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Representation of Concrete Logics and Concrete Generalized Orthomodular Posets
REPORTS ON MATHEMATICAL PHYSICS
Vol. 73 (2014)
No. 2
REPRESENTATION OF CONCRETE LOGICS AND CONCRETE GENERALIZED ORTHOMODULAR POSETS ˇ ´ 2 and E. V INCEKOV A´ 1 S. P ULMANNOV A´ 1 , Z. R IE CANOV A ˇ anikova 49, 814 73 Bratislava, Slovakia Institute, Slovak Academy of Sciences, Stef´ (e-mails: [email protected], [email protected]) 2 Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkoviˇcova 3, SK-812 19 Bratislava, Slovakia (e-mail: [email protected])
1 Mathematical
(Received October 23, 2013 – RRevised January 10, 2014) In the present paper, we deal with the question when an effect algebra, resp. a generalized effect algebra, can be represented in the projection lattice of a Hilbert space. We show that such representability is closely related to the existence of a rich set of two-valued and Jauch–Piron states, resp. generalized two-valued and Jauch–Piron states. AMS Classification: Primary 81Q10, Secondary 03G12. Keywords: quantum logic, orthomodular poset, effect algebra, generalized effect algebra, (weak) generalized orthomodular poset, concrete logic, concrete (generalized) orthomodular poset, state, generalized state, two-valued (generalized) state, order determining system of (generalized) states.
1.
Introduction
Quantum logics were introduced in early thirties as the mathematical models of quantum events. Owing to the famous Heisenberg uncertainty relations, it was recognized that the classical rules of the Kolmogorov probability theory are not satisfied by quantum mechanical measurements. Therefore it was necessary to find a suitable generalization of Boolean algebras in order to describe quantum events. Quantum logics, or orthomodular sigma-lattices from the mathematical point of view, can be viewed as a most natural nondistributive abstraction of the set of projection operators on a Hilbert space, which is the basis of the traditional von Neumann approach to quantum mechanics. Effect algebras (EAs) were introduced by Foulis and Bennett [4] in order to model unsharp quantum measurements, too. The prototype of effect algebras is the set of quantum effects, that is, self-adjoint operators between the zero and identity operator on a Hilbert space. Quantum effects play an important role in the mathematical description of quantum measurements, as the most general mathematical model of 1 supported 2 supported
by APVV-0178-11 and VEGA-grant 2/0059/12. by APVV-0178-11 and VEGA-grant 1/0297/11. [225]
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quantum observables are the normalized positive-operator-valued measures (POVMs) [2], which have their ranges in the set of quantum effects. Effect algebras are a generalization of many algebraic structures which arise in mathematical physics, as well as other branches of mathematics, in particular Boolean algebras, orthomodular posets and lattices appearing in noncommutative measure theory, and also MV-algebras in fuzzy measure theory. An account of the axiomatic approach to quantum mechanics, employing EAs and the closely related D-posets [11], can be found in [3]. Several authors have encountered, studied, or employed algebraic structures that, roughly speaking, are EAs without a largest element, i.e. generalized effect algebras (GEAs). Going back to M. H. Stone’s work [19] on generalized Boolean algebras, which was extended by M. F. Janowitz [9] to generalized orthomodular lattices, more recent works include: D. Foulis and M. Bennett [4] (positive cones in partially ordered abelian groups), J. Hedl´ıkov´a and S. Pulmannov´a [8] (generalized orthoalgebras), G. Kalmbach and Z. Rieˇcanova [10] (abelian RI-posets and abelian RI-semigroups), F. Kˆopka and F. Chovanec [11] (D-posets), A. Mayet-Ippolito [12] (generalized orthomodular posets), M. Polakoviˇc and Z. Rieˇcanova [16] (densely defined positive operators) and A. Wilce [20] (cancellative positive partial abelian semigroups). Increased interest in GEAs can be attributed to the discovery that certain systems of (possibly) unbounded positive symmetric operators on a Hilbert space, e.g. operators that represent quantum observables and states, can be organized into GEAs [17]. Recently, it has been shown in [18] that every effect algebra possessing an order determining set of states can be embedded into an effect algebra of quantum effects. In more detail, if E is an effect algebra and S is an order determining set of states on E, then there is an injective effect algebra morphism from E into the effect algebra of multiplication operators between the zero and identity operator on the complex Hilbert space #2 (S ). In [14], the question of embedding of an MV-algebra into a set of quantum effects was investigated. It was shown that for every Archimedean MV-algebra, we can choose for the order determining set of states an order determining set of extremal states. This enables us to show that there is an injective MV-algebra morphism into the effect algebra of all multiplication operators between the zero and identity operator on #2 (S ) which, as a maximal set of commuting effects, is in fact an MV-algebra. In [13], the question of representability of a generalized effect algebra in a generalized effect algebra of operators densely defined on a complex Hilbert space was studied. It was shown that a generalized effect algebra is representable in the operator generalized effect algebra GD (H) if and only if it has an order determining set of generalized states. In [6], Greechie has found an example of a finite orthomodular lattice with an order determining set of states which is not representable in the lattice L(H) of all closed subspaces (orthogonal projections, equivalently) of a separable complex Hilbert space.
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In the present paper, we deal with the question when an effect algebra, resp. a generalized effect algebra, can be represented in the projection lattice of a Hilbert space. We show that such representability is closely related to the existence of a rich set of two-valued and Jauch–Piron states, resp. generalized two-valued and Jauch–Piron states. 2.
Preliminaries
DEFINITION 2.1 ([15, Definition 1.1.1]). A quantum logic (a logic, for short) is a set L endowed with a partial order ≤ and a unary operation : L → L (called orthocomplementation) such that the following conditions are satisfied (the symbols ∨, ∧ denote the lattice-theoretic operations): (i) L possesses a least and a greatest element, 0 and 1, and 0 = 1; (ii) a ≤ b implies b ≤ a for any a, b ∈ L; (iii) (a ) = a for any a ∈ L; (iv) if (ai )i∈N is a countable subset of L such that ai ≤ aj for i = j , then the 4 supremum i∈N ai exists in L; (v) if a, b ∈ L and a ≤ b, then b = a ∨ (b ∧ a ). Elements a, b in L are called orthogonal if a ≤ b . A family {ai : i ∈ I } of elements of L is orthogonal if they are pairwise orthogonal. We will sometimes write a ⊥ b if a and b are orthogonal. In other words, quantum logics are σ -orthomodular posets (σ -OMPs, for short). If property (iv) holds only for finite subsets, then L is an orthomodular poset (OMP). Property (v) is the orthomodular law. An OMP which is a lattice is an orthomodular lattice (OML). A σ -OMP which is a lattice, is in fact a σ -lattice, hence a σ -orthomodular lattice (σ -OML). EXAMPLE 2.1. A Boolean algebra (B; ∨, ∧, , 0, 1) is an OML with the Boolean complement as an orthocomplementation. Indeed, distributivity is stronger than the orthomodular law. Similarly, a Boolean σ -algebra is a σ -OML, hence a quantum logic. A logic which is a Boolean algebra will be called a Boolean logic. Elements a, b in an OMP L are compatible if there are mutually orthogonal elements a1 , b1 , c in L such that a = a1 ∨ c, b = b1 ∨ c. The following theorem gives an intrinsic characterization of Boolean logics. THEOREM 2.1 ([15, Proposition 1.3.13]). A logic L is a Boolean σ -algebra if and only if every pair {a, b} ⊂ L is compatible. In the same way, an OMP L is a Boolean algebra iff3 every pair {a, b} ⊂ L is compatible. EXAMPLE 2.2. The lattice L(H ) of all projection operators on a Hilbert space H (real or complex), or equivalently, the lattice of all closed subspaces of H , is a logic, 3 As
usual, ‘iff” stands for ‘if and only if”.
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with the relation ≤ given by inclusion and with the orthocomplementation given by the formation of the orthocomplement in H . EXAMPLE 2.3. Let ! be a nonempty set and be a collection of subsets of ! satisfying the following conditions: 1. ∅ ∈ ; 2. if A ∈ then ! \ A ∈ ; 3. if {A7 i :∈ N} ⊆ is a countable family of mutually disjoint subsets of !, then i∈N Ai ∈ ; then is a logic, which is called a concrete logic. That is, a logic is called concrete if it admits a representation as a collection of subsets of a set ! [15]. Sometimes we write L = (!, ) for a concrete logic L. Similarly, if the axiom (3) holds only for finite subsets, then we speak about a concrete OMP. DEFINITION 2.2 ([15, Definition 2.1.1]). A state (resp. a two-valued state) on a logic L is a mapping s : L → [0, 1] (resp. a mapping s : L → {0, 1}) such that (i) s(1) = 1; (ii) if 4(ai )i∈N is a sequence of mutually orthogonal elements in L, then s( i∈N ai ) = i∈N s(ai ). The state in the above definition is σ -additive. If property (ii) holds only for finite sequences, namely has the form (ii) if a ⊥ b then s(a ∨ b) = s(a) + s(b), then s is finitely additive. In physics, we deal mostly with σ -additive states. Let S denote a (nonempty) set of states on a logic L. The set S is called order determining (or ordering) if s(a) ≤ s(b) for all s ∈ S (L) implies a ≤ b. The following characterization of concrete logics was proved in [7] (see also [15, Theorem 2.2.1]). THEOREM 2.2. (i) A logic L is isomorphic (as a logic) to a concrete logic if and only if it admits an order determining set S of two-valued σ -additive states. (ii) An OMP is isomorphic (as an OMP) to a concrete OMP if and only if it admits an order determining set S of finitely additive two-valued states. 3.
