The geometry of Boolean algebra

Chris Heunen

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Boolean algebra: example

∅

{, ,

}

{ } { } { }

{,

} {,

} {,

}

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Boole’s algebra

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Boolean algebra 6= Boole’s algebra

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Boolean algebra = Jevon’s algebra

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Boole’s algebra isn’t Boolean algebra

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Contextuality

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Orthoalgebra: definition

An orthoalgebra is a set A withI a partial binary operation ⊕ : A×A→ A

I a unary operation ¬ : A→ A

I distinguished elements 0, 1 ∈ A

such thatI ⊕ is commutative and associativeI ¬a is the unique element with a⊕ ¬a = 1I a⊕ a is defined if and only if a = 0

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Orthoalgebra: example

•

•

• • • • •

• • • • •

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Orthodomain: definition

Given a piecewise Boolean algebra A,its orthodomain BSub(A)

is the collection of its Boolean subalgebras,partially ordered by inclusion.

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Orthodomain: example

Example: if A is

•

•

• • • • •

• • • • •

then BSub(A) is• •

• • • • •

•

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Orthoalgebra: pitfalls

I subalgebras of a Boolean orthoalgebra need not be BooleanI intersection of two Boolean subalgebras need not be BooleanI two Boolean subalgebras might have no meetI two Boolean subalgebras might have upper bound but no join

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Different kinds of atoms

If A =

∅

1 2 3 4

12 13 14 23 24 34

123 124 134 234

1234

, then BSub(A) = · · ·

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Different kinds of atoms

∅

1234

∅

A

1 234∅

A

2 134∅

A

3 124∅

A

4 123∅

A

12 34∅

A

13 24∅

A

14 23

∅

A

1 2 34

234 134 12

∅

A

1 3 24

234 124 13

∅

A

1 4 23

234 123 34

∅

A

2 3 14

134 124 23

∅

A

2 4 13

134 123 24

∅

A

3 4 12

124 123 34

A

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Principal pairsReconstruct pairs (x,¬x) of A:

I principal ideal subalgebra of A is of the form

0

1

x

¬x

I they are the elements p of BSub(A) that aredual modular and (p ∨m) ∧ n = p ∨ (m ∧ n) for n ≥ patom or relative complement a ∧m = a, a ∨m = A for atom a

Reconstruct elements x of A:

I principal pairs of A are (p, q) with atomic meetp

pq

q

0

1

x¬x

Theorem: A ' Pp(BSub(A)) for Boolean algebra A of size ≥ 4D ' BSub(Pp(D)) for Boolean domain D of size ≥ 2

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Principal pairsReconstruct pairs (x,¬x) of A:

I principal ideal subalgebra of A is of the form

0

1

x

¬x

I they are the elements p of BSub(A) that aredual modular and (p ∨m) ∧ n = p ∨ (m ∧ n) for n ≥ patom or relative complement a ∧m = a, a ∨m = A for atom a

Reconstruct elements x of A:

I principal pairs of A are (p, q) with atomic meetp

pq

q

0

1

x¬x

Theorem: A ' Pp(BSub(A)) for Boolean algebra A of size ≥ 4D ' BSub(Pp(D)) for Boolean domain D of size ≥ 2

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Principal pairsReconstruct pairs (x,¬x) of A:

I principal ideal subalgebra of A is of the form

0

1

x

¬x

I they are the elements p of BSub(A) that aredual modular and (p ∨m) ∧ n = p ∨ (m ∧ n) for n ≥ patom or relative complement a ∧m = a, a ∨m = A for atom a

Reconstruct elements x of A:

I principal pairs of A are (p, q) with atomic meetp

pq

q

0

1

x¬x

Theorem: A ' Pp(BSub(A)) for Boolean algebra A of size ≥ 4D ' BSub(Pp(D)) for Boolean domain D of size ≥ 2

