Signature Gr\"obner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gr\"obner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gr\"obner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gr\"obner basis as well as a Gr\"obner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gr\"obner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalize reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gr\"obner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.


Introduction
Gröbner bases have become a fundamental and multi-purpose tool in computational algebra. They were initially introduced [Buc65] to answer questions about ideals of multivariate (commutative) polynomials, and they were subsequently generalized to several noncommutative settings, including Weyl polynomials [Gal85] encoding, for example, differential equations, and noncommutative polynomials in the free algebra [Mor85], which can model matrix identities and, more generally, identities of linear operators. In the latter setting, Gröbner bases and the associated reduction machinery are notably useful for simplifying and proving operator identities [HW94, HSW98, HRR19, CHRR20, SL20, RRHP21].
The main theoretical results needed to adapt the concept of Gröbner bases to the free algebra are due to Bergman [Ber78], who used the abstract concept of reduction systems to generalize the ideas from the commutative setting. This development was independent of the commutative theory. Around the same time, also Bokut' [Bok76] proved statements that are essentially equivalent to the ones by Bergman. Bokut' attributes his results to Shirshov, who had published similar results in the context of Lie algebras [Shi62]. The first explicit algorithm for computing noncommutative Gröbner bases was proposed by Mora [Mor85] a few years later, who adapted Buchberger's algorithm to the free algebra. Later, Mora also managed to unify the theory of Gröbner bases for commutative and noncommutative polynomial rings via a generalization of the Gaussian elimination algorithm [Mor94]. Recently, also the F4 algorithm [Fau99] has been adapted to this setting [Xiu12]. However, contrary to the commutative or the Weyl case, not all ideals in the free algebra admit a finite Gröbner basis. Instead, the algorithms always enumerate a Gröbner basis, with termination if and only if a finite Gröbner basis exists w.r.t. the chosen monomial order.
In the case of commutative polynomials, the latest generation of Gröbner basis algorithms are the so-called signature-based algorithms, heralded by the F5 algorithm [Fau02]. This class of algorithms was the subject of extensive research in the past 20 years, a survey of which can be found in [EF17]. Those algorithms compute, in addition to a Gröbner basis, some information on how the polynomials in that basis were computed. Using this information, the algorithms are able to identify relations between the computed polynomials, and use them to predict and avoid reductions to zero and redundant computations. This yields a significant performance improvement.
Additionally, it was observed recently [SW11,GVW15] that the data of signatures is enough to reconstruct the cofactors of the Gröbner basis, that is, the coordinates of the elements of the basis in terms of the input polynomials. Similarly, one can also compute a basis of the syzygy module of the generators. Alternatively, those operations can be realised using the classical theory and algorithms for Gröbner bases of modules (one can see [BW93,Ch. 10] for a textbook exposition with references, or [AL94, Ch. 3]), but signatures allow to reduce the cost of those computations to that of a Gröbner basis of an ideal.
Algorithms for computing Gröbner bases with signatures have also been developed in the case of Weyl algebras [SWMZ12], with the same application to the computation of coordinates of Gröbner basis elements, and of syzygies. Furthermore, in [Kin14] a noncommutative version of the F5 algorithm for right modules over quotients of path algebras was described. In this setting, the information encoded by the signatures is used to efficiently compute bases of Loewy layers. Finally, in the context of the free algebra, recent work [CLV] has independently introduced similar definitions as in Section 3, and proved lower bounds for the complexity of the set of leading monomials of the module of syzygies, in the sense of the Chomsky hierarchy.
The problem of computing the cofactors of a Gröbner basis is also central when working with noncommutative polynomials. In particular, when proving operator identities, this information allows to construct a proof certificate for a given identity, which can be checked easily and independently of how it was obtained, see for example [Hof20]. Like in the commutative case, the classical theory and algorithms for computing Gröbner bases in modules can also be used to obtain such information [BK06,Mor16], but those algorithms are significantly more expensive than a mere Gröbner basis computation. Furthermore, for most ideals in the free algebra, the module of syzygies is not finitely generated and does not admit a finite Gröbner basis, which makes those algorithms in fact only enumeration procedures.
In this paper, we show how to define and compute signature Gröbner bases for noncommu-2 tative polynomials in the free algebra, and we show how to use them to reconstruct the module representation of elements of the ideal, and a basis of the syzygy module of the generators. We also generalize some classical signature-based criteria such as the syzygy criterion, the F5 criterion and the singular criterion, in order to use signatures to accelerate the algorithms. More precisely, for introductory purposes, we first present Algorithm 1. This algorithm is impractical because it has to perform expensive module computations. Replacing those computations with signature manipulations naturally leads to Algorithm 2, which also includes the signature-based criteria. This algorithm can be considered as a generic template for signature-based algorithms in the free algebra. Finally, we introduce Algorithms 3 and 4 which allow to reconstruct the output of Algorithm 1 using only the signatures. A difficulty specific to the case of noncommutative polynomials is that some ideals may not have a finite signature Gröbner basis, even if the ideal has a finite Gröbner basis. This is unavoidable, but we prove that the algorithms nonetheless correctly enumerate a signature Gröbner basis.
Additionally, as already mentioned, the module of syzygies of the generators usually does not admit a finite Gröbner basis. More precisely, the module spanned by the leading terms of the so-called "trivial syzygies" (sometimes called Koszul syzygies, or principal syzygies) is typically not finitely generated. On the other hand, a strength of signature-based algorithms is precisely that they make it possible to identify those trivial syzygies, and in particular this set of trivial syzygies admits a finite and effective representation in terms of a signature Gröbner basis. In classical cases, avoiding those trivial syzygies is the crux of the F5 criterion, and in the noncommutative case, it allows the algorithm to enumerate a basis of the syzygy module by only considering the non-trivial syzygies.
If our algorithm terminates, it computes a finite signature Gröbner basis, and in particular, a finite Gröbner basis and a finite and effective description of the module of syzygies of the input polynomials. We conjecture (see Conjecture 46) that conversely, the existence of a finite Gröbner basis of the ideal and of a finite description of the module of syzygies of the generators, implies the existence of a finite signature Gröbner basis.
We also provide a toy implementation 1 of the algorithms presented in this paper in the Mathematica package OperatorGB [HRR19,Hof20]. For an overview on other available software packages in the realm of noncommutative Gröbner bases, see [LSZ20] and references therein.
We show experimentally that the use of signatures allows to drastically reduce the number of S-polynomials considered and reduced to zero. However, first timings indicate that our implementation cannot compete with the standard noncommutative Buchberger algorithm. So, while conceptually our algorithm is fairly simple and very similar to Buchberger's algorithm, implementing it in an efficient way seems to be a highly non-trivial task. However, we are optimistic that, as in the commutative case, noncommutative signature-based algorithms can lead to an acceleration of Gröbner basis computations in the free algebra. In this work, we focus on building a theoretical foundation for this kind of algorithms and on highlighting possible uses of the additional information provided by the signatures.

