A lattice formulation of the F4 completion procedure
aa r X i v : . [ c s . S C ] J a n A lattice formulationof the noncommutative F procedure Cyrille Chenavier ∗ Abstract
We introduce a new procedure for constructing noncommutative Gröbner bases using alattice formulation of completion. This leads to a lattice description of the noncommutative F procedure. Our procedure is based on the lattice structure of reduction operators whichprovides a lattice description of the confluence property. We relate reduction operatorsto noncommutative Gröbner bases, we show the Diamond Lemma for reduction operatorsand we deduce the lattice interpretation of the F procedure. Finally, we illustrate ourprocedure with a complete example. Keywords: lattice structure, noncommutative F procedure, reduction operators. Contents
The objective of the paper is to introduce a new procedure for constructing noncommutativeGröbner bases which turns out to be a lattice formulation of the noncommutative F procedure.This formulation is based on a description of the completion procedure using linear algebratechniques and is motivated by the development of effective methods in homological algebrausing such techniques [1, 2, 9, 13, 14, 18].The F procedure is an improvement of the Buchberger’s one where several S -polynomialsare reduced into normal forms simultaneously. Improvements and optimisations of Buchberger’s ∗ Université Paris-Est Marne-la-Vallée, [email protected]. F completion procedure was also introduced for polynomial ideals [10], it is adapted tothe noncommutative case [19] and an implementation of this adaptation can be found in thesystem MAGMA.Our lattice formulation of F uses the approach due to Bergman [3] who described reduc-tion systems over noncommutative algebras by reduction operators . The latter admit a latticestructure inducing lattice formulations of confluence and completion that we present now. Lattice formulations of confluence and completion.
A reduction operator relative toa well-ordered set ( G, < ) is an idempotent linear endomorphism T of the K -vector space K G spanned by G such that for every g / ∈ im ( T ) , T ( g ) is a linear combination of elements of G strictly smaller than g . We denote by RO ( G, < ) the set of reduction operators relative to ( G, < ) .From [8, Proposition 2.1.14], the kernel map induces a bijection between RO ( G, < ) andsubspaces of K G , so that RO ( G, < ) admits a lattice structure defined in terms of kernels: i. T (cid:22) T if ker ( T ) ⊆ ker ( T ) , ii. T ∧ T = ker − (ker ( T ) + ker ( T )) , iii. T ∨ T = ker − (ker ( T ) ∩ ker ( T )) .Given a subset F of RO ( G, < ) , we denote by ∧ F the lower-bound of F , that is the reductionoperator whose kernel is the sum of kernels of elements of F . We get the following latticeformulation of confluence: F is said to be confluent if the image of ∧ F is equal to the intersectionof images of elements of F . Recall from [8, Corollary 2.3.9] that F is confluent if and only ifthe reduction relation on K G defined by v −→ T ( v ) for every T ∈ F and every v / ∈ im ( T ) is confluent. Moreover, recall from [8, Theorem 3.2.6] that the completion of F is done by theoperator C F = ( ∧ F ) ∨ (cid:0) ∨ F (cid:1) , where F is a subset of RO ( G, < ) defined from F and ∨ F isthe upper-bound of F , that is F ∪ { C F } is a confluent subset of RO ( G, < ) .In Section 3, the operator C F is used to reduce simultaneously several S -polynomials intonormal forms using a triangular process such as the F procedure does. For that, we introduce presentations by operators which relate reduction operators to noncommutative Gröbner bases. Reduction operators and presentations of algebras.
A presentation by operator of anassociative A is a triple ( X, <, S ) , where X is a set, < is a monomial order on the set ofnoncommutative monomials X ∗ and S is a reduction operator relative to ( X ∗ , < ) such that A is isomorphic to the quotient of the free algebra over X by the two-sided ideal spanned by ker ( S ) .In order to describe all the reductions induced by S we consider the "extensions" of S ,that is the operators which applied to a monomial w w w gives w S ( w ) w . The presentation ( X, <, S ) is said to be confluent if the set of extensions of S is a confluent subset of RO ( G, < ) .From [8, Proposition 3.3.10], the presentation ( X, <, S ) is confluent if and only if the set ofelements w − S ( w ) with w / ∈ im ( S ) is a noncommutative Gröbner basis of I (ker ( S )) . Thislink between reduction operators and noncommutative Gröbner bases enables us to show theDiamond Lemma in terms of reduction operators in Proposition 2.2.8.2ur procedure for constructing confluent presentations by operators, and thus noncommu-tative Gröbner bases, is given in Section 3.1. At the step number d of the procedure, we reducethe S -polynomials of the current presentation (cid:0) X, <, S d (cid:1) into normal forms using a set ofreduction operators F d . The operator at the step d + 1 is S d +1 = S d ∧ C F d . Denoting by S thelower-bound of all the operators S d , the triple (cid:0) X, <, S (cid:1) is called the completed presentation of A . The main result of the paper is Theorem 3.2.5 which asserts that a completed presen-tation is confluent. In Section 3.3, we show how to implement our procedure with a completeexample as an illustration. Organisation of the paper
Section 2.1 is a recollection of results from [8]: we recall the definitions and properties of re-duction operators, their confluence and completion used in the sequel. In Section 2.2, we definepresentations by operators, the confluence property of such presentations, we formulate and weshow the Diamond Lemma for reduction operators. In Section 3.1, we write our completionprocedure and define completed presentations. In Section 3.2, we show that a completed pre-sentation is confluent. In Section 3.3, we illustrate our completion procedure with a completeexample based on the computation of lattice operations of reduction operators.
