct. The same is the case with the σ j .
Proof. This is due to the fact that partial derivatives have no relations among each other. Precisely: Let ∆ be the monoid generated (in End k (K)) by ∆. Then ∆ ∼ = N m and A m (k) is a free K-module with basis ∆ .
For computing the Hilbert function of a finitely gnerated difference-differential module, Winkler and Zhou introduced the concept of relative Gröbner bases (w.r.t. generalized term orders on N m × Z n ) [37]. Our approach is to present a difference-differential ring as a quotient of another ring that allows computing a Gröbner basis.
We use the notation y k s = (δ k • σ s )(y) (k ∈ N m , s ∈ Z n ).
For the free D-module F = D (E) , the product D×F −→ F can then be written explicitly δ k σ l · yδ r σ s e = u≤πk k u y k−u l δ u+r σ l+s e (k, r ∈ N m ; l, s ∈ Z n ; y ∈ K; e ∈ E).
This formula specializes to the basic structure formulae δ k σ l · y = u≤πk k u y k−u l δ u σ l and δ k σ l · δ r σ s = δ k+r σ l+s .
Then O/I ∼ = D.
This representaion allows the construction of Gröbner bases for submodules of finitely generated free modules over D out of Gröbner bases for the corresponding modules over O. Cf. also Lemma 3.1 and [15].
Consider the epimorphism π : O −→ D. We can write down generically an inverse image u ∈ O for arbitrary a ∈ D: If a = (k,l)∈N m ×Z n a k,l δ k σ l then set u = a = (k,l)∈N m ×Z n a k,l ∂ k1 1 · · · ∂ km m ζ |l1| 1 · · · ζ |ln| n where ζ j = η j if l j ≥ 0, and ζ j = φ j for l < 0. Let be given a left ideal N = D · {g 1 , . . . , g q } ⊆ D. Chosen inverse images h 1 , . . . , h q for g 1 , . . . , g q , the left ideal π −1 (N ) ⊆ O is generated by the set Let ρ be a Gröbner reduction for π −1 (N ), computed by stepwise extending H, and a ∈ D. Choose an inverse image u of a. Then a ∈ N iff u ρ G G 0 .

Conclusion
Given a module M and a submodule N ⊂ M , the goal of reduction is to compute a set of normal forms that allow to decide the membership problem for N . This is what the classical Gröbner basis computation provides for ideals in a polynomial ring over a field, and what similar adapted concepts yield for certain submodules over particular rings. Usually the process of computing such a basis is by iteration of a reduction step that produces a new element g out of an element f given as an input. In order to make sense, such a reduction has to obey certain properties. Clearly its iteration has to stop after finitely many steps and it should output a unique normal form as its result.
We have formulated these requirements in a list of four axioms that are appropriate for describing reduction relations for modules over a ring with basis.
The axioms are carefully dicussed and examples are provided for illustration. Depending on the ring there are several concepts that can be used to define a reduction. Among them we discuss possible term orders, filtrations and rank functions.
In the presence of a term order we consider three types of reductions: leading term reduction, T-reduction and classical reduction. Each of them is dicussed in detail and their interrelations are discribed. Then we discuss the case when the submodule N is finitely generated, describe the appropriate concept of an S-polynomial and demonstrate the equivalence of the full reduction concept with Gröbner bases.
The last part of the paper is devoted to examples that examine the introduced concepts.