Description and computation of syzygies for presentations of algebraic structures has been investigated by methods from homological algebra, Koszul duality and Gröbner bases theory. In homological algebra, the constructive methods using syzygies are initiated in the works of Koszul  and Tate  who describe free resolutions by mean of higher-order syzygies. Koszul duality, introduced by Priddy  and extended by Berger , is inspired by these works: for homogeneous associative algebras, a candidate for the space of syzygies, that is for constructing a minimal resolution, is the Koszul dual.
For commutative algebras, methods for computing syzygies are based on Gröbner bases: the module of syzygies for a Gröbner basis is spanned by -polynomials of critical pairs , that is the overlapping of two reductions, also called rewriting rules, on a term. Conversely, a critical pair whose -polynomial reduces into zero leads to a syzygy. This correspondence between syzygies and critical pairs has applications in two directions: improvements of Buchberger’s completion algorithm are based on the computation of syzygies [14, 21] and construction of free resolutions of commutative algebras are based on the computation of a Gröbner basis . The construction of free resolutions using rewriting theory for computing syzygies also appear for other algebraic structures, such as associative algebras [1, 8] or monoids [15, 16, 17].
In this paper, we give a method based on the lattice of reduction operators for computing syzygies for rewriting systems whose underlying set of terms is a vector space. Description of rewriting systems by mean of reduction operators was initiated in the works of Bergman for noncommutative Gröbner bases and exploited by Berger for studying homological properties of quadratic algebras [2, 3, 4]. Using reduction operators enables us to deduce a lattice criterion for detecting useless reductions during the completion procedure. As pointed out by Lazard , the completion procedure is interpreted as Gaussian elimination, which leads to use linear algebra techniques for studying completion. In particular, the and algorithms [11, 12] are based on such techniques and adaptations of Buchberger, or algorithms to various algebraic contexts were introduced, such as associative algebras [22, 27], invariant rings , tropical Gröbner bases  or operads , for instance.
We consider a vector space equipped with a well-ordered basis . For instance, if
is a polynomial algebra (respectively a tensor algebra, an invariant ring or an operad),is a set of monomials (respectively words, orbit sums of monomials or trees) and is an admissible order on . In our examples, we consider the case where is finite-dimensional and is a totally ordered basis of .
In this work, we describe linear rewriting systems by reduction operators. A reduction operator relative to is an idempotent linear endomorphism of such that for every , is a linear combination of elements of strictly smaller than . We denote by the set of reduction operators relative to .
Recall from [7, Proposition 2.1.14] that the kernel map induces a bijection between and subspaces of . Hence, admits a lattice structure, where the order , the lower-bound and the upper-bound are defined by
Given a subset of , we denote by the lower-bound of , that is the reduction operator whose kernel is the sums of kernels of elements of . We have the following lattice formulation of confluence: a subset of is said to be confluent if the image of is equal to the intersection of images of elements of . Recall from [7, Corollary 2.3.9] that is confluent if and only if the rewrite relation on defined by , for every and every , is confluent.
Upper-bound of reduction operators and syzygies.
In 2.1.3, we define the syzygies for a finite set of reduction operators as being the elements of the kernel of the application , mapping to . The set of syzygies for is denoted by . In 3.3, we interpret syzygies for presentations of algebras in terms of syzygies for a set of reduction operators.
In Lemma 2.2.3, we show that for every integer , is isomorphic to a supplement of in . In Proposition 2.2.4, we give an explicit description of this supplement using the operator . Using these two intermediate results, we obtain a procedure for constructing a basis of : we construct inductively bases of using the supplement of defined from . The correctness of this procedure is proven in Theorem 2.2.2.
Application to completion.
A completion of a set of reduction operators is a confluent set containing . In Section 3, we present a procedure for completing taking into account useless reductions, that is the reductions which do not change the final result of a completion procedure. This notion is formally defined in Definition 3.1.1.