Throughout the paper, K denotes a commutative field. Given a set G , we denote by K G thevector space spanned by G . Given a well-order < on G , the leading generator of a nonzeroelement v ∈ K G is written lg ( v ) . We extend the order < on G into a partial order on K G inthe following way: we have u < v if u = 0 and v = 0 or if lg ( u ) < lg ( v ) . Definition 2.1.1. A reduction operator relative to ( G, < ) is an idempotent endomorphism T of K G such that for every g ∈ G , we have T ( g ) ≤ g . We denote by RO ( G, < ) the set ofreduction operators relative to ( G, < ) . Given T ∈ RO ( G, < ) , a generator g ∈ G is saidto be a T-normal form or T-reducible according to T ( g ) = g or T ( g ) = g , respectively. Wedenote by nf ( T ) the set of T -normal forms and by red ( T ) the set of T -reducible generators. Lattice structure, confluence and completion.
Recall from [8, Proposition 2.1.14] thatthe restriction of the kernel map T ker ( T ) to RO ( G, < ) is a bijection. Using the inverse ker − , the set RO ( G, < ) admits a lattice structure for the operations i. T (cid:22) T if ker ( T ) ⊆ ker ( T ) , ii. T ∧ T = ker − (ker ( T ) + ker ( T )) , iii. T ∨ T = ker − (ker ( T ) ∩ ker ( T )) .Recall from [8, Lemma 2.1.18] that we have the following implication T (cid:22) T = ⇒ nf ( T ) ⊆ nf ( T ) a . (1) a In [8], the notation red ( T ) stands for reduced generators and correspond to nf ( T ) in the present paper. Thenotation red ( T ) of the present paper corresponds to nred ( T ) of [8] which means nonreduced generators . F of RO ( G, < ) , we denote by nf ( F ) and ∧ F the set of normalforms for each T ∈ F and the lower-bound of F , respectively. From (1), nf ( ∧ F ) is includedin nf ( T ) for every T ∈ F , so that nf ( ∧ F ) is included in nf ( F ) . We writeobs ( F ) = nf ( F ) \ nf ( ∧ F ) . (2)The set F is said to be confluent if obs ( F ) is the empty set. In Section 3.2, we use two char-acterisations of the confluence property in terms of reduction operators. First, recall from [8,Theorem 2.2.5] that F is confluent if and only if it has the Church-Rosser property , that is forevery v ∈ K G , there exist T , · · · , T r ∈ F such that ( ∧ F ) ( v ) = ( T r ◦ · · · ◦ T ) ( v ) . Moreover,from [8, Proposition 2.2.12], F is confluent if and only if it is locally confluent , that is for every v ∈ K G and for every ( T, T ′ ) ∈ F × F , there exist v ′ ∈ K G and T , · · · , T r , T ′ , · · · , T ′ k ∈ F such that v ′ = ( T r ◦ · · · ◦ T ) ( T ( v )) and v ′ = ( T ′ k ◦ · · · ◦ T ′ ) ( T ′ ( v )) . Finally, we recall how aset of reduction operators is completed into a confluent one. Definition 2.1.2. A complement of F is an element C of RO ( G, < ) such that i. ( ∧ F ) ∧ C = ∧ F , ii. obs ( F ) ⊆ red ( C ) .The F-complement is the operator C F = ( ∧ F ) ∨ (cid:0) ∨ F (cid:1) , where ∨ F is equal to ker − ( K nf ( F )) .Recall from [8, Proposition 3.2.2] that a reduction operator C satisfying ( ∧ F ) ∧ C = ∧ F isa complement of F if and only if F ∪ { C } is confluent. Recall from [8, Theorem 3.2.6] that the F -complement is a complement of F . In this section, we relate the confluence property for reduction operators to noncommutativeGröbner bases and we prove the Diamond Lemma for reduction operators.Given a set X , we denote by X ∗ the set of noncommutative monomials over X and weidentify the free algebra over X with K X ∗ , equipped with the multiplication induced by con-catenation of monomials. A monomial order over X ∗ is a well-founded total strict order < on X ∗ such that the following conditions are fulfilled: i. < w for every monomial w different from 1, ii. for every w , w , w, w ′ ∈ X ∗ such that w < w ′ , we have w ww < w w ′ w .For any f ∈ K X ∗ , the leading monomial of f is written lm ( f ) instead of lg ( f ) . Definition 2.2.1. A presentation by operator of an associative algebra A is a triple ( X, <, S ) where i. X is a set and < is a monomial order on X ∗ , ii. S is a reduction operator relative to ( X ∗ , < ) such that A is isomorphic to K X ∗ /I (ker( S )) ,where I (ker( S )) is the two-sided ideal spanned by ker ( S ) .4e fix an algebra A together with a presentation by operator ( X, <, S ) of A . For everyinteger n , we denote by X ( n ) and X ( ≤ n ) the set of monomials of length n and of length smalleror equal to n , respectively. For every integers n and m such that ( n, m ) is different from (0 , ,we consider the reduction operator S n,m = Id K X ( ≤ n + m − ⊕ (cid:16) Id K X ( n ) ⊗ S ⊗ Id K X ( m ) (cid:17) . Explicitly, for every w ∈ X ∗ , S n,m ( w ) is defined by: if the length of w is strictly smaller than n + m , then S n,m ( w ) = w , else we let w = w w w where w and w have length n and m ,respectively and we have S n,m ( w ) = w S ( w ) w . We also let S , = S . Definition 2.2.2.
