Piecewise Boolean algebra. Chris Heunen
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1 Piecewise Boolean algebra Chris Heunen 1 / 33
2 Boolean algebra: example {,, } {, } {, } {, } { } { } { } 2 / 33
3 Boolean algebra: definition A Boolean algebra is a set B with: a distinguished element 1 B; a unary operations : B B; a binary operation : B B B; such that for all x, y, z B: x (y z) = (x y) z; x y = y x; x 1 = x; x = (x y) (x y) Sets of independent postulates for the algebra of logic Transactions of the American Mathematical Society 5: , / 33
4 Boolean algebra: definition A Boolean algebra is a set B with: a distinguished element 1 B; a unary operations : B B; a binary operation : B B B; such that for all x, y, z B: x (y z) = (x y) z; x y = y x; x 1 = x; x x = x; x x = 1 = 1 x; ( x is a complement of x) x y = 1 x y = x (0 = 1 is the least element) Sets of independent postulates for the algebra of logic Transactions of the American Mathematical Society 5: , / 33
5 Boole s algebra 4 / 33
6 Boolean algebra Boole s algebra 5 / 33
7 Boolean algebra Boole s algebra 5 / 33
8 Boolean algebra = Jevon s algebra 6 / 33
9 Boolean algebra = Jevon s algebra 6 / 33
10 Boole s algebra isn t Boolean algebra 7 / 33
11 Piecewise Boolean algebra: definition A piecewise Boolean algebra is a set B with: a reflexive symmetric binary relation B 2 ; a (partial) binary operation : B; a (total) function : B B; an element 1 B with {1} B ; such that every S B with S 2 is contained in a T B with T 2 where (T,,, 1) is a Boolean algebra. 8 / 33
12 Piecewise Boolean algebra: example 9 / 33
13 Piecewise Boolean algebra quantum logic Subsets of a set Subspaces of a Hilbert space The logic of quantum mechanics Annals of Mathematics 37: , / 33
14 Piecewise Boolean algebra quantum logic Subsets of a set Subspaces of a Hilbert space An orthomodular lattice is: A partial order set (B, ) with min 0 and max 1 that has greatest lower bounds x y; an operation : B B such that x = x, and x y implies y x ; x x = 1; if x y then y = x (y x ) The logic of quantum mechanics Annals of Mathematics 37: , / 33
15 Piecewise Boolean algebra quantum logic Subsets of a set Subspaces of a Hilbert space An orthomodular lattice is not distributive: ( or ) and ( and ) ( or and ) The logic of quantum mechanics Annals of Mathematics 37: , / 33
16 Piecewise Boolean algebra quantum logic Subsets of a set Subspaces of a Hilbert space biscuit coffee tea nothing The logic of quantum mechanics Annals of Mathematics 37: , / 33
17 Piecewise Boolean algebra quantum logic Subsets of a set Subspaces of a Hilbert space biscuit coffee tea nothing However: fine when within orthogonal basis (Boolean subalgebra) The logic of quantum mechanics Annals of Mathematics 37: , / 33
18 Boole s algebra Boolean algebra Quantum measurement is probabilistic (state α 0 + β 1 gives outcome 0 with probability α 2 ) 11 / 33
19 Boole s algebra Boolean algebra Quantum measurement is probabilistic (state α 0 + β 1 gives outcome 0 with probability α 2 ) A hidden variable for a state is an assignment of a consistent outcome to any possible measurement (homomorphism of piecewise Boolean algebras to {0, 1}) 11 / 33
20 Boole s algebra Boolean algebra Quantum measurement is probabilistic (state α 0 + β 1 gives outcome 0 with probability α 2 ) A hidden variable for a state is an assignment of a consistent outcome to any possible measurement (homomorphism of piecewise Boolean algebras to {0, 1}) Theorem: hidden variables cannot exist (if dimension n 3, there is no homomorphism Sub(C n ) {0, 1} of piecewise Boolean algebras.) The problem of hidden variables in quantum mechanics Journal of Mathematics and Mechanics 17:59 87, / 33
21 Piecewise Boolean domains: definition Given a piecewise Boolean algebra B, its piecewise Boolean domain Sub(B) is the collection of its Boolean subalgebras, partially ordered by inclusion. 12 / 33
22 Piecewise Boolean domains: example Example: if B is then Sub(B) is 13 / 33
