We present three results stating when a concrete (= set-representable) quantum
logic with covering properties (generalization of compatibility) has to be a Boolean algebra.
These results complete and generalize some previous results [3, 5] and answer partially a
question posed in [2]

Tkadlec, Josef

Mathematica Bohemica

Digital Library of the Czech Technical University in Prague

Mathematica Bohemica 123 (1998), 213–218CONCRETE QUANTUM LOGICS WITH GENERALISEDCOMPATIBILITYJosef Tkadlec, Praha(Received April 16, 1997)Abstract. We present three results stating when a concrete (= set-representable) quantumlogic with covering properties (generalization of compatibility) has to be a Boolean algebra.These results complete and generalize some previous results [3, 5] and answer partially aquestion posed in [2].Keywords: Boolean algebra, concrete quantum logic, covering, Jauch–Piron state, ortho-completenessMSC 1991 : 03G12, 81P101. Basic notionsLet us recall the main notion we shall deal with in this paper.Definition 1.1. A concrete logic is a pair (X,L), where X 6= ∅ and L ⊂ expXsuch that(1) ∅ ∈ L;(2) Ac = X \A ∈ L whenever A ∈ L;(3)⋃M ∈ L whenever M ⊂ L is a finite set of mutually disjoint elements.A concrete σ-logic is a concrete logic (X,L) such that(3σ)⋃M ∈ L whenever M ⊂ L is a countable set of mutually disjoint elements.Let us note that the above definition is not given in the most efficient way. In-deed, since ∅ is a finite set of mutually disjoint elements and ⋃ ∅ = ∅, condition (1)follows from condition (3). Moreover, it is obvious that condition (3) follows fromcondition (3σ).The author acknowledges the support by the grant 201/96/0117 of the Czech Grant Agency.1The following lemma will be useful in the sequel. First, let us observe that ifA,B ∈ L and A ⊂ B, then B \A = (A ∪Bc)c ∈ L for every concrete logic (X,L).Lemma 1.2. Let (X,L) be a concrete σ-logic and let Ai ∈ L (i = 1, 2, . . .) besuch that A1 ⊃ A2 ⊃ · · ·. Then⋂∞i=1Ai ∈ L.P r o o f . The elements Ai \ Ai+1 ∈ L (i = 1, 2, . . .) are mutually orthogonal,hence⋃∞i=1(Ai \Ai+1) ∈ L and⋂∞i=1Ai = A1 \⋃∞i=1(Ai \Ai+1) ∈ L. 2. Covering propertiesDefinition 2.1. Let (X,L) be a concrete logic, Y ⊂ X and let n be a naturalnumber. A covering of Y is a set M ⊂ L such that Y = ⋃M . A covering M is ann-covering if cardM ≤ n.We say that (X,L) has the n-covering property (finite covering property , resp.) iffor every A,B ∈ L there is an n-covering (finite covering, resp.) of A ∩B.It is well-known that a concrete logic (X,L) is a Boolean algebra if and only ifA ∩B ∈ L for every A,B ∈ L, i.e., if and only if (X,L) has the 1-covering property.Thus, the notions of n-covering property (finite covering property), introduced in [3],are generalizations of compatibility in Boolean algebras.The next lemma will be used in the sequel.Lemma 2.2. Let (X,L) be a concrete logic with the finite covering property.Then for every finite set F ⊂ L there is a finite covering G ⊂ L of ⋂F .P r o o f . Let us proceed by induction. First, if F is a one-element subset of L(empty set, resp.), then we can put G = F (G = {X}, resp.).Now, let us suppose that there is a natural number n ≥ 1 such that the lemmaholds for every F ⊂ L with cardF = n. Let F ⊂ L with cardF = n + 1 and letA ∈ F . According to the previous assumption, there is a finite covering G ⊂ L of⋂(F \ {A}). According to the finite covering property, for every B ∈ G there is afinite covering GB ⊂ L of A∩B. Thus,⋃B∈GGB ⊂ L is a finite covering of⋂F .Before we present the main result of this section, let us prove the following tech-nical lemma.2Lemma 2.3. Let (X,L) be a concrete σ-logic and let m,n ≥ 2 be natural num-bers such that m ≤ n+ 1. Let us suppose that for every set F ⊂ L with cardF ≤ nthere is an m-coverig G ⊂ L of ⋂F . Then for every set F ⊂ L with cardF ≤ n thereis an (m− 1)-covering G ⊂ L of ⋂F .P r o o f . Let F ⊂ L with cardF ≤ n. Let us define by induction sequences(Ai1, . . . , Ain) ∈ Ln, (Bi0, . . . , Bin) ∈ Ln+1 (i = 1, 2, . . .) as follows: Let (A11, . . . , A1n)be such that F = {A11, . . . , A1n}. If (Ai1, . . . , Ain) ∈ Ln is defined for a natural num-ber i ≥ 1 then let us take (Bi0, . . . , Bin) ∈ Ln+1 such that Bij = ∅ for j ≥ m and⋂nj=1Aij =⋃nj=0Bij and let us put Ai+1,j = Aij \Bij (j ∈ {1, . . . , n}).Let us denoteB0 =∞⋂i=1Bi0, Bj =∞⋃i=1Bij , j ∈ {1, . . . , n}.It is easy to see that the elements B1j , B2j , . . . (j ∈ {1, . . . , n}) are mutually disjoint,hence Bj ∈ L for every j ∈ {1, . . . , n}. Moreover, Bm = · · · = Bn = ∅. Further,Bi0 ⊃n⋂j=1Ai+1,j ⊃ Bi+1,0 (i = 1, 2, . . .).Hence, according to Lemma 1.2, B0 ∈ L, too. Since⋂F = B0 ∪B1 ∪ · · · ∪Bm−1and since B0 ∪B1 ∈ L (B0 ∩B1 = ∅), the proof is complete. Theorem 2.4. Let (X,L) be a concrete σ-logic. Let us suppose that there isa natural number n ≥ 2 such that for any set F ⊂ L with cardF ≤ n there is an(n+ 1)-covering of⋂F . Then (X,L) is a Boolean algebra.P r o o f . Using Lemma 2.3 n-times, we obtain that (X,L) has the 1-coveringproperty, i.e., (X,L) is a Boolean algebra. Corollary 2.5. Every concrete σ-logic with the 3-covering property is a Booleanalgebra.This corollary generalizes [3, Proposition 4.6], where an analogous result is statedfor concrete σ-logics with the 2-covering property.33. Covering properties and Jauch–Piron statesDefinition 3.1. Let (X,L) be a concrete logic. A state on (X,L) is a mappings: L→ [0, 1] such that(1) s(X) = 1;(2) s(⋃M) =∑A∈M s(A) whenever M ⊂ L is a finite set of mutually disjointelements.A state s on (X,L) is called Jauch–Piron if for every A,B ∈ L with s(A) = s(B) = 1there is a C ∈ L such that C ⊂ A ∩B and s(C) = 1.It is easy to see that s(∅) = 0 and s(Ac) = 1−s(A) for every state s on a concretelogic (X,L) and for every A ∈ L \ {∅}. Further, for every concrete logic (X,L), everypoint x ∈ X carries a two-valued state sx on (X,L) defined bysx(A) ={1, if x ∈ A,0, if x /∈ A.Before we present the main result of this section, we need the following definition.Definition 3.2. Let (X,L) be a concrete logic and let M,N ⊂ L be two coveringsof Y ⊂ X. We say that N is a coarsing of M if for every A ∈ M there is a B ∈ Nsuch that A ⊂ B.Theorem 3.3. Let (X,L) be a concrete logic such that every state on it is Jauch–Piron. Let us suppose that for every A,B ∈ L every covering of A ∩ B admits acountable coarsing. Then L is a Boolean algebra.P r o o f . It suffices to prove that A∩B ∈ L for every A,B ∈ L. Let A,B ∈ L. IfA∩B = ∅, the proof is complete. Let us suppose that A∩B 6= ∅. Then SA,B = {s; sis a state on (X,L) with s(A) = s(B) = 1} is nonempty (every point x ∈ A ∩ Bcarries a two-valued state sx ∈ SA,B). Since every state on (X,L) is Jauch–Piron,for every s ∈ SA,B there is a Cs ∈ L such that s(Cs) = 1. Let us take a countablecoarsing M of the covering {Cs; s ∈ SA,B} of A ∩ B, a countable set Y ⊂ A ∩ Bsuch that Y ∩ (C \D) 6= ∅ for every C,D ∈M with C \D 6= ∅ and, finally, a state sthat is a σ-convex combination (with non-zero coefficients) of all sy (y ∈ Y ). Sinces ∈ SA,B, there is a Ds ∈ M such that s(Ds) = 1. Thus, Ds ⊃ Y and thereforeA ∩B = ⋃M = Ds ∈ L. Theorem 3.3 seems to be independent of the previous results in [3, 4, 7], never-theless it has corollaries that were obtained using quite a different techniques. (Letus note that a unifying look at these attempts is presented in [8].) The followingcorollary was obtained (in a more general form) in [4].4Corollary 3.4. Every countable concrete logic such that every state on it isJauch–Piron is a Boolean algebra.The next corollary of Theorem 3.3 was obtained (in a more general form) in [7].Corollary 3.5. Let (X,L) be a concrete logic such that every state on it is Jauch–Piron. Let us suppose that (X,L) contains only countably many maximal Booleansubalgebras and these are complete. Then (X,L) is a Boolean algebra.P r o o f . It is easy to see that for every A,B ∈ L every covering of A∩B admitsa countable coarsing. 