(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices and lattices of
finite closure systems: some (lower)
bounded lattices
Nathalie Caspard
LACL, University Paris 12 Val-de-Marne, France
and CAMS, EHESS, Paris, France
ICFCA’07, Clermont-Ferrand
1/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Sketch of the talk
2/103
1
(Lower) bounded lattices and the doubling operation
2
Finite Coxeter lattices
Coxeter lattices
The class HH of lattices
All lattices of HH are bounded
Finite Coxeter lattices are in HH
3
The lattice of finite closure systems
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Outline
3/103
1
(Lower) bounded lattices and the doubling operation
2
Finite Coxeter lattices
Coxeter lattices
The class HH of lattices
All lattices of HH are bounded
Finite Coxeter lattices are in HH
3
The lattice of finite closure systems
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
(Lower) bounded lattices
Definition (McKenzie [10], 1972)
A homomorphism α : L → L0 is called lower bounded if the
inverse image of each element of L0 is either empty or has a
minimum.
A lattice is lower bounded if it is the lower bounded
homomorphic image of a free lattice.
An upper bounded lattice is defined dually and a lattice is
bounded if it is lower and upper bounded.
4/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The doubling construction, Day [6], 1970
k
i
j
f
e
g
c
b
a
5/103
h
d
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The doubling construction, Day [6], 1970
k
i
j
f
e
g
c
b
a
6/103
h
d
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The doubling construction, Day [6], 1970
k
i
f
e
g
c
b
a
7/103
j
j
h
d
1
g ×
f
0
c
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The doubling construction, Day [6], 1970
k
i
j
j
f
e
g
h
j1
1
j0
g ×
f
g0
f0
c
b
a
8/103
d
g1
f1
0
c
c1
c0
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The doubling construction, Day [6], 1970
k
j1
i
j0
j
f
e
g
c
b
h
c0
k
i
j1
j0
g0
f0
f1
g1
e
c1
b
g1
f1
c1
d
a
c0
a
9/103
g0
f0
d
h
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Generalisation to lower pseudo-intervals
k
i
j
f
e
g
c
b
a
10/103
h
d
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Generalisation to lower pseudo-intervals
k
i1
i
j
i0
f
e
g
c
b
d
f0
e0
b0
g0
c0
a
a0
11/103
f1
e1
h
b1
g1
c1
a1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Generalisation to lower pseudo-intervals
k
i1
i
j
i0
f
e
g
f0
e0
d
c
b
b0
k
a
g0
c0
a0
j
i1
i0
f1
e1
f0
e0
b0
g0
c0
a0
12/103
f1
e1
h
b1
g1
c1
a1
h
b1
g1
c1
a1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Generalisation to convex sets
k
k
i1
i
j
j
i0
e
f
b
c
d
f0
e0
b0
g0
c0
a
a
13/103
f1
e
h
g
b1
d0
h
g1
c1
d1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterisation of bounded lattices
Theorem (Day [7], 1979)
Let L be a lattice. The following are equivalent :
L is bounded,
it can be constructed starting from 2 by a finite sequence of
interval doublings.
14/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterisation of lower bounded lattices
Theorem (Day [7], 1979)
Let L be a lattice. The following are equivalent :
L is lower bounded,
it can be constructed starting from 2 by a finite sequence of
lower pseudo-intervals.
15/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterisation of upper bounded lattices
Theorem (Day [7], 1979)
Let L be a lattice. The following are equivalent :
L is upper bounded,
it can be constructed starting from 2 by a finite sequence of
upper pseudo-intervals.
16/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
An example of bounded lattice
1
0
17/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
An example of bounded lattice
11
1
10
01
0
00
18/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
An example of bounded lattice
111
11
1
10
01
101
011
100
010
0
00
000
19/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Perm(3) is bounded
321
11
1
10
01
231
312
213
132
0
00
123
20/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Permutohedron on 4 elements :
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Permutohedron on 4 elements : bounded too
4321
4231
3421
3241
4312
4213
2431
4132
3412
4123
2341
3214
3142
2143
2314
1342
3124
2134
1324
1234
21/103
1432
2413
1243
1423
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Permutohedron on 5 elements :
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Permutohedron on 5 elements : bounded again
5, 4, 3, 2, 1
5, 3, 4, 2, 1
5, 3, 2, 4, 1
