Algebras of incidence structures: representing regular double - - PowerPoint PPT Presentation

▶
algebras of incidence structures representing regular
SMART_READER_LITE
LIVE PREVIEW

Algebras of incidence structures: representing regular double - - PowerPoint PPT Presentation

Dec 12, 2023 270 likes •971 views

Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University Victorian Algebra Conference 2015 Chris Taylor Algebras of incidence structures VAC 2015 1 / 24 Chris Taylor Algebras of

slide-1
SLIDE 1

Algebras of incidence structures: representing regular double p-algebras

Christopher Taylor

La Trobe University

Victorian Algebra Conference 2015

Chris Taylor Algebras of incidence structures VAC 2015 1 / 24

slide-2
SLIDE 2

Chris Taylor Algebras of incidence structures VAC 2015 1 / 24

slide-3
SLIDE 3

Boolean lattices

Theorem

Let L be a finite lattice. Then the following are equivalent.

1

L is a boolean lattice,

2

L ∼ = P(B) for some finite set B,

3

L ∼ = 2n for some n ≥ 0.

Chris Taylor Algebras of incidence structures VAC 2015 2 / 24

slide-4
SLIDE 4

Boolean lattices

Theorem

Let L be a finite lattice. Then the following are equivalent.

1

L is a boolean lattice,

2

L ∼ = P(B) for some finite set B,

3

L ∼ = 2n for some n ≥ 0.

Theorem

Let B be a boolean lattice. Then the following are equivalent.

1

B ∼ = P(X) for some set X.

2

B is complete and atomic.

3

B is complete and completely distributive.

Chris Taylor Algebras of incidence structures VAC 2015 2 / 24

slide-5
SLIDE 5

Some other classifications

Birkhoff’s duality for finite distributive lattices Stone’s duality for boolean algebras Priestley’s duality for bounded distributive lattices Every finite cyclic group is isomorphic to Zn for some n ∈ ω Every finite abelian group is isomorphic to n

i=0 Zqi where each qi

is a power of a prime

Chris Taylor Algebras of incidence structures VAC 2015 3 / 24

slide-6
SLIDE 6

Graphs

A graph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-7
SLIDE 7

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-8
SLIDE 8

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-9
SLIDE 9

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-10
SLIDE 10

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-11
SLIDE 11

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-12
SLIDE 12

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-13
SLIDE 13

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-14
SLIDE 14

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-15
SLIDE 15

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-16
SLIDE 16

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-17
SLIDE 17

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-18
SLIDE 18

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-19
SLIDE 19

Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures VAC 2015 4 / 24

slide-20
SLIDE 20

The lattice of subgraphs

Let G = V, E be a graph. The set of all subgraphs of G induces a bounded distributive lattice, which we will call S(G), where V1, E1 ∨ V2, E2 = V1 ∪ V2, E1 ∪ E2 V1, E1 ∧ V2, E2 = V1 ∩ V2, E1 ∩ E2. Note that we permit the empty graph.

Chris Taylor Algebras of incidence structures VAC 2015 5 / 24

slide-21
SLIDE 21

The lattice of subgraphs

Let G = V, E be a graph. The set of all subgraphs of G induces a bounded distributive lattice, which we will call S(G), where V1, E1 ∨ V2, E2 = V1 ∪ V2, E1 ∪ E2 V1, E1 ∧ V2, E2 = V1 ∩ V2, E1 ∩ E2. Note that we permit the empty graph.

Theorem (Reyes & Zolfaghari, 1996)

Let G be a graph. Then S(G) naturally forms a double-Heyting algebra.

Chris Taylor Algebras of incidence structures VAC 2015 5 / 24

slide-22
SLIDE 22

Graph complements

Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

slide-23
SLIDE 23

Graph complements

Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

slide-24
SLIDE 24

Graph complements

֒ →

Complement

Chris Taylor Algebras of incidence structures VAC 2015 6 / 24

slide-25
SLIDE 25

Graph complements

֒ →

Complement

Chris Taylor Algebras of incidence structures VAC 2015 7 / 24

slide-26
SLIDE 26

Graph complements

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 7 / 24

slide-27
SLIDE 27

Graph complements

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 8 / 24

slide-28
SLIDE 28

Graph complements

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 8 / 24

slide-29
SLIDE 29

Pseudocomplementation

Let L be a lattice and let x ∈ L. Then x has a pseudocomplement if there exists a largest element x∗ ∈ L such that x ∧ x∗ = 0.

Chris Taylor Algebras of incidence structures VAC 2015 9 / 24

slide-30
SLIDE 30

Pseudocomplementation

Let L be a lattice and let x ∈ L. Then x has a pseudocomplement if there exists a largest element x∗ ∈ L such that x ∧ x∗ = 0. Example: The lattice of open sets of a topological space X. If U is an open set, then U∗ = int(X \ U).

