The geometry of Boolean algebra Chris Heunen 1 / 22 Boolean - - PowerPoint PPT Presentation

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 = 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 with

◮ a partial binary operation ⊕: A × A → A ◮ a unary operation ¬: A → A ◮ distinguished elements 0, 1 ∈ A

such that

◮ ⊕ is commutative and associative ◮ ¬a is the unique element with a ⊕ ¬a = 1 ◮ 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

◮ subalgebras of a Boolean orthoalgebra need not be Boolean ◮ intersection of two Boolean subalgebras need not be Boolean ◮ two Boolean subalgebras might have no meet ◮ 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 13 / 22

Principal pairs

Reconstruct pairs (x, ¬x) of A:

◮ principal ideal subalgebra of A is of the form 1 x ¬x ◮ they are the elements p of BSub(A) that are

dual modular and (p ∨ m) ∧ n = p ∨ (m ∧ n) for n ≥ p atom or relative complement a ∧ m = a, a ∨ m = A for atom a

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Principal pairs

Reconstruct pairs (x, ¬x) of A:

◮ principal ideal subalgebra of A is of the form 1 x ¬x ◮ they are the elements p of BSub(A) that are

dual modular and (p ∨ m) ∧ n = p ∨ (m ∧ n) for n ≥ p atom or relative complement a ∧ m = a, a ∨ m = A for atom a Reconstruct elements x of A:

◮ principal pairs of A are (p, q) with atomic meet p p q q 1 x ¬x

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Principal pairs

Reconstruct pairs (x, ¬x) of A:

◮ principal ideal subalgebra of A is of the form 1 x ¬x ◮ they are the elements p of BSub(A) that are

dual modular and (p ∨ m) ∧ n = p ∨ (m ∧ n) for n ≥ p atom or relative complement a ∧ m = a, a ∨ m = A for atom a Reconstruct elements x of A:

◮ principal pairs of A are (p, q) with atomic meet p p q q 1 x ¬x

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

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Directions

If A is 1 v w x y z ¬v ¬w ¬x ¬y ¬z then BSub(A) is

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Directions

If A is 1 v w x y z ¬v ¬w ¬x ¬y ¬z

• r

1 v w x ¬v ¬w ¬x 1 ¬x y z x ¬y ¬z then BSub(A) is

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Directions

If A is 1 v w x y z ¬v ¬w ¬x ¬y ¬z

• r

1 v w x ¬v ¬w ¬x 1 ¬x y z x ¬y ¬z then BSub(A) is

• A direction for a Boolean domain is a map d: D → D2 with

◮ d(1) = (p, q) is a principal pair ◮ d(m) = (p ∧ m, q ∧ m)

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Directions

If A is 1 v w x y z ¬v ¬w ¬x ¬y ¬z

• r

1 v w x ¬v ¬w ¬x 1 ¬x y z x ¬y ¬z then BSub(A) is

• A direction for a orthodomain is a map d: D → D2 with

◮ if a ≤ m then d(m) is a principal pair with meet a in m ◮ d(m) = {(m, m) ∧ f(n) | a ≤ n} ◮ 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:

◮ A ≃ Dir(BSub(A)) for orthoalgebra A

whose blocks have > 4 elements

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

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Orthohypergraphs

An orthohypergraph is consists of a set of points, a set of lines, and a set of planes. A line is a set of 3 points, and a plane is a set of 7 points where the restriction of the lines to these 7 points is as: Every orthoalgebra/orthodomain gives rise to an orthohypergraph:

◮ points are Boolean subalgebras of size 4 ◮ lines are Boolean subalgebras of size 8 ◮ planes are Boolean subalgebras of size 16

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Projective geometry

◮ Any two lines intersect in at most one point. ◮ Any two points lie on a line or plane. ◮ 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:

◮ none defined point image isomorphism ◮ none defined point image line image isomorphism ◮ If lines l, m intersect in point p, and lines α(l) = α(m) in plane t′ intersect in edge point α(p), then l, m lie in plane t that is mapped isomorphically to t′:

l m p α(p) α(l) α(m) 19 / 22

Orthodomains and orthohypergraphs

Theorem: functor that sends orthoalgebra to its orthohypergraph:

◮ is essentially surjective on objects ◮ is injective on objects except on 1- and 2-element orthoalgebras ◮ is full on proper morphisms ◮ is faithful on proper morphisms

So for all intents and purposes is equivalence

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Conclusion

◮ Orthoalgebra: Boolean algebra as Boole intended ◮ Orthodomain: shape of parts enough to determine whole ◮ Orthohypergraph: (projective) geometry of contextuality

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References

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

Boolean subalgebras of orthoalgebras

◮ Heunen, ICALP 2014

Piecewise Boolean algebras and their domains

◮ Van den Berg, Heunen, Appl. Cat. Str. 2012

Noncommutativity as a colimit

◮ Gr¨

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

◮ Kochen, Specker, J. Math. Mech. 1967

The problem of hidden variables in quantum mechanics

◮ Sachs, Canad. J. Math. 1962

The lattice of subalgebras of a Boolean algebra

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