3, 5, 1, 2, 4
5, 3, 4, 1, 2
5, 4, 3, 1, 2
3, 5, 4, 2, 1
4, 3, 5, 2, 1
4, 5, 3, 2, 1
5, 4, 2, 3, 1
4, 5, 3, 1, 2
5, 4, 1, 3, 2
3, 5, 2, 4, 1
5, 3, 2, 1, 4
5, 3, 1, 4, 2
3, 5, 4, 1, 2
3, 4, 5, 2, 1
4, 3, 5, 1, 2
5, 1, 4, 3, 2
4, 3, 2, 5, 1
5, 2, 3, 4, 1
4, 5, 1, 3, 2
4, 5, 2, 3, 1
5, 4, 1, 2, 3
4, 5, 2, 1, 3
4, 2, 5, 3, 1
5, 2, 4, 3, 1
5, 2, 4, 1, 3
2, 5, 4, 3, 1
3, 5, 2, 1, 4
5, 3, 1, 2, 4
3, 5, 1, 4, 2
3, 2, 5, 4, 1
5, 1, 3, 4, 2
3, 4, 5, 1, 2
3, 4, 2, 5, 1
4, 3, 1, 5, 2
1, 5, 4, 3, 2
4, 3, 2, 1, 5
2, 5, 3, 4, 1
4, 1, 5, 3, 2
5, 2, 3, 1, 4
5, 1, 4, 2, 3
4, 5, 1, 2, 3
4, 2, 3, 5, 1
5, 2, 1, 4, 3
4, 2, 5, 1, 3
2, 4, 5, 3, 1
2, 5, 4, 1, 3
3, 1, 5, 4, 2
3, 2, 5, 1, 4
5, 1, 3, 2, 4
1, 5, 3, 4, 2
3, 2, 4, 5, 1
3, 4, 1, 5, 2
3, 4, 2, 1, 5
4, 3, 1, 2, 5
2, 3, 5, 4, 1
4, 1, 3, 5, 2
1, 4, 5, 3, 2
1, 5, 4, 2, 3
2, 5, 3, 1, 4
4, 1, 5, 2, 3
5, 2, 1, 3, 4
5, 1, 2, 4, 3
2, 4, 3, 5, 1
4, 2, 3, 1, 5
2, 5, 1, 4, 3
4, 2, 1, 5, 3
3, 1, 5, 2, 4
3, 2, 1, 5, 4
1, 3, 5, 4, 2
3, 1, 4, 5, 2
1, 5, 3, 2, 4
3, 2, 4, 1, 5
3, 4, 1, 2, 5
2, 3, 5, 1, 4
2, 3, 4, 5, 1
4, 1, 3, 2, 5
1, 4, 3, 5, 2
1, 4, 5, 2, 3
5, 1, 2, 3, 4
1, 5, 2, 4, 3
2, 5, 1, 3, 4
2, 4, 3, 1, 5
4, 1, 2, 5, 3
4, 2, 1, 3, 5
2, 1, 5, 4, 3
2, 4, 1, 5, 3
3, 2, 1, 4, 5
3, 1, 4, 2, 5
2, 3, 1, 5, 4
1, 3, 4, 5, 2
2, 3, 4, 1, 5
1, 5, 2, 3, 4
4, 1, 2, 3, 5
2, 1, 5, 3, 4
1, 4, 2, 5, 3
1, 2, 5, 4, 3
2, 4, 1, 3, 5
2, 1, 4, 5, 3
2, 3, 1, 4, 5
1, 3, 4, 2, 5
1, 4, 2, 3, 5
1, 2, 5, 3, 4
3, 1, 2, 5, 4
1, 3, 5, 2, 4
1, 3, 2, 5, 4
3, 1, 2, 4, 5
1, 3, 2, 4, 5
1, 4, 3, 2, 5
2, 1, 3, 5, 4
1, 2, 3, 5, 4
2, 1, 3, 4, 5
1, 2, 3, 4, 5
22/103
5, 4, 2, 1, 3
1, 2, 4, 3, 5
1, 2, 4, 5, 3
2, 1, 4, 3, 5
2, 4, 5, 1, 3
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
In fact...
Permutohedron is bounded
23/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
In fact...
Permutohedron is bounded
And in fact...
All finite Coxeter lattices are bounded
23/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Outline
24/103
1
(Lower) bounded lattices and the doubling operation
2
Finite Coxeter lattices
Coxeter lattices
The class HH of lattices
All lattices of HH are bounded
Finite Coxeter lattices are in HH
3
The lattice of finite closure systems
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
What is a Coxeter group ?
Definition
A group W is a Coxeter group if W has a set of generators
S ⊂ W , subject only to relations of the form
0
(ss0 )m(s,s ) = e
where m(s, s) = 1 for any s in S (all generators have order 2),
and m(s, s0 ) = m(s0 , s) ≥ 2 for s =
6 s0 in S.
25/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
List of all finite irreducible Coxeter groups
1
The four infinite families :
An (symmetric groups),
Bn ,
Dn ,
and In (dihedral groups).
26/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
List of all finite irreducible Coxeter groups
1
The four infinite families :
An (symmetric groups),
Bn ,
Dn ,
and In (dihedral groups).
2
26/103
and the six isolated groups : E6 , E7 , E8 , F4 , H3 and H4 .
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Coxeter graph of finite irreducible Coxeter
groups
An
E6
4
Bn (n ≥ 2)
Dn (n ≥ 4)
In (n ≥ 5)
E7
E8
n
H3
H4
27/103
4
F4
5
5
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
The lattice structure of Coxeter groups
Cayley graph of a group
ws
s
w
28/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
The lattice structure of Coxeter groups
Cayley graph of a group ordered by the (right) weak order
ws
s
w
If `(w) < `(ws).
29/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
The lattice structure of Coxeter groups
Cayley graph of a group ordered by the (right) weak order
ws
s
w
If `(w) < `(ws).
Theorem (Björner, 1984)
The weak order on any finite Coxeter group is a (autodual)
lattice.
29/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Finite Coxeter lattices are bounded
Sketch of the proof :
30/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Finite Coxeter lattices are bounded
Sketch of the proof :
1 Defining a new class of lattices : HH,
30/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Finite Coxeter lattices are bounded
Sketch of the proof :
1 Defining a new class of lattices : HH,
2
30/103
Showing that lattices of HH are bounded,
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Finite Coxeter lattices are bounded
Sketch of the proof :
1 Defining a new class of lattices : HH,
30/103
2
Showing that lattices of HH are bounded,
3
Showing that finite Coxeter lattices are in HH.