The set of all the operators S n,m with ( n, m ) ∈ N , is called the reductionfamily of ( X, <, S ) . The presentation ( X, <, S ) is said to be confluent if its reduction familyis a confluent subset of RO ( X ∗ , < ) .Recall from [8, Proposition 3.3.10] that ( X, <, S ) is confluent if and only if the set ofelements w − S ( w ) with w ∈ red ( S ) is a noncommutative Gröbner basis of I (ker( S )) , thatis red ( S ) spans leading monomials of I as a monomial ideal. Example 2.2.3.
Let X = { x, y, z } and let < be the deg-lex order induced by x < y < z .Consider the algebra presented by ( X, <, S ) where S is defined on the basis X ∗ by S ( yz ) = x , S ( zx ) = xy and S ( w ) = w for every monomial w different from yz and zx . We have yxy − xx = ( yxy − yzx ) − ( xx − yzx )= ( yS ( zx ) − yzx ) − ( S ( yz ) x − yzx )= A + B where A = ( S , − Id K X ∗ ) ( yzx ) and B = ( Id K X ∗ − S , ) ( yzx ) . Hence, yxy − xx belongsto ker ( ∧ F ) where F is the reduction family of the presentation, so that yxy is ∧ F -reducible.Moreover, yxy belongs to nf ( F ) , so that yxy belongs to obs ( F ) and F is not confluent. Thus, ( X, <, S ) is not a confluent presentation of A .In Section 3.1 we formulate our procedure for constructing confluent presentations by op-erators using critical branchings that we introduce in Definition 2.2.4. These branchings areanalogous to ambiguities for Gröbner bases. An ambiguity with respect to < of a subset R of K X ∗ is a tuple b = ( w , w , w , f, g ) where w , w , w are monomials such that w = 1 , f, g belong to R and one of the following two conditions is fulfilled:1. w w = lm ( f ) and w w = lm ( g ) .2. w w w = lm ( f ) and w = lm ( g ) .The S -polynomial of b is written sp ( b ) , that is sp ( b ) = f w − w g or sp ( b ) = f − w gw according to b is of the form 1 or 2, respectively. The ambiguity b is said to be solvable relativeto < if there exists a decompositionsp ( b ) = n X i =1 λ i w i f i w ′ i , (3)where, for every i ∈ { , · · · , n } , λ i is a non-zero scalar, w i , w ′ i are monomials and f i is anelement of R such that w i lm ( f i ) w ′ i < w w w . The Diamond Lemma [3, Theorem 1.2] asserts5hat R is a noncommutative Gröbner basis of I ( R ) if and only if every critical branching of R with respect to < is solvable relative to < .Our purpose is to formulate and to prove the Diammond Lemma for reduction operators.Until the end of the section, we fix some notations: A is an associative algebra and ( X, <, S ) is a presentation by operator of A . For every pair of integers ( n, m ) , we consider the operator S n,m defined such as the beginning of the section. We denote by R the set of elements w − S ( w ) with w ∈ red ( S ) . Definition 2.2.4. A critical branching of ( X, <, S ) is a triple b = ( w, ( n, m ) , ( n ′ , m ′ )) where w is a monomial and ( n, m ) and ( n ′ , m ′ ) are couples of integers such that i. w belongs to red ( S n,m ) ∩ red (cid:0) S n ′ ,m ′ (cid:1) , ii. n = 0 or n ′ = 0 , iii. m = 0 or m ′ = 0 , iv. n + n ′ + m + m ′ is strictly smaller than the length of w .The S-polynomial of b is SP ( b ) = S n,m ( w ) − S n ′ ,m ′ ( w ) and the source of b is the monomial w . Remark 2.2.5.
The roles of ( n, m ) and ( n ′ , m ′ ) being symmetric, we do not distinguish ( w, ( n, m ) , ( n ′ , m ′ )) and ( w, ( n ′ , m ′ ) , ( n, m )) . Definition 2.2.6.
Let w ∈ X ∗ and let f ∈ K X ∗ . We say that f admits a ( S, w ) - typedecomposition if it admits a decomposition f = n X i =1 λ i w i ( w i − S ( w i )) w i , where, for every i ∈ { , · · · , n } , λ i is a non-zero scalar, w i , w i and w i are monomials such that w i belongs to red ( S ) and w i w i w i < w . Lemma 2.2.7.