Effect algebras and their representations It turns out that OMPs form a special subclass of a more general class of effect algebras. DEFINITION 3.1 ([4]). An effect algebra (EA) is a partial algebra (E; ⊕, 0, 1) where E is a nonempty set, 0, 1 are special elements of E and the partial operation ⊕ satisfies the following properties: (E1) If a ⊕ b is defined then b ⊕ a is defined and a ⊕ b = b ⊕ a. (E2) If a ⊕ b and (a ⊕ b) ⊕ c is defined, then b ⊕ c and a ⊕ (b ⊕ c) is defined and (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
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(E3) For every a ∈ E there is a unique a ∈ E with the property a ⊕ a = 1. (E4) If a ⊕ 1 is defined then a = 0. EXAMPLE 3.1. The most important example of effect algebras is an effect algebra of Hilbert space effects, i.e. the set E (H ) of self-adjoint operators between 0 and I . The partial operation ⊕ is defined for a, b ∈ E (H ) iff a + b ≤ I , and in this case a ⊕ b := a + b. Clearly, the projection lattice L(H ) is a subset of E (H ). We say that elements a, b ∈ E are orthogonal (a ⊥ b) if a ⊕ b is defined. An effect algebra is partially ordered by the relation a ≤ b if there is c ∈ E with a ⊕ c = b. If such c exists, it is unique, and we may write c = b a. Owing to associativity (E2), we need not write parentheses in expressions like a1 ⊕ a2 ⊕ a3 , and owing to commutativity (E1), the ⊕-sums do not depend on the ordering of their summands. The finite sums a1 ⊕ a2 ⊕ · · · ⊕ an can be defined by recurrence. A finite sequence a1 , a2 , . . . , an of elements (not necessarily all different) is called orthogonal if the sum a1 ⊕ · · · ⊕ an is defined in E. An infinite subfamily {ai : i ∈ I } of E is called orthogonal if every finite subfamily 4 of it is orthogonal, and we say that it is summable if the element ⊕i∈I ai := F ⊆I ⊕i∈F ai , where the supremum goes over all finite subsets F ⊆ I , exists in E. An effect algebra E is σ -orthocomplete (orthocomplete) if any countable (arbitrary) orthogonal family admits a sum. A state on an EA E is a mapping s : E → [0, 1] such that (1) s(1) = 1 and (2) s(a ⊕ b) = s(a) + s(b) whenever a ⊕ b is defined. A state s is σ -additive if s(⊕i∈N ai ) = i∈N s(ai ) whenever ⊕i∈N ai exists. Similarly as for OMPs, a set S of states on E is order determining or ordering if s(a) ≤ s(b) for all s ∈ S implies a ≤ b. A characterization of OMPs among effect algebras is the following [3, Theorem 1.5.5]. THEOREM 3.1. An effect algebra (E; ⊕, 0, 1) can be organized into an OMP such that a ⊕ b = a ∨ b provided a ⊥ b if and only if the coherence law: p ⊥ q, q ⊥ r, r ⊥ p &⇒ ∃p ⊕ q ⊕ r holds. Notice that by a morphism of effect algebras L, E we mean a mapping φ : L → E such that for all a, b ∈ L with a ⊥ b it holds φ(a ⊕ b) = φ(a) ⊕ φ(b). If such φ is a bijection and φ −1 is also an effect algebra morphism, then φ is an isomorphism of effect algebras. By an embedding of an effect algebra E1 into an effect algebra E2 we mean a mapping φ : E1 → E2 such that a ⊥ b iff φ(a) ⊥ φ(b) and φ(a ⊕ b) = φ(a) ⊕ φ(b). Clearly, the range of φ is a sub-effect algebra of E2 . Moreover, φ : E1 → E2 is a σ -embedding if ⊕i∈N ai exist iff ⊕i∈N φ(ai ) exists and φ(⊕i∈N ai ) = ⊕i∈N φ(ai ). The range of φ is then an orthocomplete sub-EA of E2 . An embedding is called a lattice embedding if it preserves existing lattice operations, i.e. φ(a ∨ b) = φ(a) ∨ φ(b) (φ(a ∧ b) = φ(a) ∧ φ(b)) whenever a ∨ b (a ∧ b) exists.
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Recall that, for any set M, the set l2 (M) = {(xm )m∈M : xm ∈ C,
|xm |2 < ∞}
m∈M
with the inner product (xm )m∈M , (ym )m∈M = m∈M x¯m ym , is a Hilbert space. The subspace Elin (M) = {(xm )m∈M ∈ l2 (M)|xm = 0 for all but finitely many m ∈ M} is dense in l2 (M). In [18, Theorem 4] the following representation theorem was proved. THEOREM 3.2. Every effect algebra (E; ⊕, 0, 1) with an order determining set S of states can be embedded into the Hilbert space effect algebra E (#2 (S )) of self-adjoint operators between the null and identity operator on #2 (S ). We note that the embedding is obtained as follows. Define φ : E → E (#2 (S )) by putting φ(e)(xs )s∈S = (s(e)xs )s∈S for every x = (xs )s∈S ∈ #2 (S ). We see that, for every e ∈ E, the operator φ(e) acts on the elements in #2 (S ) as a multiplication by the function fe : S → [0, 1] defined by fe (s) := s(e), s ∈ S . Clearly, the multiplication operators mutually commute. The effect algebra E (#2 (S )) is in fact orthocomplete, and we can extend the representation theorem as follows. COROLLARY 3.1. Let E be a σ -orthocomplete effect algebra with an ordering set of σ -additive states S . Then there is a σ -embedding of E into the Hilbert space effect algebra E (#2 (S )) of self-adjoint operators between the null and identity operator on #2 (S ). Let us denote by Tf the multiplication operator corresponding to the function f : S → [0, 1]. Then Tf is an effect, so that it is self-adjoint, and Tf is a projection iff it is idempotent, that is, Tf2 = Tf . This implies that f (s)2 = f (s), and hence f (s) ∈ {0, 1} for every s ∈ S . This yields the following statement. THEOREM 3.3. An effect algebra E with an ordering set of states S admits a representation according to Theorem 3.2 in the projection lattice of #2 (S ) if and only if the states in S are two-valued. LEMMA 3.1. An effect algebra E with an ordering set of two-valued states is an OMP. Proof : Assume that a1 ⊕ a2 , a2 ⊕ a3 , a3 ⊕ a1 exist. For every s ∈ S , if s(ai ) = 1 for one of i ∈ {1, 2, 3} then s admits value 0 on the remaining two ai s. Therefore s(a1 ⊕ a2 ) ≤ 1 − s(a3 ) = s(a3 ) for every s ∈ S , hence a1 ⊕ a2 ⊕ a3 exists. By Theorem 3.1, E is an OMP. THEOREM 3.4. A σ -orthocomplete effect algebra E with an ordering set S of σ -additive states admits a representation in the projection lattice of #2 (S ) if and only if E is a concrete logic.
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Proof : By Theorem 3.3, E admits a representation in the set of projections in #2 (S ) if and only if the states in S are two-valued. By Lemma 3.1, E is an OMP, and since E is σ -orthocomplete, it is a logic. Finally, by Theorem 2.2, E is a concrete logic. A state s on E is called prime if whenever a ∧ b exists, we have s(a ∧ b) = min{s(a), s(b)}. A state s on E is called Jauch–Piron if s(a) = 1 = s(b) for a, b ∈ E implies that there is c ∈ E, c ≤ a, c ≤ b with s(c) = 1. (Equivalently, if s(a) = 0 = s(b), then there is c ∈ E such that a ≤ c, b ≤ c and s(c) = 0). It is easy to see that if E is a lattice, then a two-valued state s is Jauch–Piron iff it is prime. According to [15, Theorem 2.5.3a (ii)], every lattice ordered concrete Jauch–Piron logic (logic in which every state is Jauch–Piron) is Boolean. Recall that a block B in an OML L is a maximal set of pairwise commuting elements of L, which is a Boolean subalgebra of L (e.g. [15]). It follows that for any elements a, b ∈ B, the lattice operations a ∨ b, a ∧ b in B coincide with the operations in the whole L. Let E be an effect algebra which admits an order determining set S of twovalued Jauch–Piron states. Let φ : E → E (#2 (S )) be the representation of E. Let Mf denote the multiplication operator which acts as multiplication by the function f : Mf (xs )s∈S = (f (s)xs )s∈S . Denote by Em the set {Mf : 0 ≤ f ≤ 1}. Then Em is a sub-effect algebra of E (#2 (S )), and we have φ(E) ⊆ Em . Moreover, the set Pm := {Mf : f (s) ∈ {0, 1}∀s ∈ S } ⊆ Em is the set of all multiplication projections. For every e ∈ E, φ(e) = Mfe , where fe (s) = s(e), s ∈ S . Since the states s ∈ S are two-valued, Mfe ∈ Pm for all e ∈ E. Since the states are Jauch–Piron, we have for a, b ∈ E, fa∧b (s) = s(a ∧ b) = min(s(a), s(b)) = min(fa (s), fb (s)), s ∈ S . Now Mmin(fa ,fb ) is the infimum of Mfa , Mfb in the set of all multiplication projections Pm , and we have Mmin(fa ,fb ) = Mfa∧b . Since the set of all multiplication operators is a maximal set of pairwise commuting operators on #2 (S ) ([14]), Pm is a block in the OML of all projections on #2 (S ). This yields φ(a ∧ b) = φ(a) ∧ φ(b), where the latter infimum is taken in the lattice of projections P (l2 (S )) on #2 (S ). This yields the following theorem. THEOREM 3.5. A lattice logic (i.e. σ -OML) L with an ordering set of states S admits a lattice embedding into the projection lattice of #2 (S ) if and only if the states in S are two-valued Jauch–Piron states. Consequently, L is Boolean. 4.
Generalized concrete OMPs
DEFINITION 4.1 ([3, Definition 1.2.1]). A generalized effect algebra (GEA) (E, +, 0) is a set E with a partial binary operation + and an element 0 ∈ E such that: (gea1) x + y = y + x if one side is defined (commutativity); (gea2) x + (y + z) = (x + y) + z if one side is defined (associativity); (gea3) if x + y = x + z, then y = z (cancellativity); (gea4) if x + y = 0, then x = y = 0 (positivity);
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(gea5) x + 0 = x for all x ∈ E. Elements x, y ∈ E are said to be orthogonal iff x + y exists. On a GEA E we may define a binary relation by x ≤ y ⇔ ∃z ∈ E : x + z = y and then an operation − by y − x = z ⇔ x + z = y. Clearly, y − x is defined iff x ≤ y. The relation ≤ is a partial ordering in E with respect to which 0 is the smallest element in E. A GEA E becomes an effect algebra iff there is a greatest element 1 ∈ E. A subset Q ⊆ E of a GEA E is called a sub-generalized effect algebra (subGEA) of E iff 0 ∈ Q and if for elements x, y, z ∈ E with x + y = z at least two are in Q then x, y, z ∈ Q. Then Q is a GEA in its own right. Let (E1 ; +1 , 01 ) and (E2 ; +2 , 02 ) be generalized effect algebras. A map f : E1 → E2 is called a morphism iff f (a +1 b) = f (a) +2 f (b) for any a, b ∈ E1 with a +1 b defined. Notice that if f is a morphism, then for all a, b ∈ E1 , a ≤ b implies f (a) ≤ f (b). A morphism f : E1 → E2 is called an isomorphism iff f is a bijection and the inverse mapping is also a morphism. A morphism f : E1 → E2 is called an embedding iff f (E1 ) is a sub-GEA of E2 and f : E1 → f (E1 ) is an isomorphism. A morphism f : E1 → E2 is called order reflecting iff f (a) ≤ f (b) implies a ≤ b for all a, b ∈ E1 . A morphism f : E1 → E2 is called orthogonality reflecting iff f (a) ⊥ f (b) implies a ⊥ b. A morphism f : E1 → E2 is an embedding iff it is order and orthogonality reflecting. A morphism of effect algebras is order reflecting iff it is orthogonality reflecting, and an order reflecting morphism is an embedding. In what follows we consider special subclasses of GEAs, namely weak generalized orthomodular posets, generalized orthomodular posets and generalized Boolean algebras. The notion of a (weak) generalized orthomodular poset has been introduced in [12]. DEFINITION 4.2 ([15, Definition 1.5.12]). A weak generalized orthomodular poset (WGOMP) is a poset (P ; ≤, 0) such that (wgomp1) for every a ∈ P , the interval ([0, a], ≤,$a , 0, a) is an OMP; (wgomp2) a, b, c ∈ P : a ≤ b ≤ c ⇒ a $b = b ∧ a $c ; elements a, b ∈ P are said to be orthogonal (a ⊥ b) if there exists an element c of P such that a, b ≤ c and a ≤ b$c , (wgomp3) a ⊥ b ⇒ a ∨ b exists; (wgomp4) a ⊥ b, b ⊥ c, c ⊥ a ⇒ a ∨ b ⊥ c.