We first remark that the vector space admits as a basis the set of all , where , and is at position . Using a well-order on this basis, we consider the set of reduction operators obtaining from removing the reductions
where is the leading term of an element of for the order . Formally, the operators are defined in the following way:
We call the set , the reduction of . In 3.2.4, we construct inductively a set of reduction operators which leads to a completion of . We call the set the incremental completion of . In Theorem 3.2.5, we show that the reductions (1) are useless in the sense that completes :
Theorem 3.2.5. Let F be a set of reduction operators, let be the reduction of and let C be the incremental completion of . Then, is a completion of F.
Moreover, a consequence of our method for constructing the basis of is that its leading terms are the elements such that does not belong to the image of . Hence, we obtain the following lattice criterion: the reductions , where , are useless reductions.
Useless reductions and construction of commutative Gröbner bases.
In Section 3.3, we relate the confluence property and the completion procedure for reduction operators to the construction of commutative Gröbner bases. We consider a set of variables as well as an ideal of spanned by a set of polynomials . Given an admissible order on the set of monomials, we consider the reduction operator whose kernel is the ideal spanned by . In Proposition 3.3.4, we show that is a Gröbner basis of if and only if the set of reduction operators associated to is confluent. This characterisation of Gröbner bases enables us to interpret the completion of a set of reduction operators as a procedure for constructing commutative Gröbner bases. Hence, the criterion of Section 3.2 enables us to detect useless reductions during the construction of commutative Gröbner bases. In Example 3.3.6, we illustrate with an example how to use this criterion.
In Section 2.1 we recall the definition and the lattice structure of reduction operators. We interpret the upper-bound of two reduction operators in terms of syzygies. In Section 2.2, we construct a basis of syzygies using the lattice structure of reduction operators. In particular, we characterise leading terms of syzygies using the lattice structure. In Section 2.3, we illustrate how our basis is constructed. In Section 3.1, we recall how works the completion in terms of reduction operators. In Section 3.2, we exploit the relationship between syzygies and useless reductions as well as our construction of a basis of syzygies to provide a lattice criterion for rejecting useless reductions during a completion procedure. In Section 3.3, we show how to use this criterion during the construction of commutative Gröbner bases.
Acknowledgement. This work was supported by the Sorbonne-Paris-Cité IDEX grant Focal and the ANR grant ANR-13-BS02-0005-02 CATHRE.
2 Computation of syzygies
In this section, we define syzygies for a set of reduction operators and we compute these syzygies using the lattice structure of reduction operators.
2.1 Syzygies for a set of reduction operators
Conventions and notations.
We fix a commutative field as well as a well-ordered set . We denote by the vector space spanned by .
For every , we denote by the support of , that is the set of elements of which belongs to the decomposition of . The greatest element of is denoted by and the coefficient of in is denoted by . The notations and are the abbreviations of leading term and leading coefficient of , respectively. Given a subset of , we denote by the set of leading terms of elements of : . We extend the order on into a partial order on in the following way: we have if and or if .
Let be a subspace of . A reduced basis of is a basis of such that the following two conditions are fulfilled:
for every , is equal to 1,
given two different elements and of , does not belong to the support of .
Recall from [7, Theorem 2.1.13] that admits a unique reduced basis.
A reduction operator relative to is an idempotent endomorphism of such that for every , we have . We denote by the set of reduction operators relative to . Given , a term is said to be a T-normal form or T-reducible according to or , respectively. We denote by the set of -normal forms and by the set of -reducible terms.
Kernels of reduction operators.
Let . The kernel of admits as a basis the set of elements , where belongs to . Hence, every admits a unique decomposition
The decomposition (2) is called the T-decomposition of .
Let be the set of subspaces of . Recall from [7, Proposition 2.1.14] that the kernel map induces a bijection between and . The inverse map is denoted by . Explicitly, for every , let be the unique reduced basis of . Then, is defined on the basis by:
where is the unique element of with leading term .