Preliminaries
For the convenience of the reader, we recall the most important aspects of the theory of Gröbner bases in the free algebra and of Gröbner bases of submodules of the free bimodule in this section. Additionally, we introduce the notion of signatures in this noncommutative setting as a straightforward generalization of signatures from the commutative case.
We fix a finite set of indeterminates X = {x 1 , . . . , x n } and denote by X the free monoid over X containing all words (or monomials) of the form w = x i 1 . . . x i k including the empty word 1. The quantity k is called the length of w.
For a field K, we let be the free algebra generated by X over K. We consider the elements in K X as noncommutative polynomials with coefficients in K and indeterminates in X, where indeterminates commute with coefficients but not with each other. For a given set of polynomials F ⊆ K X , we denote by (F) the (two-sided) ideal generated by F, that is The set F is called a set of generators of (F). An ideal I ⊆ K X is said to be finitely generated if there exists a finite set of generators F ⊆ K X such that I = (F). We agree upon the convention to write ( f 1 , . . . , f r ) instead of ({ f 1 , . . . , f r }) if the elements of F = { f 1 , . . . , f r } are given explicitly.
Remark 1. If |X| > 1, the free algebra K X is not Noetherian, i.e., there exist ideals in K X which are not finitely generated. One prominent example is the ideal (xy i x | i ∈ N) ⊆ K x, y , which has no finite set of generators.
Definition 2. A monomial ordering on X is a well-ordering that is compatible with the multiplication in X , that is, w w ′ implies awb aw ′ b for all a, b, w, w ′ ∈ X .
An example of a monomial ordering on X is the degree lexicographic ordering deglex , where two words w, w ′ ∈ X are first compared by their length and ties are broken by comparing the variables in w and w ′ from left to right using the lexicographic ordering x 1 ≺ lex · · · ≺ lex x n .
In what follows, we fix a monomial ordering on X . Then, every non-zero f ∈ K X has a unique representation of the form f = c 1 w 1 + · · · + c d w d with c 1 , . . . , c d ∈ K \ {0} and w 1 , . . . , w d ∈ X such that w 1 ≻ · · · ≻ w d .
Definition 3. Let f = c 1 w 1 +· · ·+c d w d ∈ K X \{0} with c 1 , . . . , c d ∈ K \{0} and w 1 , . . . , w d ∈ X such that w 1 ≻ · · · ≻ w d . Then, w 1 is called the leading monomial of f , denoted by lm( f ). The coefficient c i of w i is denoted by coeff( f, w i ) for i = 1, . . . , d. We call c 1 the leading coefficient of f , abbreviated as lc( f ). If lc( f ) = 1, then f is called monic. Furthermore, the leading term lt( f ) of f is lt( f ) = lc( f ) · lm( f ). Finally, the set {w 1 , . . . , w d } is called the support of f and denoted by supp( f ).
We use the convention that the leading term and leading coefficient of the zero polynomial are 0. The leading monomial of the zero polynomial is undefined. 4 In the following, we briefly recall the most important results about Gröbner bases in K X . For a more extensive treatment of this subject, we refer to the recent surveys [Xiu12,Mor16,Hof20]. The main concept needed to discuss and compute noncommutative Gröbner bases is polynomial reduction.
Definition 4. Let f, f ′ , g ∈ K X with g 0. We say that f reduces to f ′ by g if there exist a, b ∈ X such that a lm(g)b ∈ supp( f ) and In this case, we write f → g f ′ .
Based on this concept, for a set G ⊆ K X , we define a reduction relation We denote by * → G the reflexive, transitive closure of → G . Using this reduction relation, we can now define Gröbner bases in K X .
Definition 5. Let I ⊆ K X be an ideal and G ⊆ I. Then, G is a Gröbner basis of I if f * → G 0 for all f ∈ I. Furthermore, G is called reduced if all elements in G are monic and no g ∈ G is reducible by G \ {g}.
We note that not all finitely generated ideals in K X have a finite Gröbner basis as witnessed by the following example. For this example, we recall the well-known facts that the reduced Gröbner basis G of an ideal I ⊆ K X is unique (w.r.t. a fixed monomial ordering) and that I has a finite Gröbner basis if and only if G is finite (see for example [Xiu12,Prop. 3.3.17, Cor. 3.3.18]).
Example 6. Let K be a field and X = {x, y}. The principal ideal I = (xyx − xy) ⊆ K X does not have a finite Gröbner basis for any monomial ordering on X . In fact, the reduced Gröbner basis of I is given by the infinite set G = {xy n x − xy n | n ≥ 1}. To see this, we first note that g n = xy n x − xy n ∈ I for all n ≥ 1, which follows inductively from g 1 = xyx − xy ∈ I and g n+1 = xyg n + g 1 (y n − y n x). Consequently, we get that G ⊆ I. Furthermore, since xyx = (xy) · x ≻ (xy) · 1 = xy, all elements in G are monic. Also, no element in G can be reduced by any other element, which shows that G is reduced. Then, one can verify that G is indeed a Gröbner basis using the noncommutative analogue of Buchberger's S-polynomial criterion, also known as Bergman's diamond lemma [Ber78, Thm. 1.2]. We note that each pair g i , g j ∈ G leads to an S-polynomial s i, j = xy i+ j x − xy i xy j , which can be reduced to zero as follows: xy i+ j x − xy i xy j → g i xy i+ j x − xy i+ j → g i+ j 0.
Even though ideal membership is undecidable in the free algebra, a noncommutative analog of Buchberger's algorithm can be used to enumerate a (possibly infinite) Gröbner basis.
We also need the notion of Gröbner bases of sub-bimodules of a free K X -bimodule. Hence, we shall briefly recall this concept in the following. For further details on this topic, we refer to [Xiu12,Mor16]. We note that when speaking about a (sub)module, we always mean a (sub-)bimodule.
Given a monomial ordering , an example of a module ordering is the term-over-position ordering top , where a 1 ε i 1 b 1 top a 2 ε i 2 b 2 for two module monomials a 1 ε i 1 b 1 , a 2 ε i 2 b 2 ∈ M(Σ) if one of the following conditions holds: 1. a 1 b 1 ≺ a 2 b 2 ; 2. a 1 b 1 = a 2 b 2 and a 1 ≺ a 2 ; 3. a 1 b 1 = a 2 b 2 and a 1 = a 2 and i 1 ≤ i 2 .
As for polynomials, we use the convention that the signature coefficient and the signature term of the zero element in Σ are 0. The signature of the zero element remains undefined. By a slight abuse of notation, for a set F ⊆ Σ, we denote by s(F) the set of signatures of all non-zero α ∈ F, i.e., s(F) = {s(α) | 0 α ∈ F}.
Remark 9. Note that we define the signature s(α) of a module element α ∈ Σ. This is different to the original definition of a signature. Initially, signatures were looked at from a polynomial point of view and therefore signatures of polynomials were defined [Fau02]. The notion of signature which we use was first introduced in [GGV10].