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DEFINITION 4.3. A WGOMP is a GOMP (generalized orthomodular poset), if it moreover satisfies the condition: (gomp) a ⊥ c, b ⊥ c and existence of a ∨ b imply a ∨ b ⊥ c. By Theorem 1.5.13 in [3] we have the following result. THEOREM 4.1. Let (P ; +, 0) be a GEA. For every a ∈ P define x $a := a −x (x ∈ P , x ≤ a). Then P is a WGOMP if and only if the following conditions are satisfied: (W1) If a, b ∈ P and a ⊥ b, then a + b is the supremum of a and b. (W2) If a, b, c ∈ P are such that a ⊥ b, b ⊥ c, c ⊥ a, then a + b ⊥ c. On a GEA E we define a generalized state as in [13]. DEFINITION 4.4. A map s : E → R+ 0 on a GEA E is called a generalized state, if s is a morphism (that is, s(a + b) = s(a) + s(b), where R+ 0 is considered as a GEA with its usual addition operation). We say that a generalized state s is Jauch–Piron iff s(a) = s(b) = 0 for some a, b ∈ E imply that there exists an element c of E such that c ≥ a, c ≥ b and s(c) = 0. We say that a generalized state s is two-valued iff s(a) ∈ {0, 1} for all a ∈ E. A set of generalized states S on a GEA E is called order determining (or ordering), if s(a) ≤ s(b) ∀s ∈ S ⇒ a ≤ b for any elements a, b ∈ E; S is said to be bounded iff, for any element a ∈ E, there is a positive real number ca such that s(a) ≤ ca ∀s ∈ S . A set of two-valued generalized states S is called orthogonality determining, iff it satisfies s(a) + s(b) ≤ 1 ∀ s ∈ S ⇒ a ⊥ b. If a set of two-valued generalized states is order determining and orthogonality determining, we will call it an OOD system (set) for short. EXAMPLE 4.1. To see that there are systems of two-valued generalized states on a GEA, which are order but not orthogonality determining, or which are orthogonality but not order determining, let us consider a GEA E in Fig. 1. Let first S be the set of two-valued generalized states: S = {s1 , s2 }, where s1 (0) = 0, s2 (0) = 0,
s1 (a) = 1, s2 (a) = 0,
s1 (b) = 0, s2 (b) = 1,
s1 (c) = 1, s2 (c) = 1,
s1 (d) = 1, s2 (d) = 1.
Then the system S is orthogonality determining, but it is not order determining, because s1 (c) = s1 (d), s2 (c) = s2 (d), but c = d. Now let the set S of two-valued
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s c =a+b C C sd a s Cs b @ A @A @ s 0A Fig. 1.
generalized states be S = {s1 , s2 , s3 , s4 }, where s1 (0) = 0, s2 (0) = 0, s3 (0) = 0, s4 (0) = 0,
s1 (a) = 1, s2 (a) = 0, s3 (a) = 1, s4 (a) = 0,
s1 (b) = 0, s2 (b) = 1, s3 (b) = 0, s4 (b) = 0,
s1 (c) = 1, s2 (c) = 1, s3 (c) = 1, s4 (c) = 0,
s1 (d) = 1, s2 (d) = 0, s3 (d) = 0, s4 (d) = 1.
Then this system is order determining, but not orthogonality determining. Indeed, si (b) + si (d) ≤ 1 for all i = 1, 2, 3, 4, but b ⊥ d. If we add s5 to the system S , where s5 (0) = 0, s5 (a) = 0, s5 (b) = 1, s5 (c) = 1, s5 (d) = 1, then the system is OOD. LEMMA 4.1. If a GEA E is upward directed (in particular, if E is a lattice or E is an effect algebra) then a system S of generalized two-valued states is order determining if and only if it is orthogonality determining. Proof : Let S be an order determining system of two-valued states. Assume that for a, b ∈ E, s(a) + s(b) ≤ 1 ∀s ∈ S . There is c ∈ E with a, b ≤ c. Then s(c) = 0 implies s(a) = s(b) = 0, whence s(a) + s(b) ≤ s(c) for all s ∈ S . It follows that s(a) ≤ s(c) − s(b) = s(c − b) for all s, whence a ≤ c − b = b$c , hence a ⊥ b. Now let S be orthogonality determining, and assume s(a) ≤ s(b) ∀s ∈ S . Let c ∈ E be such that a, b ≤ c. Then s(c − b) ≤ s(c − a) = s(c) − s(a) implies s(c − b) + s(a) ≤ s(c) ≤ 1 for all s, and so (c − b) + a is defined, and from a ⊥ b$c in [0, c] we get a ≤ b. LEMMA 4.2. A GEA with an OOD set of two-valued generalized states is a WGOMP. Proof : We need to show properties (W1) and (W2) from Theorem 4.1. (W1): If a ⊥ b, then we have a, b ≤ a + b; let us assume there is another c : a, b ≤ c. Then we have s(a) ≤ s(c), s(b) ≤ s(c), but also s(a)+s(b) = s(a +b) ≤ 1. Thus we see that s(a) and s(b) cannot be both 1 and if s(a) = s(b) = 0, then s(a + b) = 0 and of course, s(a + b) ≤ s(c). If, let us say, s(a) = 1, then s(a + b) = 1 = s(c) and s(a + b) ≤ s(c) too. While this holds for every state s ∈ S and the system S is ordering, we have a + b ≤ c and thus a + b = a ∨ b.
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(W2): If a ⊥ b, b ⊥ c and c ⊥ a, then either s(a) = s(b) = s(c) = 0 or exactly one of s(a), s(b), s(c) is 1. In either case, s(a) + s(b + c) ≤ 1, hence by OOD, a + b + c exists. The following notion of a concrete WGOMP generalizes the notion of a concrete OMP, resp. a concrete logic, [15]. DEFINITION 4.5. A system of subsets of a nonempty set X is called a concrete WGOMP iff it satisfies the following properties: (gcl1) A, B ∈ ; A ∩ B = ∅ ⇒ A ∪ B ∈ . (gcl2) A, B ∈ ; A ⊆ B ⇒ B \ A ∈ . Clearly, a concrete WGOMP is a concrete OMP iff X ∈ . In the next lemma we show that a concrete WGOMP is indeed a WGOMP. LEMMA 4.3. Every system of sets satisfying properties (gcl1) and (gcl2) is a WGOMP. Proof : (wgomp1) Define the relative complement of B ∈ [0, A] by B $A = B C ∩A. So if B ⊆ C ⊆ A, then C C ⊆ B C and C $A ⊆ B $A . Also, for every B ∈ [0, A]: (B $A )$A = (B C ∩A)C ∩A = (B ∪AC )∩A = (B ∩A)∪∅ = B. Further, given two disjoint sets B1 , B2 ∈ [0, A], we evidently have B1 ∪ B2 ∈ [0, A]. And finally, if B ⊆ C ⊆ A, then (C ∩B $A )∪B = (C ∩B C ∩A)∪B = (C ∪B)∩(B C ∪B)∩(A∪B) = C ∩X∩A = C. This proves that every interval [0, A] of the concrete WGOMP is an OMP. (wgomp2) Let us have A ⊆ B ⊆ C. Then B ∩A$C = B ∩AC ∩C = B ∩AC = A$B . While A ⊥ B means that A ∩ B = ∅ here, conditions (wgomp3) and (wgomp4) are trivial. THEOREM 4.2. A GEA L is (isomorphic to) a concrete WGOMP iff there exists an OOD system of two-valued generalized states on L. Proof : If there exists an OOD system of two-valued generalized states on L, then L is a WGOMP (Lemma 4.2). Let us consider an OOD system S of two-valued generalized states on L and assign the set S(a) := {s ∈ S : s(a) = 1} to every element a ∈ L. We show that the system := {S(a) : a ∈ L} is a concrete WGOMP. If a, b ∈ L are such that S(a) ∩ S(b) = ∅, then we have s(a) = 1 ⇒ s(b) = 0 and s(b) = 1 ⇒ s(a) = 0 which means that s(a) + s(b) ≤ 1 for every s ∈ S and thus a ⊥ b and S(a + b) = {s : s(a ∨ b) = 1} = {s : s(a) = 1} ∪ {s : s(b) = 1} = S(a) ∪ S(b), which proves (gcl1). If S(a) ⊆ S(b), then s(a) = 1 ⇒ s(b) = 1 for every s ∈ S . We thus have s(a) ≤ s(b) ∀ s ∈ S , so that a ≤ b. But then S (b−a) = {s : s(b−a) = s(b)−s(a) = 1} = {s : s(b) = 1 and s(a) = 0} = {s : s(b) = 1}\{s : s(a) = 1} = S(b)\S(a), therefore (gcl2) holds.
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To prove that the mapping a *→ S(a) of L to is an isomorphism, observe that it is a bijection, and a ⊥ b iff S(a) ∩ S(b) = ∅, and S(a + b) = S(a) ∪ S(b). On the other hand, let us have a concrete WGOMP ⊆ 2X and let us assign to every element x ∈ X a mapping sx , sx (A) = 1 if x ∈ A
and
sx (A) = 0 otherwise.