23 Piecewise Boolean domains: theorems Can reconstruct B from Sub(B) (B = colim Sub(B)) (the parts determine the whole) Noncommutativity as a colimit Applied Categorical Structures 20(4): , / 33
24 Piecewise Boolean domains: theorems Can reconstruct B from Sub(B) (B = colim Sub(B)) (the parts determine the whole) Sub(B) determines B (B = B Sub(B) = Sub(B )) (shape of parts determines whole) Noncommutativity as a colimit Applied Categorical Structures 20(4): , 2012 Subalgebras of orthomodular lattices Order 28: , / 33
25 Piecewise Boolean domains: as complex as graphs State space = Hilbert space Sharp measurements = subspaces (projections) Jointly measurable = overlapping or orthogonal (commute) 15 / 33
26 Piecewise Boolean domains: as complex as graphs State space = Hilbert space Sharp measurements = subspaces (projections) Jointly measurable = overlapping or orthogonal (commute) (In)compatibilities form graph: r p s q t 15 / 33
27 Piecewise Boolean domains: as complex as graphs State space = Hilbert space Sharp measurements = subspaces (projections) Jointly measurable = overlapping or orthogonal (commute) (In)compatibilities form graph: r s p Theorem: Any graph can be realised as sharp measurements on some Hilbert space. q t Quantum theory realises all joint measurability graphs Physical Review A 89(3):032121, / 33
28 Piecewise Boolean domains: as complex as graphs State space = Hilbert space Sharp measurements = subspaces (projections) Jointly measurable = overlapping or orthogonal (commute) (In)compatibilities form graph: r s p Theorem: Any graph can be realised as sharp measurements on some Hilbert space. Corollary: Any piecewise Boolean algebra can be realised on some Hilbert space. q t Quantum theory realises all joint measurability graphs Physical Review A 89(3):032121, 2014 Quantum probability quantum logic Springer Lecture Notes in Physics 321, / 33
29 Piecewise Boolean domains: as complex as hypergraphs State space = Hilbert space Unsharp measurements = positive operator-valued measures Jointly measurable = marginals of larger POVM 16 / 33
30 Piecewise Boolean domains: as complex as hypergraphs State space = Hilbert space Unsharp measurements = positive operator-valued measures Jointly measurable = marginals of larger POVM (In)compatibilities now form hypergraph: r s p q t 16 / 33
31 Piecewise Boolean domains: as complex as hypergraphs State space = Hilbert space Unsharp measurements = positive operator-valued measures Jointly measurable = marginals of larger POVM (In)compatibilities now form abstract simplicial complex: r s p q t 16 / 33
32 Piecewise Boolean domains: as complex as hypergraphs State space = Hilbert space Unsharp measurements = positive operator-valued measures Jointly measurable = marginals of larger POVM (In)compatibilities now form abstract simplicial complex: r s p Theorem: Any abstract simplicial complex can be realised as POVMs on a Hilbert space. q t All joint measurability structures are quantum realizable Physical Review A 89(5):052126, / 33
33 Piecewise Boolean domains: as complex as hypergraphs State space = Hilbert space Unsharp measurements = positive operator-valued measures Jointly measurable = marginals of larger POVM (In)compatibilities now form abstract simplicial complex: r s p Theorem: Any abstract simplicial complex can be realised as POVMs on a Hilbert space. Corollary: Any interval effect algebra can be realised on some Hilbert space. q t All joint measurability structures are quantum realizable Physical Review A 89(5):052126, 2014 Hilbert space effect-representations of effect algebras Reports on Mathematical Physics 70(3): , / 33
34 Piecewise Boolean domains: partition lattices What does Sub(B) look like when B is an honest Boolean algebra? 17 / 33
35 Piecewise Boolean domains: partition lattices What does Sub(B) look like when B is an honest Boolean algebra? Boolean algebras are dually equivalent to Stone spaces The theory of representations of Boolean algebras Transactions of the American Mathematical Society 40:37 111, / 33