One immediate consequence of the definition of signatures is the following lemma.
Remark 12. Gröbner bases of submodules can also be defined using a notion of reduction. The two definitions are equivalent, and the proof is the same as in the classical case of ideals. In our case, the present definition will be the more useful one for modules.

Signature and labelled Gröbner bases
The aim of this section is to introduce the notion of signature Gröbner bases of ideals in the free algebra. As an intermediate notion, we define the concept of labelled Gröbner bases, which are Gröbner bases keeping track of the construction of each element in terms of the generators. As in the commutative case, they are defined using a more restrictive notion of polynomial reduction, called s-reduction. Moreover, we also define and characterize noncommutative minimal signature/labelled Gröbner bases. We note that all notions introduced here are straightforward 6 generalization of the same notions for commutative polynomials. The key differences between the commutative and noncommutative case will become apparent in Section 4. For the rest of this paper, we fix a finite indexed family of polynomials ( f 1 , . . . , f r ) ∈ K X r generating an ideal I = ( f 1 , . . . , f r ). Furthermore, we fix a monomial ordering on X and a module ordering M on M(Σ). We additionally require the following two conditions: 1. and M have to be compatible in the sense that for all a, b ∈ X and i = 1, . . . , r.
Example 13. The module ordering top with underlying monomial ordering deglex is fair. Furthermore, in that case, the orderings deglex and top are compatible. On the other hand, if the rank of the module is at least 2, any position-over-term module ordering pot , that is, any order- We note that the requirement for a fair module ordering is a particularity of the noncommutative case. To compute Gröbner bases (without signatures) in the free algebra using Buchberger's algorithm, a so-called fair selection strategy has to be used. Such a selection strategy ensures that every S-polynomial that is formed is eventually processed. Using a non-fair selection strategy can cause the algorithm to run indefinitely, even if the ideal admits a finite Gröbner basis (w.r.t. the used monomial ordering). Furthermore, in such cases, the infinite set produced by the algorithm fails to be a Gröbner basis, see Example 47.6.27 and the subsequent discussion in [Mor16].
Transferring the idea of a fair selection strategy to the case of signature-based algorithms leads to our definition of a fair module ordering. Using such a fair ordering guarantees that the selection strategy used by our algorithm is fair. It is not clear whether or to what extent this requirement can be weakened.
From now on, we shall denote both orders M and by the same symbol . It will be clear from the context which ordering is meant, as we will denote elements from Σ by Greek letters and elements from K X by Roman letters. We note that all results that follow from here on depend (implicitly) on and are to be understood w.r.t. our fixed monomial and module ordering.
Elements in the free K X -bimodule Σ encode elements of the ideal I via the K X -module homomorphism with a i , b i ∈ X and j i ∈ {1, . . . , r}.
We adapt the notation from [SW11] and denote by f [α] a pair ( f, α) ∈ K X × Σ with f = α. We refer to f [α] as a labelled polynomial. By the definition of ·, we always have f [α] ∈ I × Σ. Furthermore, as done in [SW11], we denote by f (σ) a pair ( f, σ) ∈ K X × M(Σ) such that there exists f [α] ∈ I [Σ] with s(α) = σ. We call such a pair a signature polynomial. Additionally, we denote by I [Σ] and I (Σ) the set of all labelled polynomials, respectively, the set of all signature polynomials, that is, Remark 14. We note that different families of generators of the same ideal I ⊆ K X always lead to different sets I [Σ] . To be more precise, given I [Σ] , the family of generators of I used in the construction can be recovered: It is still true that different families of generators of I can lead to different sets I (Σ) , but it is not necessarily the case. So to be precise, we should only speak of the set of signature polynomials I (Σ) w.r.t. the family of generators ( f 1 , . . . , f r ). However, whenever the generators f 1 , . . . , f r are clear from the context, we shall omit this part and only speak of I (Σ) .
The motivation for using the notation f [α] (resp. f (σ) ) is that the polynomial f , rather than the module element α (resp. the module monomial σ), is the main object of interest. By abuse of language, we call s(α) (resp. σ) the signature of the labelled polynomial f [α] (resp. of the signature polynomial f (σ) ) and we denote it by s( f [α] ) (resp. s( f (σ) )).
The reason for introducing both labelled polynomials and signature polynomials is that the former allow to present the theory in a simpler fashion and lead to simpler proofs. However, in an actual implementation of a signature-based algorithm, keeping track of the full module representation stored in a labelled polynomial causes a significant overhead in terms of memory consumption and overall computation time. Fortunately, we will see that all theoretical results only depend on information encoded in signature polynomials. Consequently, when implementing a signature-based algorithm, one would only work with signature polynomials. This reduces the computational overhead. Additionally, we note that the reconstruction techniques discussed in Section 5 allow to efficiently recover all information encoded in labelled polynomials from signature polynomials.
In what follows, we will sometimes work with sets of polynomials, sometimes with sets of labelled polynomials and sometimes with sets of signature polynomials. To be able to better distinguish between these cases, we denote subsets of K X by capital letters (e.g. G ⊆ K X ) and subsets of I [Σ] and I (Σ) by capital letters with the additional exponent [Σ] and (Σ) respectively (e.g. G [Σ] ⊆ I [Σ] and G (Σ) ⊆ I (Σ) ).
Computations in I [Σ] can be defined naturally. In particular, for [α+β] , and With these operations the set [β] if and only if f = g and α = β.
In order to discuss signature Gröbner bases, we need to adapt the notion of polynomial reduction to labelled polynomials. This leads to the following definition of s-reduction.
• s(aγb) s(α), and In this case, we write Remark 16. In terms of polynomials, the first condition means that we can do usual polynomial reduction. This implies that either f ′ = 0 or lm( f ′ ) lm( f ). The second condition ensures that this inequality also transfers over to Σ, i.e., that either α ′ = 0 or s(α ′ ) s(α). Furthermore, if the s-reduction is regular, then s(α) = s(α ′ ).
Those observations allow to generalize the definition to signature polynomials. More precisely, given f (α) , g (γ) ∈ I (Σ) , it is possible to test whether f (α) is s-reducible by g (γ) . Furthermore, if the s-reduction is regular, it is possible to compute its remainder as a signature polynomial.
We capture some useful facts about s-reduction that will be needed later. This first lemma follows immediately from the definition.
The following lemma appears to be folklore, it is for example used in the proof of [RS12, Lemma 9]. We include a proof for completeness.
is also top s-reducible by G [Σ] . In fact, the same element from The outcome of classical polynomial reduction depends on more than just the leading term of the polynomial that is reduced. Polynomials which share the same leading term can still reduce to different elements. In case of regular s-reductions, certain assumptions on the set of reducers G [Σ] imply that all labelled polynomials with the same signature term yield the same regular s-reduced result. This fact is captured in the following lemma which is a noncommutative analogue of [RS12, Lemma 2].