To see that sx is a generalized state on for every x ∈ X, let us consider sx (A+B), that is, A, B ∈ ; A ∩ B = ∅ and sx (A + B) = sx (A ∪ B). Then sx (A ∪ B) = 0 iff x ∈ A and x ∈ B, thus sx (A) = 0 = sx (B) ⇒ sx (A) + sx (B) = sx (A ∪ B) and sx (A ∪ B) = 1 iff sx (A) = 1 or sx (B) = 1, thus again sx (A) + sx (B) = sx (A ∪ B). Now we check that the system (sx )x∈X is OOD. First, let sx (A) ≤ sx (B) ∀ x ∈ X. Then for every x ∈ X it holds x ∈ A ⇒ x ∈ B and therefore A ⊆ B. Second, let sx (A) + sx (B) ≤ 1 ∀ x ∈ X. Then sx (A) = 1 = sx (B) cannot occur and thus there is no x ∈ X such that x ∈ A and x ∈ B, which means A ∩ B = ∅, therefore A ⊥ B. THEOREM 4.3. If a WGOMP has an OOD set of two-valued generalized states S , which are Jauch–Piron, then it is a GOMP. Proof : To prove (gomp), let us consider a ⊥ c, b ⊥ c and a ∨ b exists. Then for every state s in S we have s(a) + s(c) ≤ 1 and s(b) + s(c) ≤ 1. There are three possibilities. If s(a) = 1, then s(c) = 0 and 1 = s(a) ≤ s(a ∨ b) ≤ 1. Thus s(a ∨ b) + s(c) = 1. If s(b) = 1, we get the same result. Let now s(a) = s(b) = 0. Then by Jauch–Piron property, there is some d with d ≥ a, d ≥ b and s(d) = 0. But then a ∨ b ≤ d and so 0 ≤ s(a ∨ b) ≤ s(d) = 0. Therefore s(a ∨ b) + s(c) ≤ 1 for any value of s(c), which means that a ∨ b ⊥ c. In [13] (Theorem 4.1), a representation of a GEA with an order determining set of generalized states in a generalized effect algebra of operators on a complex Hilbert space was shown. We will shortly describe some details of the representation. Let H be a complex Hilbert space and D ⊆ H be a linear subspace dense in H. A linear operator Q on a dense subspace D of H is positive if, for all x ∈ D, 0 ≤ x, Qx . Let
GD (H) := {A : D → H : A is a positive linear operator defined on D.} Then GD (H) is a generalized effect algebra where 0 is the null operator and + is the usual sum of operators defined on D (cf. [17]). Moreover, for any x ∈ D, the mapping ωx : GD (H) → R+ defined by ωx (A) = x, Ax is a generalized state on GD (H), called a generalized vector state. In [13, Definition 2.5], the following definition is introduced. Let (E; +, 0) be a generalized effect algebra. Assume further that H is a complex Hilbert space and let D ⊆ H be a linear subspace dense in H. We say that E is representable (in positive linear operators) iff there is an order reflecting morphism φ : E → GD (H).
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THEOREM 4.4. [13, Theorem 3.1 (iv), (v)] Let E be a generalized effect algebra and let S be a set of generalized states on E. Then there exists a morphism φS : E → GElin (S ) (l2 (S )) such that, for all s ∈ S , there is a vector xs ∈ l2 (S ) satisfying s = ωxs ◦ φS . Moreover, (i) if S is order determining, then φS is order reflecting (φS (a) ≤ φS (b) ⇒ a ≤ b); and (ii) if S is bounded, then for any element a ∈ E, φS (a) is a bounded operator. The morphism φS : E → GElin (S ) (l2 (S )) is defined as φS (a)(x) = (s(a)xs )s∈S for any a ∈ E and x = (xs )s∈S ∈ Elin (S ). That is, the operator φS (a) acts on Elin (S ) as a multiplication operator by the function fa : S → R+ , fa (s) = s(a). THEOREM 4.5. A generalized effect algebra E has an order determining set of two-valued generalized states S if and only if it admits a representation in the projection lattice P (l2 (S )) of l2 (S ). If, moreover, the states in S are Jauch–Piron, then φS (a ∨ b) = φS (a) ∨ φS (b) whenever a ∨ b exists in E. Proof : Assume that E has an ordering set of two-valued generalized states. For two-valued states, evidently ∀ a ∃ ca ≥ 0 (namely ca = 1) such that ∀s ∈ S , s(a) ≤ ca . Therefore φS (a) are bounded operators by Theorem 4.4 (ii). Moreover, for every a ∈ E, the operator φS (a) is positive, hence self-adjoint, and φS (a)2 (xs )s∈S = (fa2 (s)xs )s∈S = (fa (s)xs )s∈S , as fa (s) = s(a) ∈ {0, 1}. Thus φS (a) is a projection for every a ∈ E. Conversely, if E admits a representation such that for all a ∈ E, φS (a) is a projection, then φS (a)2 = φS (a) implies s(a) ∈ {0, 1}. Hence E admits an order determining set of two-valued states. Assume that, in addition, the states s ∈ S are Jauch–Piron. It is easy to see that if s is Jauch–Piron and a ∨ b exists, then s(a ∨ b) = max{s(a), s(b)}. Therefore φS (a ∨ b)(xs )s∈S = (s(a ∨ b)xs )s∈S = (max{s(a), s(b)}xs )s∈S . Similarly as in the case of effect algebras (see remarks before Theorem 3.5), we have φS (a ∨ b) = φS (a) ∨ φS (b) with the latter supremum in P (l2 (S )). THEOREM 4.6. A GEA E admits a representation in a projection lattice P (l2 (S )) of l2 (S ) which is an embedding if and only if E is a concrete WGOMP. If, in addition, the states in S are Jauch–Piron, then E is a GOMP and the representation preserves all existing suprema. Proof : If E admits a representation in a projection lattice P (l2 (S )) of l2 (S ), then E admits an order determining set of generalized two-valued states by Theorem 4.5. Assume that this representation is an embedding, then it is orthogonality reflecting and hence for all a, b ∈ E, φ(a) + φ(b) ≤ I implies a + b exists in E. But φ(a) + φ(b) ≤ I iff s(a) + s(b) ≤ 1 for all s ∈ S , which implies that S is orthogonality determining. Hence S is OOD, so by Theorem 4.2, E is a concrete WGOMP.
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Conversely, if E is a concrete WGOMP, then it has an OOD system of generalized two-valued states. By Theorem 4.5, E admits a representation φ in P (l2 (S )). Then φ is order reflecting by definition, and since φ(a) + φ(b) ≤ I iff s(a) + s(b) ≤ 1 for all s ∈ S implies, since S is OOD, that a + b is defined, φ is also orthogonality reflecting, hence an embedding. The rest follows from Theorems 4.5 and 4.3. COROLLARY 4.1. A lattice ordered WGOMP (GOMP) with an order determining system of two-valued Jauch–Piron generalized states admits a lattice embedding into the projection lattice of l2 (S ). Proof : By the last theorem we see that the representation preserves suprema, we need only to prove, it also preserves all existing infima. If a ∧ b exists, then it equals (a $a∨b ∨ b$a∨b )$a∨b = a ∨ b − ((a ∨ b − a) ∨ (a ∨ b − b)) and thus φ(a ∧ b) = φ(a ∨ b) − ((φ(a ∨ b) − φ(a)) ∨ (φ(a ∨ b) − φ(b))) = φ(a) ∧ φ(b). A generalized Boolean algebra (GBA) is a distributive lattice L with smallest element 0 such that, for p, q ∈ L with q ≤ p, the interval L[q, p] := {r ∈ L : q ≤ r ≤ p} is complemented, i.e. if r ∈ L[q, p], there exists s ∈ L[q, p] such that r ∨ s = p and r ∧ s = q [5, p. 77]. One easily proves that, if L is a distributive lattice with 0, then L is a GBA iff, for every p ∈ L, the interval L[0, p] is complemented, i.e. L[0, p] is a Boolean algebra. Every BA is a GBA; in fact, if B is a BA, b ∈ B, and c ∈ B[0, b], then b ∧ c is a complement of c in B[0, b]. Clearly, a BA is the same thing as a GBA with a largest element. Notice that the notion of a generalized orthomodular lattice (GOML) was introduced in [9]. A GOML can be described as a lattice ordered GOMP [12]. THEOREM 4.7. Every lattice ordered GEA E with an order determining system S of two-valued Jauch–Piron generalized states is a generalized Boolean algebra. Proof : By Lemma 4.1, the system S is OOD. By Lemma 4.2, E is a WGOMP, and by Theorem 4.2, E is a concrete WGOMP. Finally, since the states in S are Jauch–Piron, E is a concrete GOMP, which is lattice ordered, hence a GOML. We show that every interval [0, a], a ∈ E, is a Boolean algebra. Assume that E is isomorphic with a GOML ⊆ S (see proof of Theorem 4.2). Let A, X, Y ∈ and X, Y ⊆ A. It is easy to see that the set {sp : p ∈ A} is an order determining set of two-valued states for the OMP [∅, A]. Assume that X ∪ Y = A and let p ∈ A, p ∈ / X ∪ Y . Then sp (X) = 0, sp (Y ) = 0 and by the Jauch–Piron property, sp (X ∨ Y ) = 0, hence p ∈ / X ∨ Y . It follows that X ∨ Y ⊆ X ∪ Y . Since the converse inclusion is obvious, we have X ∨ Y = X ∪ Y , and hence X ∪ Y ∈ [∅, A]. This implies that [∅, A] is a Boolean algebra. REFERENCES [1] J. Blank, P. Exner and M. Havl´ıcˇ ek: Hilbert Space Operators in Quantum Physics (Second edition), Springer 2008.
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[2] P. Busch, P. J. Lahti and P. Mittelstaedt: The Quantum Theory of Measurement, Lecture Notes in Physics, m2, Springer, Berlin/Heidelberg/New York 1991. [3] A. Dvureˇcenskij and S. Pulmannov´a: New Trends in Quantum Structures, Kluwer Academic Publishers, Dordrecht 2000. [4] D. J. Foulis and M. K. Bennett: Effect algbebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325–1346. [5] G. Gr¨atzer: General Lattice Theory, Mathematische Reihe Band 52, Birkh¨auser, Basel 1978. [6] R. J. Greechie: Another nonstandard quantum logic (And how I found it), in Mathematical Foundations of Quantum Theory, A.R. Marlow Ed. Academic Press, New York 1978, 71-85. [7] S. Gudder: Stochastic Methods in Quantum Mechanics, Elsevier, North Holland, Amsterdam 1979. [8] J. Hedl´ıkov´a and S. Pulmannov´a: Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996), 247–279. [9] M. F. Janowitz: A note on generalized orthomodular lattices, J. Natur. Sci. and Math. 8 (1968), 89–94. [10] G. Kalmbach and Z. Rieˇcanov´a: An axiomatization for abelian relative inverses, Demonstratio Math. 27 (1994), 535–537. [11] F. Kˆopka and F. Chovanec: D-posets, Math. Slovaca 44 (1994), 21–34. [12] A. Mayet-Ippolito: Generalized orthomodular posets, Demonstratio Math. 24 (1991), 263–274. [13] J. Paseka: On realization of generalized effect algebras, Rep. Math. Phys., to appear. [14] S. Pulmannov´a: Representations of MV-algebras by Hilbert space effects, Int. J. Theor. Phys. 52 (2013), 2163–2170. [15] P. Pt´ak and S. Pulmannov´a: Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht 1991. [16] M. Polakoviˇc and Z. Rieˇcanov´a: Generalized effect algebras of positive operators densely defined on Hilbert spaces, Int. J. Theor. Phys. 50 (2011), 1167–1174. [17] Z. Rieˇcanov´a, M. Zajac and S. Pulmannov´a: Effect algebras of positive operators densely defined on Hilbert space, Rep. Math. Phys. 68 (2011), 261–270. [18] Z. Rieˇcanov´a and M. Zajac: Hilbert space-representations of effect algebras, Rep. Math. Phys. 70 (2012), 283–290. [19] M. H. Stone: Postulates for Boolean algebras and generalized Boolean algebras, Amer. J. Math. 57 (1935), 703–732. [20] A. Wilce: Perspectivity and congruence in partial abelian semigroups, Math. Slovaca 48 (1998), 117–135.
