the core label order of a congruence-uniform latticehmuehle/files/talks/aaa_corelabel.pdf · the...

The CoreLabel Order ofa Congruence-

UniformLattice

Henri Muhle

The CoreLabel Order

LatticeProperty ofCLO(L)

Characterizationof CertainCongruence-UniformLattices

The Core Label Order of aCongruence-Uniform Lattice

Henri Muhle

TU Dresden

March 01, 2019AAA’97, TU Wien

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UniformLattice

Henri Muhle

The CoreLabel Order



Outline

1 The Core Label Order

2 Lattice Property of CLO(L)

3 Characterization of Certain Congruence-UniformLattices

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UniformLattice

Henri Muhle

The CoreLabel Order



Outline




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UniformLattice

Henri Muhle

The CoreLabel Order



The Doubling Construction

P = (P,≤) finite poset; X ⊆ P

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Henri Muhle

The CoreLabel Order




P = (P,≤) finite poset; X ⊆ P

P≤Xdef= {p ∈ P | p ≤ x for some x ∈ X}

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UniformLattice

Henri Muhle

The CoreLabel Order




P = (P,≤) finite poset; X ⊆ P; 2 =({0, 1},≤

)P≤X

def= {p ∈ P | p ≤ x for some x ∈ X}

doubling of P by X: subposet of P × 2 induced by

P[X]def=(

P≤X × {0})]((

(P \ P≤X) ∪X)× {1}

)

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UniformLattice

Henri Muhle

The CoreLabel Order





)P≤X



P[X]def=(

P≤X × {0})]((

(P \ P≤X) ∪X)× {1}

)

P≤X

P \ P≤X

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UniformLattice

Henri Muhle

The CoreLabel Order





)P≤X



P[X]def=(

P≤X × {0})]((

(P \ P≤X) ∪X)× {1}

)

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UniformLattice

Henri Muhle

The CoreLabel Order





)order convex: for all x, y, z ∈ P with x ≤ y ≤ z, thenx, z ∈ X implies y ∈ X

Theorem (A. Day; 1970)

If L = (L,≤) is a finite lattice and X ⊆ L is order convex, thenL[X] is a lattice, too.

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UniformLattice

Henri Muhle

The CoreLabel Order



Congruence-Uniform Lattices

L = (L,≤) finite lattice

congruence uniform: can be obtained from thesingleton by a sequence of doublings by intervalslabel edges according to creation step λ

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UniformLattice

Henri Muhle

The CoreLabel Order






1

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UniformLattice

Henri Muhle

The CoreLabel Order






11 2

2 1

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UniformLattice

Henri Muhle

The CoreLabel Order






1 2

31

2

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UniformLattice

Henri Muhle

The CoreLabel Order






1 2

34 1

4

24

3

2

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UniformLattice

Henri Muhle

The CoreLabel Order






1 2

34

5

1

4

45

35

5

3

4

4 23

2

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UniformLattice

Henri Muhle

The CoreLabel Order




L = (L,≤) finite lattice; x ∈ L

nucleus: x↓def=

∧y∈L:ylx

y

1 2

34

5

1

4

45

35

5

3

4

4 23

2

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UniformLattice

Henri Muhle

The CoreLabel Order





nucleus: x↓def=

∧y∈L:ylx

y

1 2

34

5

1

4

45

35

5

3

4

4 23

2

x→

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UniformLattice

Henri Muhle

The CoreLabel Order





nucleus: x↓def=

∧y∈L:ylx

y

1 2

34

5

1

4

45

35

5

3

4

4 23

2

x→

x↓ →

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UniformLattice

Henri Muhle

The CoreLabel Order





nucleus: x↓def=

∧y∈L:ylx

y

core: interval [x↓, x] in L

1 2

34

5

1

4

45

35

5

3

4

4 23

2

x→

x↓ →

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UniformLattice

Henri Muhle

The CoreLabel Order





nucleus: x↓def=

∧y∈L:ylx

y


core labels: Ψ(x) def={

λ(u, v) | x↓ ≤ u l v ≤ x}

1 2

34

5

1

4

45

35

5

3

4

4 23

2

x→

x↓ →

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UniformLattice

Henri Muhle

The CoreLabel Order





nucleus: x↓def=

∧y∈L:ylx

y


core labels: Ψ(x) def={

λ(u, v) | x↓ ≤ u l v ≤ x}

1 2

34

5

1

4

45

35

5

3

4

4 23

2

x→

x↓ →

Ψ(x) = {3, 4, 5}

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UniformLattice

Henri Muhle

The CoreLabel Order





core label order:CLO(L) def

=({

Ψ(x) | x ∈ L}

,⊆)

1 2

34

5

1

4

45

35

5

3

4

4 23

2

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UniformLattice

Henri Muhle

The CoreLabel Order






=({

Ψ(x) | x ∈ L}

,⊆)

1 2

34

5

1

4

45

35

5

3

4

4 23

2

∅

{1} {2} {4} {3} {5}

{2, 4} {3, 4} {4, 5} {3, 5}

{3, 4, 5}{1, 2, 3, 5}

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UniformLattice

Henri Muhle

The CoreLabel Order






=({

Ψ(x) | x ∈ L}

,⊆)

CLO(L) is not necessarily a meet-semilattice

1 2 3

2 3

4

1 2

1 3

3 2 1

∅

{1} {2} {4} {3}

{1, 2, 4} {1, 3} {2, 3, 4}

{1, 2, 3, 4}

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UniformLattice

Henri Muhle

The CoreLabel Order






=({

Ψ(x) | x ∈ L}

,⊆)

CLO(L) is not necessarily a meet-semilattice

Question (N. Reading; 2016)

For which congruence-uniform lattices L is CLO(L) a lattice,too?