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Directions

If A is

0

1

v w x y z

¬v ¬w ¬x ¬y ¬z

or

0

1

v w x

¬v ¬w ¬x

0

1

¬x y z

x ¬y ¬z

then BSub(A) is

• •

• • • • •

•

A direction for a is a map d : D → D2 withI

I

I if m, n cover a, d(m) = (a, m), d(n) = (n, a), then m ∨ n exists

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Directions

If A is

0

1

v w x y z

¬v ¬w ¬x ¬y ¬z

or

0

1

v w x

¬v ¬w ¬x

0

1

¬x y z

x ¬y ¬z

then BSub(A) is

• •

• • • • •

•

A direction for a is a map d : D → D2 withI

I

I if m, n cover a, d(m) = (a, m), d(n) = (n, a), then m ∨ n exists

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Directions

If A is

0

1

v w x y z

¬v ¬w ¬x ¬y ¬z

or

0

1

v w x

¬v ¬w ¬x

0

1

¬x y z

x ¬y ¬z

then BSub(A) is

• •

• • • • •

•

A direction for a Boolean domain is a map d : D → D2 withI d(1) = (p, q) is a principal pairI d(m) = (p ∧m, q ∧m)

I if m, n cover a, d(m) = (a, m), d(n) = (n, a), then m ∨ n exists

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Directions

If A is

0

1

v w x y z

¬v ¬w ¬x ¬y ¬z

or

0

1

v w x

¬v ¬w ¬x

0

1

¬x y z

x ¬y ¬z

then BSub(A) is

• •

• • • • •

•

A direction for a orthodomain is a map d : D → D2 withI if a ≤ m then d(m) is a principal pair with meet a in m

I d(m) =∨{(m, m) ∧ f(n) | a ≤ n}

I if m, n cover a, d(m) = (a, m), d(n) = (n, a), then m ∨ n exists

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Orthoalgebras and orthodomains

Lemma: If an atom in an orthodomain has a direction,then it has exactly two directions

Theorem:I A ' Dir(BSub(A)) for orthoalgebra A

whose blocks have > 4 elementsI D ' BSub(Dir(D)) for orthodomain D

that has enough directions and is tall

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Orthohypergraphs

An orthohypergraph is consists of a set of points, a set of lines, anda set of planes. A line is a set of 3 points, and a plane is a set of 7points where the restriction of the lines to these 7 points is as:

Every orthoalgebra/orthodomain gives rise to an orthohypergraph:I points are Boolean subalgebras of size 4I lines are Boolean subalgebras of size 8I planes are Boolean subalgebras of size 16

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Orthohypergraphs

An orthohypergraph is consists of a set of points, a set of lines, anda set of planes. A line is a set of 3 points, and a plane is a set of 7points where the restriction of the lines to these 7 points is as:

Every orthoalgebra/orthodomain gives rise to an orthohypergraph:I points are Boolean subalgebras of size 4I lines are Boolean subalgebras of size 8I planes are Boolean subalgebras of size 16

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Projective geometry

I Any two lines intersect in at most one point.I Any two points lie on a line or plane.I For orthomodular posets: if it looks like a plane, it is a plane.

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Orthohypergraph morphisms

Morphism of orthohypergraphs is partial function such that:

I

none defined point image isomorphism

I

none defined point image line image isomorphism

I If lines l,m intersect in point p, and lines α(l) 6= α(m) in plane t′ intersect in edgepoint α(p), then l,m lie in plane t that is mapped isomorphically to t′:

l

m

p α(p)α(l)

α(m)

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Orthodomains and orthohypergraphs

Theorem: functor that sends orthoalgebra to its orthohypergraph:I is essentially surjective on objectsI is injective on objects except on 1- and 2-element orthoalgebrasI is full on proper morphismsI is faithful on proper morphisms

So for all intents and purposes is equivalence

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Conclusion

I Orthoalgebra: Boolean algebra as Boole intended

I Orthodomain: shape of parts enough to determine whole

I Orthohypergraph: (projective) geometry of contextuality

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ReferencesI Harding, Heunen, Lindenhovius, Navara, arXiv:1711.03748

Boolean subalgebras of orthoalgebras

I Heunen, ICALP 2014Piecewise Boolean algebras and their domains

I Van den Berg, Heunen, Appl. Cat. Str. 2012Noncommutativity as a colimit

I Gratzer, Koh, Makkai, Proc. Amer. Math. Soc. 1972On the lattice of subalgebras of a Boolean algebra,

I Kochen, Specker, J. Math. Mech. 1967The problem of hidden variables in quantum mechanics

I Sachs, Canad. J. Math. 1962The lattice of subalgebras of a Boolean algebra

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