. Then, the following hold.
Proof. The proof of the commutative version of this lemma [RS12, Lemma 2] carries over to the noncommutative setting.
Using the notion of s-reduction, we now define a labelled Gröbner basis of the module I [Σ] .
Remark 21. Since different families of generators of the ideal I lead to different modules I [Σ] , they also lead to different labelled Gröbner bases (see also Example 31).
We recall that we present all the relevant theory for our signature-based algorithm in terms of labelled polynomials keeping in mind that in an actual implementation one would work only with signature polynomials. Consequently, such an implementation would not compute a labelled Gröbner basis but instead a signature Gröbner basis as defined below.
In the theoretical sections, which are this section and the next one, we focus on the notion of labelled Gröbner bases and present all theoretical results in terms of this concept. However, one should always keep in mind that all relevant results also transfer over to signature Gröbner bases. In Section 5, we then shift our focus to a more application-oriented point of view and present our signature-based algorithm in terms of signature polynomials. Additionally, we show how to reconstruct a labelled Gröbner basis from a signature Gröbner basis.
It follows directly from the definition that a labelled Gröbner basis of I [Σ] is by no means unique. In fact, if . Furthermore, the set I [Σ] is always a labelled Gröbner basis of I [Σ] . Thus, we can immediately deduce the following corollary.
Corollary 23. For every finite family of generators of an ideal I ⊆ K X , the module I [Σ] has a (possibly infinite) labelled Gröbner basis.
We also provide the following equivalent characterization of labelled Gröbner bases, which will turn out to be useful later.
Proof. The proof is an adaptation of the proof of the commutative version of this statement without signatures [BW93, Theorem 5.35]. One only has to replace commutative polynomials by labelled (noncommutative) polynomials and polynomial reduction by s-reduction.
The following proposition relates labelled Gröbner bases to Gröbner bases and is an immediate consequence of Lemma 17.
Although a labelled Gröbner basis G [Σ] of I [Σ] is not unique in general, we can demand certain additional properties from G [Σ] in order to at least obtain a labelled Gröbner basis which is as small as possible. We call such a labelled Gröbner basis a minimal labelled Gröbner basis of I [Σ] .
We also extend this definition to signature Gröbner bases.
be a labelled Gröbner basis (up to signature σ) such that A minimal labelled Gröbner basis is minimal in the following sense.  and a, b ∈ X such that lm(g) = lm(ahb) and s(γ) s(aδb).
Combining these two statements yields We note that, starting with a finite family of generators ( f 1 , . . . , f r ) ∈ K X r of an ideal I ⊆ K X , the module I [Σ] always has a minimal labelled Gröbner basis, which is finite if and only if I [Σ] has a finite labelled Gröbner basis w.r.t. the family of generators f 1 , . . . , f r . This follows from the following proposition, which tells us that we can obtain a minimal labelled Gröbner basis from a labelled Gröbner basis by removing all elements that are top s-reducible. 11 By Proposition 28, R 1 ⊆ R 2 and R 2 ⊆ R 1 , so R 1 = R 2 . We claim that the cardinality of G [Σ] 1 and G [Σ] 2 is equal to that of R 1 . Indeed, assume that it is not the case. W.l.o.g. we can assume that G [Σ] 1 is larger than R 1 , so by the pigeonhole principle, there exist g [γ] and h [δ] in G [Σ] 1 , distinct, such that lm(g) = lm(h) and s(γ) = s(δ). Then by definition, h [δ] is top s-reducible by g [γ] , which contradicts the minimality of G [Σ] 1 . However, we cannot expect I [Σ] to have a finite (minimal) labelled Gröbner basis for any finitely generated ideal I ⊆ K X , as there are finitely generated ideals that simply do not have a finite Gröbner basis, and consequently, also no finite labelled Gröbner basis. Unfortunately, the condition that an ideal I = ( f 1 , . . . , f r ) has a finite Gröbner basis is also not sufficient to ensure that I [Σ] has a finite labelled Gröbner basis w.r.t. the family of generators f 1 , . . . , f r .
Example 31. We give an example of an ideal with a finite Gröbner basis, but no finite labelled Gröbner basis. The construction and the proof of the claims rely on notions introduced in Section 4, and will be deferred until that point.
Let K be a field and X = {x, y}. We consider the ideal We equip K X with deglex where we order the indeterminates as x ≺ lex y and we use top as a module ordering. Then, with g n = yx n+2 y and certain α, γ n ∈ Σ such that s(α) = ε 1 y and s(γ n ) = yε 3 y n . So, Corollary 30 implies that I [Σ] does not have a finite labelled Gröbner basis. The obstruction to having a finite labelled Gröbner basis is that f [α] 4 cannot be used to s-reduce any of the 12 n . Indeed, since s(α) ≻ ε 3 and yx n s(α) ≻ s(γ n ), the reductions would cause the signatures to increase, and would not be s-reductions.
If instead we consider the family of generators generating the module I [Σ ′ ] , where Σ ′ = (K X ⊗ K X ) 4 denotes the free K X -bimodule of rank 4, then the finite set

Regular S-polynomials
The objective of this section is to state an adaptation of the noncommutative version of Buchberger's algorithm to include signatures. To this end, we need to adapt the notion of S-polynomials to the case of labelled polynomials. We first extend the notion of ambiguities from [Ber78] from polynomials to labelled polynomials. Recall that we fixed an indexed family of generators ( f 1 , . . . , f r ) ∈ K X r of an ideal I = ( f 1 , . . . , f r ) ⊆ K X as well as a monomial and a module ordering. an overlap ambiguity of G [Σ] . We define its S-polynomial sp(a) to be Ag [β] .
Similarly, if f g, lm( f ) = ABC and lm(g) = B for some words A, B, C ∈ X , then we call the tuple an inclusion ambiguity of G [Σ] . We define its S-polynomial sp(a) to be Disregarding the module labelling in the definition recovers the usual constructions of the noncommutative version of Buchberger's algorithm. We recall a few classical observations on this construction, which might be unfamiliar to a reader more used to the commutative case. First, two labelled polynomials can have more than one ambiguity with each other. Furthermore, an element can also form overlap ambiguities with itself.
In the noncommutative case, following [Ber78], we only form S-polynomials in the presence of an ambiguity. This is an analogue of Buchberger's coprime criterion (sometimes called GCD criterion) [Mor94, Cor. 5.8]. Contrary to the commutative case, the criterion is embedded in the definition of an S-polynomial. This ensures that two polynomials can only give rise to finitely many S-polynomials. This is necessary to ensure that Buchberger's algorithm terminates whenever a finite Gröbner basis exists.
An ambiguity a = (ABC, in case that a is an overlap ambiguity, respectively if in case that a is an inclusion ambiguity. If an ambiguity is not singular, it is called regular. We call an S-polynomial regular (resp. singular), if the respective ambiguity is regular (resp. singular). Similarly to the case of reductions, it is possible to compute regular S-polynomials of signature polynomials.
