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

The geometry of Boolean algebra Chris Heunen 1 / 22

Boolean algebra: example � � , , � � � � � � , , , � � � � � � ∅ 2 / 22

Boole’s algebra 3 / 22

Boolean algebra � = Boole’s algebra 4 / 22

Boolean algebra = Jevon’s algebra 5 / 22

Boole’s algebra isn’t Boolean algebra 6 / 22

Contextuality 7 / 22

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 8 / 22

Orthoalgebra: example • • • • • • • • • • • • 9 / 22

Orthodomain: definition Given a piecewise Boolean algebra A , its orthodomain BSub( A ) is the collection of its Boolean subalgebras, partially ordered by inclusion. 10 / 22

Orthodomain: example Example: if A is • • • • • • • • • • • • then BSub( A ) is • • • • • • • • 11 / 22

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 12 / 22

Different kinds of atoms 1234 123 124 134 234 If A = , then BSub( A ) = · · · 12 13 14 23 24 34 1 2 3 4 ∅ 13 / 22

Different kinds of atoms A A A A A A A 234 134 12 234 124 13 234 123 34 134 124 23 134 123 24 124 123 34 1 2 34 1 3 24 1 4 23 2 3 14 2 4 13 3 4 12 ∅ ∅ ∅ ∅ ∅ ∅ A A A A A A A 1 234 2 134 3 124 4 123 12 34 13 24 14 23 ∅ ∅ ∅ ∅ ∅ ∅ ∅ 1234 ∅ 13 / 22

Principal pairs 1 Reconstruct pairs ( x, ¬ x ) of A : x ◮ principal ideal subalgebra of A is of the form ¬ x 0 ◮ 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 14 / 22

Principal pairs 1 Reconstruct pairs ( x, ¬ x ) of A : x ◮ principal ideal subalgebra of A is of the form ¬ x 0 ◮ 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 1 Reconstruct elements x of A : q p x ◮ principal pairs of A are ( p, q ) with atomic meet ¬ x p q 0 14 / 22

Principal pairs 1 Reconstruct pairs ( x, ¬ x ) of A : x ◮ principal ideal subalgebra of A is of the form ¬ x 0 ◮ 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 1 Reconstruct elements x of A : q p x ◮ principal pairs of A are ( p, q ) with atomic meet ¬ x p q 0 Theorem : A ≃ Pp(BSub( A )) for Boolean algebra A of size ≥ 4 D ≃ BSub(Pp( D )) for Boolean domain D of size ≥ 2 14 / 22

Directions 1 ¬ y ¬ v ¬ w ¬ x ¬ z If A is y v w x z 0 • • • • • • • then BSub( A ) is • 15 / 22

Directions 1 1 1 ¬ y ¬ y ¬ v ¬ w ¬ x ¬ z ¬ v ¬ w ¬ x ¬ z x If A is or y y v w x z v w x ¬ x z 0 0 0 • • • • • • • then BSub( A ) is • 15 / 22

Directions 1 1 1 ¬ y ¬ y ¬ v ¬ w ¬ x ¬ z ¬ v ¬ w ¬ x ¬ z x If A is or y y v w x z v w x ¬ x z 0 0 0 • • • • • • • then BSub( A ) is • A direction for a Boolean domain is a map d : D → D 2 with ◮ d (1) = ( p, q ) is a principal pair ◮ d ( m ) = ( p ∧ m, q ∧ m ) 15 / 22

Directions 1 1 1 ¬ y ¬ y ¬ v ¬ w ¬ x ¬ z ¬ v ¬ w ¬ x ¬ z x If A is or y y v w x z v w x ¬ x z 0 0 0 • • • • • • • then BSub( A ) is • A direction for a orthodomain is a map d : D → D 2 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 15 / 22

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 16 / 22

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: 17 / 22

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 17 / 22

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. 18 / 22

Orthohypergraph morphisms Morphism of orthohypergraphs is partial function such that: ◮ point image none defined isomorphism ◮ point image line image isomorphism none defined ◮ 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 ′ : m α ( m ) α ( l ) l p α ( p ) 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 20 / 22

Conclusion ◮ Orthoalgebra: Boolean algebra as Boole intended ◮ Orthodomain: shape of parts enough to determine whole ◮ Orthohypergraph: (projective) geometry of contextuality 21 / 22

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 22 / 22