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Coxeter lattices
Finite Coxeter lattices are bounded
Sketch of the proof :
1 Defining a new class of lattices : HH,
30/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Hat, antihat and 2-facet
Definition
a Hat (y, x, z)∧ :
x
y
z
an antiHat (y, x, z)∨ :
y
z
x
31/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Hat, antihat and 2-facet
Definition
a Hat (y, x, z)∧ :
x
y
z
an antiHat (y, x, z)∨ :
y
z
x
31/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Hat, antihat and 2-facet
Definition
a Hat (y, x, z)∧ :
x
y
z
an antiHat (y, x, z)∨ :
y
z
x
a 2-facet F y,x,z :
x
y
z
y∧z
31/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Definition of a 2-facet labelling
t6
t1
t5
t4
t7
t3
t2
t1
t6
Fig.: Example of a 2-facet labelling
32/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Definition of a 2-facet labelling
t6
t1
t5
t4
t7
t3
t2
t1
t6
Fig.: Example of a 2-facet labelling
33/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Definition of a 2-facet labelling
t6
t1
t5
t4
t7
t3
t2
t1
t6
Fig.: Example of a 2-facet labelling
34/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Definition of a 2-facet labelling
t6
t7
t1
t1
t5
t4
t2
t5
t3
t7
t3
t6
t1
t2
t4
t2
t1
t6
t4
t1
Fig.: Another example of a 2-facet labelling
35/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
Definition of a 2-facet labelling
t6
t7
t1
t1
t5
t4
t2
t5
t3
t7
t3
t6
t1
t2
t4
t2
t1
t6
t4
t1
Fig.: Another example of a 2-facet labelling
36/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
Definition
t6
t1
t5
t4
t7
t3
t2
t1
37/103
t6
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
Definition
r(t6 )
r(t1 )
r(t5 )
r(t4 )
r(t7 )
r(t3 )
r(t2 )
r(t1 )
r(t6 )
This is a function r from T = {t1 , ..., ti , ...tp } to R
38/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
Definition
r(t6 )
r(t1 )
r(t5 )
r(t4 )
r(t7 )
r(t3 )
r(t2 )
r(t1 )
r(t6 )
This is a function r from T = {t1 , ..., ti , ...tp } to R such that :
39/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
Definition
r(t6 )
r(t1 )
r(t5 )
r(t4 )
r(t7 )
r(t3 )
r(t2 )
r(t1 )
r(t6 )
This is a function r from T = {t1 , ..., ti , ...tp } to R such that :
So : r(t1 ) < r(t2 ) < r(t3 )
and r(t6 ) < r(t5 ) < r(t4 )
and r(t1 ), r(t6 ) < r(t7 )
39/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
t7
t1
t2
t5
t3
t6
t1
t4
t2
t4
40/103
t1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
t7
t1
t2
t5
t3
t6
t1
t4
t2
t4
Here r(t1 ) < r(t5 ), r(t3 )
41/103
t1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
2-facet rank function on a 2-facet labelling
t7
t1
t2
t5
t3
t6
t1
t4
t2
t4
t1
Here r(t1 ) < r(t5 ), r(t3 ) and r(t2 ) < r(t6 ), r(t3 )
42/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
On semidistributivity
Definition
A lattice is semidistributive if, for all x, y, z ∈ L :
x ∧ y = x ∧ z implies x ∧ y = x ∧ (y ∨ z)
x ∨ y = x ∨ z implies x ∨ y = x ∨ (y ∧ z)
43/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
On semidistributivity
Definition
A lattice is semidistributive if, for all x, y, z ∈ L :
x ∧ y = x ∧ z implies x ∧ y = x ∧ (y ∨ z)
x ∨ y = x ∨ z implies x ∨ y = x ∨ (y ∧ z)
Proposition (Day, Nation, Tschantz [8], 1989)
Bounded lattices are semidistributive.
43/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
The class HH of lattices
Definition
A finite lattice L is in the class HH if it satisfies :
44/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The class HH of lattices
The class HH of lattices
Definition
A finite lattice L is in the class HH if it satisfies :
1
2
44/103
L is semidistributive,
to every hat (y, x, z)∧ of L is associated an anti-hat
(y 0 , y ∧ z, z 0 )∨ of L such that [y ∧ z, x] is a 2-facet,
3
to every anti-hat (y, x, z)∨ of L is associated a hat
(y 0 , y ∨ z, z 0 )∧ of L such that [x, y ∨ z] is a 2-facet,
4
there exists a 2-facet labelling T on the (covering) edges of
L and a 2-facet rank function r on T .
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
First part of the theorem
All lattices of HH are bounded
How do we prove this ?
45/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Recalling arrow relations...
m+
j ↓ m : j ∧ m = j−
m+
m
m
j
m
j−
j
j ↑ m : j ∨ m = m+
46/103
j
jlm
j−
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Characterising semidistributivity with arrow
relations
Proposition (Day [7], 1979)
A lattice L is semidistributive if and only if the relation l on
J × M induces a bijection between J and M .