36 Piecewise Boolean domains: partition lattices What does Sub(B) look like when B is an honest Boolean algebra? Boolean algebras are dually equivalent to Stone spaces Sub(B) becomes a partition lattice / /23 13/2 12/3 1/2/3 123/4 124/3 13/24 12/34 14/23 134/2 1/234 12/3/4 13/2/4 14/2/3 1/23/4 1/3/24 1/2/34 1/2/3/4 The theory of representations of Boolean algebras Transactions of the American Mathematical Society 40:37 111, 1936 On the lattice of subalgebras of a Boolean algebra Proceedings of the American Mathematical Society 36: 87 92, / 33
37 Piecewise Boolean domains: partition lattices What does Sub(B) look like when B is an honest Boolean algebra? Boolean algebras are dually equivalent to Stone spaces Sub(B) becomes a partition lattice / /23 13/2 12/3 1/2/3 123/4 124/3 13/24 12/34 14/23 134/2 1/234 12/3/4 13/2/4 14/2/3 1/23/4 1/3/24 1/2/34 1/2/3/4 Idea: every downset in Sub(B) is a partition lattice (upside-down)! The theory of representations of Boolean algebras Transactions of the American Mathematical Society 40:37 111, 1936 On the lattice of subalgebras of a Boolean algebra Proceedings of the American Mathematical Society 36: 87 92, / 33
38 Piecewise Boolean domains: characterisation Lemma: Piecewise Boolean domain D gives functor F : D Bool that preserves subobjects; F is a piecewise Boolean diagram. (Sub(F (x)) = x, and B = colim F ) Piecewise Boolean algebras and their domains ICALP Proceedings, Lecture Notes in Computer Science 8573: , / 33
39 Piecewise Boolean domains: characterisation Lemma: Piecewise Boolean domain D gives functor F : D Bool that preserves subobjects; F is a piecewise Boolean diagram. (Sub(F (x)) = x, and B = colim F ) F Piecewise Boolean algebras and their domains ICALP Proceedings, Lecture Notes in Computer Science 8573: , / 33
40 Piecewise Boolean domains: characterisation Lemma: Piecewise Boolean domain D gives functor F : D Bool that preserves subobjects; F is a piecewise Boolean diagram. (Sub(F (x)) = x, and B = colim F ) F Theorem: A partial order is a piecewise Boolean domain iff: it has directed suprema; it has nonempty infima; each element is a supremum of compact ones; each downset is cogeometric with a modular atom; each element of height n 3 covers ( ) n+1 2 elements. a set of atoms has a sup iff each finite subset does Piecewise Boolean algebras and their domains ICALP Proceedings, Lecture Notes in Computer Science 8573: , / 33
41 Orthoalgebras This is almost a piecewise Boolean domain D: a b c d e f a a b c d e f a That is of the form D = Sub(B) for this B: 1 0 abc cde ef a a b c d e f a But B is not a piecewise Boolean algebra: {a, c, e} not in one block 0 19 / 33
42 Piecewise Boolean domains: higher order Scott topology turns directed suprema into topological convergence (closed sets = downsets closed under directed suprema) Lawson topology refines it from dcpos to continuous lattices (basic open sets = Scott open minus upset of finite set) 20 / 33
43 Piecewise Boolean domains: higher order Scott topology turns directed suprema into topological convergence (closed sets = downsets closed under directed suprema) Lawson topology refines it from dcpos to continuous lattices (basic open sets = Scott open minus upset of finite set) If B 0 is piecewise Boolean algebra, Sub(B 0 ) is algebraic dcpo and complete semilattice, 20 / 33
44 Piecewise Boolean domains: higher order Scott topology turns directed suprema into topological convergence (closed sets = downsets closed under directed suprema) Lawson topology refines it from dcpos to continuous lattices (basic open sets = Scott open minus upset of finite set) If B 0 is piecewise Boolean algebra, Sub(B 0 ) is algebraic dcpo and complete semilattice, hence a Stone space under Lawson topology! Continuous lattices and domains Cambridge University Press, / 33
