From binary relations to Tamari lattice

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Motivations Weak order on binary relations Subposet and sublattice

From binary relations to Tamari lattice

Viviane Pons

with Gr´ egory Chˆ atel and Vincent Pilaud Universit´ e Paris-Sud

Discrete Structure Days

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Posets and lattice Bubble sort on permutations Géométrie, combinatoire, algèbre

What is an order?

Natural order on integers: 1 < 2 < 3 < . . . Partial orders?

The closet-poset!

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What is an order?

Natural order on integers: 1 < 2 < 3 < . . .

Partial orders?

The closet-poset!

(4)

What is an order?

Natural order on integers: 1 < 2 < 3 < . . . Partial orders?

The closet-poset!

(5)

What is an order?

Natural order on integers: 1 < 2 < 3 < . . . Partial orders?

The closet-poset!

(6)

Properties:

I Antisymmetric

I Transitive

I Lattice?

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Properties:

I Antisymmetric

I Transitive

I Lattice?

(8)

Properties:

I Antisymmetric

I Transitive

I Lattice?

(9)

Properties:

I Antisymmetric

I Transitive

I Lattice?

(10)

Non lattice example:

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Bubble sort on permutations 251436

215436 251346

215346

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Bubble sort on permutations 251436

215436 251346

215346

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Bubble sort on permutations

251436 215436

251346

215346

(14)

Bubble sort on permutations

251436 215436

251346

215346

(15)

Bubble sort on permutations

251436 215436 251346

215346

(16)

Bubble sort on permutations

251436 215436 251346

215346

(17)

Weak Order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

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Weak Order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(19)

Weak Order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(20)

Weak Order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(21)

Weak Order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231

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Triple interpretation

Geometry

Algebra

F

21

. F

12

= F

₂₁₁₂

= F

2134

+ F

2314

+ F

2341

+ F

3214

+ F

3241

+ F

3421

Combinatorics

1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

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Triple interpretation

Geometry

Algebra

P . P = P + P + P + P + P + P

Combinatorics

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Back to permutations Binary relations

The graph of a permutation 4312

1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

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The graph of a permutation 4312

1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

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The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(27)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(28)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(29)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(30)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(31)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(32)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(33)

The graph of a permutation

4312 1 2 3 4

3412

1 2 3 4

3142 1 2 3 4

(34)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(35)

The graph of a permutation

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(36)

Binary Relations on integers Let R be a relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

For size n: 2 ⁿ⁽ⁿ⁻¹⁾ possible binary relations.

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Binary Relations on integers Let R be a relation of size n.

1 2 ... i ... j ... k ... n

i R j

k R i i R k

For size n: 2 ⁿ⁽ⁿ⁻¹⁾ possible binary relations.

(38)

Binary Relations on integers Let R be a relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i

i R k

For size n: 2 ⁿ⁽ⁿ⁻¹⁾ possible binary relations.

(39)

Binary Relations on integers Let R be a relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

For size n: 2 ⁿ⁽ⁿ⁻¹⁾ possible binary relations.

(40)

Binary Relations on integers Let R be a relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

For size n: 2 ⁿ⁽ⁿ⁻¹⁾ possible binary relations.

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Weak order on binary relations Let R be a binary relation

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j }

Let R and S be two binary relations.

R 4 S ⇔ R ^Inc ⊇ S ^Inc and R ^Dec ⊆ S ^Dec

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Weak order on binary relations Let R be a binary relation

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j } Let R and S be two binary relations.

R 4 S ⇔ R ^Inc ⊇ S ^Inc and R ^Dec ⊆ S ^Dec

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Binary relations of size 2

1 2

1 2 1 2

1 2

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Binary relations of size 2

1 2

1 2 1 2

1 2

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Binary relations of size 2

1 2

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Binary relations of size 2

1 2

1 2 1 2

1 2

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Binary relations of size 2

1 2

1 2 1 2

1 2

(48)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 31 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(49)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(50)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

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Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

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Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

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Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

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Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

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The lattice of posets Subposets and sublattices

We want to keep binary relations which are both

I antisymmetric

I transitive

(posets)

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Antisymmetry

1 2

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Antisymmetry

1 2

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1 2 3

1 2 3 1 2 3

1 2 3

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Sublattice ?

Let R and S be antisymmetric, is R ∧ S also antisymmetric ?

1 2 3 4 1 2 3 4

1 2 3 4

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Sublattice.

Let R and S be antisymmetric, is R ∧ S also antisymmetric ?

... i ... j ... ... i ... j ...

... i ... j ...

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Sublattice.

Let R and S be antisymmetric, is R ∧ S also antisymmetric ? ... i ... j ... ... i ... j ...

... i ... j ...

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Sublattice.

Let R and S be antisymmetric, is R ∧ S also antisymmetric ? ... i ... j ... ... i ... j ...

... i ... j ...

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Sublattice.

Let R and S be antisymmetric, is R ∧ S also antisymmetric ? ... i ... j ... ... i ... j ...

... i ... j ...

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Sublattice.

Let R and S be antisymmetric, is R ∧ S also antisymmetric ? ... i ... j ... ... i ... j ...

... i ... j ...

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Transitivity

Non transitive transitive

1 2 3 4 1 2 3 4

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1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

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Sublattice ?

No! But still a lattice

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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Sublattice ? No!

But still a lattice

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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Sublattice ? No!

But still a lattice

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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Sublattice ? No! But still a lattice

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(71)

Back to permutations

4312 1 2 3 4

We have i R j iif the number i appears before j in the permutations.

The relation is then

I antisymmetric

I transitive

I and a total order

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Back to permutations

4312 1 2 3 4

We have i R j iif the number i appears before j in the permutations.

The relation is then

I antisymmetric

I transitive

I and a total order

(73)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(74)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(75)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(76)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(77)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(78)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(79)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(80)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

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