In Section 2.2, we need the following lemma:
Let V be a subspace of . We have an isomorphism:
Let . The operator being a linear map, we have an isomorphism between . Moreover, it is also a projector, so that we have . The latter is equal to , which proves Lemma 2.1.2. ∎
We deduce from the bijection induced by the kernel map that admits a lattice structure, where the order , the lower-bound and the upper-bound are defined by
Given a subset of , the lower-bound of is written :
Moreover, recall from [7, Lemma 2.1.18] that implies that is included in . Passing to the complement, we obtain
Let be a finite subset of . The vector space is denoted . We consider the linear map defined by
for every .
The elements of are called the syzygies for , and the set of syzygies for is denoted by .
In Section 2.2, we construct a basis of . This construction requires to relate syzygies to the upper-bound of reduction operators. This link is given by the following proposition:
Let be a pair of reduction operators. We have an isomorphism:
Since is equal to , the map (4) is well-defined. Moreover, it is injective since is equal to if and only if is equal to . Finally, it is surjective since belongs to if and only if and in this case, belongs to .
2.2 Construction of a basis of syzygies
Throughout the section, we fix a set of reduction operators.
For every and for every , we denote by
where is at position . The set of all ’s is a basis of . Moreover, we let if or if and . Such defined, is a well-order, so that is a vector space equipped with a well-ordered basis.
By definition of syzygies, we have an isomorphism of vector spaces . From Lemma 2.1.2, admits as a basis the set
where is the set of leading terms of elements of for the order . Hence, every admits a unique decomposition
where, for every index in the sum, belongs to . The decomposition (6) in called the canonical decomposition of with respect to .
Procedure for constructing a basis of .
For every integer such that , we consider the reduction operator
For every , we denote by
The vector belongs to and to , so that it admits a canonical decomposition relative to as well as well as a -decomposition. Let
be these two decompositions. We let:
We define by induction sets in the following way: and for every ,
With the previous notations, is a basis of .
Moreover, we abuse notations in the following ways:
given two integers and such that , we still denote by and their images by the natural projection of on ,
using the injection , we consider that we have , for every integer such that .
Let i be an integer such that .
We have .
For every , we have
First, we show i. An element belongs to the kernel of if and only if is equal to . Hence, v belongs to the the kernel of if and only if it belongs to the image of .
Let us show ii. Let
be the canonical decomposition of with respect to . Every index of the sum (11) is strictly smaller than , so that we have
Moreover, letting the canonical the -decomposition of , we have
Hence, we have
Let i be an integer such that . We have the following direct sum decomposition:
The set of all , where belongs to , is a basis of , so that the set of pairs , where belongs to , is a basis of from Proposition 2.1.4. The morphism is surjective, so that we have . Hence, from ii. of Lemma 2.2.3, induces an isomorphism between the vector space spanned by elements , where belongs to , and . In particular, is a supplement of in . From i. of Lemma 2.2.3, is equal to , which proves Proposition 2.2.4.
Now, we can show Theorem 2.2.2.
Proof of theorem 2.2.2.
We show by induction that for every integer such that , the set obtained in 10 of the procedure is a basis of . If is equal to , there is nothing to prove since is reduced to . Let be an integer such that and assume by induction hypothesis that is a basis of . From Proposition 2.2.4
is a basis of . Hence, is a basis of .
We deduce the following lattice description of the set of leading terms of syzygies:
Let be a finite set of reduction operators. We have
In this section we illustrate the construction of with an example. For that, we use the implementation of the lattice structure of reduction operators available online222https://pastebin.com/Ds5haArH.
We consider . We let , where the operators are defined by their matrices with respect to the basis :
The vector space is spanned by the following eight vectors:
We simplify notations:
In particular, we have . Moreover, as done in the previous section, we let , for .
We have .
The set is reduced to and is equal to . We have
and its -decomposition is