There is a one-to-one correspondence b ˜ b between critical branchings of ( X, <, S ) and ambiguities of R with respect to < . Moreover, a critical branching b of sourcew admits a ( S, w ) -type decomposition if and only if ˜ b is solvable relative to < .Proof. Let us show the first part of the lemma. Let b = ( w, ( n, m ) , ( n ′ , m ′ )) be a criticalbranching of ( X, <, S ) . In order to define ˜ b , we distinguish four cases depending on the valuesof n and m : Case 1: ( n, m ) = (0 , . We write w = w w w , where the lengths of w and w are equalto n ′ and m ′ , respectively. By definition of a critical branching, w and w belong to red ( S ) and we let ˜ b = (cid:16) w , w , w , w − S ( w ) , w ( w − S ( w )) w (cid:17) . By definition of a criticalbranching, n + n ′ + m + m ′ = n ′ + m ′ is strictly smaller than the length of w . In particular, w is not the empty word, so that the tuple ˜ b is an ambiguity of R with respect to < of theform 2. 6 ase 2: n = 0 and m = 0 . By definition of a critical branching, m ′ = 0 . If n ′ isalso equal to , we have ( n ′ , m ′ ) = (0 , , so that we exchange the roles of ( n, m ) and ( n ′ , m ′ ) and we recover the first case. If n ′ = 0 , we write w = w w w , where thelengths of w and w are equal to n ′ and m , respectively. In particular, b being a criticalbranching, the monomials w w and w w belong to red ( S ) and w is different from . Hence, ˜ b = (cid:16) w , w , w , w w − S ( w w ) , w w − S ( w w ) (cid:17) , is an ambiguity of R with respect to < . Case 3: n = 0 and m = 0 . By definition of a critical branching, n ′ is equal to . Exchangingthe roles of ( n, m ) and ( n ′ , m ′ ) , we recover the second case. Case 4: n = 0 and m = 0 . By definition of a critical branching, the pair ( n ′ , m ′ ) is equalto (0 , . Exchanging the roles of ( n, m ) and ( n ′ , m ′ ) , we recover the first case.We have a well-defined map b ˜ b between critical branchings of ( X, <, S ) and am-biguities of R with respect to < . Now, we define the inverse map ˜ b b . Let ˜ b =( w , w , w , f, g ) be an ambiguity of R with respect to < and let w = w w w . • If ˜ b is an ambiguity of the form 1, let n and m ′ be the lengths of w and w , respectively.The word w being non-empty, n + m ′ is strictly smaller than the length of w , so that b = ( w, ( n, , (0 , m ′ )) is a critical branching of ( X, <, S ) . • If ˜ b is of the form 2, let n and m be the lengths of n and m , respectively. Then, b =( w, ( n, m ) , (0 , is a critical branching of ( X, <, S ) .Such defined, the two composites of b ˜ b and ˜ b b are identities.Let us show the second part of the lemma. Given a critical branching b , sp ( b ) and sp (cid:16) ˜ b (cid:17) are equal. Letting w the source of w , a ( S, w ) -type decomposition of sp ( b ) is precisely adecomposition of the from (3). That shows the second part of the lemma.The Diamond Lemma for reduction operators is formulated as follows: Proposition 2.2.8.
The presentation ( X, <, S ) is confluent if and only if for every criticalbranching b of source w, SP ( b ) admits a ( S, w ) -type decomposition.Proof. The two-sided ideal I ( R ) spanned by R is equal to I (ker( S )) . Hence, from [8, Proposi-tion 3.3.10], ( X, <, S ) is confluent if and only if R is a noncommutative Gröbner basis of I ( R ) .From the Diamond Lemma, the presentation ( X, <, S ) is confluent if and only if every ambi-guity of R with respect to < is solvable relative to < . Thus, from Lemma 2.2.7, ( X, <, S ) isconfluent if and only if for every critical branching b of source w the S -polynomial sp ( b ) admitsa ( S, w ) -type decomposition. Example 2.2.9.
Considering the presentation of Example 2.2.3, we have one critical branching b = ( yzx, (1 , , (0 , and we have sp ( b ) = yxy − xx . This S -polynomial does notadmit a ( S, yzx ) -type decomposition so that we recover that the presentation is not confluent.7 Completion procedure
In Section 3.1, we formulate our procedure for constructing confluent presentations by operatorsand we show the correctness of this procedure in Section 3.2. Throughout Section 3, we fix thefollowing notations: i. A is an algebra and ( X, <, S ) is a presentation by operator of A . ii. Given a reduction operator T ∈ RO ( X ∗ , < ) and a pair of integers ( n, m ) , the operator T n,m is defined such as the beginning of Section 2.2. iii. For every f ∈ K X ∗ , we write T ( f ) = ker − ( K f ) . Explicitly, ( T ( f )) ( lm ( f )) is equalto lm ( f ) − / lc ( f ) f and all other monomial is a normal form for T ( f ) . Moreover, wewrite supp ( f ) the support of f , that is the set monomials occurring in the decompositionof f with a nonzero coefficient. iv. Given a subset E ⊆ K X ∗ , we write lm ( E ) the set of leading monomials of elements of E . Our procedure requires a function called normalisation with inputs a finite set E ⊂ K X ∗ and a reduction operator U ∈ RO ( X ∗ , < ) and with output a finite set of reduction operators.Then, normalisation ( E, U ) is defined as follows:1. Let M = (cid:16)S f ∈ E supp ( f ) (cid:17) \ lm ( E ) and F = { T ( f ) | f ∈ E } .2. while ∃ w ww ∈ M such that w ∈ red ( U ) , i. we add T ( w ( w − U ( w )) w ) to F , ii. we remove w ww from M , iii. we add supp ( w U ( w ) w ) to M .3. normalisation ( E, U ) is the set F obtained when the loop while is over.The loop while is terminating beacause E is finite and < is a monomial order.We formulate our completion procedure. We assume that the presentation ( X, <, S ) is finite , that is X is finite and ker( S ) is finite-dimensional. In particular, the set of criticalbranchings of ( X, <, S ) is finite. 8 lgorithm 1 Completion procedure
Initialisation : • d := 0 , • S d := S , • Q d := ∅ and P d := n critical branchings of (cid:0) X, <, S d (cid:1) o , • E d := n w − S dn,m ( w ) | ( w, ( n, m ) , ( n ′ , m ′ )) ∈ P d o . while Q d = P d do F d := normalisation ( E d , S d ) ; S d +1 := S d ∧ C F d ; Q d +1 := P d ; d = d + 1 ; P d := n critical branchings of (cid:0) X, <, S d (cid:1) o ; E d := n w − S dn,m ( w ) | ( w, ( n, m ) , ( n ′ , m ′ )) ∈ P d \ Q d o ; end while This first and the last instruction of the loop while make sense because we have the fol-lowing:
Lemma 3.1.1.