47/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Characterising semidistributivity with arrow
relations
Proposition (Day [7], 1979)
A lattice L is semidistributive if and only if the relation l on
J × M induces a bijection between J and M .
Notation
In any semidistributive lattice L, we can denote by (j, mj ) – or
by (jm , m) – the elements of JL × ML which are bijective for
the relation l.
47/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Relations on the edges of the lattices of HH
j
t7
i
t1
t2
h
g
t5
f
t3
e
t6
t1
d
t4
t2
c
b
t1
t4
a
48/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Relations on the edges of the lattices of HH
j
t7
i
t1
t2
h
g
t5
f
t3
e
t6
t1
d
t4
t2
c
b
t1
t4
a
We write : bd ≺t2 gi
48/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Relations on the edges of the lattices of HH
j
t7
i
t1
t2
h
g
t5
f
t3
e
t6
t1
d
t4
t2
c
b
t1
t4
a
We write : bd ≺t2 gi
and ab ≺t4 ce
48/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Relations on the edges of the lattices of HH
j
t7
i
t1
t2
h
g
t5
f
t3
e
t6
t1
d
t4
t2
c
b
t1
t4
a
We write : bd ≺t2 gi
and ab ≺t4 ce
and ac ≺t1 be ≺t1 hi
48/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Relations on the edges of the lattices of HH
j
t7
i
t1
t2
h
g
t5
f
t3
e
t6
t1
d
t4
t2
c
b
t1
t4
a
We write : bd ≺t2 gi
and ab ≺t4 ce
and ac ≺t1 be ≺t1 hi
and so : ac ≤t1 hi.
48/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Using the ≤t relations
Theorem
Let m be meet-irreducible in L ∈ HH and let (m, m+ ) be
labelled by t.
The set Em = {(x, y) : (x, y) ≤t (m, m+ )} is not empty and has
a least element (u, v).
Moreover v is a join-irreducible, v − = u and v l m.
49/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
m+
m
50/103
t
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
m+
m
t
t
t
t
t
51/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
m+
m
t
t
t
t
v = jm
t
−
u = v − = jm
52/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Lemma
Let L ∈ HH and T a 2-facet labelling of L. There exists a label
t ∈ T such that for any hat (y, x, z)∧ whose arc (y, x) or (z, x)
is labelled by t, F (y,x,z) is a diamond.
53/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Lemma
Let L ∈ HH and T a 2-facet labelling of L. There exists a label
t ∈ T such that for any hat (y, x, z)∧ whose arc (y, x) or (z, x)
is labelled by t, F (y,x,z) is a diamond.
More precisely :
If
x
t
y
and if r(t) is maximum in r(T )
53/103
z
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Lemma
Let L ∈ HH and T a 2-facet labelling of L. There exists a label
t ∈ T such that for any hat (y, x, z)∧ whose arc (y, x) or (z, x)
is labelled by t, F (y,x,z) is a diamond.
More precisely :
If
x
t
y
z
and if r(t) is maximum in r(T ) then :
x
t
y
z
t
y∧z
53/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
”Disconstructing” an interval to construct a
second lemma
Definition
Let L be a lattice and I ⊆ L an interval of L. We say that I is
contractible (in L) if L can be obtained from a lattice L0 by the
doubling of an interval I0 ⊆ L0 (with I = I0 × 2).
54/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
”Disconstructing” an interval to construct a
second lemma
Definition
Let L be a lattice and I ⊆ L an interval of L. We say that I is
contractible (in L) if L can be obtained from a lattice L0 by the
doubling of an interval I0 ⊆ L0 (with I = I0 × 2).
Lemma
Let L ∈ HH, j ∈ JL and t the label of the arcs (j − , j) and
(mj , m+
j ).
Assume all 2-facets contained in [j − , m+
j ] and which have one
edge labelled by t are isomorphic with diamonds.
54/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
”Disconstructing” an interval to construct a
second lemma
Definition
Let L be a lattice and I ⊆ L an interval of L. We say that I is
contractible (in L) if L can be obtained from a lattice L0 by the
doubling of an interval I0 ⊆ L0 (with I = I0 × 2).
Lemma
Let L ∈ HH, j ∈ JL and t the label of the arcs (j − , j) and
(mj , m+
j ).
Assume all 2-facets contained in [j − , m+
j ] and which have one
edge labelled by t are isomorphic with diamonds.
Then the interval Ij,mj = [j − , m+
j ] is contractible.
54/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
55/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
m+
t
m
56/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
m+
t
m
t
t
t
t
j
t
j−
57/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
m+
t
m
t
t
t
t
j
t
j−
58/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
59/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Illustration of the lemma
60/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
At last...
Theorem
The class HH of lattices is closed for the contraction of a
contractible interval w.r.t. a label whose 2-facet rank function is
maximal.
Hence the result : lattices of HH are bounded !
61/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
At last...
Theorem
The class HH of lattices is closed for the contraction of a
contractible interval w.r.t. a label whose 2-facet rank function is
maximal.
Hence the result : lattices of HH are bounded !
61/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Not all bounded lattices are in HH
62/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
All lattices of HH are bounded
Not all bounded lattices are in HH
A bounded lattice that does not belong to HH
WHY ? ? ?
62/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Second part of the theorem
Finite Coxeter lattices are in HH
How do we prove this ?