45 Piecewise Boolean domains: higher order Scott topology turns directed suprema into topological convergence (closed sets = downsets closed under directed suprema) Lawson topology refines it from dcpos to continuous lattices (basic open sets = Scott open minus upset of finite set) If B 0 is piecewise Boolean algebra, Sub(B 0 ) is algebraic dcpo and complete semilattice, hence a Stone space under Lawson topology! It then gives rise to a new Boolean algebra B 1. Repeat: B 2, B 3,... (Can handle domains of Boolean algebras with Boolean algebra!) Continuous lattices and domains Cambridge University Press, 2003 Domains of commutative C*-subalgebras Logic in Computer Science, ACM/IEEE Proceedings , / 33
46 Piecewise Boolean diagrams: topos Consider contextual sets over piecewise Boolean algebra B assignment of set S(C) to each C Sub(B) such that C D implies S(C) S(D) 21 / 33
47 Piecewise Boolean diagrams: topos Consider contextual sets over piecewise Boolean algebra B assignment of set S(C) to each C Sub(B) such that C D implies S(C) S(D) They form a topos T (B)! category whose objects behave a lot like sets in particular, it has a logic of its own! 21 / 33
48 Piecewise Boolean diagrams: topos Consider contextual sets over piecewise Boolean algebra B assignment of set S(C) to each C Sub(B) such that C D implies S(C) S(D) They form a topos T (B)! category whose objects behave a lot like sets in particular, it has a logic of its own! There is one canonical contextual set B B(C) = C 21 / 33
49 Piecewise Boolean diagrams: topos Consider contextual sets over piecewise Boolean algebra B assignment of set S(C) to each C Sub(B) such that C D implies S(C) S(D) They form a topos T (B)! category whose objects behave a lot like sets in particular, it has a logic of its own! There is one canonical contextual set B B(C) = C T (B) believes that B is an honest Boolean algebra! A topos for algebraic quantum theory Communications in Mathematical Physics 291:63 110, / 33
50 Operator algebra C*-algebras: main examples of piecewise Boolean algebras. 22 / 33
51 Operator algebra -algebras: main examples of piecewise Boolean algebras. 22 / 33
52 Operator algebra -algebras: main examples of piecewise Boolean algebras. Example: C(X) = {f : X C continuous} Theorem: Every commutative -algebra is of this form. Normierte Ringe Matematicheskii Sbornik 9(51):3 24, / 33
53 Operator algebra -algebras: main examples of piecewise Boolean algebras. Example: C(X) = {f : X C continuous} Theorem: Every commutative -algebra is of this form. Example: B(H) = {f : H H continuous linear} Theorem: Every -algebra embeds into one of this form. Normierte Ringe Matematicheskii Sbornik 9(51):3 24, 1941 On the imbedding of normed rings into operators on a Hilbert space Mathematicheskii Sbornik 12(2): , / 33
54 Operator algebra -algebras: main examples of piecewise Boolean algebras. Example: C(X) = {f : X C continuous} Theorem: Every commutative -algebra is of this form. Example: B(H) = {f : H H continuous linear} Theorem: Every -algebra embeds into one of this form. piecewise Boolean algebras projections -algebras Normierte Ringe Matematicheskii Sbornik 9(51):3 24, 1941 On the imbedding of normed rings into operators on a Hilbert space Mathematicheskii Sbornik 12(2): , / 33
55 Operator algebra -algebras: main examples of piecewise Boolean algebras. Example: C(X) = {f : X C continuous} Theorem: Every commutative -algebra is of this form. Example: B(H) = {f : H H continuous linear} Theorem: Every -algebra embeds into one of this form. piecewise Boolean algebras projections -algebras Normierte Ringe Matematicheskii Sbornik 9(51):3 24, 1941 On the imbedding of normed rings into operators on a Hilbert space Mathematicheskii Sbornik 12(2): , 1943 Active lattices determine AW*-algebras Journal of Mathematical Analysis and Applications 416: , / 33
56 Operator algebra: same trick A (piecewise) -algebra A gives a dcpo Sub(A). 23 / 33