Let d be an integer.1. The kernels of S d and C F d are finite-dimensional.2. The set Q d is included in P d .Proof. We show Point 1 by induction on d . The kernel of S = S is finite-dimensionalby hypotheses. Let d ∈ N and assume that the kernel of S d is finite-dimensional. Let M d = S f ∈ E d supp ( f ) be the union of words appearing in E d . The elements of F d are onlyacting on M d , so that we have the inclusion ker (cid:0) C F d (cid:1) ⊂ K M d . (4)The kernel of S d being finite-dimensional by induction hypothesis, the set of critical branchingsof (cid:0) X, <, S d (cid:1) is finite. Hence, E d and M d are finite sets, so that ker (cid:0) C F d (cid:1) is finite-dimensionalfrom (4). Moreover, by definition of ∧ , ker (cid:0) S d +1 (cid:1) is equal to ker (cid:0) S d (cid:1) + ker (cid:0) C F d (cid:1) , so that ker (cid:0) S d +1 (cid:1) is finite-dimensional.Let us show Point 2. By construction, Q d is equal to P d − , that is Q d is the set of criticalbranchings of (cid:0) X, <, S d − (cid:1) . Let ( w, ( n, m ) , ( n ′ , m ′ )) be such a critical branching, so thatwe have w ∈ red (cid:18)(cid:16) S d − (cid:17) n,m (cid:19) ∩ red (cid:18)(cid:16) S d − (cid:17) n ′ ,m ′ (cid:19) . (5)9oreover, by construction, we have S d (cid:22) S d − . Hence, from implication (1) (see page 3), wehave red (cid:16) S d − (cid:17) ⊂ red (cid:16) S d (cid:17) . (6)From (5) and (6), w belongs to red (cid:0) S dn,m (cid:1) ∩ red (cid:0) S dn ′ ,m ′ (cid:1) , so that ( w, ( n, m ) , ( n ′ , m ′ )) is acritical branching of (cid:0) X, <, S d +1 (cid:1) , that is it belongs to P d . Thus, Q d is included in P d . Remark 3.1.2.
Our procedure requires to compute lower-bound of reduction operators relativeto ( X ∗ , < ) . In Section 3.3, we give the implementation of ker − for totally ordered finite sets,so that it cannot be used for a set of monomials. However, from Lemma 3.1.1, the kernels of S d and C F d are finite-dimensional, so that these two operators can be computed by restrictions overfinite-dimensional subspaces of K X ∗ . We illustrate how works such computations in Section 3.3.Our procedure has no reason to terminate since there exist finitely presented algebras withno finite Gröbner basis [17, Section 1.3]. If the procedure terminates after d iterations of theloop while , we let S n = S d for every integer n ≥ d , so that the sequence (cid:0) S d (cid:1) d ∈ N iswell-defined if the procedure terminates or not. We let S = ^ d ∈ N S d . Definition 3.1.3.
The triple (cid:0)
X, <, S (cid:1) is called the completed presentation of ( X, <, S ) .The purpose of the next section is to show that the completed presentation of ( X, <, S ) is a confluent presentation of A , that is our procedure computes a noncommutative Gröbnerbasis. In this section, we say reduction operator instead of reduction operator relative to ( X ∗ , < ) . Lemma 3.2.1.
Let w ∈ X ∗ and let T and T ′ be two reduction operators such that T ′ (cid:22) T .1. Let ( n, m ) be a pair of integers such that w is T n,m -reducible. Then, (cid:0) T n,m − T ′ n,m (cid:1) ( w ) admits a ( T ′ , w ) -type decomposition.2. Let f ∈ K X ∗ admitting a ( T, w ) -type decomposition. Then, f admits a ( T ′ , w ) -typedecomposition.Proof. Let us show Point 1. We let w = w ( n ) w ′ w ( m ) , where w ( n ) and w ( m ) have length n and m , respectively. Let T ( w ′ ) = k X i =1 λ i w i , (7)be the decomposition of T ( w ′ ) with respect to the basis X ∗ . By hypotheses, T ′ is smaller than T , that is ker ( T ) ⊆ ker ( T ′ ) , so that T ′ ◦ T is equal to T ′ . Hence, we have (cid:0) T n,m − T ′ n,m (cid:1) ( w ) = w ( n ) (cid:0) T ( w ′ ) − T ′ ( w ′ ) (cid:1) w ( m ) = w ( n ) (cid:0) T ( w ′ ) − T ′ (cid:0) T ( w ′ ) (cid:1)(cid:1) w ( m ) . (cid:0) T n,m − T ′ n,m (cid:1) ( w ) = k X i =1 λ i w ( n ) (cid:0) w i − T ′ ( w i ) (cid:1) w ( m ) . (8)By hypotheses, w is T n,m -reducible, so that w ′ is T -reducible and each w i is strictly smallerthan w ′ for < . The strict order < being monomial, each w ( n ) w i w ( m ) is strictly smaller than w ( n ) w ′ w ( m ) = w , so that (8) is a ( T ′ , w ) -type decomposition of (cid:0) T n,m − T ′ n,m (cid:1) ( w ) .Let us show Point 2. Let f = n X i =1 λ i w i ( w i − T ( w i )) w i , (9)be a ( T, w ) -type decomposition of f . Letting A = n X i =1 λ i w i (cid:0) w i − T ′ ( w i ) (cid:1) w i and B = n X i =1 λ i w i (cid:0) T ( w i ) − T ′ ( w i ) (cid:1) w i ,f is equal to A − B . The decomposition (9) being ( T, w ) -type, each w ′ i = w i w i w i is strictlysmaller than w , so that A is ( T ′ , w ) -type. For every i ∈ { , · · · , n } , let n i and m i bethe lengths of w i and w i , respectively, so that we have B = P ni =1 λ i (cid:0) T n i ,m i − T ′ n i ,m i (cid:1) ( w ′ i ) .Each w i being T -reducible, each w ′ i is T n i ,m i -reducible. Hence, from Point 1 of the lemma, each (cid:0) T n i ,m i − T ′ n i ,m i (cid:1) ( w ′ i ) admits a ( T ′ , w ′ i ) -type decomposition, so that it admits a ( T ′ , w ) -typedecomposition since w ′ i is strictly smaller than w . Hence, B admits a ( T ′ , w ) -type decomposi-tion, so that f also admits such a decomposition. Notation.
