Motivations
Weak order on binary relations
Subposet and sublattice
Binary relations lattice
Grégory Châtel
joint work with Joel Gay, Vincent Pilaud et Viviane Pons
Université Paris-Est Marne-la-Vallée
SLC 76 05/04/2016
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
215436
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
215436
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
215436
Grégory Châtel
251346
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Transpositions
251436
215436
251346
215346
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Weak order
4321
3421 4231 4312
321
3241 2431 3412 4213 4132
231
312
3214 2341 3142 2413 4123 1432
2314 3124 2143 1342 1423
213
132
2134 1324 1243
123
1234
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Weak order
4321
3421 4231 4312
321
3241 2431 3412 4213 4132
231
312
3214 2341 3142 2413 4123 1432
2314 3124 2143 1342 1423
213
132
2134 1324 1243
123
1234
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Weak order
4321
3421 4231 4312
321
3241 2431 3412 4213 4132
231
312
3214 2341 3142 2413 4123 1432
2314 3124 2143 1342 1423
213
132
2134 1324 1243
123
1234
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Weak order
4321
3421 4231 4312
321
3241 2431 3412 4213 4132
231
312
3214 2341 3142 2413 4123 1432
2314 3124 2143 1342 1423
213
132
2134 1324 1243
123
1234
2413 ∧ 4213 = 2413
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Weak order
4321
3421 4231 4312
321
3241 2431 3412 4213 4132
231
312
3214 2341 3142 2413 4123 1432
2314 3124 2143 1342 1423
213
132
2134 1324 1243
123
1234
2413 ∧ 4213 = 2413
2413 ∨ 4213 = 4231
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Triple interpretation
Combinatorial interpretation
4321
Geometry
3241
3214
3421
4231
4312
2431
3412
4213
2341
2314
3142
2413
2143
1342
2134
1324
1243
1234
F21 . F12 = F2112
= F2134 + F2314 + F2341 + F3214 + F3241 + F3421
Grégory Châtel
4123
3124
Algebra
Binary relations lattice
4132
1432
1423
Motivations
Weak order on binary relations
Subposet and sublattice
Weak order on permutations
Geometry, combinatorics, algebra
Triple interpretation
Combinatorial interpretation
Geometry
Algebra
P
.P
=P
+P
+P
Grégory Châtel
+P
+P
Binary relations lattice
+P
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
Grégory Châtel
2
3
4
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
2
3
4
3412
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
2
3
4
3412
1
2
3
4
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Permutation graph
4312
1
2
3
4
3412
1
2
3
4
3142
1
2
3
4
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Binary relations on integer
Let R be a relation of size n.
1
2
...
i
...
Grégory Châtel
j
...
k
Binary relations lattice
...
n
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Binary relations on integer
Let R be a relation of size n.
1
2
...
i
...
j
...
k
i Rj
Grégory Châtel
Binary relations lattice
...
n
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Binary relations on integer
Let R be a relation of size n.
1
2
i Rj
...
i
...
j
...
k
k Ri
Grégory Châtel
Binary relations lattice
...
n
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Binary relations on integer
Let R be a relation of size n.
1
2
i Rj
...
i
...
j
...
k
k Ri
Grégory Châtel
Binary relations lattice
...
i Rk
n
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Binary relations on integer
Let R be a relation of size n.
1
2
i Rj
...
i
...
j
...
k
k Ri
There is 2n(n−1) binary relations.
Grégory Châtel
Binary relations lattice
...