In the following, we collect some useful results about S-polynomials. We start by relating the signature of a regular S-polynomial to the signatures of the two input elements. This first proposition is an immediate consequence of the definition of a regular ambiguity. Proof. Let a = (ABC, A, C, f [α 1 ] 1 , f [α 2 ] 2 ) be a regular ambiguity between f [α] and some element According to Proposition 34, we have s(sp(a)) s(α). Assume for contradiction that s(sp(a)) = s(α). Then, a cannot be an overlap ambiguity as otherwise where the strict inequality follows from the fact that A, C 1. So, a must be an inclusion ambiguity. Note that then g [γ] f [α] , as a labelled polynomial cannot form inclusion ambiguities with itself. Therefore, g [γ] ∈ G [Σ] . Furthermore, by definition of a regular S-polynomial we know s(α) = s(sp(a)) = max{s(α 1 ), s(Aα 2 C)} s (Aα 2 C).

Characterization of labelled Gröbner bases
As in the commutative case, the design and the proof of correctness of the algorithm will rely on a signature variant of Buchberger's characterization of Gröbner bases, stating that if all regular S-polynomials s-reduce to zero, then one has a labelled Gröbner basis.
A particularity of the noncommutative case is that we will need to handle trivial syzygies separately in the proof process. This is because the noncommutative definition of S-polynomials (Definition 33) effectively contains the restrictions granted by Buchberger's coprime criterion, eliminating some trivial syzygies. Without those restrictions, the algorithms (even without signatures) would rarely terminate, because the module of trivial syzygies is in general not finitely generated.
In order to prove the noncommutative characterization of labelled Gröbner bases, we prove several lemmas. Leaving aside the provisions for trivial syzygies, we first prove Lemma 36, which states that, given two labelled polynomials with the same leading monomial and different signatures, one can find a regular S-polynomial whose signature divides the larger signature. This lemma is technical and useful for the rest of the proofs. It ensures that it suffices to consider regular S-polynomials.
Then, we prove Lemma 37, which states that given one regular S-polynomial q [ρ] , one can find another regular S-polynomial p [π] such that s(π) and s(ρ) have a common multiple in M(Σ) and such that p [π] satisfies certain additional conditions concerning its s-reducibility. This allows us to prove Lemma 38, which makes the same statement, but starting from any polynomial. This last lemma is a noncommutative analogue of Lemma 9 in [RS12,appendix]. Just like in the commutative case, it is the cornerstone of the proof of the final Theorem 39, which is the wanted characterization.
We are now going to construct p [π] ∈ I [Σ] such that there exist a, b ∈ X with s(aπb) = s(a ′ ρb ′ ) and 1. π is a trivial syzygy between two elements in G [Σ] , or 2. p [π] is a regular S-polynomial with p = 0 or lm(apb) ≺ lm(a ′ qb ′ ).
We are done if π is a trivial syzygy or if ap ′[π ′ ] b is not regular top s-reducible where p ′[π ′ ] is the result of regular s-reducing p [π] . Otherwise we can repeat this process to construct a third labelled polynomial with the same properties. This process must terminate at some point since ≺ is a well-ordering.

π is a trivial syzygy between two elements in G
is the result of regular s-reducing p [π] ; Proof. The proof follows the same structure as that of [RS12, Lemma 9, appendix]: first, we combine leading terms to show that there exists L [λ] ∈ I [Σ] with signature dividing that of δ, and then, starting from that element, we construct p [π] as wanted. 18 Considering leading terms. Let s(δ) = a 1 ε i b 1 for some 1 ≤ i ≤ r and a 1 , b 1 ∈ X . By our assumption on G [Σ] there exists g Hence, with a 1 , b 1 , g [γ 1 ] 1 from above, we have Now, we apply Lemma 36 to a 1 g [γ 1 ] 1 b 1 and a 2 g [γ 2 ] 2 b 2 , which yields a ′ , b ′ ∈ X and q [ρ] ∈ I [Σ] such that s(a ′ ρb ′ ) = s(a 1 γ 1 b 1 ) = s(δ) and such that one of the following conditions holds: 1. ρ is a trivial syzygy between g [γ 1 ] 1 and g [γ 2 ] 2 ; 2. q [ρ] is a regular S-polynomial with q = 0 or lm(a ′ qb ′ ) ≺ lm(a 1 g 1 b 1 ); If ρ is a trivial syzygy, we can set p [π] = 0 [ρ] , a = a ′ , b = b ′ and are done. Otherwise, we note that by assumption all u [µ] ∈ I [Σ] with s(µ) ≺ s(δ) = s(a ′ ρb ′ ) s-reduce to zero by G [Σ] . Hence, we can apply Lemma 37 to the regular S-polynomial q [ρ] which gives p [π] ∈ I [Σ] and a, b ∈ X such that s(aπb) = s(a ′ ρb ′ ) = s(δ) and such that one of the following conditions holds: 1. π is a trivial syzygy between two elements in G These are the desired a, b and p [π] .
We can now finally state and prove the following theorem.
Theorem 39. Let σ ∈ M(Σ) be a module monomial and let G [Σ] ⊆ I [Σ] be such that for all ε i ≺ σ there exists g Assume that all regular S-polynomials p [π] of G [Σ] with s(π) ≺ σ regular s-reduce to some p ′[π ′ ] by G [Σ] such that π ′ is a syzygy or p ′[π ′ ] is singular top s-reducible. Then, G [Σ] is a labelled Gröbner basis of I [Σ] up to signature σ.
Remark 40. The notion of being singular top s-reducible is equivalent to what is in the (commutative) literature also called sig-redundant (see [EP11]) and included in the concept of super top reductions in [GVW15]. Additionally, a regular s-reduced element being singular top s-reducible corresponds to the notion of not being primitive s-irreducible in [AP11]. and a, b ∈ X such that s(aπb) = s(δ) and such that 1. π is a (trivial) syzygy (between two elements in G [Σ] ), or 2. p [π] is a regular S-polynomial of G [Σ] and is the result of regular s-reducing p [π] .
We distinguish between the two possible cases.
Case 2: p [π] is a regular S-polynomial of G [Σ] . Then, by construction we know that ap ′[π ′ ] b is not regular top s-reducible where p ′[π ′ ] is the result of regular s-reducing p [π] . By assumption, π ′ is a syzygy or p ′[π ′ ] is singular top s-reducible. In the first case, we can reuse the arguments from above to reach the same contradiction. Hence, we can assume that p ′ 0 and that p ′[π ′ ] is singular top s-reducible. We denote f [α] = sc(δ) sc(π ′ ) ap ′[π ′ ] b and note that this element is regular top s-reduced since ap ′[π ′ ] b is regular top s-reduced. Since also h [δ] is regular top s-reduced and st(δ) = st(α), Lemma 19 yields that lt(h) = lt( f ). So, anything that top s-reduces ap ′[π ′ ] b also top s-reduces h [δ] . We note that ap ′ Example 31 (continuing from p. 12). Recall that in Example 31 we considered the ideal over a field K in the variables X = {x, y}. We also defined f 4 = xxy and used deglex , where x ≺ lex y, as a monomial ordering and top as a module ordering. We claimed that a minimal labelled Gröbner basis of I [Σ] , w.r.t. the family of generators f 1 , f 2 , f 3 , is given by with g n = yx n+2 y and certain α, γ n ∈ Σ such that s(α) = ε 1 y and s(γ n ) = yε 3 y n . We now prove that G and f [ε 3 ] 3 , the first hypothesis of the theorem is satisfied. Then, we verify that all regular S-polynomials top s-reduce to 0 or to a singular top s-reducible element. It is a straightforward, if tedious, calculation, which is detailed in Appendix A.
