[PPT] - Syzygies among reduction operators Cyrille Chenavier INRIA Lille - PowerPoint Presentation

SLIDE 1

1/14 Motivations Reduction operators Lattice description of syzygies

Syzygies among reduction operators

Cyrille Chenavier

INRIA Lille - Nord Europe Équipe GAIA

October 2, 2018

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 2

2/14 Motivations Reduction operators Lattice description of syzygies

Plan

I. Motivations

⊲ Various notions of syzygy ⊲ Computation of syzygies

II. Reduction operators

⊲ Linear algebra, syzygies and useless reductions ⊲ Reduction operators and labelled reductions

III. Lattice description of syzygies

⊲ Lattice structure of reduction operators ⊲ Construction of a basis of syzygies ⊲ A lattice criterion for rejecting useless reductions

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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3/14 Motivations Reduction operators Lattice description of syzygies

Plan

I. Motivations

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 4

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 5

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them?

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 6

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0,

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 7

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field?

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 8

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 9

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

◮ A method for studying these problems:

⊲ compute a generating set for the associated notion of syzygy

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 10

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

◮ A method for studying these problems:

⊲ compute a generating set for the associated notion of syzygy (two-dimensional cell

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

◮ A method for studying these problems:

⊲ compute a generating set for the associated notion of syzygy (two-dimensional cell, homological syzygy

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 12

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

◮ A method for studying these problems:

⊲ compute a generating set for the associated notion of syzygy (two-dimensional cell, homological syzygy, identity among relations

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 13

4/14 Motivations Reduction operators Lattice description of syzygies

Various notions of syzygy

◮ Consider the following questions:

⊲ Standardisation problems: given two vertices in an abstract rewriting system v

v′

how to choose a "standard" path between them? ⊲ Construction of free resolutions: given an augmented algebra A X | R and A[S]

d

− → A[R]

d

− → A[X]

d

− → A

ε

− → K − → 0, how to extend the beginning of a resolution of the ground field? ⊲ Detecting useless critical pairs: how to obtain a criterion for rejecting useless critical pairs during the completion procedure?

◮ A method for studying these problems:

⊲ compute a generating set for the associated notion of syzygy (two-dimensional cell, homological syzygy, identity among relations, · · · ).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 18

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 19

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 20

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 21

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R. ⊲ Let S be the set of critical pairs of R.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 22

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R. ⊲ Let S be the set of critical pairs of R. ⊲ Reduce S and R (algebraic Morse theory, homotopical reduction, · · · ).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 23

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R. ⊲ Let S be the set of critical pairs of R. ⊲ Reduce S and R (algebraic Morse theory, homotopical reduction, · · · ).

◮ Our goals:

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 24

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R. ⊲ Let S be the set of critical pairs of R. ⊲ Reduce S and R (algebraic Morse theory, homotopical reduction, · · · ).

◮ Our goals:

⊲ Compute syzygies of abstract rewriting systems using the lattice structure of reduction

perators.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 25

5/14 Motivations Reduction operators Lattice description of syzygies

Computation of syzygies

◮ Consider an algebra A presented by X | R.

⊲ Question: how are the syzygies of X | R generated?

◮ If A is quadratic, a candidate is the Koszul dual A!

X ∗ | R⊥ .

◮ Methods from rewriting theory:

⊲ If R is a Gröbner basis, the syzygies are spanned by critical pairs of R. ⊲ If R is a not a Gröbner basis, apply the completion-reduction procedure:

⊲ Complete R into a Gröbner basis R. ⊲ Let S be the set of critical pairs of R. ⊲ Reduce S and R (algebraic Morse theory, homotopical reduction, · · · ).

◮ Our goals:

⊲ Compute syzygies of abstract rewriting systems using the lattice structure of reduction

perators.