63/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
A strong result
Proposition (L.C.d.P.-B., 1994)
Finite Coxeter lattices are semidistributive.
64/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
A strong result
Proposition (L.C.d.P.-B., 1994)
Finite Coxeter lattices are semidistributive.
Proposition (Duquenne and Cherfouh, 1994)
Permutohedron is semidistributive.
64/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Reflections as elements and edge labels
Definition
TW = {t ∈ W : ∃s ∈ S, ∃w ∈ W such that t = wsw−1 }
is the set of the reflections of the Coxeter group W .
65/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Reflections as elements and edge labels
Definition
TW = {t ∈ W : ∃s ∈ S, ∃w ∈ W such that t = wsw−1 }
is the set of the reflections of the Coxeter group W .
Two labellings of the edges : the g-labelling
ws
s
w
65/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Reflections as elements and edge labels
Definition
TW = {t ∈ W : ∃s ∈ S, ∃w ∈ W such that t = wsw−1 }
is the set of the reflections of the Coxeter group W .
Two labellings of the edges : the g-labelling and the r-labelling
ws
s
t = wsw−1
w
66/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Properties of the reflections
Proposition (L.C.d.P.-B.)
Two ”opposite” edges of a 2-facet of a Coxeter lattice are
labelled by the same reflection.
67/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Properties of the reflections
Proposition (L.C.d.P.-B.)
Two ”opposite” edges of a 2-facet of a Coxeter lattice are
labelled by the same reflection.
t1
t2
t3
t3
t2
t4
67/103
t4
t1
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Properties of the reflections
Proposition (L.C.d.P.-B.)
Two ”opposite” edges of a 2-facet of a Coxeter lattice are
labelled by the same reflection.
t1
t4
t2
t3
t3
t2
t4
t1
Corollary
The r-labelling on the edges of any finite Coxeter lattice is a
2-facet labelling.
67/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Properties of the length function
Theorem (L.C.d.P.-B.)
The length function ` on every Coxeter lattice LW is a 2-facet
rank function when defined on the r-labelling of the edges of LW .
68/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Properties of the length function
Theorem (L.C.d.P.-B.)
The length function ` on every Coxeter lattice LW is a 2-facet
rank function when defined on the r-labelling of the edges of LW .
So :
Theorem
Every Coxeter lattice is in the class HH and therefore is
bounded.
68/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Two additional results
Theorem
Let LW be a Coxeter lattice and WH a parabolic subgroup of W .
There exists a series of interval contractions that leads from LW
to the lattice LWH of its parabolic subgroup WH .
69/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finite Coxeter lattices are in HH
Two additional results
Theorem
Let LW be a Coxeter lattice and WH a parabolic subgroup of W .
There exists a series of interval contractions that leads from LW
to the lattice LWH of its parabolic subgroup WH .
Proposition
There exists a particular interval doubling series from a given
Coxeter lattice generated by n generators to the Coxeter lattice
of the same family, generated by n + 1 generators.
69/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Outline
70/103
1
(Lower) bounded lattices and the doubling operation
2
Finite Coxeter lattices
Coxeter lattices
The class HH of lattices
All lattices of HH are bounded
Finite Coxeter lattices are in HH
3
The lattice of finite closure systems
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Definition
A closure system C on S : a subset of 2S which contains S and
is closed under set intersection.
Example (S = {1, 2, 3, 4})
1234
234
134
34
14
1
4
∅
71/103
3
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The lattice (Mn , ⊆) of closure systems on a
finite set S
Example (n=2)
2S
{∅, 1, S}
{∅, 2, S}
{∅, S}
{1, S}
{2, S}
{S}
72/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Structures cryptomorphic with :
closure operators,
finite lattices,
full implicational systems (or full systems of dependencies).
73/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Structures cryptomorphic with :
closure operators,
finite lattices,
full implicational systems (or full systems of dependencies).
Theorem
The lattice (Mn , ⊆) of closure systems is lower bounded.
How do we prove this ?
73/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Two dependence relations on the
join-irreducibles of Mn
The dependence relation δ (Monjardet [11], 1990),
The strong dependence relation δd (Day [7], 1979).
Definition
74/103
0
1
jδj 0 if j = j or if ∃x ∈ L with j < j 0 ∨ x, j 6≤ x and j 0 6≤ x.
2
jδd j 0 if j = j or if ∃x ∈ L with j < j 0 ∨ x and j 6≤ j
0
0−
∨ x.
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Two dependence relations on the
join-irreducibles of Mn
The dependence relation δ (Monjardet [11], 1990),
The strong dependence relation δd (Day [7], 1979).
Definition
0
1
jδj 0 if j = j or if ∃x ∈ L with j < j 0 ∨ x, j 6≤ x and j 0 6≤ x.
2
jδd j 0 if j = j or if ∃x ∈ L with j < j 0 ∨ x and j 6≤ j
0
In particular, we have δd ⊆ δ.
74/103
0−
∨ x.
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterising δ and δd with the arrow
relations
Proposition
75/103
1
jδj 0 ⇐⇒ ∃m ∈ M : j ↑ m and j 0 6≤ m.
2
jδd j ⇐⇒ ∃m ∈ M : j ↑ m and j ↓ m.
0
0
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Some results
Proposition
In any lattice L, the following are equivalent :
76/103
1
L is atomistic,
2
∀ j ∈ J, ∀ m ∈ M, j 6≤ m implies j ↓ m,
3
δd = δ.