For every integer d , let F d be the reduction family of (cid:0) X, <, S d (cid:1) , that is F d isequal to n (cid:0) S d (cid:1) n,m | ( n, m ) ∈ N o . Lemma 3.2.2.
Let d be an integer, let ( w, ( n, m ) , ( n ′ , m ′ )) ∈ P d \ Q d and let f be theS-polynomial of ( w, ( n, m ) , ( n ′ , m ′ )) .1. ( ∧ F d ) ( f ) is equal to .2. f admits a (cid:0) S d +1 , w (cid:1) -type decomposition.Proof. Let us show Point 1. The two elements w − (cid:0) S d (cid:1) n,m ( w ) and w − (cid:0) S d (cid:1) n ′ ,m ′ ( w ) belongto E d by construction of the latter. Hence, by definition of the function normalisation , theoperators T = T (cid:16) w − (cid:0) S d (cid:1) n,m ( w ) (cid:17) and T = T (cid:16) w − (cid:0) S d (cid:1) n ′ ,m ′ ( w ) (cid:17) belong to F d , so that f = ( w − S dn,m ( w )) − ( w − S dn ′ ,m ′ ( w )) belongs to the kernel of T ∧ T . The latter is includedin the kernel of ∧ F d , which shows Point 1.Let us show Point 2. The operator C F d being a complement of F d , we have ∧ (cid:0) F d ∪ (cid:8) C F d (cid:9)(cid:1) = ∧ F d , (10)and F d ∪ (cid:8) C F d (cid:9) is confluent (see the paragraph after Definition 2.1.2), that is it has the Church-Rosser property (see the paragraph before Definition 2.1.2). Hence, from Point 1 of the lemmaand Relation (10), there exist T , · · · , T r ∈ F d ∪ (cid:8) C F d (cid:9) such that ( T r ◦ · · · ◦ T ) ( f ) = 0 . (11)11e let f = ( Id K X ∗ − T ) ( f ) and for every k ∈ { , · · · , r } , f k = ( Id K X ∗ − T k ) ( T k − ◦ · · · ◦ T ( f )) .From (11), we have f = r X k =1 f k . (12)The tuple ( w, ( n, m ) , ( n ′ , m ′ )) being a critical branching of (cid:0) X, <, S d (cid:1) , w belongs tored (cid:16)(cid:0) S d (cid:1) n,m (cid:17) ∩ red (cid:16)(cid:0) S d (cid:1) n ′ ,m ′ (cid:17) , so that the leading monomial of f is strictly smaller than w .Moreover, each T i is either of the form T (cid:0) w ( w − S d ( w )) w (cid:1) , or is equal to C F d . Hence,each f i admits a (cid:0) S d , w (cid:1) -type decomposition or a (cid:0) C F d , w (cid:1) -type decomposition. The reductionoperators S d and C F d being smaller than S d +1 , each f i admits a (cid:0) S d +1 , w (cid:1) -type decompositionfrom Point 2 of Lemma 3.2.1, so that f admits a (cid:0) S d +1 , w (cid:1) -type decomposition from (12). Proposition 3.2.3.
Let d be an integer. For every ( w, ( n, m ) , ( n ′ , m ′ )) ∈ Q d , the S -polynomial (cid:0) S d (cid:1) n,m ( w ) − (cid:0) S d (cid:1) n ′ ,m ′ ( w ) admits a (cid:0) S d , w (cid:1) -type decomposition.Proof. We show the proposition by induction on d . The set Q being empty, Proposition 3.2.3holds for d = 0 . Assume that for every ( w, ( n, m ) , ( n ′ , m ′ )) ∈ Q d , S dn,m ( w ) − S dn ′ ,m ′ ( w ) admits a ( S d , w ) -type decomposition. We let A = (cid:16) S d (cid:17) n ′ ,m ′ ( w ) − (cid:16) S d +1 (cid:17) n ′ ,m ′ ( w ) ,B = (cid:16) S d (cid:17) n,m ( w ) − (cid:16) S d +1 (cid:17) n,m ( w ) ,C = (cid:16) S d (cid:17) n,m ( w ) − (cid:16) S d (cid:17) n ′ ,m ′ ( w ) . We have (cid:16) S d +1 (cid:17) n,m ( w ) − (cid:16) S d +1 (cid:17) n ′ ,m ′ ( w ) = A − B + C. By construction, S d +1 is smaller than S d . Moreover, ( w, ( n, m ) , ( n ′ , m ′ )) being a criticalbranching, w belongs to red (cid:16)(cid:0) S d (cid:1) n,m (cid:17) ∩ red (cid:16)(cid:0) S d (cid:1) n ′ ,m ′ (cid:17) . Hence, from Point 1 of Lemma 3.2.1, A and B admit a (cid:0) S d +1 , w (cid:1) -type decomposition. It remains to show that C admits a (cid:0) S d +1 , w (cid:1) -type decomposition. By construction, Q d +1 is equal to P d , so that it contains Q d from Point 2of Lemma 3.1.1. If ( w, ( n, m ) , ( n ′ , m ′ )) does not belong to Q d , C admits a (cid:0) S d +1 , w (cid:1) -typedecomposition from Point 2 of Lemma 3.2.2. If ( w , ( n, m ) , ( n ′ , m ′ )) belongs to Q d , C admitsa (cid:0) S d , w (cid:1) -type decomposition by induction hypothesis. Hence, from Point 2 of Lemma 3.2.1, C admits a (cid:0) S d +1 , w (cid:1) -type decomposition.Recall that the lower-bound of the operators S d is written S . The last lemma we need toprove Theorem 3.2.5 is Lemma 3.2.4.