Chris Taylor Algebras of incidence structures VAC 2015 9 / 24

slide-31
SLIDE 31

Pseudocomplementation

Let L be a lattice and let x ∈ L. Then x has a pseudocomplement if there exists a largest element x∗ ∈ L such that x ∧ x∗ = 0. Example: The lattice of open sets of a topological space X. If U is an open set, then U∗ = int(X \ U). Let L be a lattice and let x ∈ L. Then x has a dual pseudocomplement if there exists a smallest element x+ ∈ L such that x ∨ x+ = 1.

Chris Taylor Algebras of incidence structures VAC 2015 9 / 24

slide-32
SLIDE 32

Pseudocomplementation

Let L be a lattice and let x ∈ L. Then x has a pseudocomplement if there exists a largest element x∗ ∈ L such that x ∧ x∗ = 0. Example: The lattice of open sets of a topological space X. If U is an open set, then U∗ = int(X \ U). Let L be a lattice and let x ∈ L. Then x has a dual pseudocomplement if there exists a smallest element x+ ∈ L such that x ∨ x+ = 1. Example: The lattice of closed sets of a topological space X. If C is a closed set, then U+ = cl(X \ C).

Chris Taylor Algebras of incidence structures VAC 2015 9 / 24

slide-33
SLIDE 33

Pseudocomplementation

Let L be a lattice and let x ∈ L. Then x has a pseudocomplement if there exists a largest element x∗ ∈ L such that x ∧ x∗ = 0. Example: The lattice of open sets of a topological space X. If U is an open set, then U∗ = int(X \ U). Let L be a lattice and let x ∈ L. Then x has a dual pseudocomplement if there exists a smallest element x+ ∈ L such that x ∨ x+ = 1. Example: The lattice of closed sets of a topological space X. If C is a closed set, then U+ = cl(X \ C).

Definition

An algebra A = A; ∨, ∧, 0, 1, ∗, + is a double p-algebra if A; ∨, ∧, 0, 1 is a bounded lattice, and ∗ and + are the pseudocomplement and dual pseudocomplement respectively.

Chris Taylor Algebras of incidence structures VAC 2015 9 / 24

slide-34
SLIDE 34

The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra

  • edges. Formally, for a graph G = V, E and a subgraph H = V ′, E′:

H∗ = V\V ′, {e ∈ E\E′ | (∀x ∈ e) x ∈ V\V ′}

Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

slide-35
SLIDE 35

The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra

  • edges. Formally, for a graph G = V, E and a subgraph H = V ′, E′:

H∗ = V\V ′, {e ∈ E\E′ | (∀x ∈ e) x ∈ V\V ′}

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

slide-36
SLIDE 36

The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra

  • edges. Formally, for a graph G = V, E and a subgraph H = V ′, E′:

H∗ = V\V ′, {e ∈ E\E′ | (∀x ∈ e) x ∈ V\V ′}

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 10 / 24

slide-37
SLIDE 37

The algebra of subgraphs

Dual pseudocomplement

Just add the missing vertices back. Formally, for a graph G = V, E and a subgraph H = V ′, E′: H+ = V\V ′ ∪ {v ∈ V | (∃e ∈ E\E′) v ∈ e}, E\E′

Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

slide-38
SLIDE 38

The algebra of subgraphs

Dual pseudocomplement

Just add the missing vertices back. Formally, for a graph G = V, E and a subgraph H = V ′, E′: H+ = V\V ′ ∪ {v ∈ V | (∃e ∈ E\E′) v ∈ e}, E\E′

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

slide-39
SLIDE 39

The algebra of subgraphs

Dual pseudocomplement

Just add the missing vertices back. Formally, for a graph G = V, E and a subgraph H = V ′, E′: H+ = V\V ′ ∪ {v ∈ V | (∃e ∈ E\E′) v ∈ e}, E\E′

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 11 / 24

slide-40
SLIDE 40

Pseudocomplements are not bijective

Boolean lattices: no two elements share a complement Double p-algebras: not true!

Chris Taylor Algebras of incidence structures VAC 2015 12 / 24

slide-41
SLIDE 41

Pseudocomplements are not bijective

Boolean lattices: no two elements share a complement Double p-algebras: not true!

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 12 / 24

slide-42
SLIDE 42

Pseudocomplements are not bijective

Boolean lattices: no two elements share a complement Double p-algebras: not true!