⊲ Deduce a lattice criterion for rejecting useless reductions during the completion procedure.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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6/14 Motivations Reduction operators Lattice description of syzygies

Plan

II. Reduction operators

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 27

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 28

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
INRIA Lille - Nord Europe, équipe GAIA

Syzygies among reduction operators

SLIDE 29

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
⊲ The reductions C and F are useless!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 30

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

N

a

d

E

F
⊲ The reductions C and F are useless!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 31

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
c

B

b

N

a

d

E

⊲ The reductions C and F are useless!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 32

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
⊲ The reductions C and F are useless!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 33

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 34

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 35

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 36

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 37

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 38

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 39

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 40

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 41

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.
iii. Compute a row echelon basis of the syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 42

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.
iii. C’ − B’ − A’

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.
iii. Compute a row echelon basis of the syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 43

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.
iii. C’ − B’ − A’

and F’ − E’ − D’ + A’. ⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.
iii. Compute a row echelon basis of the syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 44

7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.
iii. C’ − B’ − A’

and F’ − E’ − D’ + A’. ⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.
iii. Compute a row echelon basis of the syzygies.
iv. Remove the reductions corresponding to leading terms of syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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7/14 Motivations Reduction operators Lattice description of syzygies

Linear algebra, syzygies and useless reductions

◮ We consider abstract rewriting systems (V , −

→) such that V is a vector space and every reduction is labelled.

⊲ e.g., V is spanned by the letters {a, b, c, d, e} submitted to the reductions e

D

A
C
c

B

b

a d

E

F
i. A’ = e − c, B’ = c − b, C’ = e − b, · · ·
ii. A ⊏ B ⊏ · · · ⊏ F.
iii. C’ − B’ − A’

and F’ − E’ − D’ + A’.

iv. C and F.

⊲ The reductions C and F are useless!

◮ How to detect useless reductions using syzygies and linear algebra?

i. Replace each reduction by the difference between its source and its target.
ii. Introduce a terminating order on labels.
iii. Compute a row echelon basis of the syzygies.
iv. Remove the reductions corresponding to leading terms of syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 48

8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

⊲ The set of reduction operators is denoted by RO (G, <).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

⊲ The set of reduction operators is denoted by RO (G, <). Labelled reductions. ⊲ A reduction operator T induces the labelled reductions v

ℓT,v

− → T(v).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

⊲ The set of reduction operators is denoted by RO (G, <). Labelled reductions. ⊲ A reduction operator T induces the labelled reductions v

ℓT,v

− → T(v). ⊲ The labels of reductions induced by F = {T1, · · · , Tn} ⊂ RO (G, <) are ordered by: ℓTi ,u ⊏ ℓTj ,v := (i < j) ∨ (i = j ∧ lt (u) < lt (v)) .

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

⊲ The set of reduction operators is denoted by RO (G, <). Labelled reductions. ⊲ A reduction operator T induces the labelled reductions v

ℓT,v

− → T(v). ⊲ The labels of reductions induced by F = {T1, · · · , Tn} ⊂ RO (G, <) are ordered by: ℓTi ,u ⊏ ℓTj ,v := (i < j) ∨ (i = j ∧ lt (u) < lt (v)) .

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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8/14 Motivations Reduction operators Lattice description of syzygies

Reduction operators and labelled reductions

◮ We fix V is a vector space equipped with a well-ordered basis (G, <).

Definition. ⊲ An endomorphism T of V is a reduction operator if

⊲ T is a projector, ⊲ ∀g ∈ G, we have either T(g) = g or lt (T(g)) < g.

⊲ The set of reduction operators is denoted by RO (G, <). Labelled reductions. ⊲ A reduction operator T induces the labelled reductions v

ℓT,v

− → T(v). ⊲ The labels of reductions induced by F = {T1, · · · , Tn} ⊂ RO (G, <) are ordered by: ℓTi ,u ⊏ ℓTj ,v := (i < j) ∨ (i = j ∧ lt (u) < lt (v)).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

T3

T1
T2
c

T2

b

a d

T4

T5
INRIA Lille - Nord Europe, équipe GAIA

Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

T3

T1
T2
c

T2

b

a d

T4

T5
◮ We obtain the following order on labels:

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

T3

T2
ℓT1,e
c

T2

b

a d

T4

T5
◮ We obtain the following order on labels:

ℓT1,e

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

T3

T2
ℓT1,e
c

ℓT2,c

b

a d

T4

T5
◮ We obtain the following order on labels:

ℓT1,e ⊏ ℓT2,c

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

T3

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

T4

T5
◮ We obtain the following order on labels:

ℓT1,e ⊏ ℓT2,c ⊏ ℓT2,e

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

T4

T5
◮ We obtain the following order on labels:

ℓT1,e ⊏ ℓT2,c ⊏ ℓT2,e ⊏ ℓT3,e

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

T5

ℓT4,d
◮ We obtain the following order on labels:

ℓT1,e ⊏ ℓT2,c ⊏ ℓT2,e ⊏ ℓT3,e ⊏ ℓT4,d

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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9/14 Motivations Reduction operators Lattice description of syzygies

Example

◮ G =

a < b < c < d < e
and

e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
◮ We obtain the following order on labels:

ℓT1,e ⊏ ℓT2,c ⊏ ℓT2,e ⊏ ℓT3,e ⊏ ℓT4,d ⊏ ℓT5,d.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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10/14 Motivations Reduction operators Lattice description of syzygies

Plan

III. Lattice description of syzygies

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F). Example. e

T3

T1
T2
c

T2

b

a d

T4

T5
INRIA Lille - Nord Europe, équipe GAIA

Syzygies among reduction operators

SLIDE 64

11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F). Example. e

T3

T1
T2
c

T2

b

a d

T4

T5
syz (T1, · · · , T5) = K{s1, s2} where

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F). Example. e

T3

T 1
T 2
c

T 2

b

a d

T4

T5
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b) − (c − b), 0, 0, 0)

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F). Example. e

T 3

T 1
T2
c

T2

b

a d

T 4

T 5
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b) − (c − b), 0, 0, 0) s2 = (e − c, 0, −(e − a), −(d − c), d − a)

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F).

Notation. We denote by ui,g = (0, · · · , 0, g − Ti(g), 0, · · · , 0) .

Example. e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b) − (c − b), 0, 0, 0) s2 = (e − c, 0, −(e − a), −(d − c), d − a)

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F).

Notation. We denote by ui,g = (0, · · · , 0, g − Ti(g), 0, · · · , 0) .

Example. e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b) − (c − b), 0, 0, 0) = u2,e s2 = (e − c, 0, −(e − a), −(d − c), d − a)

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F).

Notation. We denote by ui,g = (0, · · · , 0, g − Ti(g), 0, · · · , 0) .

Example. e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b)−(c − b), 0, 0, 0) = u2,e − u2,c s2 = (e − c, 0, −(e − a), −(d − c), d − a)

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F).

Notation. We denote by ui,g = (0, · · · , 0, g − Ti(g), 0, · · · , 0) .

Example. e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
syz (T1, · · · , T5) = K{s1, s2} where

s1 = (−(e − c), (e − b) − (c − b), 0, 0, 0) = u2,e − u2,c − u1,e s2 = (e − c, 0, −(e − a), −(d − c), d − a)

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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11/14 Motivations Reduction operators Lattice description of syzygies

Definition of syzygies

Syzygies. ⊲ The space of syzygies of F = {T1, · · · , Tn} ⊂ RO (G, <) is the kernel of ker (T1) × · · · × ker (Tn) − → V , (v1, · · · , vn) − → v1 + · · · + vn. ⊲ The space of syzygies of F is denoted by syz (F).

Notation. We denote by ui,g = (0, · · · , 0, g − Ti(g), 0, · · · , 0) .

