Binary relations lattice
Gr´egory Chˆatel joint work with Joel Gay, Vincent Pilaud et Viviane Pons
Universit´eParis-Est Marne-la-Vall´ee
SLC 76 - April 2016
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 1 / 22 2413 ∨ 4213 = 4231
2413 ∧ 4213 = 2413
Right 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 2 / 22 2413 ∨ 4213 = 4231
2413 ∧ 4213 = 2413
Right 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 2 / 22 2413 ∨ 4213 = 4231
2413 ∧ 4213 = 2413
Right 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 2 / 22 2413 ∨ 4213 = 4231
Right 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 2 / 22 Right 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 2 / 22 Right weak order on permutation graphs
1 2 3 321
1 2 3 1 2 3 231 312
213 132 1 2 3 1 2 3
123 1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 3 / 22 i R j k R i i R k
There is 2n(n−1) binary relations.
Binary relations on integer Let R be a relation of size n.
1 2 ... i ... j ... k ... n
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 4 / 22 k R i i R k
There is 2n(n−1) binary relations.
Binary relations on integer Let R be a relation of size n.
1 2 ... i ... j ... k ... n
i R j
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 4 / 22 i R k
There is 2n(n−1) binary relations.
Binary relations on integer Let R be a relation of size n.
1 2 ... i ... j ... k ... n
i R j k R i
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 4 / 22 There is 2n(n−1) binary relations.
Binary relations on integer Let R be a relation of size n.
1 2 ... i ... j ... k ... n
i R j k R i i R k
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 4 / 22 Binary relations on integer Let R be a relation of size n.
1 2 ... i ... j ... k ... n
i R j k R i i R k
There is 2n(n−1) binary relations.
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 4 / 22 Let R and S be two binary relations,
Inc Inc Dec Dec R 4 S ⇔ R ⊇ S and R ⊆ S
Partial order on relations Let R be a binary relation
RInc = {i R j, i < j} RDec = {j R i, i < j}
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 5 / 22 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,
Inc Inc Dec Dec R 4 S ⇔ R ⊇ S and R ⊆ S
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 5 / 22 1 2
1 2 1 2
1 2
Relations of size 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2
1 2 1 2
Relations of size 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2
1 2
Relations of size 2
1 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2
Relations of size 2
1 2 1 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 6 / 22 Relations of size 2
1 2
1 2 1 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 6 / 22 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 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 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 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 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 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 1 2 3 1 2 3 1 2 3
1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 7 / 22 1 2 3 4
1 2 3 4
Meet and Join
1 2 3 4 1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 1 2 3 4
Meet and Join
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 1 2 3 4
Meet and Join
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 1 2 3 4
Meet and Join
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 1 2 3 4
Meet and Join
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 Meet and Join
1 2 3 4
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 8 / 22 Posets are AWESOME
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 9 / 22 Objects viewed as posets
4312 1 2 3 4
2
1 4 1 2 3 4 5
3 5
+ − − 1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 10 / 22 Intervals viewed as posets
3421
3241 1 2 3 4 3214 2341
2314
3 1
[ 1 , 3 ] 1 2 3
2 2
[+ − −−, + − ++] 1 2 3 4 5
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 11 / 22 We want to keep the relations that are: antisymmetric transitives (posets)
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 12 / 22 Antisymmetry
1 2
1 2 1 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 13 / 22 1 2
Antisymmetry
1 2
1 2
1 2
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 13 / 22 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
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 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 14 / 22 Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
1 2 3 4 1 2 3 4
1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 15 / 22 ... i ... j ...
... i ... j ...... i ... j ...
Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 16 / 22 ... i ... j ...
Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...... i ... j ...
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 16 / 22 Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...... i ... j ...
... i ... j ...
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 16 / 22 Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...... i ... j ...
... i ... j ...
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 16 / 22 Sublattice? If R and S are antisymmetric, is R ∧ S also antisymmetric?
... i ... j ...... i ... j ...
... i ... j ...
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 16 / 22 Transitivity
Non transitive Transitive
1 2 3 4 1 2 3 4
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 17 / 22 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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 18 / 22 1 2 3 1 2 3
1 2 3
1 2 3 1 2 3
Sublattice?
1 2 3 1 2 3 1 2 3 1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 19 / 22 1 2 3
1 2 3 1 2 3
Sublattice?
1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 19 / 22 1 2 3 1 2 3
Sublattice?
1 2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3
1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 19 / 22 Sublattice?
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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 19 / 22 a j b, i a j b, i
1 2 3 1 2 3
Transitive decreasing deletion j i
a ≤ j, i ≤ b a b
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 20 / 22 a j b, i a j b, i
1 2 3 1 2 3
Transitive decreasing deletion j i
a ≤ j, i ≤ b a b
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 20 / 22 a j b, i a j b, i
Transitive decreasing deletion j i
a ≤ j, i ≤ b a b
1 2 3 1 2 3
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 20 / 22 Transitive decreasing deletion j i
a ≤ j, i ≤ b a b
1 a 2 j 3b, i 1 a 2 j 3b, i
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 20 / 22 Transitive decreasing deletion j i
a ≤ j, i ≤ b a b
1 a 2 j 3b, i 1 a 2 j 3b, i
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 20 / 22 Poset lattice
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Weak Order Element Poset (WOEP) ' Right weak order
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Weak Order Element Poset (WOEP) ' Right weak order
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Weak Order Interval Poset (WOIP)
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Weak Order Face Poset (WOFP)
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Tamari Order Element Poset (TOEP) ' Tamari order
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Tamari Order Interval Poset (TOIP)
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Tamari Order Face Poset (TOFP)
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Boolean Order Element Poset (BOEP) ' Boolean order
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Boolean Order Interval Poset (BOIP = BOFP)
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 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
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 21 / 22 Results A lattice on posets. A set of rules on posets that automatically yields sublattices.
Work in progress ”Correction” algorithms for the subposets that are not sublattices. Study associated Hopf algebras. Study associated polytopes. Study the same problems in the Coxeter group framework.
Gr´egory Chˆatel Binary relations lattice SLC 76 - April 2016 22 / 22