Vol. 73 (2014)
No. 2
REPRESENTATION OF CONCRETE LOGICS AND CONCRETE GENERALIZED ORTHOMODULAR POSETS ˇ ´ 2 and E. V INCEKOV A´ 1 S. P ULMANNOV A´ 1 , Z. R IE CANOV A ˇ anikova 49, 814 73 Bratislava, Slovakia Institute, Slovak Academy of Sciences, Stef´ (e-mails: [email protected], [email protected]) 2 Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkoviˇcova 3, SK-812 19 Bratislava, Slovakia (e-mail: [email protected])
1 Mathematical
(Received October 23, 2013 – RRevised January 10, 2014) In the present paper, we deal with the question when an effect algebra, resp. a generalized effect algebra, can be represented in the projection lattice of a Hilbert space. We show that such representability is closely related to the existence of a rich set of two-valued and Jauch–Piron states, resp. generalized two-valued and Jauch–Piron states. AMS Classification: Primary 81Q10, Secondary 03G12. Keywords: quantum logic, orthomodular poset, effect algebra, generalized effect algebra, (weak) generalized orthomodular poset, concrete logic, concrete (generalized) orthomodular poset, state, generalized state, two-valued (generalized) state, order determining system of (generalized) states.
1.
Introduction
Quantum logics were introduced in early thirties as the mathematical models of quantum events. Owing to the famous Heisenberg uncertainty relations, it was recognized that the classical rules of the Kolmogorov probability theory are not satisfied by quantum mechanical measurements. Therefore it was necessary to find a suitable generalization of Boolean algebras in order to describe quantum events. Quantum logics, or orthomodular sigma-lattices from the mathematical point of view, can be viewed as a most natural nondistributive abstraction of the set of projection operators on a Hilbert space, which is the basis of the traditional von Neumann approach to quantum mechanics. Effect algebras (EAs) were introduced by Foulis and Bennett [4] in order to model unsharp quantum measurements, too. The prototype of effect algebras is the set of quantum effects, that is, self-adjoint operators between the zero and identity operator on a Hilbert space. Quantum effects play an important role in the mathematical description of quantum measurements, as the most general mathematical model of 1 supported 2 supported
by APVV-0178-11 and VEGA-grant 2/0059/12. by APVV-0178-11 and VEGA-grant 1/0297/11. [225]
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quantum observables are the normalized positive-operator-valued measures (POVMs) [2], which have their ranges in the set of quantum effects. Effect algebras are a generalization of many algebraic structures which arise in mathematical physics, as well as other branches of mathematics, in particular Boolean algebras, orthomodular posets and lattices appearing in noncommutative measure theory, and also MV-algebras in fuzzy measure theory. An account of the axiomatic approach to quantum mechanics, employing EAs and the closely related D-posets [11], can be found in [3]. Several authors have encountered, studied, or employed algebraic structures that, roughly speaking, are EAs without a largest element, i.e. generalized effect algebras (GEAs). Going back to M. H. Stone’s work [19] on generalized Boolean algebras, which was extended by M. F. Janowitz [9] to generalized orthomodular lattices, more recent works include: D. Foulis and M. Bennett [4] (positive cones in partially ordered abelian groups), J. Hedl´ıkov´a and S. Pulmannov´a [8] (generalized orthoalgebras), G. Kalmbach and Z. Rieˇcanova [10] (abelian RI-posets and abelian RI-semigroups), F. Kˆopka and F. Chovanec [11] (D-posets), A. Mayet-Ippolito [12] (generalized orthomodular posets), M. Polakoviˇc and Z. Rieˇcanova [16] (densely defined positive operators) and A. Wilce [20] (cancellative positive partial abelian semigroups). Increased interest in GEAs can be attributed to the discovery that certain systems of (possibly) unbounded positive symmetric operators on a Hilbert space, e.g. operators that represent quantum observables and states, can be organized into GEAs [17]. Recently, it has been shown in [18] that every effect algebra possessing an order determining set of states can be embedded into an effect algebra of quantum effects. In more detail, if E is an effect algebra and S is an order determining set of states on E, then there is an injective effect algebra morphism from E into the effect algebra of multiplication operators between the zero and identity operator on the complex Hilbert space #2 (S ). In [14], the question of embedding of an MV-algebra into a set of quantum effects was investigated. It was shown that for every Archimedean MV-algebra, we can choose for the order determining set of states an order determining set of extremal states. This enables us to show that there is an injective MV-algebra morphism into the effect algebra of all multiplication operators between the zero and identity operator on #2 (S ) which, as a maximal set of commuting effects, is in fact an MV-algebra. In [13], the question of representability of a generalized effect algebra in a generalized effect algebra of operators densely defined on a complex Hilbert space was studied. It was shown that a generalized effect algebra is representable in the operator generalized effect algebra GD (H) if and only if it has an order determining set of generalized states. In [6], Greechie has found an example of a finite orthomodular lattice with an order determining set of states which is not representable in the lattice L(H) of all closed subspaces (orthogonal projections, equivalently) of a separable complex Hilbert space.
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In the present paper, we deal with the question when an effect algebra, resp. a generalized effect algebra, can be represented in the projection lattice of a Hilbert space. We show that such representability is closely related to the existence of a rich set of two-valued and Jauch–Piron states, resp. generalized two-valued and Jauch–Piron states. 2.
Preliminaries
DEFINITION 2.1 ([15, Definition 1.1.1]). A quantum logic (a logic, for short) is a set L endowed with a partial order ≤ and a unary operation : L → L (called orthocomplementation) such that the following conditions are satisfied (the symbols ∨, ∧ denote the lattice-theoretic operations): (i) L possesses a least and a greatest element, 0 and 1, and 0 = 1; (ii) a ≤ b implies b ≤ a for any a, b ∈ L; (iii) (a ) = a for any a ∈ L; (iv) if (ai )i∈N is a countable subset of L such that ai ≤ aj for i = j , then the 4 supremum i∈N ai exists in L; (v) if a, b ∈ L and a ≤ b, then b = a ∨ (b ∧ a ). Elements a, b in L are called orthogonal if a ≤ b . A family {ai : i ∈ I } of elements of L is orthogonal if they are pairwise orthogonal. We will sometimes write a ⊥ b if a and b are orthogonal. In other words, quantum logics are σ -orthomodular posets (σ -OMPs, for short). If property (iv) holds only for finite subsets, then L is an orthomodular poset (OMP). Property (v) is the orthomodular law. An OMP which is a lattice is an orthomodular lattice (OML). A σ -OMP which is a lattice, is in fact a σ -lattice, hence a σ -orthomodular lattice (σ -OML). EXAMPLE 2.1. A Boolean algebra (B; ∨, ∧, , 0, 1) is an OML with the Boolean complement as an orthocomplementation. Indeed, distributivity is stronger than the orthomodular law. Similarly, a Boolean σ -algebra is a σ -OML, hence a quantum logic. A logic which is a Boolean algebra will be called a Boolean logic. Elements a, b in an OMP L are compatible if there are mutually orthogonal elements a1 , b1 , c in L such that a = a1 ∨ c, b = b1 ∨ c. The following theorem gives an intrinsic characterization of Boolean logics. THEOREM 2.1 ([15, Proposition 1.3.13]). A logic L is a Boolean σ -algebra if and only if every pair {a, b} ⊂ L is compatible. In the same way, an OMP L is a Boolean algebra iff3 every pair {a, b} ⊂ L is compatible. EXAMPLE 2.2. The lattice L(H ) of all projection operators on a Hilbert space H (real or complex), or equivalently, the lattice of all closed subspaces of H , is a logic, 3 As
usual, ‘iff” stands for ‘if and only if”.
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with the relation ≤ given by inclusion and with the orthocomplementation given by the formation of the orthocomplement in H . EXAMPLE 2.3. Let ! be a nonempty set and be a collection of subsets of ! satisfying the following conditions: 1. ∅ ∈ ; 2. if A ∈ then ! \ A ∈ ; 3. if {A7 i :∈ N} ⊆ is a countable family of mutually disjoint subsets of !, then i∈N Ai ∈ ; then is a logic, which is called a concrete logic. That is, a logic is called concrete if it admits a representation as a collection of subsets of a set ! [15]. Sometimes we write L = (!, ) for a concrete logic L. Similarly, if the axiom (3) holds only for finite subsets, then we speak about a concrete OMP. DEFINITION 2.2 ([15, Definition 2.1.1]). A state (resp. a two-valued state) on a logic L is a mapping s : L → [0, 1] (resp. a mapping s : L → {0, 1}) such that (i) s(1) = 1; (ii) if 4(ai )i∈N is a sequence of mutually orthogonal elements in L, then s( i∈N ai ) = i∈N s(ai ). The state in the above definition is σ -additive. If property (ii) holds only for finite sequences, namely has the form (ii) if a ⊥ b then s(a ∨ b) = s(a) + s(b), then s is finitely additive. In physics, we deal mostly with σ -additive states. Let S denote a (nonempty) set of states on a logic L. The set S is called order determining (or ordering) if s(a) ≤ s(b) for all s ∈ S (L) implies a ≤ b. The following characterization of concrete logics was proved in [7] (see also [15, Theorem 2.2.1]). THEOREM 2.2. (i) A logic L is isomorphic (as a logic) to a concrete logic if and only if it admits an order determining set S of two-valued σ -additive states. (ii) An OMP is isomorphic (as an OMP) to a concrete OMP if and only if it admits an order determining set S of finitely additive two-valued states. 3.