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UniformLattice

Henri Muhle

The CoreLabel Order



Outline




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UniformLattice

Henri Muhle

The CoreLabel Order



Spherical Lattices

L = (L,≤) finite latticeµL Mobius function Mobius Function

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Henri Muhle

The CoreLabel Order



Spherical Lattices

L = (L,≤) finite lattice; 0, 1 least/greatest elementµL Mobius function Mobius Function

Mobius invariant: µ(L) def= µL(0, 1)

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The CoreLabel Order



Spherical Lattices



spherical: µ(L) ∈ {−1, 1}

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Henri Muhle

The CoreLabel Order



Spherical Lattices




Theorem (T. McConville; 2015)If L is congruence uniform, then µ(L) ∈ {−1, 0, 1}.

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UniformLattice

Henri Muhle

The CoreLabel Order



Spherical Lattices




Theorem ( ; 2019)Let L be congruence uniform. If CLO(L) is a lattice, then L isspherical.

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Henri Muhle

The CoreLabel Order



The Intersection Property


intersection property: for all x, y ∈ L exists z ∈ L suchthat Ψ(x) ∩Ψ(y) = Ψ(z) Intersection Property

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UniformLattice

Henri Muhle

The CoreLabel Order






Theorem ( ; 2019)Let L be congruence uniform. Then, CLO(L) is ameet-semilattice if and only if L has the intersection property.

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Henri Muhle

The CoreLabel Order






Corollary ( ; 2019)

Let L be congruence uniform. Then, CLO(L) is a lattice if andonly if L is spherical and has the intersection property.

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UniformLattice

Henri Muhle

The CoreLabel Order






Question ( ; 2019)Which congruence-uniform lattices have the intersectionproperty?

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UniformLattice

Henri Muhle

The CoreLabel Order



Outline




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Henri Muhle

The CoreLabel Order



A Characterization of Boolean Lattices

M finite set; ℘(M) power set of M

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Henri Muhle

The CoreLabel Order





Boolean lattice: Bool(M)def=(℘(M),⊆

)

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Henri Muhle

The CoreLabel Order






)

TheoremEvery finite Boolean lattice is congruence uniform.

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UniformLattice

Henri Muhle

The CoreLabel Order






)

1 2 3

23

1 3

12

3 2 1

∅

{1} {2} {3}

{1, 2} {1, 3} {2, 3}

{1, 2, 3}

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UniformLattice

Henri Muhle

The CoreLabel Order






)

Theorem ( ; 2019)Let L be congruence uniform. Then, L ∼= Bool(M) for somefinite set M if and only if L ∼= CLO(L).

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UniformLattice

Henri Muhle

The CoreLabel Order



Canonical Join Representations


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UniformLattice

Henri Muhle

The CoreLabel Order




L = (L,≤) finite lattice; x ∈ Ljoin representation: X ⊆ L such that x =

∨X

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UniformLattice

Henri Muhle

The CoreLabel Order





∨X

X refines Y: L≤X ⊆ L≤Y

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UniformLattice

Henri Muhle

The CoreLabel Order





∨X


canonical join representation: minimum with respectto refinement Γ(x)

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UniformLattice

Henri Muhle

The CoreLabel Order




L = (L,≤) congruence-uniform lattice; x ∈ Ljoin representation: X ⊆ L such that x =

∨X



Theorem (A. Day; 1979)Every element of a congruence-uniform lattice admits a canonicaljoin representation.

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UniformLattice

Henri Muhle

The CoreLabel Order





∨X



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UniformLattice

Henri Muhle

The CoreLabel Order





∨X



2

4

42

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UniformLattice

Henri Muhle

The CoreLabel Order





∨X


canonical join representation:Γ(x) =

{λ(y, x) | y l x

}

2

4

42

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UniformLattice

Henri Muhle

The CoreLabel Order



The Canonical Join Complex


Proposition (N. Reading; 2015)For any finite lattice, the set of canonical join representations isclosed under taking subsets.

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UniformLattice

Henri Muhle

The CoreLabel Order





canonical join complex: simplicial complex whosefaces are canonical join representations Can(x)

Proposition (N. Reading; 2015)For any finite lattice, the set of canonical join representations isclosed under taking subsets.