Then, we prove that G [Σ] is minimal. Looking at the leading terms, the only possible reductions would be using f [α] 4 to reduce f [ε 3 ] 3 or g n . But as s(α) ≻ ε 3 and yx n s(α) ≻ s(γ n ), those reductions would not be s-reductions. So none of the elements of G [Σ] is s-reducible modulo the others, and G [Σ] is a minimal labelled Gröbner basis.
The claim that I [Σ ′ ] with the family of generators f 1 , f 2 , f 3 , f 4 has minimal labelled Gröbner basis } is proved along the same lines in Appendix A.

Effective description of the module of syzygies
Similarly to Buchberger's classical characterization of Gröbner bases, Theorem 39 allows us to state a first, non-optimized version of a signature-based algorithm for noncommutative polynomials, by ensuring that regular S-polynomials which are not trivial syzygies regular s-reduce to zero or to a singular top s-reducible normal form.
The fact that we need to handle at least some trivial syzygies separately is a crucial difference to the commutative case: in the commutative case, Buchberger's coprime criterion and the F5 criterion allow to eliminate some (resp. all) trivial syzygies, but signature-based algorithms terminate even without the criteria.
By contrast, in the noncommutative case, the module of trivial syzygies is in general not finitely generated, which requires handling trivial syzygies separately. In doing so, we are able to obtain an effective description of the module of syzygies. More precisely, we state the following fact about syzygies. Proof. Since 0 [µ] is top s-reduced by G [Σ] , Lemma 38 yields the existence of p [π] ∈ I [Σ] and a, b ∈ X such that s(µ) = s(aπb) and such that one of the following conditions holds: 1. π is a trivial syzygy between two elements in G is the result of regular s-reducing p [π] ; If π is a trivial syzygy, we are done. Otherwise, since neither 0 [µ] nor cap ′[π ′ ] b, with c = sc(µ) sc(π ′ ) , are regular top s-reducible and st(µ) = st(caπ ′ b), Lemma 19 yields that lt(cap ′ b) = lt(0) = 0, and consequently, also p ′ = 0.
Lemma 41 allows us to describe more precisely the syzygy module S = Syz( f 1 , . . . , f r ). Consider the set H triv of trivial trivial syzygies of G [Σ] Note that the set of signatures of H triv contains all the elements of the form max{s(γ 1 )m lm(g 2 ), lm(g 1 )m s(γ 2 )}, . It may happen that this set contains infinitely many module monomials which do not divide each other, and indeed this will be the case for all sufficiently non-trivial ideals. It implies that for such ideals, S does not admit a finite Gröbner basis.
However, Lemma 41 shows that a Gröbner basis of S is given by adding to H triv all the syzygies found by regular s-reducing to zero all regular S-polynomials of G [Σ] . Furthermore, if G [Σ] is finite, the set of signatures of syzygies in H triv can be enumerated using the description (1), and altogether, we obtain an effective description of the syzygy module of f 1 , . . . , f r .

Algorithm
The algorithm, incorporating both the computation of the labelled Gröbner basis and of the aforementioned description of the syzygy module of f 1 , . . . , f r , is given in Algorithm 1. We note that we state this algorithm only for theoretical consideration. In an actual implementation, one would replace all computations with labelled polynomials in Algorithm 1 by computations Algorithm 1 LabelledGB Input: ( f 1 , . . . , f r ) ∈ K X r generating an ideal I Output (if the algorithm terminates): with signature polynomials. In Section 5, we state with Algorithm 2 an optimized version of Algorithm 1 incorporating this.
We note that we cannot expect Algorithm 1 to always terminate since, as already mentioned, there are polynomials in K X generating a module I [Σ] which does not have a finite labelled Gröbner basis. However, the following theorem ensures that the algorithm always correctly enumerates a labelled Gröbner basis of the module I [Σ] defined by the input ( f 1 , . . . , f r ) ∈ K X r , and a Gröbner basis of the syzygy module Syz( f 1 , . . . , f r ).
In this sense, Algorithm 1 enumerates a labelled Gröbner basis of I [Σ] and a Gröbner basis of the syzygy module Syz( f 1 , . . . , f r ).
In order to prove this theorem, we first state the following useful lemma which ensures that Algorithm 1 cannot "get stuck" at a certain signature indefinitely.
Lemma 43. During the execution of Algorithm 1, elements from P are processed in ascending order w.r.t. their signatures and every possible signature is eventually processed.
Proof. Note that the set P is finite at all times. Hence, we can associate to P the tuple P ′ of all signatures of P sorted in increasing order, i.e., if P = {p [π 1 ] 1 , . . . , p [π n ] n } with s(π 1 ) · · · s(π n ), then P ′ = (s(π 1 ), . . . , s(π n )). We claim that P ′ strictly increases lexicographically between each run of line 5. If the algorithm does not reach line 12 after choosing p [π] ∈ P in line 5, this statement follows since p [π] is removed but nothing is added to P. Otherwise, the statement follows from Lemma 35, which implies that the signatures of the elements added to P in line 12 are strictly larger than s(π). Note that Lemma 35 is applicable here since the algorithm only reaches line 12 if the normal form p ′[π ′ ] computed in line 7 is not top s-reducible by G [Σ] (before p ′[π ′ ] is added to G [Σ] ). Then, P ′ strictly increasing between each run of line 5 and the fact that p [π] is always chosen to have minimal signature among all elements in P show that elements from P are processed in ascending order w.r.t. their signature. Furthermore, this, together with the fairness of the module ordering, also implies that every element in P will be removed eventually.
as input to Algorithm 1, after processing the signature ε 1 y, the set 4 }, and the set of corresponding polynomials is a Gröbner basis of I = ( f 1 , f 2 , f 3 ) The following corollary is an immediate consequence of the last observation.
Combining this corollary with Corollary 30, we see that Algorithm 1 terminates whenever I [Σ] admits a finite labelled Gröbner basis w.r.t. the family of generators f 1 , . . . , f r .
Corollary 45. Let ( f 1 , . . . , f r ) ∈ K X r be such that the corresponding module I [Σ] has a finite labelled Gröbner basis. Then, Algorithm 1 terminates when given f 1 , . . . , f r as input.