֒ →

Chris Taylor Algebras of incidence structures VAC 2015 12 / 24

slide-43
SLIDE 43

Regular double p-algebras

Let A be an algebra. We say that A is congruence regular if, for all α, β ∈ Con(A), we have ((∃x ∈ A) x/α = x/β) = ⇒ α = β. Example: groups

Chris Taylor Algebras of incidence structures VAC 2015 13 / 24

slide-44
SLIDE 44

Regular double p-algebras

Let A be an algebra. We say that A is congruence regular if, for all α, β ∈ Con(A), we have ((∃x ∈ A) x/α = x/β) = ⇒ α = β. Example: groups

Theorem (Varlet, 1972)

Let A be a double p-algebra. Then the following are equivalent.

1

A is congruence regular.

2

(∀a, b ∈ A) if a∗ = b∗ and a+ = b+ then a = b.

3

(∀a, b ∈ A) a ∧ a+ ≤ b ∨ b∗.

Chris Taylor Algebras of incidence structures VAC 2015 13 / 24

slide-45
SLIDE 45

A well-behaved structure

Theorem

Let G = V, E be a graph. Then S(G) is (the underlying lattice of) a regular double p-algebra.

Chris Taylor Algebras of incidence structures VAC 2015 14 / 24

slide-46
SLIDE 46

A well-behaved structure

Theorem

Let G = V, E be a graph. Then S(G) is (the underlying lattice of) a regular double p-algebra.

Proof.

Let A = AV, AE and B = BV, BE be subgraphs of G. Recall that for a subgraph H = V ′, E′, H∗ = V\V ′, {e ∈ E\E′ | (∀x ∈ e) x ∈ V\V ′} (1) H+ = V\V ′ ∪ {v ∈ V | (∃e ∈ E\E′) v ∈ e}, E\E′. (2) Assume A∗ = B∗ and A+ = B+. Then from (1) we have V\AV = V\BV and from (2) we have E\AE = E\BE. Hence, A = B.

Chris Taylor Algebras of incidence structures VAC 2015 14 / 24

slide-47
SLIDE 47

Some results from the literature

Theorem (Reyes & Zolfaghari, 1996)

Let G be a graph. Then S(G) naturally forms a double-Heyting algebra.

Chris Taylor Algebras of incidence structures VAC 2015 15 / 24

slide-48
SLIDE 48

Some results from the literature

Theorem (Reyes & Zolfaghari, 1996)

Let G be a graph. Then S(G) naturally forms a double-Heyting algebra.

Theorem (Katriˇ nák, 1973)

Let A be a regular double p-algebra. Then A is term-equivalent to a double-Heyting algebra via the term x → y = (x∗ ∨ y∗∗)∗∗ ∧ [(x ∨ x∗)+ ∨ x∗ ∨ y ∨ y∗], and its dual.

Chris Taylor Algebras of incidence structures VAC 2015 15 / 24

slide-49
SLIDE 49

Are graphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 16 / 24

slide-50
SLIDE 50

Are graphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 16 / 24

slide-51
SLIDE 51

Are graphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 16 / 24

slide-52
SLIDE 52

Are graphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 16 / 24

slide-53
SLIDE 53

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-54
SLIDE 54

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-55
SLIDE 55

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-56
SLIDE 56

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-57
SLIDE 57

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-58
SLIDE 58

Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures VAC 2015 17 / 24

slide-59
SLIDE 59

Incidence structures

Definition

An incidence structure is a triple P, L, I where P is a set of points, L is a set of lines and I ⊆ P × L is an incidence relation describing which points are incident to which lines.

Example

Let P = {1, 2, 3}, L = {x, y, z, a, b}, and let I = {1, 2, 3} × {x, y} ∪ {1, 2} × {z} ∪ {(1, a), (1, b)}

Chris Taylor Algebras of incidence structures VAC 2015 18 / 24

slide-60
SLIDE 60

Incidence structures

Example

Let P = {1, 2, 3}, L = {x, y, z, a, b}, and let I = {1, 2, 3} × {x, y} ∪ {1, 2} × {z} ∪ {(1, a), (1, b)} 1 2 3 z a b y x

Chris Taylor Algebras of incidence structures VAC 2015 19 / 24

slide-61
SLIDE 61

Point-preserving substructures

Definition

Let G = P, L, I be an incidence structure. A point-preserving substructure of G is a pair P′, L′ such that

1

P′ ⊆ P and L′ ⊆ L,

2

for all ℓ ∈ L′, if (p, ℓ) ∈ I then p ∈ P′. The incidence relation is defined implicitly from G.