1. The sequence ( I d ) d ∈ N of ideals spanned by ker (cid:0) S d (cid:1) is constant.2. Red (cid:0) S (cid:1) is equal to S d ∈ N Red (cid:0) S d (cid:1) . roof. Let us show Point 1. By definition of the function normalisation , the kernel of eachelement of F d is included in I d . In particular, ker ( ∧ F d ) = P T ∈ F d ker ( T ) is also includedin I d . Moreover, C F d being a complement of F d , it is smaller than ∧ F d , that is its kernel isincluded in the one of ∧ F d . In particular, ker (cid:0) C F d (cid:1) is included in I d , so that ker (cid:0) S d +1 (cid:1) , whichby definition is equal to ker (cid:0) S d (cid:1) + ker (cid:0) C F d (cid:1) , is also included in I d . Hence, the sequence ( I d ) d ∈ N is not increasing. Moreover, the sequence (cid:0) S d (cid:1) d ∈ N is not increasing by construction,which means that (cid:0) ker (cid:0) S d (cid:1)(cid:1) d ∈ N is not decreasing. Hence, ( I d ) d ∈ N constant.Let us show Point 2. The equality we want to prove means that the set F = (cid:8) S d | d ∈ N (cid:9) is confluent. From Newman’s Lemma (see the paragraph before Definition 2.1.2) in terms ofreduction operators, it is sufficient to show that F is locally confluent. Let f ∈ K X ∗ and let d and d ′ be two integers which we assume to satisfy d ≥ d ′ . In particular, we have S d ′ (cid:22) S d ,so that S d ◦ S d ′ is equal to S d ′ . Hence, (cid:16) S d ◦ S d ′ (cid:17) ( f ) and S d ( f ) are equal, so that F is locallyconfluent. Theorem 3.2.5.
Let A be an algebra and let ( X, <, S ) be a presentation by operator of A .The completed presentation of ( X, <, S ) is a confluent presentation of A .Proof. Let S be the lower-bound of the operators S d .First, we show that (cid:0) X, <, S (cid:1) is a presentation of A . From Point 1 of Lemma 3.2.4, theideal spanned by the kernels of the operators S d is equal to the ideal I spanned by the kernel of S = S . In particular, the ideal spanned by ker (cid:0) S (cid:1) = P d ∈ N ker (cid:0) S d (cid:1) is equal to I . Hence, ( X, <, S ) being a presentation of A , (cid:0) X, <, S (cid:1) is also a presentation of A .Let us show that this presentation is confluent. From the Diamond Lemma, it is sufficientto show that for each critical branching b = ( w, ( n, m ) , ( n ′ , m ′ )) of (cid:0) X, <, S (cid:1) , the S -polynomial sp ( b ) admits a (cid:0) S, w (cid:1) -type decomposition. From Point 2 of Lemma 3.2.4, thereexist integers d and d ′ such that w ∈ red (cid:16)(cid:0) S d (cid:1) n,m (cid:17) ∩ red (cid:16)(cid:16) S d ′ n ′ ,m ′ (cid:17)(cid:17) . Without lost ofgeneralities, we may assume that d is greater or equal to d ′ , so that b is a critical branching of (cid:0) X, <, S d (cid:1) , that is it belongs to P d = Q d +1 . We let A d = (cid:16) S d +1 (cid:17) n ′ ,m ′ ( w ) − S n ′ ,m ′ ( w ) ,B d = (cid:16) S d +1 (cid:17) n,m ( w ) − S n,m ( w ) ,C d = (cid:16) S d +1 (cid:17) n,m ( w ) − (cid:16) S d +1 (cid:17) n ′ ,m ′ ( w ) . We have sp ( b ) = A d − B d + C d . (13)From Proposition 3.2.3, b being an element of Q d +1 , C d admits a (cid:0) S d +1 , w (cid:1) -type decomposition,so that it admits a (cid:0) S, w (cid:1) -type decomposition from Point 2 of Lemma 3.2.1. Moreover, S d +1 being smaller than S d , w belongs to red (cid:16)(cid:0) S d +1 (cid:1) n,m (cid:17) ∩ red (cid:16)(cid:0) S d +1 (cid:1) n ′ ,m ′ (cid:17) . The operator S being smaller than S d +1 , A d and B d also admit a (cid:0) S, w (cid:1) -type decomposition from Point 1 ofLemma 3.2.1. Hence, from (13), sp ( b ) admits a (cid:0) S, w (cid:1) -type decomposition.13 xample 3.2.6.