Proof. Since I [Σ] has a finite labelled Gröbner basis, by Corollary 44, the algorithm will eventually compute a minimal labelled Gröbner basis G [Σ] of I [Σ] , which must be finite as well by Corollary 30. At each run of the loop, only finitely many S-polynomials are added to P, so P has finite cardinality. Since G [Σ] is a labelled Gröbner basis, all the remaining elements in P will regular s-reduce to 0 or to a singular top s-reducible normal form, so no new polynomials will be added to P and the algorithm will terminate.
Note that if the algorithm terminates, or equivalently if I [Σ] admits a finite labelled Gröbner basis, then it has finite output G [Σ] and H. This output is such that the polynomial part of elements of G [Σ] forms a (finite) Gröbner basis of the ideal I, and that H ∪ H triv is a (usually infinite, but with a finite data representation) Gröbner basis of the module Syz( f 1 , . . . , f r ).
We conjecture that also the converse holds. , and a finite subsetH ⊆ Syz( f 1 , . . . , f r ) such that: } is a Gröbner basis of I; • all elements ofG [Σ] s-reduce to 0 modulo G [Σ] n ; • for allσ ∈H, there exists σ ∈ H n such that s(σ) = s(σ); n is a labelled Gröbner basis of I [Σ] , and in particular, I [Σ] has a finite labelled Gröbner basis.
Note thatG [Σ] need not be a labelled Gröbner basis, but merely a set of labelled polynomials which, without the module representations, forms a Gröbner basis. Put differently, the statement is equivalent to saying that I admits a finite Gröbner basis G, and that the module Syz( f 1 , . . . , f r ) has a Gröbner basis given by adding a finite set to the set of trivial syzygies of G (expressed in the module Σ).
In the commutative case where all ideals have a finite signature Gröbner basis, the analogue of this conjecture would give a characterization of signature Gröbner bases in terms of a Gröbner basis of the ideal and of its module of syzygies. To the best of our knowledge, no such characterization is proved in the commutative case.

S-polynomial elimination
In the commutative case, it is well known that additional criteria can be used to detect s-reductions to zero. So far, we have already seen that we can immediately discard all singular S-polynomials and remove a regular S-polynomial if it leads to a singular top s-reducible normal form. In this section, we adapt some other well-known techniques from the commutative case to our setting, namely the syzygy criterion, the F5 criterion and the singular criterion. In Algorithm 2, we include these criteria to show how to use them in practice.
Proof. Let σ ∈ Σ be a syzygy and a, b ∈ X such that s(π) = a s(σ)b. Now, consider Then, s(τ) ≺ s(π). Since G [Σ] is a labelled Gröbner basis up to signature s(π), the labelled polynomial p [τ] s-reduces to zero by G [Σ] . Thus, using the same reductions, we see that p [π] regular s-reduces to zero by G [Σ] .
Hence, we can immediately discard an S-polynomial p [π] during the computation of a labelled Gröbner basis if its signature s(π) is divisible by the signature of a syzygy. Clearly, we obtain syzygies whenever we s-reduce an S-polynomial to zero but there are also syzygies known prior to any computations. Recall that for all labelled polynomials f [α] , g [β] ∈ I [Σ] we have the trivial syzygies αmg − f mβ, for all monomials m ∈ X . This means that for any family of generators f 1 , . . . , f r ∈ K X , we immediately obtain the trivial syzygies for all 1 ≤ i, j ≤ r and all m ∈ X , which we can use to eliminate S-polynomials. Additionally, whenever we add a new element g [γ] to G [Σ] during the executing of Algorithm 1, we get the new trivial syzygies γmg ′ − gmγ ′ and γ ′ mg − g ′ mγ for all g ′[γ ′ ] ∈ G [Σ] and all m ∈ X . Identifying those trivial syzygies leads to the F5 criterion.

Computation of signature Gröbner bases and reconstruction
So far, Algorithm 1 keeps but does not exploit all the information encoded in the full module representation of the polynomials. As indicated earlier, keeping track of the full module representation, however, causes a significant overhead in terms of memory consumption and overall computation time. Consequently, in an actual implementation of Algorithm 1, one would only keep track of the signatures of each polynomial, and thereby, work with signature polynomials. In doing so, instead of computing a labelled Gröbner basis and a Gröbner basis of Syz( f 1 , . . . , f r ), the algorithm only computes a signature Gröbner basis (Definition 22) and a Gröbner basis of the module generated by s (Syz( f 1 , . . . , f r )). Additionally, to obtain an efficient implementation, one would also exploit the elimination criteria discussed in the previous section. Note that these criteria only depend on information encoded in signature polynomials. Incorporating these changes leads to Algorithm 2, which is an optimized version of Algorithm 1.
Theorem 51. Algorithm 2 is correct. Furthermore, if ( f 1 , . . . , f r ) ∈ K X r is such that the corresponding module I [Σ] has a finite labelled Gröbner basis, then Algorithm 2 terminates when given f 1 , . . . , f r as input.
Proof. Follows from the correctness of Algorithm 1, Corollary 45 and Section 4.5.
In the following, we discuss how to recover the information that is lost when Algorithm 2 is used instead of Algorithm 1. In particular, this means reconstructing a labelled Gröbner basis from a signature Gröbner basis and reconstructing a Gröbner basis of Syz( f 1 , . . . , f r ) from one of the module generated by s (Syz( f 1 , . . . , f r )). To this end, we adapt the reconstruction methods
We let G (Σ) ⊆ I (Σ) and H ⊆ M(Σ) be the output of Algorithm 2 when given the family of generators f 1 , . . . , f r ∈ K X as input. Recall that the algorithm does not necessarily terminate. As such, G (Σ) will either be the full output of Algorithm 2 assuming termination, or the partial output after interrupting the computation. In the latter case, the set G (Σ) is only a signature Gröbner basis up to a certain signature σ ∈ M(Σ) and H together with the signatures of the trivial syzygies does not necessarily form a Gröbner basis of the module generated by s (Syz( f 1 , . . . , f r )).
In this general setting, the goal of this section is twofold. First of all, starting from G (Σ) we want to reconstruct a labelled Gröbner basis G [Σ] (up to signature σ). Secondly, for each element β ∈ H, we want to find a module element α ∈ Syz( f 1 , . . . , f r ) such that s(α) = β.
In situations where Algorithm 2 terminates, i.e., when G (Σ) is a signature Gröbner basis and H together with the signatures of the trivial syzygies forms a Gröbner basis of the module generated by s (Syz( f 1 , . . . , f r )), achieving both of these goals allows us to also recover a Gröbner basis of Syz( f 1 , . . . , f r ). The algorithms which we describe in this section are a direct adaptation of the procedure outlined in [GVW15].
Our first goal can be achieved by the following algorithm. We note that no matter whether Algorithm 2 terminates by itself or whether we interrupt the computation, the sets G (Σ) and H are always finite.