Chris Taylor Algebras of incidence structures VAC 2015 20 / 24

slide-62
SLIDE 62

Point-preserving substructures

Definition

Let G = P, L, I be an incidence structure. A point-preserving substructure of G is a pair P′, L′ such that

1

P′ ⊆ P and L′ ⊆ L,

2

for all ℓ ∈ L′, if (p, ℓ) ∈ I then p ∈ P′. The incidence relation is defined implicitly from G. Let S(G) denote the set of all point-preserving substructures of a structure G. This induces a double p-algebra in a similar way to graphs, where P′, L′∗ = P\P′, {ℓ ∈ L\L′ | (∀p ∈ P) (p, ℓ) ∈ I = ⇒ p ∈ P\P′} P′, L′+ = P\P′ ∪ {p ∈ P | (∃ℓ ∈ L\L′) (p, ℓ) ∈ I}, L\L′.

Chris Taylor Algebras of incidence structures VAC 2015 20 / 24

slide-63
SLIDE 63

The main result (finite version)

Theorem

Let L be a finite lattice. Then the following are equivalent.

1

L is a boolean lattice,

2

L ∼ = P(B) for some set B,

3

L ∼ = 2n for some n ≥ 0.

Chris Taylor Algebras of incidence structures VAC 2015 21 / 24

slide-64
SLIDE 64

The main result (finite version)

Theorem

Let L be a finite lattice. Then the following are equivalent.

1

L is a boolean lattice,

2

L ∼ = P(B) for some set B,

3

L ∼ = 2n for some n ≥ 0.

Theorem (T., 2015)

Let L be a finite lattice. Then the following are equivalent.

1

L is (the underlying lattice of) a regular double p-algebra,

2

L ∼ = S(G) for some incidence structure G,

3

L ∼ = 2n × S(G) for some n ≥ 0 and some incidence structure G.

Chris Taylor Algebras of incidence structures VAC 2015 21 / 24

slide-65
SLIDE 65

The main result

Theorem

Let B be a boolean lattice. Then the following are equivalent.

1

B ∼ = P(X) for some set X.

2

B is complete and atomic.

3

B is complete and completely distributive.

Chris Taylor Algebras of incidence structures VAC 2015 22 / 24

slide-66
SLIDE 66

The main result

Theorem

Let B be a boolean lattice. Then the following are equivalent.

1

B ∼ = P(X) for some set X.

2

B is complete and atomic.

3

B is complete and completely distributive.

Theorem (T., 2015)

Let A be a regular double p-algebra. Then the following are equivalent.

1

A ∼ = P(B) × S(G) for some set B and some incidence structure G.

2

A ∼ = S(G) for some incidence structure G.

3

A is complete, completely distributive and doubly atomic.

Chris Taylor Algebras of incidence structures VAC 2015 22 / 24

slide-67
SLIDE 67

Behind the scenes

For a complete lattice L we say that L satisfies the join infinite distributivity law (JID) if the following identity holds: x ∧

  • Y =
  • {x ∧ y | y ∈ Y}.

Dually we say that A satisfies the meet infinite distributivity law (MID) if the following identity holds: x ∨

  • Y =
  • {x ∨ y | y ∈ Y}.

Theorem (T., 2015)

Let A be a complete doubly atomic regular double p-algebra. If A satisfies (JID), and for all atoms a ∈ A and all coatoms c ∈ A we have a ≤ c, then there is an incidence structure G such that S(G) ∼ = A. Furthermore, G has no isolated points and no empty lines.

Chris Taylor Algebras of incidence structures VAC 2015 23 / 24

slide-68
SLIDE 68

The main result

Theorem (T., 2015)

Let A be a regular double p-algebra. The following are equivalent.

1

A ∼ = P(B) × S(G) for some set B and some incidence structure G with no isolated points and no empty lines.

2

A ∼ = P(B) × S(G) for some set B and some incidence structure G.

3

A ∼ = S(G) for some incidence structure G.

4

A is complete, completely distributive and doubly atomic.

5

A is complete, satisfies (JID) and (MID) and is doubly atomic.

Chris Taylor Algebras of incidence structures VAC 2015 24 / 24

slide-69
SLIDE 69

The main result

Theorem (T., 2015)

Let A be a regular double p-algebra. The following are equivalent.

1

A ∼ = P(B) × S(G) for some set B and some incidence structure G with no isolated points and no empty lines.

2

A ∼ = P(B) × S(G) for some set B and some incidence structure G.

3

A ∼ = S(G) for some incidence structure G.

4

A is complete, completely distributive and doubly atomic.

5

A is complete, satisfies (JID) and (MID) and is doubly atomic.

Theorem (T., 2015)

Let A be a regular double p-algebra. Then there is an incidence structure G such that A is isomorphic to a subalgebra of S(G).

Chris Taylor Algebras of incidence structures VAC 2015 24 / 24