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  • 1
    Electronic Resource
    Electronic Resource
    Jauch-Piron property (everywhere!) in the logicoalgebraic foundation of quantum theories (1993)
    Springer
    International journal of theoretical physics 32 (1993), S. 1985-1991 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Jauch-Piron property of states on a quantum logic is seen to be of considerable importance within the foundation of quantum theories. In this survey we summarize and comment on recent results on the Jauch-Piron property. We also pose a few open problems whose solution may help in further developing quantum theories and noncommutative measure theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00979520
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  • 2
    Electronic Resource
    Electronic Resource
    Concrete quantum logics with covering properties (1992)
    Springer
    International journal of theoretical physics 31 (1992), S. 843-854 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL∈ C n , if for any pairA, B∈L we can find a collection {C i ∶ i∈I}, where cardI〈n andC i ∈L for anyi∈I, such thatA ∩B=∪ i∈l C i . Thus, in a certain sense, ifL∈C n , then “the rate of noncompatibility” of an arbitrary pairA,B∈L is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC n and exhibit typical examples. In particular, for a givenn we construct examples ofL∈ C n+1\C n . Further, we discuss the relation of the classesC n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00678549
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  • 3
    Electronic Resource
    Electronic Resource
    States on orthoalgebras (1995)
    Springer
    International journal of theoretical physics 34 (1995), S. 1439-1465 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The orthoalgebras, introduced by Foulis and Randall and studied by various authors, have recently become a significant mathematical structure of the logicoalgebraic foundation of quantum theories. In this paper we give a coherent account of states (=finitely additive measures) on orthoalgebras. In the first section we review basic properties of (and constructions with) orthoalgebras and develop a useful “pasting technique” (Theorem 1.12 and Proposition 1.16) applied later in this paper (and possibly elsewhere, too). We also exhibit orthoalgebras with rather interesting and “exotic” state spaces (Example 1.20 and Proposition 1.21). In the second section we construct orthoalgebras with preassigned state space properties. We prove a state representation theorem (Theorem 2.1) and obtain an orthoalgebraic version of Shultz's theorem (Theorem 2.7). In the third section we make a thorough analysis of the extension problem for states on orthoalgebras. We first study the orthoalgebras whose state spaces are finite dimensional. For these orthoalgebras we find a necessary and sufficient condition to allow extensions of states over larger orthoalgebras (Theorem 3.4). Then we prove that all Hilbertian orthoalgebras as well as all Boolean orthoalgebras allow extensions of states over larger orthoalgebras (Theorems 3.10 and 3.12).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00676255
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  • 4
    Electronic Resource
    Electronic Resource
    Quantum logics with the Radon-Nikodym property (1988)
    Springer
    Order 4 (1988), S. 387-395 
    Details
    ISSN: 1572-9273
    Keywords: Primary: 81B10 ; secondary: 03G12 ; 28A25 ; Central observables on a quantum logic ; absolute continuity of states ; Radon-Nikodym theorem for logics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Following the definition of S. Gudder and J. Zerbe, we say that a logicL has the Radon-Nikodym property (or, in short,L is an RN logic) if the following condition holds: Ifs, t are states onL ands is absolutely continuous with respect tot, then there is a central observablex such that $$s(a) = \int {_a x dt}$$ for allaεL. We first consider general RN logics. We establish their basic properties and show that they are closed under the formation of epimorphisms and products. Then we take up the RN property for concrete logics. We first show that in many cases the concrete RN logics have to be ‘fully compatible’ (= Boolean σ-algebras). In contrast to that, we show that there are concrete RN logics with an arbitrary degree of noncompatibility. This extends the result of Navara and Pták and answers in full the question posed by Gudder and Zerbe.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00714479
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  • 5
    Electronic Resource
    Electronic Resource
    Concrete Quantum Logics (2000)
    Springer
    International journal of theoretical physics 39 (2000), S. 827-837 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Concrete quantum logics are quantum logics which allow for a set representation.They seem to be of significant conceptual value within quantum axiomatics andthey play an important role in the theory of orthomodular structures asset-representable orthomodular posets or lattices and they also sometimes constitutea “domain” for investigations in “noncommutative.” measure theory. This paperpresents a survey of recent results on this class of logics. Stress is put on thealgebraic and measure-theoretic aspects. Several open questions relevant tothe logicoalgebraic foundation of quantum theories are posed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1003626929648
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  • 6
    Electronic Resource
    Electronic Resource
    Introduction (1995)
    Springer
    International journal of theoretical physics 34 (1995), S. 1163-1163 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00676226
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  • 7
    Electronic Resource
    Electronic Resource
    Quantum Logics with Given Centers and Variable State Spaces (1998)
    Springer
    International journal of theoretical physics 37 (1998), S. 139-145 
    Details
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We ask which logics with a given center allowfor enlargements with an arbitrary state space. We showthat these are precisely those logics the center ofwhich possesses a two-valued state and the state space of which is nonempty. This extends theresults of Binder as well as our previous results andsupplements the results of Foulis and Ptak and ofNavara, Ptak, and Rogalewicz. We also comment on some related questions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1026629726261
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  • 8
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    On the Radon–Nikodym property for σ‐classes (1983)
    American Institute of Physics
    Details
    Publication Date: 1983-06-01
    Print ISSN: 0022-2488
    Electronic ISSN: 1089-7658
    Topics: Mathematics , Physics
    Published by American Institute of Physics
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  • 9
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    On the Farkas lemma and the Horn–Tarski measure-extension theorem (2015)
    Elsevier
    Details
    Publication Date: 2015-09-01
    Print ISSN: 0024-3795
    Electronic ISSN: 1873-1856
    Topics: Mathematics
    Published by Elsevier
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  • 10
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    Almost Boolean orthomodular posets (1989)
    Elsevier
    Details
    Publication Date: 1989-09-01
    Print ISSN: 0022-4049
    Electronic ISSN: 1873-1376
    Topics: Mathematics
    Published by Elsevier
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