i Rk
n
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Partial order on relations
Let R be a binary relation
RInc = {i R j, i < j}
RDec = {j R i, i < j}
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Partial order on relations
Let R be a binary relation
RInc = {i R j, i < j}
RDec = {j R i, i < j}
Let R and S be two binary relations,
R4S⇔
RInc ⊇ SInc and RDec ⊆ SDec
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Relations of size 2
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Relations of size 2
1
Grégory Châtel
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Relations of size 2
1
2
1
Grégory Châtel
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Relations of size 2
1
2
1
Grégory Châtel
1
2
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Relations of size 2
1
1
2
1
Grégory Châtel
2
1
2
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
1
1
1
2
31
2
2
3
Back to the permutations
Generalization to binary relations
2
3
1
2
3
1
2
31
2
31
2
3
1
2
3
1
2
3
2
31
2
31
2
31
31
2
3
1
2
3
1
2
31
2
31
2
31
2
31
2
31
2
3
2
31
2
1
31
31
2
31
2
31
2
31
2
31
31
2
31
2
31
2
31
2
31
2
31
31
2
31
2
2
31
2
2
31
2
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
31
2
1
31
2
3
1
2
31
2
31
2
31
2
3
1
2
3
1
2
3
1
Grégory Châtel
2
3
Binary relations lattice
1
2
3
1
2
3
2
31
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
3
4
Grégory Châtel
1
2
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
3
4
1
2
1
2
3
4
Grégory Châtel
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
3
4
1
2
1
2
3
4
Grégory Châtel
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
3
4
1
2
1
2
3
4
Grégory Châtel
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
3
4
1
2
1
2
3
4
Grégory Châtel
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Back to the permutations
Generalization to binary relations
Meet and Join
1
2
1
2
3
4
3
4
1
2
1
2
3
4
Grégory Châtel
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
We want to keep the relations that are:
I
antisymmetric
I
transitives
(posets)
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Antisymmetry
1
1
2
1
Grégory Châtel
2
1
2
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Antisymmetry
1
1
2
2
1
Grégory Châtel
2
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
1
2
3
1
2
3
1
2
3
Poset lattice
Suborders and sublattices
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
3
3
3
2
2
2
1
1
1
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
1
2
3
Grégory Châtel
2
3
Binary relations lattice
1
2
3
1
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
1
2
3
1
4
1
2
Grégory Châtel
3
2
4
Binary relations lattice
3
4
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...
Grégory Châtel
... i ... j ...
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...
... i ... j ...
... i ... j ...
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...
... i ... j ...
... i ... j ...
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...
... i ... j ...
... i ... j ...
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitivity
Non transitive
1
2
3
Transitive
4
Grégory Châtel
1
2
3
Binary relations lattice
4
Motivations
Weak order on binary relations
Subposet and sublattice
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
2
3
3
3
1
Poset lattice
Suborders and sublattices
1
1
2
3
1
2
3
1
2
3
3
1
Grégory Châtel
2
3
2
1
2
2
1
3
1
1
1
2
2
3
2
3
2
3
3
1
3
3
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
1
2
3
1
2
3
Grégory Châtel
1
2
3
Binary relations lattice
1
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
1
2
3
1
1
2
2
3
3
1
2
3
1
1
Grégory Châtel
Binary relations lattice
2
3
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
1
2
3
1
1
2
3
1
2
3
2
3
1
2
3
1
1
Grégory Châtel
Binary relations lattice
2
3
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Sublattice?
1
2
3
1
1
2
3
1
2
3
1
2
3
2
3
Grégory Châtel
1
2
3
1
1
2
3
1
2
3
Binary relations lattice
2
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitive decreasing deletion
j
a
Grégory Châtel
i
b
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitive decreasing deletion
j
a
Grégory Châtel
i
b
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitive decreasing deletion
j
i
a
1
2
3
Grégory Châtel
b
1
2
Binary relations lattice
3
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitive decreasing deletion
j
i
a
1a
2j
3b, i
Grégory Châtel
b
1a
2j
Binary relations lattice
3b, i
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Transitive decreasing deletion
j
i
a
1a
2j
3b, i
Grégory Châtel
b
1a
2j
Binary relations lattice
3b, i
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Back to the permutations
4312
1
2
3
4
We have i R j iff the number i is on the left of j in the permutation.
The relation is
I
antisymmetric
I
transitive
I
and total
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Back to the permutations
4312
1
2
3
4
We have i R j iff the number i is on the left of j in the permutation.
The relation is
I
antisymmetric
I
transitive
I
and total
Grégory Châtel
Binary relations lattice
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
1
1
1
2
3
1
1
2
1
2
3
2
2
2
1
3
3
2
1
1
2
3
1
2
3
1
2
3
1
3
1
Grégory Châtel
3
3
3
1
2
2
2
1
3
2
1
1
2
2
3
3
Binary relations lattice
1
3
2
3
2
3
2
3
3
1
Motivations
Weak order on binary relations
Subposet and sublattice
Poset lattice
Suborders and sublattices
Results
I
A set of rules on posets that automatically yields sublattices.
Work in progress
I
Study associated Hopf algebras.
I
Study associated polytopes.
I
Study the same problems in the Coxeter group framework.
Grégory Châtel
Binary relations lattice