57 Operator algebra: same trick A (piecewise) -algebra A gives a dcpo Sub(A). Can characterize partial orders Sub(A) arising this way. Involves action of unitary group U(A). Characterizations of categories of commutative C*-subalgebras Communications in Mathematical Physics 331(1): , / 33
58 Operator algebra: same trick A (piecewise) -algebra A gives a dcpo Sub(A). Can characterize partial orders Sub(A) arising this way. Involves action of unitary group U(A). If Sub(A) = Sub(B), then A = B as Jordan algebras. Except C 2 and M 2. Characterizations of categories of commutative C*-subalgebras Communications in Mathematical Physics 331(1): , 2014 Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras Journal of Mathematical Analysis and Applications, 383: , / 33
59 Operator algebra: same trick A (piecewise) -algebra A gives a dcpo Sub(A). Can characterize partial orders Sub(A) arising this way. Involves action of unitary group U(A). If Sub(A) = Sub(B), then A = B as Jordan algebras. Except C 2 and M 2. If Sub(A) = Sub(B) preserves U(A) Sub(A) Sub(A), then A = B as -algebras. Needs orientation! Characterizations of categories of commutative C*-subalgebras Communications in Mathematical Physics 331(1): , 2014 Isomorphisms of ordered structures of abelian C*-subalgebras of C*-algebras Journal of Mathematical Analysis and Applications, 383: , 2011 Active lattices determine AW*-algebras Journal of Mathematical Analysis and Applications 416: , / 33
60 Scatteredness A space is scattered if every nonempty subset has an isolated point. Precisely when each continuous f : X R has countable image. Example: {0, 1, 1 2, 1 3, 1 4, 1 5,...}. Inductive Limits of Finite Dimensional C*-algebras Transactions of the American Mathematical Society 171: , 1972 Scattered C*-algebras Mathematica Scandinavica 41: , / 33
61 Scatteredness A space is scattered if every nonempty subset has an isolated point. Precisely when each continuous f : X R has countable image. Example: {0, 1, 1 2, 1 3, 1 4, 1 5,...}. A -algebra A is scattered if X is scattered for all C(X) Sub(A). Precisely when each self-adjoint a = a A has countable spectrum. Example: K(H) + 1 H Inductive Limits of Finite Dimensional C*-algebras Transactions of the American Mathematical Society 171: , 1972 Scattered C*-algebras Mathematica Scandinavica 41: , / 33
62 Scatteredness A space is scattered if every nonempty subset has an isolated point. Precisely when each continuous f : X R has countable image. Example: {0, 1, 1 2, 1 3, 1 4, 1 5,...}. A -algebra A is scattered if X is scattered for all C(X) Sub(A). Precisely when each self-adjoint a = a A has countable spectrum. Example: K(H) + 1 H Nonexample: C(Cantor) is approximately finite-dimensional Nonexample: C([0, 1]) is not even approximately finite-dimensional Inductive Limits of Finite Dimensional C*-algebras Transactions of the American Mathematical Society 171: , 1972 Scattered C*-algebras Mathematica Scandinavica 41: , / 33
63 Scatteredness Theorem: the following are equivalent for a -algebra A: Sub(A) is algebraic Sub(A) is continuous Sub(A) is meet-continuous Sub(A) is quasi-algebraic Sub(A) is quasi-continuous Sub(A) is atomistic A is scattered A characterization of scattered C*-algebras and application to crossed products Journal of Operator Theory 63(2): , 2010 Domains of commutative C*-subalgebras Logic in Computer Science, ACM/IEEE Proceedings , / 33
64 Back to quantum logic For -algebra C(X), projections are clopen subsets of X. Can characterize in order-theoretic terms: (if X 3) closed subsets of X = ideals of C(X) = elements of Sub(C(X)) clopen subsets of X = good pairs of elements of Sub(C(X)) Compactifications and functions spaces Georgia Institute of Technology, / 33
65 Back to quantum logic For -algebra C(X), projections are clopen subsets of X. Can characterize in order-theoretic terms: (if X 3) closed subsets of X = ideals of C(X) = elements of Sub(C(X)) clopen subsets of X = good pairs of elements of Sub(C(X)) Each projection of -algebra A is in some maximal C Sub(A). Can recover poset of projections from Sub(A)! (if dim(z(a)) 3) Compactifications and functions spaces Georgia Institute of Technology, 1996 C(A) Radboud University Nijmegen, / 33