Effect algebras and their representations It turns out that OMPs form a special subclass of a more general class of effect algebras. DEFINITION 3.1 ([4]). An effect algebra (EA) is a partial algebra (E; ⊕, 0, 1) where E is a nonempty set, 0, 1 are special elements of E and the partial operation ⊕ satisfies the following properties: (E1) If a ⊕ b is defined then b ⊕ a is defined and a ⊕ b = b ⊕ a. (E2) If a ⊕ b and (a ⊕ b) ⊕ c is defined, then b ⊕ c and a ⊕ (b ⊕ c) is defined and (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
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(E3) For every a ∈ E there is a unique a ∈ E with the property a ⊕ a = 1. (E4) If a ⊕ 1 is defined then a = 0. EXAMPLE 3.1. The most important example of effect algebras is an effect algebra of Hilbert space effects, i.e. the set E (H ) of self-adjoint operators between 0 and I . The partial operation ⊕ is defined for a, b ∈ E (H ) iff a + b ≤ I , and in this case a ⊕ b := a + b. Clearly, the projection lattice L(H ) is a subset of E (H ). We say that elements a, b ∈ E are orthogonal (a ⊥ b) if a ⊕ b is defined. An effect algebra is partially ordered by the relation a ≤ b if there is c ∈ E with a ⊕ c = b. If such c exists, it is unique, and we may write c = b a. Owing to associativity (E2), we need not write parentheses in expressions like a1 ⊕ a2 ⊕ a3 , and owing to commutativity (E1), the ⊕-sums do not depend on the ordering of their summands. The finite sums a1 ⊕ a2 ⊕ · · · ⊕ an can be defined by recurrence. A finite sequence a1 , a2 , . . . , an of elements (not necessarily all different) is called orthogonal if the sum a1 ⊕ · · · ⊕ an is defined in E. An infinite subfamily {ai : i ∈ I } of E is called orthogonal if every finite subfamily 4 of it is orthogonal, and we say that it is summable if the element ⊕i∈I ai := F ⊆I ⊕i∈F ai , where the supremum goes over all finite subsets F ⊆ I , exists in E. An effect algebra E is σ -orthocomplete (orthocomplete) if any countable (arbitrary) orthogonal family admits a sum. A state on an EA E is a mapping s : E → [0, 1] such that (1) s(1) = 1 and (2) s(a ⊕ b) = s(a) + s(b) whenever a ⊕ b is defined. A state s is σ -additive if s(⊕i∈N ai ) = i∈N s(ai ) whenever ⊕i∈N ai exists. Similarly as for OMPs, a set S of states on E is order determining or ordering if s(a) ≤ s(b) for all s ∈ S implies a ≤ b. A characterization of OMPs among effect algebras is the following [3, Theorem 1.5.5]. THEOREM 3.1. An effect algebra (E; ⊕, 0, 1) can be organized into an OMP such that a ⊕ b = a ∨ b provided a ⊥ b if and only if the coherence law: p ⊥ q, q ⊥ r, r ⊥ p &⇒ ∃p ⊕ q ⊕ r holds. Notice that by a morphism of effect algebras L, E we mean a mapping φ : L → E such that for all a, b ∈ L with a ⊥ b it holds φ(a ⊕ b) = φ(a) ⊕ φ(b). If such φ is a bijection and φ −1 is also an effect algebra morphism, then φ is an isomorphism of effect algebras. By an embedding of an effect algebra E1 into an effect algebra E2 we mean a mapping φ : E1 → E2 such that a ⊥ b iff φ(a) ⊥ φ(b) and φ(a ⊕ b) = φ(a) ⊕ φ(b). Clearly, the range of φ is a sub-effect algebra of E2 . Moreover, φ : E1 → E2 is a σ -embedding if ⊕i∈N ai exist iff ⊕i∈N φ(ai ) exists and φ(⊕i∈N ai ) = ⊕i∈N φ(ai ). The range of φ is then an orthocomplete sub-EA of E2 . An embedding is called a lattice embedding if it preserves existing lattice operations, i.e. φ(a ∨ b) = φ(a) ∨ φ(b) (φ(a ∧ b) = φ(a) ∧ φ(b)) whenever a ∨ b (a ∧ b) exists.
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Recall that, for any set M, the set l2 (M) = {(xm )m∈M : xm ∈ C,
|xm |2 < ∞}
m∈M
with the inner product (xm )m∈M , (ym )m∈M = m∈M x¯m ym , is a Hilbert space. The subspace Elin (M) = {(xm )m∈M ∈ l2 (M)|xm = 0 for all but finitely many m ∈ M} is dense in l2 (M). In [18, Theorem 4] the following representation theorem was proved. THEOREM 3.2. Every effect algebra (E; ⊕, 0, 1) with an order determining set S of states can be embedded into the Hilbert space effect algebra E (#2 (S )) of self-adjoint operators between the null and identity operator on #2 (S ). We note that the embedding is obtained as follows. Define φ : E → E (#2 (S )) by putting φ(e)(xs )s∈S = (s(e)xs )s∈S for every x = (xs )s∈S ∈ #2 (S ). We see that, for every e ∈ E, the operator φ(e) acts on the elements in #2 (S ) as a multiplication by the function fe : S → [0, 1] defined by fe (s) := s(e), s ∈ S . Clearly, the multiplication operators mutually commute. The effect algebra E (#2 (S )) is in fact orthocomplete, and we can extend the representation theorem as follows. COROLLARY 3.1. Let E be a σ -orthocomplete effect algebra with an ordering set of σ -additive states S . Then there is a σ -embedding of E into the Hilbert space effect algebra E (#2 (S )) of self-adjoint operators between the null and identity operator on #2 (S ). Let us denote by Tf the multiplication operator corresponding to the function f : S → [0, 1]. Then Tf is an effect, so that it is self-adjoint, and Tf is a projection iff it is idempotent, that is, Tf2 = Tf . This implies that f (s)2 = f (s), and hence f (s) ∈ {0, 1} for every s ∈ S . This yields the following statement. THEOREM 3.3. An effect algebra E with an ordering set of states S admits a representation according to Theorem 3.2 in the projection lattice of #2 (S ) if and only if the states in S are two-valued. LEMMA 3.1. An effect algebra E with an ordering set of two-valued states is an OMP. Proof : Assume that a1 ⊕ a2 , a2 ⊕ a3 , a3 ⊕ a1 exist. For every s ∈ S , if s(ai ) = 1 for one of i ∈ {1, 2, 3} then s admits value 0 on the remaining two ai s. Therefore s(a1 ⊕ a2 ) ≤ 1 − s(a3 ) = s(a3 ) for every s ∈ S , hence a1 ⊕ a2 ⊕ a3 exists. By Theorem 3.1, E is an OMP. THEOREM 3.4. A σ -orthocomplete effect algebra E with an ordering set S of σ -additive states admits a representation in the projection lattice of #2 (S ) if and only if E is a concrete logic.
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Proof : By Theorem 3.3, E admits a representation in the set of projections in #2 (S ) if and only if the states in S are two-valued. By Lemma 3.1, E is an OMP, and since E is σ -orthocomplete, it is a logic. Finally, by Theorem 2.2, E is a concrete logic. A state s on E is called prime if whenever a ∧ b exists, we have s(a ∧ b) = min{s(a), s(b)}. A state s on E is called Jauch–Piron if s(a) = 1 = s(b) for a, b ∈ E implies that there is c ∈ E, c ≤ a, c ≤ b with s(c) = 1. (Equivalently, if s(a) = 0 = s(b), then there is c ∈ E such that a ≤ c, b ≤ c and s(c) = 0). It is easy to see that if E is a lattice, then a two-valued state s is Jauch–Piron iff it is prime. According to [15, Theorem 2.5.3a (ii)], every lattice ordered concrete Jauch–Piron logic (logic in which every state is Jauch–Piron) is Boolean. Recall that a block B in an OML L is a maximal set of pairwise commuting elements of L, which is a Boolean subalgebra of L (e.g. [15]). It follows that for any elements a, b ∈ B, the lattice operations a ∨ b, a ∧ b in B coincide with the operations in the whole L. Let E be an effect algebra which admits an order determining set S of twovalued Jauch–Piron states. Let φ : E → E (#2 (S )) be the representation of E. Let Mf denote the multiplication operator which acts as multiplication by the function f : Mf (xs )s∈S = (f (s)xs )s∈S . Denote by Em the set {Mf : 0 ≤ f ≤ 1}. Then Em is a sub-effect algebra of E (#2 (S )), and we have φ(E) ⊆ Em . Moreover, the set Pm := {Mf : f (s) ∈ {0, 1}∀s ∈ S } ⊆ Em is the set of all multiplication projections. For every e ∈ E, φ(e) = Mfe , where fe (s) = s(e), s ∈ S . Since the states s ∈ S are two-valued, Mfe ∈ Pm for all e ∈ E. Since the states are Jauch–Piron, we have for a, b ∈ E, fa∧b (s) = s(a ∧ b) = min(s(a), s(b)) = min(fa (s), fb (s)), s ∈ S . Now Mmin(fa ,fb ) is the infimum of Mfa , Mfb in the set of all multiplication projections Pm , and we have Mmin(fa ,fb ) = Mfa∧b . Since the set of all multiplication operators is a maximal set of pairwise commuting operators on #2 (S ) ([14]), Pm is a block in the OML of all projections on #2 (S ). This yields φ(a ∧ b) = φ(a) ∧ φ(b), where the latter infimum is taken in the lattice of projections P (l2 (S )) on #2 (S ). This yields the following theorem. THEOREM 3.5. A lattice logic (i.e. σ -OML) L with an ordering set of states S admits a lattice embedding into the projection lattice of #2 (S ) if and only if the states in S are two-valued Jauch–Piron states. Consequently, L is Boolean. 4.
Generalized concrete OMPs
DEFINITION 4.1 ([3, Definition 1.2.1]). A generalized effect algebra (GEA) (E, +, 0) is a set E with a partial binary operation + and an element 0 ∈ E such that: (gea1) x + y = y + x if one side is defined (commutativity); (gea2) x + (y + z) = (x + y) + z if one side is defined (associativity); (gea3) if x + y = x + z, then y = z (cancellativity); (gea4) if x + y = 0, then x = y = 0 (positivity);
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(gea5) x + 0 = x for all x ∈ E. Elements x, y ∈ E are said to be orthogonal iff x + y exists. On a GEA E we may define a binary relation by x ≤ y ⇔ ∃z ∈ E : x + z = y and then an operation − by y − x = z ⇔ x + z = y. Clearly, y − x is defined iff x ≤ y. The relation ≤ is a partial ordering in E with respect to which 0 is the smallest element in E. A GEA E becomes an effect algebra iff there is a greatest element 1 ∈ E. A subset Q ⊆ E of a GEA E is called a sub-generalized effect algebra (subGEA) of E iff 0 ∈ Q and if for elements x, y, z ∈ E with x + y = z at least two are in Q then x, y, z ∈ Q. Then Q is a GEA in its own right. Let (E1 ; +1 , 01 ) and (E2 ; +2 , 02 ) be generalized effect algebras. A map f : E1 → E2 is called a morphism iff f (a +1 b) = f (a) +2 f (b) for any a, b ∈ E1 with a +1 b defined. Notice that if f is a morphism, then for all a, b ∈ E1 , a ≤ b implies f (a) ≤ f (b). A morphism f : E1 → E2 is called an isomorphism iff f is a bijection and the inverse mapping is also a morphism. A morphism f : E1 → E2 is called an embedding iff f (E1 ) is a sub-GEA of E2 and f : E1 → f (E1 ) is an isomorphism. A morphism f : E1 → E2 is called order reflecting iff f (a) ≤ f (b) implies a ≤ b for all a, b ∈ E1 . A morphism f : E1 → E2 is called orthogonality reflecting iff f (a) ⊥ f (b) implies a ⊥ b. A morphism f : E1 → E2 is an embedding iff it is order and orthogonality reflecting. A morphism of effect algebras is order reflecting iff it is orthogonality reflecting, and an order reflecting morphism is an embedding. In what follows we consider special subclasses of GEAs, namely weak generalized orthomodular posets, generalized orthomodular posets and generalized Boolean algebras. The notion of a (weak) generalized orthomodular poset has been introduced in [12]. DEFINITION 4.2 ([15, Definition 1.5.12]). A weak generalized orthomodular poset (WGOMP) is a poset (P ; ≤, 0) such that (wgomp1) for every a ∈ P , the interval ([0, a], ≤,$a , 0, a) is an OMP; (wgomp2) a, b, c ∈ P : a ≤ b ≤ c ⇒ a $b = b ∧ a $c ; elements a, b ∈ P are said to be orthogonal (a ⊥ b) if there exists an element c of P such that a, b ≤ c and a ≤ b$c , (wgomp3) a ⊥ b ⇒ a ∨ b exists; (wgomp4) a ⊥ b, b ⊥ c, c ⊥ a ⇒ a ∨ b ⊥ c.