In Section 3.3, we compute the completed presentation of Example 2.2.3. Itis given by the operator defined by S ( yz ) = x , S ( zx ) = xy , S ( yxy ) = xx , S ( yxx ) = xxz , S ( yxxx ) = xxxy and S ( w ) = w for all other monomial w . In this section, we compute the completed presentation of Example 3.2.6. Before that, we showhow to use Gaussian elimination to compute lattice operations and completion for reductionoperators relative to totally ordered finite sets. We use the SageMath software, written inPython.
Lattice operations and completion.
Let ( G, < ) be a totally ordered finite set. The set G being finite, the Gaussian elimination provides a unique basis B of any suspace V ⊆ K G such that for every e ∈ B , lc ( e ) is equal to 1 and, given two different elements e and e ′ of B , lg ( e ′ ) does not belong to the decomposition of e . The operator T = ker − ( V ) satisfies T ( lg ( e )) = lg ( e ) − e for every e ∈ B and T ( g ) = g if g is not a leading generator of B .Moreover, we represent the subspaces of K G by lists of generating vectors and for any list ofvectors L , let reducedBasis(L) be the basis of K L obtained by Gaussian elimination.First, we define the function operator which takes as input a list of vectors L and returnsker − ( K L ) . We deduce the functions which compute the lattice operations of RO ( G, < ) . def o p e r a t o r( G ):L = r e d u c e d B a s i s ( G ) n = len ( L [0])V = V e c t o r S p a c e( QQ , n ) v = V . zero ()G =( lg ( L [0]) -1)*[ v ]+[ L [0]] k = len ( L )for i in [1.. k -1]: G = G +( lg ( L [ i ]) - lg ( L [i -1]) -1)*[ v ]+[ L [ i ]]G = G +( n - lg ( L [k - 1 ] ) ) * [v ] return i d e n t i t y _ m a t r i x ( QQ , n ) - matrix ( G ). t r a n s p o s e () def l o w e r B o u n d ( T_1 , T_2 ):V_1 , V_2 = kernel ( T_1 . t r a n s p o s e ()) , kernel ( T_2 . t r a n s p o se ()) G_1 , G_2 = basis ( V_1 ) , basis ( V_2 )L_1 , L_2 = r e d u c e d B a s i s( G_1 ) , r e d u c e d B a s i s ( G_2 ) G = L_1 + L_2L = r e d u c e d B a s i s ( G ) return o p e r a t o r( L ) def u p p e r B o u n d ( T_1 , T_2 ):V_1 , V_2 = kernel ( T_1 . t r a n s p o s e ()) , kernel ( T_2 . t r a n s p o se ()) V = V_1 . i n t e r s e c t i o n ( V_2 )G = basis ( V ) L = r e d u c e d B a s i s ( G )return o p e r a t o r( L )
By definition of the F -complement, we need an intermediate function with input a reductionoperator T and output ker − ( K nf ( T )) . We define this function before defining the one of the14 -complement. def tilde ( T ): n , L = T . nrows () ,[]for i in [0.. n -1]: j , k =i ,n -i -1if T [i , i ]==1: L = L +[ vector ( j * [ 0 ] + [ 1 ] +k *[0])] return o p e r a t o r( L ) def c o m p l e m e n t ( L ):n ,C , T = len ( L ) ,L [0] , tilde ( L [0]) for i in [1.. n -1]: C = l o w e r B o u n d (C , L [ i ])for j in [1.. n -1]: T = u p p e r B o u n d (T , tilde ( L [ j ])) return l o w e r B o u n d (C , T ) Example.
Now, we use our implementation to compute the completed presentation of Ex-ample 2.2.3: we consider the algebra A presented by ( X, <, S ) where X = { x, y, z } , < isthe deg-lex order induced by x < y < z and S ( yz ) = x , S ( zx ) = xy and S ( w ) = w forevery monomial w different from yz and zx .Recall that S d denotes the operator of the presentation at the beginning of step d of theprocedure, P d is the set of critical branchings of (cid:0) X, <, S d (cid:1) , Q d = P d − , E d = n w − S dn,m ( w ) | ( w, ( n, m ) , ( n ′ , m ′ )) ∈ P d \ Q d o and F d = normalisation ( E d , S d ) .Moreover, we represent reduction operators by matrices. For that, we use that the operatorsappearing in the procedure act nontrivially on finite-dimensional subspaces of K X ∗ spannedby an ordered set of monomials w < w < · · · < w n .At the first step, we have d = 0 . The presentation (cid:0) X, <, S (cid:1) has one critical branching b = ( yzx, (1 , , (0 , and we have P = { b } and E = n yzx − xx, yzx − yxy o .We have F = n T , T o where the matrices of the restrictions of T and T to the subspacespanned by xx < yxy < yzx are T = and T = . that is T ( yzx ) = xx and T ( yzx ) = yxy . The matrice of C F = complement ([ T , T ]) restricted to K { xx, yxy, yzx } is , The operator S = S ∧ C F can be computed by restriction to the subspace spanned by x < xx < xy < yz < zx < yxy and the matrices of the restrictions of S and C F to15his subspace are S = and C F = . We obtain that S is the operator defined by S ( yz ) = x , S ( zx ) = xy , S ( yxy ) = xx and S ( w ) = w for every monomial w different from yz , zx and yxy .The presentation (cid:0) X, <, S (cid:1) has two new critical branchings b and b equal to ( yxyz, (2 , , (0 , and ( yxyxy, (2 , , (0 , , respectively. 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