66 Back to piecewise Boolean domains Sub(B) determines B (B = B Sub(B) = Sub(B )) (shape of parts determines whole) Caveat: not 1-1 correspondence! Subalgebras of orthomodular lattices Order 28: , / 33
67 Back to piecewise Boolean domains Sub(B) determines B (B = B Sub(B) = Sub(B )) (shape of parts determines whole) Caveat: not 1-1 correspondence! If B Boolean algebra, then Sub(B) partition lattice Caveat: not constructive, not categorical Subalgebras of orthomodular lattices Order 28: , 2011 On the lattice of subalgebras of a Boolean algebra Proceedings of the American Mathematical Society 36: 87 92, / 33
68 Different kinds of atoms If B = , then Sub(B) = / 33
69 Different kinds of atoms B B B B B B B B B B B B B B / 33
70 Principal pairs Reconstruct pairs (x, x) of B: principal ideal subalgebra of B is of the form x 1 x they are the elements p of Sub(B) that are dual modular and (p m) n = p (m n) for n p atom or relative complement a m = a, a m = B for atom a 0 29 / 33
71 Principal pairs Reconstruct pairs (x, x) of B: principal ideal subalgebra of B is of the form x 1 x they are the elements p of Sub(B) that are dual modular and (p m) n = p (m n) for n p atom or relative complement a m = a, a m = B for atom a 0 Reconstruct elements x of B: principal pairs of B are (p, q) with atomic meet x q p 1 p x q 0 29 / 33
72 Principal pairs Reconstruct pairs (x, x) of B: principal ideal subalgebra of B is of the form x 1 x they are the elements p of Sub(B) that are dual modular and (p m) n = p (m n) for n p atom or relative complement a m = a, a m = B for atom a 0 Reconstruct elements x of B: principal pairs of B are (p, q) with atomic meet x q p 1 p x q 0 Theorem: B Pp(Sub(B)) for Boolean algebra B of size 4 D Sub(Pp(D)) for Boolean domain D of size 2 29 / 33
73 Directions 1 If B is v w x y z v w x y z 0 then Sub(B) is 30 / 33
74 Directions If B is v w x y z v w x y z or v w x v w x x y z x y z then Sub(B) is 30 / 33
75 Directions If B is v w x y z v w x y z or v w x v w x x y z x y z then Sub(B) is A direction for a Boolean domain is map d: D D 2 with d(1) = (p, q) is a principal pair d(m) = (p m, q m) 30 / 33
76 Directions If B is v w x y z v w x y z or v w x v w x x y z x y z then Sub(B) is A direction for a piecewise Boolean domain is 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 30 / 33
77 Orthoalgebras Almost theorem: B Dir(Sub(B)) for orthoalgebra B of size 4 D Sub(Dir(D)) for piecewise orthodomain D of size 2 Problems: 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 Boolean subalgebras of orthoalgebras Ongoing work 31 / 33
78 Conclusion Should consider piecewise Boolean algebras Give rise to domain of honest Boolean subalgebras Complicated structure, but can characterize Shape of parts enough to determine whole Same trick works for scattered operator algebras Direction needed for almost categorical equivalence 32 / 33
79 Conclusion Should consider piecewise Boolean algebras Give rise to domain of honest Boolean subalgebras Complicated structure, but can characterize Shape of parts enough to determine whole Same trick works for scattered operator algebras Direction needed for almost categorical equivalence 32 / 33
80 Question Theorem: any Boolean algebra is isomorphic to the global sections of a sheaf on its Stone space Question: is any piecewise Boolean algebra isomorphic to the global sections of a sheaf on its Stone space? Would give logic of contextuality The theory of representations of Boolean algebras Transactions of the American Mathematical Society 40:37 111, 1936 Representations of algebras by continuous sections Bulletin of the American Mathematical Society 78(3): , 1972 The sheaf-theoretic structure of nonlocality and contextuality New Journal of Physics 13:113036, 2011? 33 / 33
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