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UniformLattice

Henri Muhle

The CoreLabel Order





canonical join complex: simplicial complex whosefaces are canonical join representations Can(x)

1 2

34

5

1

4

45

35

5

3

4

243

2

3 5

4

2

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UniformLattice

Henri Muhle

The CoreLabel Order





canonical join complex: simplicial complex whosefaces are canonical join representations Can(x)face poset: set of faces ordered by inclusion

1 2

34

5

1

4

45

35

5

3

4

243

2

3 5

4

2

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UniformLattice

Henri Muhle

The CoreLabel Order






1 2

34

5

1

4

45

35

5

3

4

243

2

∅

{1} {2} {4} {3} {5}

{1, 2} {2, 4} {3, 4} {4, 5} {3, 5}

{3, 4, 5}

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UniformLattice

Henri Muhle

The CoreLabel Order






1 2

34

5

1

4

45

35

5

3

4

243

2

∅

{1} {2} {4} {3} {5}

{1, 2, 3, 5}

{2, 4} {3, 4} {4, 5} {3, 5}

{3, 4, 5}

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UniformLattice

Henri Muhle

The CoreLabel Order



Distributive Lattices


distributive: for all x, y, z ∈ L holdsx∨ (y∧ z) = (x∨ y) ∧ (x∨ z) and the dual

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UniformLattice

Henri Muhle

The CoreLabel Order






Theorem (M. Erne, J. Heitzig, J. Reinhold; 2002)A finite lattice is distributive if and only if it can be obtained fromthe singleton lattice by a sequence of doublings by principal orderideals.

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UniformLattice

Henri Muhle

The CoreLabel Order






Theorem ( ; 2018)A congruence-uniform lattice L is distributive if and only ifCLO(L) is the face poset of Can(L).

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UniformLattice

Henri Muhle

The CoreLabel Order






Corollary ( ; 2018)Every finite distributive lattice has the intersection property.

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UniformLattice

Henri Muhle

The CoreLabel Order



Thank You.

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UniformLattice

Henri Muhle

Mobius Function

P = (P,≤) finite poset

Mobius function: the map µP : P× P→ Z given by

µP (x, y) =

1, if x = y,− ∑

x≤z<yµP (x, z), if x < y,

0, otherwise

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UniformLattice

Henri Muhle

Mobius Function

P = (P,≤) finite poset

Mobius function: the map µP : P× P→ Z given by

µP (x, y) =

1, if x = y,− ∑


0, otherwise

Theorem (G.-C. Rota; 1964)Let P = (P,≤) be a finite poset, and let f , g : P× P→ Z. Itholds f (y) = ∑x≤y g(x) if and only if g(y) = ∑x≤y g(x)µP (x, y).

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UniformLattice

Henri Muhle

Mobius Function

P = (P,≤) finite bounded poset; 0, 1 least/greatestelementMobius function: the map µP : P× P→ Z given by

µP (x, y) =

1, if x = y,− ∑


0, otherwise

Theorem (P. Hall; 1936)Let P = (P,≤) be a finite bounded poset. The reduced Eulercharacteristic of the order complex of

(P \ {0, 1},≤) equals

µP (0, 1) up to sign.

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UniformLattice

Henri Muhle



Theorem ( ; 2019)Let L be a finite lattice and let Θ be a lattice congruence of L. IfCLO(L) has the intersection property, then so does CLO(L/Θ).

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UniformLattice

Henri Muhle



join irreducible: an element j ∈ L such that wheneverj = x∨ y, then j ∈ {x, y}

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UniformLattice

Henri Muhle



join irreducible: an element j ∈ L such that wheneverj = x∨ y, then j ∈ {x, y}

Theorem ( ; 2019)Let L = (L,≤) be a finite lattice, and let x, y ∈ L such thatΨ(j) ⊆ Ψ(x) ∩Ψ(y) for some join-irreducible element j ∈ L. Ifj ∈ [x↓, x] ∩ [y↓, y]. Then L[j] is spherical if and only if L is, butCLO

(L[j]

)is not a meet-semilattice.

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UniformLattice

Henri Muhle

Semidistributive Lattices


join semidistributive: for all x, y, z ∈ L holdsx∨ y = x∨ z implies x∨ z = x∨ (y∧ z)

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UniformLattice

Henri Muhle



join semidistributive: for all x, y, z ∈ L holdsx∨ y = x∨ z implies x∨ z = x∨ (y∧ z)meet semidistributive: dual is join semidistributive

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UniformLattice

Henri Muhle



join semidistributive: for all x, y, z ∈ L holdsx∨ y = x∨ z implies x∨ z = x∨ (y∧ z)meet semidistributive: dual is join semidistributivesemidistributive: join and meet semidistributive

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UniformLattice

Henri Muhle




TheoremA finite lattice is join-semidistributive if and only if every elementadmits a canonical join representation.

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UniformLattice

Henri Muhle




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UniformLattice

Henri Muhle




Theorem (A. Day; 1979)Every congruence-uniform lattice is semidistributive.

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the core label order of a congruence-uniform latticehmuehle/files/talks/aaa_corelabel.pdf · the...

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