4. Covering properties and orthocompletenessDefinition 4.1. Let α be a cardinal number. A concrete logic (X,L) is calledα-orthocomplete if∨M ∈ L (supremum with respect to inclusion) whenever M ⊂ Lis a set of mutually disjoint elements with cardM ≤ α.It is obvious that condition (3σ) of Definition 1.1 implies that a concrete σ-logicis ω0-orthocomplete (ω0 denotes the countable cardinal) — this is usually denoted asσ-orthocompleteness.The following theorem generalizes a result from [5] and answers partially a ques-tion posed in [2].Theorem 4.2. Every c-orthocomplete (c denotes the cardinality of real numbers)concrete σ-logic with the finite covering property is a Boolean algebra.P r o o f . Let (X,L) be a concrete σ-logic with the finite covering property andlet A,B ∈ L. It suffices to prove that A ∩ B ∈ L. Let us define by induction finitesubsets Fi (i = 1, 2, . . .) of L as follows: First, F1 ⊂ L is a finite covering of A ∩ B.Now, let a finite set Fi = {A1, . . . , An} ⊂ L be defined for a natural number i ≥ 1.Let us denote by Gi the set of all intersections of the form Ae11 ∩ · · · ∩ Aenn , where(e1, . . . , en) ∈ {−1, 1}n \ {−1}n and A1j = Aj , A−1j = X \ Aj (j = 1, . . . , n). Gi isa finite set of mutually disjoint subsets of X such that⋂Fi =⋃Gi. According toLemma 2.2, for every Y ∈ Gi there is a finite covering GY ⊂ L of Y . Let us putFi+1 =⋃Y ∈Gi GY .Let us consider all sequences C1, C2, . . . such that Ci ∈ Fi (i = 1, 2, . . .) andC1 ⊃ C2 ⊃ · · ·. According to Lemma 1.2,⋂∞i=1Ci ∈ L for each such sequence. Hence,we have obtained at most the continuum of mutually disjoint elements of L suchthat their union is A ∩ B. Since their supremum exists, it is equal to A ∩ B. Thus,A ∩B ∈ L. 5Before we present a corollary of Theorem 4.2, let us recall a result connecting thecovering properties with Jauch–Piron states [3, Theorem 3.5].Theorem 4.3. Let (X,L) be a concrete logic such that every two-valued stateon it is Jauch–Piron. Then (X,L) has the finite covering property.Corollary 4.4. Every c-orthocomplete concrete σ-logic such that every two-valuedstate on it is Jauch–Piron is a Boolean algebra.P r o o f . It follows from Theorem 4.3 and Theorem 4.2. R e m a r k 4.5. The above corollary can be stated in the following (more general)way: Every c-orthocomplete quantum σ-logic with a closed full set of two-valuedJauch–Piron σ-states is a Boolean algebra. Indeed, concrete σ-logics are exactly rep-resentations of quantum σ-logics with a full set of two-valued σ-states (see e.g. [1, 6])and Theorem 4.3 can be stated for quantum logics with a closed full set of two-valuedJauch–Piron states (the set of two-valued states is closed in the product topology in[0, 1]L).The following question (posed in [2]) remains open. Here we have given thenegative answer in the case that the concrete logic in question is also c-orthocomplete.Q u e s t i o n 4.6. Is there a concrete σ-logic that is not a Boolean algebra suchthat every state on it is Jauch–Piron?References[1] S. P. Gudder : Stochastic Methods in Quantum Mechanics. North Holland, New York,1979.[2] V. Mu¨ller : Jauch–Piron states on concrete quantum logics. Int. J. Theor. Phys. 32 (1993),433–442.[3] V. Mu¨ller, P. Pta´k, J. Tkadlec: Concrete quantum logics with covering properties. Int. J.Theor. Phys. 31 (1992), 843–854.[4] M. Navara, P. Pta´k : Almost Boolean orthomodular posets. J. Pure Appl. Algebra 60(1989), 105–111.[5] P. Pta´k : Some nearly Boolean orthomodular posets. Proc. Amer. Math. Soc., to appear.[6] P. Pta´k, S. Pulmannova´: Orthomodular Structures as Quantum Logics. Kluwer, Dor-drecht, 1991.[7] J. Tkadlec: Boolean orthoposets—concreteness and orthocompleteness. Math. Bohem. 119(1994), 123–128.[8] J. Tkadlec: Conditions that force an orthomodular poset to be a Boolean algebra. TatraMt. Math. Publ. 10 (1997), 55–62.Author’s address: Josef Tkadlec, Department of Mathematics, Faculty of Electrical En-gineering, Czech Technical University, Technicka´ 2, 166 27 Praha 6, Czech Republic, e-mail:tkadlec@math.feld.cvut.cz.6

Concrete Quantum Logics with Generalised Compatibiliy

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