Algorithm 3 Sig2LabelledGB
Input: G (Σ) a finite minimal signature Gröbner basis (up to some signature µ ∈ M(Σ)) Output: G [Σ] a finite minimal labelled Gröbner basis (up to signature µ) 1: ⊲ make a copy so that we do not alter G (Σ) 3: while H (Σ) ∅ do 4: Remark 52. As will be clear from the proof of Proposition 53, the minimality condition in line 6 of Algorithm 3 is not required for the correctness of the algorithm. It is included purely for efficiency reasons with the hope of having to do less s-reductions if lm(agb) is minimal. We note that the same also holds for the minimality condition in line 4 of Algorithm 4.
be the output of Algorithm 3 given G (Σ) as input. To prove the correctness of Algorithm 3, we show that In other words, we show that the labelled polynomials in G [Σ] have the same leading monomials and signatures as the elements in the labelled Gröbner basisG [Σ] . Because then, every f [α] ∈ I [Σ] is s-reducible by G [Σ] if and only if it is s-reducible byG [Σ] and Lemma 24 yields that G [Σ] is a labelled Gröbner basis (up to signature µ). Furthermore, the minimality ofG [Σ] implies the minimality of G [Σ] .
To prove (2), we show that the following loop invariant holds whenever the algorithm reaches line 3: Once the algorithm terminates and H (Σ) = ∅ this implies (2) since the leading monomials and signatures of the elements in G (Σ) are equal to those ofG [Σ] by definition ofG [Σ] . Obviously (3) holds in the very beginning when H (Σ) = G (Σ) . So, now assume that (3) holds at some point when the algorithm reaches line 3 and let f (σ) ∈ H (Σ) be the signature polynomial that is chosen in line 4. Furthermore, let α ∈ Σ be such that is also regular top s-reduced by G [Σ] . Note that, since we only care about regular top s-reducibility, it is irrelevant whether we consider G [Σ] before or after adding g ′[γ ′ ] as s(α) = s(γ ′ ). Also, note that the loop invariant, together with the fact that σ was chosen to be minimal among all signatures in H (Σ) , implies that G [Σ] is a labelled Gröbner basis up to 28 signature σ. Hence, Lemma 19 is applicable to g ′[γ ′ ] and c f [α] with c = sc(γ ′ ) sc(α) . It yields that lt(g ′ ) = lt(c f ), and consequently, lm(g ′ ) = lm( f ). Since also s(γ ′ ) = s(aγb) = σ, the loop invariant still holds after removing f (σ) from H (Σ) and adding g ′[γ ′ ] to G [Σ] .
After recovering a labelled Gröbner basis, we can proceed with the following algorithm to also recover the syzygies whose signatures are saved in H. as required in this line is always possible. It remains to show that ag [γ] b really regular s-reduces to zero by G [Σ] . To this end, we note that it follows from the definition of G (Σ) and Proposition 53, that G [Σ] is a labelled Gröbner basis up to signature σ ′ = max H. Furthermore, by definition of H, we know that σ is the signature of a syzygy. Consequently, we can apply Proposition 47 to conclude that ag [γ] b indeed regular s-reduces to zero.

Experimental results and future work
In this section, we compare Algorithm 2 to the classical Buchberger algorithm. Since our focus is on the feasibility of signature-compatible computations and not on their efficiency, we give data about the number of S-polynomials computed and reduced as well as about the number of reductions to zero when computing (signature) Gröbner bases for certain benchmark examples. The following are taken from [LSL09].

Example
Generators of the ideal braid3 yxy − zyz, xyx − zxy, zxz − yzx, x 3 + y 3 + z 3 + xyz lp1 z 4 + yxyx − xy 2 x − 3zyxz, x 3 yxy − xyx, zyx − xyz + zxz lv2 xy + yz, x 2 + xy − yx − y 2 As done in [LSL09], we only compute truncated (signature) Gröbner bases of these homogeneous ideals. The designated degree bounds are indicated by the number after the "-" in the name of each example in Table 1. So, for example lp1-11 means that we compute a partial Gröbner basis of the example lp1 up to degree 11. Additionally, we also consider two non-homogeneous ideals derived from finite generalized triangular groups taken from [RS02, Theorem 2.12] as done in [Xiu12]. Both of these ideals have finite (signature) Gröbner bases.

Example
Generators of the ideal tri1 x 3 − 1, y 2 − 1, (yxyxyx 2 yx 2 ) 2 − 1 tri3 x 3 − 1, y 3 − 1, (yxyx 2 ) 2 − 1 For all examples, we fix deglex as a monomial ordering where we order the indeterminates as x ≺ lex y ≺ lex z and work over the coefficient field Q. As a module ordering, top is chosen. Table 1 compares the number of S-polynomials computed and reduced and the number of reductions to zero that occur while computing (truncated) (signature) Gröbner bases for the examples stated above. Algorithm 2, denoted by SigGB, is compared to a vanilla Buchberger algorithm, denoted by BB vanilla, and to an optimized Buchberger algorithm including a noncommutative version of the chain criterion as described in [Hof20, Sec. 4.5.1], denoted by BB optimized. For each example, we list in the column "S-poly" the total number of S-polynomials that are computed and reduced during the execution of the respective algorithm. Additionally, we list the total number of reductions to zero in the column "red. to 0".
We note that all algorithms are part of the OperatorGB package 2 and that a Mathematica notebook containing all computations can be obtained from the same website as the package.
As can be seen, the signature-based algorithm considers fewer S-polynomials and needs fewer reductions to zero. In two of the examples, there are even no zero reductions at all. However, in terms of absolute computation time, SigGB cannot compete with the two other algorithms. In comparison, SigGB performs worst on the tri1 benchmark example where it is about four times slower than BB vanilla and about ten times slower than BB optimized (62 sec vs. 16 sec vs. 6 sec). For other examples, such as lv2-100, the timings are closer together but still in favor of the classical Buchberger algorithm (60 sec vs. 43 sec vs. 46 sec). This is mainly because of two reasons. First of all, when using the F5 criterion, the number of checks that have to be done for each S-polynomial increases quadratically with the size of the set G (Σ) , which becomes computationally quite intense as G (Σ) grows. Additionally, the fact that we are restricted to regular s-reductions in Algorithm 2 requires an additionally check before each s-reduction. This cost also adds up for longer computations.
We will investigate whether it is possible to improve the performance of Algorithm 2 to obtain a competitive algorithm in practice. One step towards achieving this goal could be finding ways to also allow non-fair module orderings such as a position-over-term ordering. Additionally, future research will be focused on adapting the concepts developed in this paper to the noncommutative F4 algorithm.
We also plan to leverage the algorithms developed here to find short representations of ideal elements. This is particularly useful when proving operator identities, where such short representations correspond to short proofs of the statement about operators. In particular, the effective description of the syzygy module provided by a signature Gröbner basis might allow to compute the shortest proof of certain operator identities.