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Some results
Proposition
In any lattice L, the following are equivalent :
77/103
1
L is atomistic,
2
∀ j ∈ J, ∀ m ∈ M, j 6≤ m implies j ↓ m,
3
δd = δ.
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Some results
Proposition
In any lattice L, the following are equivalent :
1
L is atomistic,
2
∀ j ∈ J, ∀ m ∈ M, j 6≤ m implies j ↓ m,
3
δd = δ.
Moreover :
Proposition (Day [7], 1979)
A lattice L is lower bounded if and only if δd has no circuit.
77/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
Each CA is a closure system,
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
Each CA is a closure system,
they are exactly the atoms of Mn ,
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
Each CA is a closure system,
they are exactly the atoms of Mn ,
and any other closure system (except from {S}) is a join of
some CA .
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
Each CA is a closure system,
they are exactly the atoms of Mn ,
and any other closure system (except from {S}) is a join of
some CA .
Thus :
Proposition
The lattice Mn is atomistic
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The join-irreducibles of Mn
For A ⊂ S, we set CA = {A, S}.
Each CA is a closure system,
they are exactly the atoms of Mn ,
and any other closure system (except from {S}) is a join of
some CA .
Thus :
Proposition
The lattice Mn is atomistic and δd = δ.
78/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
What about the meet-irreducibles of Mn ?
Proposition
Let C be a closure system of Mn . The following holds :
C ∈ MMn ⇐⇒ C = CA,i = {X ⊆ S : A 6⊆ X or i ∈ X}.
79/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finally...
Proposition
Let CA and CB be two join-irreducible elements of Mn .
CA δCB ⇐⇒ A ⊆ B ⊂ S
80/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finally...
Proposition
Let CA and CB be two join-irreducible elements of Mn .
CA δCB ⇐⇒ A ⊆ B ⊂ S
So δ is an order relation.
80/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finally...
Proposition
Let CA and CB be two join-irreducible elements of Mn .
CA δCB ⇐⇒ A ⊆ B ⊂ S
So δ is an order relation.
And since δ = δd ...
80/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finally...
Proposition
Let CA and CB be two join-irreducible elements of Mn .
CA δCB ⇐⇒ A ⊆ B ⊂ S
So δ is an order relation.
And since δ = δd ...
Theorem
The lattice Mn of closure systems is lower bounded.
80/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Finally...
Proposition
Let CA and CB be two join-irreducible elements of Mn .
CA δCB ⇐⇒ A ⊆ B ⊂ S
So δ is an order relation.
And since δ = δd ...
Theorem
The lattice Mn of closure systems is lower bounded.
It is not bounded since it is not semidistributive.
80/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
My colleagues (and friends !)
Fig.: C. le Conte de Poly-Barbut, CAMS, EHESS, Paris.
81/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
My colleagues (and friends !)
Fig.: B. Leclerc and B. Monjardet, CAMS, EHESS, Paris.
82/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
The final word.
T
H
4321
N
A
K
Y
4231
4312
3421
3412
4213
4123
4132
3214
2341
2143
1423
2314
2134
1324
1234
O
2431
2413
3142
3124
83/103
3241
1243
U
1432
1342
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
No questions.. ?
84/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
N. Caspard, The lattice of permutations is bounded, International Journal of Algebra and
Computation 10(4), 481–489 (2000).
N. Caspard, A characterization for all interval doubling schemes of the lattice of permutations,
Discr. Maths. and Theoretical Comp. Sci. 3(4), 177–188 (1999).
N. Caspard, C. Le Conte de Poly-Barbut et M. Morvan, Cayley lattices of finite Coxeter
groups are bounded, Advances in Applied Mathematics, 33(1), 71–94 (2004).
N. Caspard and B. Monjardet, The lattice of Moore families and closure operators on a finite
set : a survey, Electronic Notes in Discrete Mathematics, 2 (1999).
N. Caspard et B. Monjardet, The lattice of closure systems, closure operators and implicational
systems on a finite set : a survey, Discrete Applied Mathematics, 127(2), 241–269 (2003).
A. Day, A simple solution to the word problem for lattices, Canad. Math. Bull. 13, 253–254
(1970).
85/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
A. Day, characterisations of finite lattices that are bounded-homomorphic images or sublattices
of free lattices, Canadian J. Math. 31, 69–78 (1979).
A. Day, J.B. Nation and S. Tschantz, Doubling Convex Sets in Lattices and a Generalized
Semidistributivity Condition, Order 6, 175–180 (1989).
W. Geyer, The generalized doubling construction and formal concept analysis, Algebra
Universalis 32, 341–367 (1994).
R. McKenzie, Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc 174,
1–43 (1972).
B. Monjardet, Arrowian characterizations of latticial federation consensus functions,
Mathematical Social Sciences 20(1), 51-71 (1990).
86/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalling arrow relations...
m+
j ↓ m : j ∧ m = j−
m+
m
m
j
m
j−
j
j ↑ m : j ∨ m = m+
87/103
j
jlm
j−
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
... And the A-context of a lattice
x
v
w
u
z
t
y
a
88/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
... And the A-context of a lattice
x
v
w
u
z
t
y
a
88/103
y
z
t
u
a
l
z
×
×
↓
l
t
×
↓
×
l
v
×
×
l
×
w
×
l
×
×
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
... And the A-context of a lattice
x
v
w
u
z
t
y
z
t
u
a
l
z
×
×
↓
l
t
×
↓
×
l
v
×
×
l
×
w
×
l
×
×
y
a
Any lattice has (|J|! × |M |!) tableaux to describe its A-context.
