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Algebras of incidence structures: representing regular double p-algebras Christopher Taylor La Trobe University AustMS 2015 Chris Taylor Algebras of incidence structures AustMS 2015 1 / 25 Acknowledgements Thanks to the AustMS Student

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Algebras of incidence structures: representing regular double p-algebras

Christopher Taylor

La Trobe University

AustMS 2015

Chris Taylor Algebras of incidence structures AustMS 2015 1 / 25

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Acknowledgements

Thanks to the AustMS Student Support Scheme for providing additional funding to help attend the conference.

Chris Taylor Algebras of incidence structures AustMS 2015 2 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2} {3}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2} {3} {1, 2}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2} {3} {1, 2} {1, 3}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3}

Chris Taylor Algebras of incidence structures AustMS 2015 3 / 25

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Boolean lattices

Definition

Boolean lattice: a bounded distributive lattice B = B; ∨, ∧, 0, 1 such that every x ∈ B has a (unique) complement.

Chris Taylor Algebras of incidence structures AustMS 2015 4 / 25

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Boolean lattices

Definition

Boolean lattice: a bounded distributive lattice B = B; ∨, ∧, 0, 1 such that every x ∈ B has a (unique) complement.

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 AustMS 2015 4 / 25

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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 AustMS 2015 5 / 25

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Graphs

A graph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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Graphs

A graph: A subgraph:

Chris Taylor Algebras of incidence structures AustMS 2015 6 / 25

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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 AustMS 2015 7 / 25

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Graph complements

Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

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Graph complements

Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

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Graph complements

֒ →

Complement

Chris Taylor Algebras of incidence structures AustMS 2015 8 / 25

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Graph complements

֒ →

Complement

Chris Taylor Algebras of incidence structures AustMS 2015 9 / 25

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Graph complements

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 9 / 25

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Graph complements

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 10 / 25

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Graph complements

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 10 / 25

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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 AustMS 2015 11 / 25

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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 AustMS 2015 11 / 25

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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 AustMS 2015 11 / 25

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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 AustMS 2015 11 / 25

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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 AustMS 2015 11 / 25

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The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra edges.

Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25

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The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra edges.

Dual pseudocomplement

Just add the missing vertices back

Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25

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The algebra of subgraphs

Pseudocomplement

Take the set complement of the subgraph and abandon the extra edges.

Dual pseudocomplement

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

Chris Taylor Algebras of incidence structures AustMS 2015 12 / 25

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Pseudocomplement

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 13 / 25

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Pseudocomplement

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 13 / 25

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Pseudocomplements are not bijective

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 14 / 25

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Pseudocomplements are not bijective

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 14 / 25

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Pseudocomplements are not bijective

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 14 / 25

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Pseudocomplements are not bijective

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

֒ →

Chris Taylor Algebras of incidence structures AustMS 2015 14 / 25

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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 AustMS 2015 15 / 25

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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 AustMS 2015 15 / 25

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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 AustMS 2015 16 / 25

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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 AustMS 2015 16 / 25

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Are graphs enough?

→

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Are graphs enough?

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Are graphs enough?

→

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Are graphs enough?

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Chris Taylor Algebras of incidence structures AustMS 2015 17 / 25

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Are multigraphs enough?

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Chris Taylor Algebras of incidence structures AustMS 2015 18 / 25

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Are multigraphs enough?

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Are multigraphs enough?

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Chris Taylor Algebras of incidence structures AustMS 2015 18 / 25

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Are multigraphs enough?

→

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Are multigraphs enough?

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Chris Taylor Algebras of incidence structures AustMS 2015 18 / 25

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Are multigraphs enough?

→

Chris Taylor Algebras of incidence structures AustMS 2015 18 / 25

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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 AustMS 2015 19 / 25

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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 AustMS 2015 20 / 25

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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 AustMS 2015 21 / 25

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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 AustMS 2015 21 / 25

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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 AustMS 2015 22 / 25

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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 (Taylor, 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 AustMS 2015 22 / 25

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An infinite counterexample

The finite-cofinite algebra of N is a boolean algebra.

◮ FC(N) := {S ⊆ N | S is finite or N \ S is finite}. Chris Taylor Algebras of incidence structures AustMS 2015 23 / 25

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An infinite counterexample

The finite-cofinite algebra of N is a boolean algebra.

◮ FC(N) := {S ⊆ N | S is finite or N \ S is finite}.

FC(N) is countable.

Chris Taylor Algebras of incidence structures AustMS 2015 23 / 25

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An infinite counterexample

The finite-cofinite algebra of N is a boolean algebra.

◮ FC(N) := {S ⊆ N | S is finite or N \ S is finite}.

FC(N) is countable. Every powerset lattice has cardinality 2X for some set X.

Chris Taylor Algebras of incidence structures AustMS 2015 23 / 25

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An infinite counterexample

The finite-cofinite algebra of N is a boolean algebra.

◮ FC(N) := {S ⊆ N | S is finite or N \ S is finite}.

FC(N) is countable. Every powerset lattice has cardinality 2X for some set X. Thus FC(N) is not a powerset algebra.

Chris Taylor Algebras of incidence structures AustMS 2015 23 / 25

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The characterisation of powerset algebras

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 AustMS 2015 24 / 25

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The main result

Theorem (Taylor, 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 AustMS 2015 25 / 25

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The main result

Theorem (Taylor, 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.

Theorem (Taylor, 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 AustMS 2015 25 / 25