Example. e

ℓT3,e

ℓT1,e
ℓT2,e
c

ℓT2,c

b

a d

ℓT5,d

ℓT4,d
syz (T1, · · · , T5) = K{s1, s2} where

s1 = ( − (e − c), (e − b) − (c − b), 0, 0, 0) = u2,e − u2,c − u1,e s2 = (e − c, 0, −(e − a), −(d − c), d − a) = u5,d − u4,d − u3,e + u1,e

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12/14 Motivations Reduction operators Lattice description of syzygies

Lattice structure on RO (G, <) and syzygies

Lattice structure on RO (G, <). ⊲ The map RO (G, <) − → Subspaces (V ) , T − → ker (T) is a bijection.

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12/14 Motivations Reduction operators Lattice description of syzygies

Lattice structure on RO (G, <) and syzygies

Lattice structure on RO (G, <). ⊲ The map RO (G, <) − → Subspaces (V ) , T − → ker (T) is a bijection. ⊲ RO (G, <) admits a lattice structure where:

⊲ T1 T2 if ker (T2) ⊆ ker (T1), ⊲ T1 ∧ T2 := ker−1 (ker (T1) + ker (T2)), ⊲ T1 ∨ T2 := ker−1 (ker (T1) ∩ ker (T2)).

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12/14 Motivations Reduction operators Lattice description of syzygies

Lattice structure on RO (G, <) and syzygies

Lattice structure on RO (G, <). ⊲ The map RO (G, <) − → Subspaces (V ) , T − → ker (T) is a bijection. ⊲ RO (G, <) admits a lattice structure where:

⊲ T1 T2 if ker (T2) ⊆ ker (T1), ⊲ T1 ∧ T2 := ker−1 (ker (T1) + ker (T2)), ⊲ T1 ∨ T2 := ker−1 (ker (T1) ∩ ker (T2)).

Proposition i. Let P = (T, T ′) ⊂ RO (G, <). We have a linear isomorphism ker T ∨ T ′

∼

− → syz (P) .

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12/14 Motivations Reduction operators Lattice description of syzygies

Lattice structure on RO (G, <) and syzygies

Lattice structure on RO (G, <). ⊲ The map RO (G, <) − → Subspaces (V ) , T − → ker (T) is a bijection. ⊲ RO (G, <) admits a lattice structure where:

⊲ T1 T2 if ker (T2) ⊆ ker (T1), ⊲ T1 ∧ T2 := ker−1 (ker (T1) + ker (T2)), ⊲ T1 ∨ T2 := ker−1 (ker (T1) ∩ ker (T2)).

Proposition i. Let P = (T, T ′) ⊂ RO (G, <). We have a linear isomorphism ker T ∨ T ′

∼

− → syz (P) , v − → (−v, v).

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12/14 Motivations Reduction operators Lattice description of syzygies

Lattice structure on RO (G, <) and syzygies

Lattice structure on RO (G, <). ⊲ The map RO (G, <) − → Subspaces (V ) , T − → ker (T) is a bijection. ⊲ RO (G, <) admits a lattice structure where:

⊲ T1 T2 if ker (T2) ⊆ ker (T1), ⊲ T1 ∧ T2 := ker−1 (ker (T1) + ker (T2)), ⊲ T1 ∨ T2 := ker−1 (ker (T1) ∩ ker (T2)).

Proposition i. Let P = (T, T ′) ⊂ RO (G, <). We have a linear isomorphism ker T ∨ T ′

∼

− → syz (P) . Proposition ii. Let F = {T1, · · · , Tn} ⊂ RO (G, <). For every integer 2 ≤ i ≤ n, we have a short exact sequence 0 − → syz (T1, · · · , Ti−1)

ιi

− → syz (T1, · · · , Ti)

πi

− → syz (T1 ∧ · · · ∧ Ti−1, Ti) − → 0.

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
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Syzygies among reduction operators

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

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13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

⊲ e − c = e − T1(e).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 84

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

⊲ e − c = e − T1(e). ⊲ e − c = (e − T2(e)) − (c − T2(c)).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 85

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

⊲ e − c = e − T1(e). ⊲ e − c = (e − T2(e)) − (c − T2(c)). ⊲ We get the first basis element: s1 = u2,e

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 86

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

⊲ e − c = e − T1(e). ⊲ e − c = (e − T2(e)) − (c − T2(c)). ⊲ We get the first basis element: s1 = u2,e − u2,c

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 87

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 1. We have ker (T1 ∨ T2) = K{e − c}.