88/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
On semidistributivity
Definition
A lattice is meet-semidistributive if, for all x, y, z ∈ L,
x ∧ y = x ∧ z implies x ∧ y = x ∧ (y ∨ z).
Join-semidistributive lattices are defined dually and a lattice is
semidistributive if it is meet- and join-semidistributive.
89/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
On semidistributivity
Definition
A lattice is meet-semidistributive if, for all x, y, z ∈ L,
x ∧ y = x ∧ z implies x ∧ y = x ∧ (y ∨ z).
Join-semidistributive lattices are defined dually and a lattice is
semidistributive if it is meet- and join-semidistributive.
Proposition (Day, Nation, Tschantz [8], 1989)
Bounded lattices are semidistributive.
89/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Results
Proposition (Duquenne and Cherfouh, L.C.d.P.-B., 1994)
Permutohedron is semidistributive.
90/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Results
Proposition (Duquenne and Cherfouh, L.C.d.P.-B., 1994)
Permutohedron is semidistributive.
Proposition (Day [7], 1979)
A lattice L is semidistributive if and only if the relation l on
J × M induces a bijection between J and M .
90/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Results
Proposition (Duquenne and Cherfouh, L.C.d.P.-B., 1994)
Permutohedron is semidistributive.
Proposition (Day [7], 1979)
A lattice L is semidistributive if and only if the relation l on
J × M induces a bijection between J and M .
Hence :
Given any total order on JP erm(n) , there exists a unique total
order on MP erm(n) – say L∗M – such that T = (AL , LJ , L∗M ) has
all l on the principal diagonal.
90/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
A simple idea from a strong result
Definition
Let L be a semidistributive lattice. A tableau T = (AL , LJ , LM )
of the A-context of L is a B-tableau if the following hold :
1
the |J| arrows l of T are on the principal diagonal of T ,
2
All arrows ↑. are below this diagonal and all arrows ↓. are
above.
Proposition (Geyer [9], 1994)
A lattice is bounded if and only if its A-context admits a
B-tableau.
91/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
A B-tableau of Perm(4)
J \M
1243
1324
1342
1423
2134
2314
2341
2413
3124
3412
4123
92/103
3421
l
×
×
↑
×
×
×
↑
×
×
↑
4231
×
l
↑
×
×
×
×
×
↑
↑
×
3241
↓
×
l
×
×
×
2431
×
↓
l
×
×
×
×
×
↑
↑
4312
×
×
×
×
l
↑
↑
↑
×
×
×
4213
×
↓
×
×
l
↑
×
3214
↓
×
↓
×
×
l
↓
×
↓
4132
×
×
×
×
↓
3142
↓
×
×
l
↑
×
×
l
↓
1432
×
×
×
×
↓
l
×
×
2413
×
↓
↓
l
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
A B-tableau of Perm(4)
J \M
1243
1324
1342
1423
2134
2314
2341
2413
3124
3412
4123
3421
l
×
×
↑
×
×
×
↑
×
×
↑
4231
×
l
↑
×
×
×
×
×
↑
↑
×
3241
↓
×
l
×
×
×
2431
×
↓
l
×
×
×
×
×
↑
↑
4312
×
×
×
×
l
↑
↑
↑
×
×
×
4213
×
↓
×
×
l
↑
×
3214
↓
×
↓
×
×
l
↓
×
↓
4132
×
×
×
×
↓
3142
↓
×
×
l
↑
×
×
l
↓
Here : LJ is equal to Lex(J).
In fact :
Theorem
The tableau T = (AP erm(n) , LexJ , L∗M ) of the A-context of the
lattice Perm(n) is a B-tableau.
92/103
1432
×
×
×
×
↓
l
×
×
2413
×
↓
↓
l
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
Example
α = 3241 ∈ Perm(4).
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
Example
α = 3241 ∈ Perm(4).
A(3241) = {24, 34}
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
Example
α = 3241 ∈ Perm(4).
A(3241) = {24, 34} and D(3241) = {32, 31, 21, 41}.
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
Example
α = 3241 ∈ Perm(4).
A(3241) = {24, 34} and D(3241) = {32, 31, 21, 41}.
The weak order defined on Perm(n) is characterised by :
α ≤ β ⇐⇒ A(β) ⊆ A(α)
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Recalls
Definition
A(α) : the set of agreements of α,
D(α) : the set of disagreements of α.
Example
α = 3241 ∈ Perm(4).
A(3241) = {24, 34} and D(3241) = {32, 31, 21, 41}.
The weak order defined on Perm(n) is characterised by :
α ≤ β ⇐⇒ A(β) ⊆ A(α) ⇐⇒ D(α) ⊆ D(β)
93/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of JP erm(n)
Result
α ∈ JP erm(n) if and only if there exists a unique ordered pair vu
of adjacent elements in α such that u < v.
94/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of JP erm(n)
Result
α ∈ JP erm(n) if and only if there exists a unique ordered pair vu
of adjacent elements in α such that u < v.