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DEFINITION 4.3. A WGOMP is a GOMP (generalized orthomodular poset), if it moreover satisfies the condition: (gomp) a ⊥ c, b ⊥ c and existence of a ∨ b imply a ∨ b ⊥ c. By Theorem 1.5.13 in [3] we have the following result. THEOREM 4.1. Let (P ; +, 0) be a GEA. For every a ∈ P define x $a := a −x (x ∈ P , x ≤ a). Then P is a WGOMP if and only if the following conditions are satisfied: (W1) If a, b ∈ P and a ⊥ b, then a + b is the supremum of a and b. (W2) If a, b, c ∈ P are such that a ⊥ b, b ⊥ c, c ⊥ a, then a + b ⊥ c. On a GEA E we define a generalized state as in [13]. DEFINITION 4.4. A map s : E → R+ 0 on a GEA E is called a generalized state, if s is a morphism (that is, s(a + b) = s(a) + s(b), where R+ 0 is considered as a GEA with its usual addition operation). We say that a generalized state s is Jauch–Piron iff s(a) = s(b) = 0 for some a, b ∈ E imply that there exists an element c of E such that c ≥ a, c ≥ b and s(c) = 0. We say that a generalized state s is two-valued iff s(a) ∈ {0, 1} for all a ∈ E. A set of generalized states S on a GEA E is called order determining (or ordering), if s(a) ≤ s(b) ∀s ∈ S ⇒ a ≤ b for any elements a, b ∈ E; S is said to be bounded iff, for any element a ∈ E, there is a positive real number ca such that s(a) ≤ ca ∀s ∈ S . A set of two-valued generalized states S is called orthogonality determining, iff it satisfies s(a) + s(b) ≤ 1 ∀ s ∈ S ⇒ a ⊥ b. If a set of two-valued generalized states is order determining and orthogonality determining, we will call it an OOD system (set) for short. EXAMPLE 4.1. To see that there are systems of two-valued generalized states on a GEA, which are order but not orthogonality determining, or which are orthogonality but not order determining, let us consider a GEA E in Fig. 1. Let first S be the set of two-valued generalized states: S = {s1 , s2 }, where s1 (0) = 0, s2 (0) = 0,
s1 (a) = 1, s2 (a) = 0,
s1 (b) = 0, s2 (b) = 1,
s1 (c) = 1, s2 (c) = 1,
s1 (d) = 1, s2 (d) = 1.
Then the system S is orthogonality determining, but it is not order determining, because s1 (c) = s1 (d), s2 (c) = s2 (d), but c = d. Now let the set S of two-valued
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s c =a+b C C sd a s Cs b @ A @A @ s 0A Fig. 1.
generalized states be S = {s1 , s2 , s3 , s4 }, where s1 (0) = 0, s2 (0) = 0, s3 (0) = 0, s4 (0) = 0,
s1 (a) = 1, s2 (a) = 0, s3 (a) = 1, s4 (a) = 0,
s1 (b) = 0, s2 (b) = 1, s3 (b) = 0, s4 (b) = 0,
s1 (c) = 1, s2 (c) = 1, s3 (c) = 1, s4 (c) = 0,
s1 (d) = 1, s2 (d) = 0, s3 (d) = 0, s4 (d) = 1.
Then this system is order determining, but not orthogonality determining. Indeed, si (b) + si (d) ≤ 1 for all i = 1, 2, 3, 4, but b ⊥ d. If we add s5 to the system S , where s5 (0) = 0, s5 (a) = 0, s5 (b) = 1, s5 (c) = 1, s5 (d) = 1, then the system is OOD. LEMMA 4.1. If a GEA E is upward directed (in particular, if E is a lattice or E is an effect algebra) then a system S of generalized two-valued states is order determining if and only if it is orthogonality determining. Proof : Let S be an order determining system of two-valued states. Assume that for a, b ∈ E, s(a) + s(b) ≤ 1 ∀s ∈ S . There is c ∈ E with a, b ≤ c. Then s(c) = 0 implies s(a) = s(b) = 0, whence s(a) + s(b) ≤ s(c) for all s ∈ S . It follows that s(a) ≤ s(c) − s(b) = s(c − b) for all s, whence a ≤ c − b = b$c , hence a ⊥ b. Now let S be orthogonality determining, and assume s(a) ≤ s(b) ∀s ∈ S . Let c ∈ E be such that a, b ≤ c. Then s(c − b) ≤ s(c − a) = s(c) − s(a) implies s(c − b) + s(a) ≤ s(c) ≤ 1 for all s, and so (c − b) + a is defined, and from a ⊥ b$c in [0, c] we get a ≤ b. LEMMA 4.2. A GEA with an OOD set of two-valued generalized states is a WGOMP. Proof : We need to show properties (W1) and (W2) from Theorem 4.1. (W1): If a ⊥ b, then we have a, b ≤ a + b; let us assume there is another c : a, b ≤ c. Then we have s(a) ≤ s(c), s(b) ≤ s(c), but also s(a)+s(b) = s(a +b) ≤ 1. Thus we see that s(a) and s(b) cannot be both 1 and if s(a) = s(b) = 0, then s(a + b) = 0 and of course, s(a + b) ≤ s(c). If, let us say, s(a) = 1, then s(a + b) = 1 = s(c) and s(a + b) ≤ s(c) too. While this holds for every state s ∈ S and the system S is ordering, we have a + b ≤ c and thus a + b = a ∨ b.
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(W2): If a ⊥ b, b ⊥ c and c ⊥ a, then either s(a) = s(b) = s(c) = 0 or exactly one of s(a), s(b), s(c) is 1. In either case, s(a) + s(b + c) ≤ 1, hence by OOD, a + b + c exists. The following notion of a concrete WGOMP generalizes the notion of a concrete OMP, resp. a concrete logic, [15]. DEFINITION 4.5. A system of subsets of a nonempty set X is called a concrete WGOMP iff it satisfies the following properties: (gcl1) A, B ∈ ; A ∩ B = ∅ ⇒ A ∪ B ∈ . (gcl2) A, B ∈ ; A ⊆ B ⇒ B \ A ∈ . Clearly, a concrete WGOMP is a concrete OMP iff X ∈ . In the next lemma we show that a concrete WGOMP is indeed a WGOMP. LEMMA 4.3. Every system of sets satisfying properties (gcl1) and (gcl2) is a WGOMP. Proof : (wgomp1) Define the relative complement of B ∈ [0, A] by B $A = B C ∩A. So if B ⊆ C ⊆ A, then C C ⊆ B C and C $A ⊆ B $A . Also, for every B ∈ [0, A]: (B $A )$A = (B C ∩A)C ∩A = (B ∪AC )∩A = (B ∩A)∪∅ = B. Further, given two disjoint sets B1 , B2 ∈ [0, A], we evidently have B1 ∪ B2 ∈ [0, A]. And finally, if B ⊆ C ⊆ A, then (C ∩B $A )∪B = (C ∩B C ∩A)∪B = (C ∪B)∩(B C ∪B)∩(A∪B) = C ∩X∩A = C. This proves that every interval [0, A] of the concrete WGOMP is an OMP. (wgomp2) Let us have A ⊆ B ⊆ C. Then B ∩A$C = B ∩AC ∩C = B ∩AC = A$B . While A ⊥ B means that A ∩ B = ∅ here, conditions (wgomp3) and (wgomp4) are trivial. THEOREM 4.2. A GEA L is (isomorphic to) a concrete WGOMP iff there exists an OOD system of two-valued generalized states on L. Proof : If there exists an OOD system of two-valued generalized states on L, then L is a WGOMP (Lemma 4.2). Let us consider an OOD system S of two-valued generalized states on L and assign the set S(a) := {s ∈ S : s(a) = 1} to every element a ∈ L. We show that the system := {S(a) : a ∈ L} is a concrete WGOMP. If a, b ∈ L are such that S(a) ∩ S(b) = ∅, then we have s(a) = 1 ⇒ s(b) = 0 and s(b) = 1 ⇒ s(a) = 0 which means that s(a) + s(b) ≤ 1 for every s ∈ S and thus a ⊥ b and S(a + b) = {s : s(a ∨ b) = 1} = {s : s(a) = 1} ∪ {s : s(b) = 1} = S(a) ∪ S(b), which proves (gcl1). If S(a) ⊆ S(b), then s(a) = 1 ⇒ s(b) = 1 for every s ∈ S . We thus have s(a) ≤ s(b) ∀ s ∈ S , so that a ≤ b. But then S (b−a) = {s : s(b−a) = s(b)−s(a) = 1} = {s : s(b) = 1 and s(a) = 0} = {s : s(b) = 1}\{s : s(a) = 1} = S(b)\S(a), therefore (gcl2) holds.
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To prove that the mapping a *→ S(a) of L to is an isomorphism, observe that it is a bijection, and a ⊥ b iff S(a) ∩ S(b) = ∅, and S(a + b) = S(a) ∪ S(b). On the other hand, let us have a concrete WGOMP ⊆ 2X and let us assign to every element x ∈ X a mapping sx , sx (A) = 1 if x ∈ A
and
sx (A) = 0 otherwise.
To see that sx is a generalized state on for every x ∈ X, let us consider sx (A+B), that is, A, B ∈ ; A ∩ B = ∅ and sx (A + B) = sx (A ∪ B). Then sx (A ∪ B) = 0 iff x ∈ A and x ∈ B, thus sx (A) = 0 = sx (B) ⇒ sx (A) + sx (B) = sx (A ∪ B) and sx (A ∪ B) = 1 iff sx (A) = 1 or sx (B) = 1, thus again sx (A) + sx (B) = sx (A ∪ B). Now we check that the system (sx )x∈X is OOD. First, let sx (A) ≤ sx (B) ∀ x ∈ X. Then for every x ∈ X it holds x ∈ A ⇒ x ∈ B and therefore A ⊆ B. Second, let sx (A) + sx (B) ≤ 1 ∀ x ∈ X. Then sx (A) = 1 = sx (B) cannot occur and thus there is no x ∈ X such that x ∈ A and x ∈ B, which means A ∩ B = ∅, therefore A ⊥ B. THEOREM 4.3. If a WGOMP has an OOD set of two-valued generalized states S , which are Jauch–Piron, then it is a GOMP. Proof : To prove (gomp), let us consider a ⊥ c, b ⊥ c and a ∨ b exists. Then for every state s in S we have s(a) + s(c) ≤ 1 and s(b) + s(c) ≤ 1. There are three possibilities. If s(a) = 1, then s(c) = 0 and 1 = s(a) ≤ s(a ∨ b) ≤ 1. Thus s(a ∨ b) + s(c) = 1. If s(b) = 1, we get the same result. Let now s(a) = s(b) = 0. Then by Jauch–Piron property, there is some d with d ≥ a, d ≥ b and s(d) = 0. But then a ∨ b ≤ d and so 0 ≤ s(a ∨ b) ≤ s(d) = 0. Therefore s(a ∨ b) + s(c) ≤ 1 for any value of s(c), which means that a ∨ b ⊥ c. In [13] (Theorem 4.1), a representation of a GEA with an order determining set of generalized states in a generalized effect algebra of operators on a complex Hilbert space was shown. We will shortly describe some details of the representation. Let H be a complex Hilbert space and D ⊆ H be a linear subspace dense in H. A linear operator Q on a dense subspace D of H is positive if, for all x ∈ D, 0 ≤ x, Qx . Let
GD (H) := {A : D → H : A is a positive linear operator defined on D.} Then GD (H) is a generalized effect algebra where 0 is the null operator and + is the usual sum of operators defined on D (cf. [17]). Moreover, for any x ∈ D, the mapping ωx : GD (H) → R+ defined by ωx (A) = x, Ax is a generalized state on GD (H), called a generalized vector state. In [13, Definition 2.5], the following definition is introduced. Let (E; +, 0) be a generalized effect algebra. Assume further that H is a complex Hilbert space and let D ⊆ H be a linear subspace dense in H. We say that E is representable (in positive linear operators) iff there is an order reflecting morphism φ : E → GD (H).