⊲ e − c = e − T1(e). ⊲ e − c = (e − T2(e)) − (c − T2(c)). ⊲ We get the first basis element: s1 = u2,e − u2,c − u1,c.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 88

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 2. We have ker ((T1 ∧ T2) ∨ T3) = {0}.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 89

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 2. We have ker ((T1 ∧ T2) ∨ T3) = {0}.

⊲ No new basis element!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 90

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 3. ker ((T1 ∧ T2 ∧ T3) ∨ T4) = {0}.

⊲ No new basis element!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 91

13/14 Motivations Reduction operators Lattice description of syzygies

Construction of a basis of syz (F)

◮ How to construct a basis of syz (F)?

⊲ We have syz (T1, T2) ⊆ syz (T1, T2, T3) ⊆ · · · ⊆ syz (T1, · · · , Tn). ⊲ Main step: construct a supplement of syz (T1, · · · , Ti−1) in syz (T1, · · · , Ti).

⊲ This supplement is constructed using the isomorphism syz (T1 · · · , Ti) /syz (T1, · · · , Ti−1) ≃ ker ((T1 ∧ · · · ∧ Ti−1) ∨ Ti) .

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
Step 4. We have

ker ((T1 ∧ T2 ∧ T3 ∧ T4) ∨ T5) = K{d − a}. ⊲ d−a = (d − T4(d))+(e − T3(e))−(e − T1(e)). ⊲ d − a = d − T5(d). ⊲ We get the second basis element: s2 = u5,d − u4,d − u3,e + u1,e.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 92

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <).

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 93

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 94

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

T3

T2
T1
c

T2

b

a d

T4

T5
INRIA Lille - Nord Europe, équipe GAIA

Syzygies among reduction operators

SLIDE 95

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT2,e
ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

ℓT5,d
INRIA Lille - Nord Europe, équipe GAIA

Syzygies among reduction operators

SLIDE 96

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT2,e
ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

ℓT5,d
We have:

⊲ ker (T1 ∨ T2) = K{e − c}.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 97

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT2,e
ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

ℓT5,d
We have:

⊲ ker (T1 ∨ T2) = K{e − c}.

⊲ T1 ∨ T2(e) = c.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 98

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

ℓT5,d
We have:

⊲ ker (T1 ∨ T2) = K{e − c}.

⊲ T1 ∨ T2(e) = c. ⊲ The labelled reduction ℓT2,e is useless.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 99

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

We have:

⊲ ker (T1 ∨ T2) = K{e − c}.

⊲ T1 ∨ T2(e) = c. ⊲ The labelled reduction ℓT2,e is useless.

⊲ ker ((T1 ∧ T2 ∧ T3 ∧ T4) ∨ T5) = K{d − a}.

⊲ The labelled reduction ℓT5,d is useless.

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators

SLIDE 100

14/14 Motivations Reduction operators Lattice description of syzygies

A lattice criterion for rejecting useless reductions

The criterion. Let F = {T1, · · · , Tn} ⊂ RO (G, <). ⊲ The useless reductions are labelled by leading terms of syzygies.

⊲ These labels are ℓTi ,g, where g / ∈ im ((T1 ∧ · · · ∧ Ti−1) ∨ Ti).

Example. e

ℓT3,e

ℓT1,e
c

ℓT2,c

b

a d

ℓT4,d

We have:

⊲ ker (T1 ∨ T2) = K{e − c}.

⊲ T1 ∨ T2(e) = c. ⊲ The labelled reduction ℓT2,e is useless.

⊲ ker ((T1 ∧ T2 ∧ T3 ∧ T4) ∨ T5) = K{d − a}.

⊲ The labelled reduction ℓT5,d is useless.

THANK YOU FOR YOUR ATTENTION!

INRIA Lille - Nord Europe, équipe GAIA Syzygies among reduction operators