Example
4123, 1324 ∈ JP erm(4)
94/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of JP erm(n)
Result
α ∈ JP erm(n) if and only if there exists a unique ordered pair vu
of adjacent elements in α such that u < v.
Example
4123, 1324 ∈ JP erm(4) but 1432, 4213 6∈ JP erm(4) .
94/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of JP erm(n)
Result
α ∈ JP erm(n) if and only if there exists a unique ordered pair vu
of adjacent elements in α such that u < v.
Example
4123, 1324 ∈ JP erm(4) but 1432, 4213 6∈ JP erm(4) .
So, in other words :
α ∈ JP erm(n) ⇐⇒ α = A|A = Bv|uB
with u < v and A = Bv and A = uB the two maximal linear
suborders of α compatible with 0P erm(n) = 1...i...n.
94/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of MP erm(n)
Result
α ∈ MP erm(n) if and only if there exists a unique ordered pair lp
of adjacent elements in α such that l < p.
Example
4213, 1432 ∈ MP erm(4) but 1342, 4231 6∈ MP erm(4) .
95/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Expression of the elements of MP erm(n)
Result
α ∈ MP erm(n) if and only if there exists a unique ordered pair lp
of adjacent elements in α such that l < p.
Example
4213, 1432 ∈ MP erm(4) but 1342, 4231 6∈ MP erm(4) .
So, in other words :
α ∈ MP erm(n) ⇐⇒ α = C|C = Dl|pD
with l < p and C = Dl and C = pD the two maximal linear
suborders of α compatible with 1P erm(n) = n...i...1.
95/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterising the A-context of Perm(n)
Lemma
Let γ = Bv|uB ∈ JP erm(n) and µ = Cl|pC ∈ MP erm(n) .
96/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterising the A-context of Perm(n)
Lemma
Let γ = Bv|uB ∈ JP erm(n) and µ = Cl|pC ∈ MP erm(n) .
1
γ ≤ µ ⇐⇒ D(γ) ⊆ D(µ) ⇐⇒ A(µ) ⊆ A(γ).
2
γ ↑ µ ⇐⇒ pl ∈ D(γ) and D(γ) ⊆ D(µ+ ).
3
γ ↓ µ ⇐⇒ uv ∈ A(µ) and A(µ) ⊆ A(γ − ).
4
96/103
γ l µ ⇐⇒ pl ∈ D(γ), uv ∈ A(µ), D(γ) ⊆ D(µ+ ) and
A(µ) ⊆ A(γ − ).
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Characterising the bijection between J and
M induced by l
Proposition
1. Let γ = Bu|vB be a join-irreducible and µ a meet-irreducible of Perm(n).

γ l µ ⇐⇒ µ = Cu|vC
with
`
´
C = `{x ∈ B : u < x} ∪ {x ∈ B : v < x}, > ´
C = {x ∈ B : x < u} ∪ {x ∈ B : x < v}, >
2. Let µ = Cl|pC be a meet-irreducible and γ a join-irreducible of Perm(n).

`
´
B = `{x ∈ C : x < p} ∪ {x ∈ C : x < l}, < ´
γ l µ ⇐⇒ γ = Bp|lB with
B = {x ∈ C : p < x} ∪ {x ∈ C : l < x}, <
97/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
An additional result
Theorem
Let LJ be a linear order on JP erm(n) and L∗M the ”associated”
linear order on MP erm(n) . The following are equivalent :
98/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
An additional result
Theorem
Let LJ be a linear order on JP erm(n) and L∗M the ”associated”
linear order on MP erm(n) . The following are equivalent :
1
2
98/103
T = (AP erm(n) , LJ , L∗M ) is a B-tableau of Perm(n),
LJ is a linear extension of (J, ≤P erm(n) ) and L∗M a linear
extension of (M, ≥P erm(n) ).
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Not all tableaux of Perm(n) are B-tableaux
99/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Not all tableaux of Perm(n) are B-tableaux
Proof :
2341
2314
3124
2134
99/103
3412
2413
1324
4123
1342
1243
1423
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Not all tableaux of Perm(n) are B-tableaux
Proof :
2314 4
2341
3412
4123
9
10
11
3124
5
2413
6
7
1342
1
2
3
2134
1324
1243
8
1423
Fig.: A linear extension LJ of (J, ≤P erm(4) ) for which L∗M on M is
not a linear extension of (M, ≥P erm(4) ).
100/103
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Not all tableaux of Perm(n) are B-tableaux
Proof :
2314 4
4132 5
101/103
2341
3412
4123
9
10
11
3124
5
2413
6
7
1342
1
2
3
2134
1324
1243
4312
4231
3421
1
2
3
4213
4
3142
10
8
2431
11
6
9
1432
2143
3214
8
1423
7
3241
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Definition
A closure system C on S : a subset of 2S which contains S and
is closed under set intersection.
Example (S = {1, 2, 3, 4})
1234
234
134
34
14
1
4
∅
102/103
3
(Lower) bounded lattices and the doubling operation Finite Coxeter lattices The lattice of finite closure syste
Proposition
The set of all the lattices that can be obtained from L ∈ HH by
a series of interval contractions is a distributive lattice when
ordered by the following natural order relation : L < L0 if L can
be obtained from L0 by a series of interval contractions.
103/103