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THEOREM 4.4. [13, Theorem 3.1 (iv), (v)] Let E be a generalized effect algebra and let S be a set of generalized states on E. Then there exists a morphism φS : E → GElin (S ) (l2 (S )) such that, for all s ∈ S , there is a vector xs ∈ l2 (S ) satisfying s = ωxs ◦ φS . Moreover, (i) if S is order determining, then φS is order reflecting (φS (a) ≤ φS (b) ⇒ a ≤ b); and (ii) if S is bounded, then for any element a ∈ E, φS (a) is a bounded operator. The morphism φS : E → GElin (S ) (l2 (S )) is defined as φS (a)(x) = (s(a)xs )s∈S for any a ∈ E and x = (xs )s∈S ∈ Elin (S ). That is, the operator φS (a) acts on Elin (S ) as a multiplication operator by the function fa : S → R+ , fa (s) = s(a). THEOREM 4.5. A generalized effect algebra E has an order determining set of two-valued generalized states S if and only if it admits a representation in the projection lattice P (l2 (S )) of l2 (S ). If, moreover, the states in S are Jauch–Piron, then φS (a ∨ b) = φS (a) ∨ φS (b) whenever a ∨ b exists in E. Proof : Assume that E has an ordering set of two-valued generalized states. For two-valued states, evidently ∀ a ∃ ca ≥ 0 (namely ca = 1) such that ∀s ∈ S , s(a) ≤ ca . Therefore φS (a) are bounded operators by Theorem 4.4 (ii). Moreover, for every a ∈ E, the operator φS (a) is positive, hence self-adjoint, and φS (a)2 (xs )s∈S = (fa2 (s)xs )s∈S = (fa (s)xs )s∈S , as fa (s) = s(a) ∈ {0, 1}. Thus φS (a) is a projection for every a ∈ E. Conversely, if E admits a representation such that for all a ∈ E, φS (a) is a projection, then φS (a)2 = φS (a) implies s(a) ∈ {0, 1}. Hence E admits an order determining set of two-valued states. Assume that, in addition, the states s ∈ S are Jauch–Piron. It is easy to see that if s is Jauch–Piron and a ∨ b exists, then s(a ∨ b) = max{s(a), s(b)}. Therefore φS (a ∨ b)(xs )s∈S = (s(a ∨ b)xs )s∈S = (max{s(a), s(b)}xs )s∈S . Similarly as in the case of effect algebras (see remarks before Theorem 3.5), we have φS (a ∨ b) = φS (a) ∨ φS (b) with the latter supremum in P (l2 (S )). THEOREM 4.6. A GEA E admits a representation in a projection lattice P (l2 (S )) of l2 (S ) which is an embedding if and only if E is a concrete WGOMP. If, in addition, the states in S are Jauch–Piron, then E is a GOMP and the representation preserves all existing suprema. Proof : If E admits a representation in a projection lattice P (l2 (S )) of l2 (S ), then E admits an order determining set of generalized two-valued states by Theorem 4.5. Assume that this representation is an embedding, then it is orthogonality reflecting and hence for all a, b ∈ E, φ(a) + φ(b) ≤ I implies a + b exists in E. But φ(a) + φ(b) ≤ I iff s(a) + s(b) ≤ 1 for all s ∈ S , which implies that S is orthogonality determining. Hence S is OOD, so by Theorem 4.2, E is a concrete WGOMP.
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ˇ ´ Z. RIECANOV ´ and E. VINCEKOVA ´ S. PULMANNOVA, A
Conversely, if E is a concrete WGOMP, then it has an OOD system of generalized two-valued states. By Theorem 4.5, E admits a representation φ in P (l2 (S )). Then φ is order reflecting by definition, and since φ(a) + φ(b) ≤ I iff s(a) + s(b) ≤ 1 for all s ∈ S implies, since S is OOD, that a + b is defined, φ is also orthogonality reflecting, hence an embedding. The rest follows from Theorems 4.5 and 4.3. COROLLARY 4.1. A lattice ordered WGOMP (GOMP) with an order determining system of two-valued Jauch–Piron generalized states admits a lattice embedding into the projection lattice of l2 (S ). Proof : By the last theorem we see that the representation preserves suprema, we need only to prove, it also preserves all existing infima. If a ∧ b exists, then it equals (a $a∨b ∨ b$a∨b )$a∨b = a ∨ b − ((a ∨ b − a) ∨ (a ∨ b − b)) and thus φ(a ∧ b) = φ(a ∨ b) − ((φ(a ∨ b) − φ(a)) ∨ (φ(a ∨ b) − φ(b))) = φ(a) ∧ φ(b). A generalized Boolean algebra (GBA) is a distributive lattice L with smallest element 0 such that, for p, q ∈ L with q ≤ p, the interval L[q, p] := {r ∈ L : q ≤ r ≤ p} is complemented, i.e. if r ∈ L[q, p], there exists s ∈ L[q, p] such that r ∨ s = p and r ∧ s = q [5, p. 77]. One easily proves that, if L is a distributive lattice with 0, then L is a GBA iff, for every p ∈ L, the interval L[0, p] is complemented, i.e. L[0, p] is a Boolean algebra. Every BA is a GBA; in fact, if B is a BA, b ∈ B, and c ∈ B[0, b], then b ∧ c is a complement of c in B[0, b]. Clearly, a BA is the same thing as a GBA with a largest element. Notice that the notion of a generalized orthomodular lattice (GOML) was introduced in [9]. A GOML can be described as a lattice ordered GOMP [12]. THEOREM 4.7. Every lattice ordered GEA E with an order determining system S of two-valued Jauch–Piron generalized states is a generalized Boolean algebra. Proof : By Lemma 4.1, the system S is OOD. By Lemma 4.2, E is a WGOMP, and by Theorem 4.2, E is a concrete WGOMP. Finally, since the states in S are Jauch–Piron, E is a concrete GOMP, which is lattice ordered, hence a GOML. We show that every interval [0, a], a ∈ E, is a Boolean algebra. Assume that E is isomorphic with a GOML ⊆ S (see proof of Theorem 4.2). Let A, X, Y ∈ and X, Y ⊆ A. It is easy to see that the set {sp : p ∈ A} is an order determining set of two-valued states for the OMP [∅, A]. Assume that X ∪ Y = A and let p ∈ A, p ∈ / X ∪ Y . Then sp (X) = 0, sp (Y ) = 0 and by the Jauch–Piron property, sp (X ∨ Y ) = 0, hence p ∈ / X ∨ Y . It follows that X ∨ Y ⊆ X ∪ Y . Since the converse inclusion is obvious, we have X ∨ Y = X ∪ Y , and hence X ∪ Y ∈ [∅, A]. This implies that [∅, A] is a Boolean algebra. REFERENCES [1] J. Blank, P. Exner and M. Havl´ıcˇ ek: Hilbert Space Operators in Quantum Physics (Second edition), Springer 2008.
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[2] P. Busch, P. J. Lahti and P. Mittelstaedt: The Quantum Theory of Measurement, Lecture Notes in Physics, m2, Springer, Berlin/Heidelberg/New York 1991. [3] A. Dvureˇcenskij and S. Pulmannov´a: New Trends in Quantum Structures, Kluwer Academic Publishers, Dordrecht 2000. [4] D. J. Foulis and M. K. Bennett: Effect algbebras and unsharp quantum logics. Found. Phys. 24 (1994), 1325–1346. [5] G. Gr¨atzer: General Lattice Theory, Mathematische Reihe Band 52, Birkh¨auser, Basel 1978. [6] R. J. Greechie: Another nonstandard quantum logic (And how I found it), in Mathematical Foundations of Quantum Theory, A.R. Marlow Ed. Academic Press, New York 1978, 71-85. [7] S. Gudder: Stochastic Methods in Quantum Mechanics, Elsevier, North Holland, Amsterdam 1979. [8] J. Hedl´ıkov´a and S. Pulmannov´a: Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996), 247–279. [9] M. F. Janowitz: A note on generalized orthomodular lattices, J. Natur. Sci. and Math. 8 (1968), 89–94. [10] G. Kalmbach and Z. Rieˇcanov´a: An axiomatization for abelian relative inverses, Demonstratio Math. 27 (1994), 535–537. [11] F. Kˆopka and F. Chovanec: D-posets, Math. Slovaca 44 (1994), 21–34. [12] A. Mayet-Ippolito: Generalized orthomodular posets, Demonstratio Math. 24 (1991), 263–274. [13] J. Paseka: On realization of generalized effect algebras, Rep. Math. Phys., to appear. [14] S. Pulmannov´a: Representations of MV-algebras by Hilbert space effects, Int. J. Theor. Phys. 52 (2013), 2163–2170. [15] P. Pt´ak and S. Pulmannov´a: Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht 1991. [16] M. Polakoviˇc and Z. Rieˇcanov´a: Generalized effect algebras of positive operators densely defined on Hilbert spaces, Int. J. Theor. Phys. 50 (2011), 1167–1174. [17] Z. Rieˇcanov´a, M. Zajac and S. Pulmannov´a: Effect algebras of positive operators densely defined on Hilbert space, Rep. Math. Phys. 68 (2011), 261–270. [18] Z. Rieˇcanov´a and M. Zajac: Hilbert space-representations of effect algebras, Rep. Math. Phys. 70 (2012), 283–290. [19] M. H. Stone: Postulates for Boolean algebras and generalized Boolean algebras, Amer. J. Math. 57 (1935), 703–732. [20] A. Wilce: Perspectivity and congruence in partial abelian semigroups, Math. Slovaca 48 (1998), 117–135.