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Binary Relations Lattice

Binary Relations Lattice

Binary relations lattice Gr´egory Ch^atel joint work with Joel Gay, Vincent Pilaud et Viviane Pons Universit´eParis-Est Marne-la-Vall´ee SLC 76 - April 2016 Gr´egory Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 = fi R j; i < jg RDec = fj R i; i < jg Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 5 / 22 Partial order on relations Let R be a binary relation RInc = fi R j; i < jg RDec = fj R i; i < jg Let R and S be two binary relations, Inc Inc Dec Dec R 4 S , R ⊇ S and R ⊆ S Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 5 / 22 1 2 1 2 1 2 1 2 Relations of size 2 Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2 1 2 1 2 Relations of size 2 1 2 Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2 1 2 Relations of size 2 1 2 1 2 Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 6 / 22 1 2 Relations of size 2 1 2 1 2 1 2 Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 6 / 22 Relations of size 2 1 2 1 2 1 2 1 2 Gr´egory Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel Binary relations lattice SLC 76 - April 2016 8 / 22 Posets are AWESOME Gr´egory Ch^atel 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 Ch^atel 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 Ch^atel Binary relations lattice SLC 76 - April 2016 11 / 22 We want to keep the relations that are: antisymmetric transitives (posets) Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 12 / 22 Antisymmetry 1 2 1 2 1 2 1 2 Gr´egory Ch^atel Binary relations lattice SLC 76 - April 2016 13 / 22 1 2 Antisymmetry 1 2 1 2 1 2 Gr´egory Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel 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 Ch^atel Binary relations lattice SLC 76 - April 2016 16 / 22 Sublattice? If R and S are antisymmetric, is R ^ S also antisymmetric? ..

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