zbMATH — the first resource for mathematics

Examples, problems, and results in effect algebras. (English) Zbl 0868.03028
Various unsolved problems and conjectures in the theory of effect algebras (or D-posets) concerning mainly sharp and principal elements, the existence of suprema (or infima) in Hilbert space effect algebras, tensor products and interval algebras are discussed. Also some examples, counterexamples and results are included that motivate or partially solve these problems.
Reviewer: H.Länger (Wien)

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
Full Text: DOI
[1] Beltrametti, E., and Cassinelli, G. (1981).The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts. · Zbl 0491.03023
[2] Bennett, M. K., and Foulis, D. (n.d.-a). Phi-symmetric effect algebras,Foundations of Physics. · Zbl 0883.03048
[3] Bennett, M. K., and Foulis, D. (n.d.-b). Interval algebras and unsharp quantum logics. · Zbl 1213.06004
[4] Busch, P., Lahti, P., and Mittelstaedt, P. (1991).The Quantum Theory of Measurements, Springer-Verlag, Berlin. · Zbl 0868.46051
[5] Cattaneo, G., and Nisticò, G. (1985). Complete effect-preparation structures: Attempt of an unification of two different approaches to axiomatic quantum mechanics,Nuovo Cimento 90B, 1661–175.
[6] Davies, E. B. (1976).Quantum Theory of Open Systems, Academic Press, New York. · Zbl 0388.46044
[7] Dvurečenskij, A. (1995). Tensor products of difference posets,Transactions of the American Mathematical Society,347, 1043–1057. · Zbl 0859.03031
[8] Dvurečenskij, A., and Pulmannová, S. (1994). Difference posets, effects and quantum measurements,International Journal of Theoretical Physics,33, 819–850. · Zbl 0806.03040
[9] Foulis, D. (1989). Coupled physical systems,Foundations of Physics,19, 905–922.
[10] Foulis, D., and Bennett, M. K. (1994). Effect algebra and unsharp quantum logics,Foundations of Physics,24, 1331–1352. · Zbl 1213.06004
[11] Foulis, D., Greechie, R., and Bennett, M. K. (1994). Sums and products of interval algebras,International Journal of Theoretical Physics,33, 2119–2136. · Zbl 0815.06015
[12] Fuchs, L. (1963).Partially Ordered Algebraic Systems, Pergamon Press, Oxford. · Zbl 0137.02001
[13] Giuntini, R., and Greuling, H. (1989). Toward a formal language for unsharp properties,Foundations of Physics,19, 931–945.
[14] Goodearl, K. (1986).Partially Ordered Abelian Groups, American Mathematical Society, Providence, Rhode Island. · Zbl 0589.06008
[15] Greechie, R. (1971). Orthomodular lattices admitting no states,Journal of Combinatorial Theory,10, 119–132. · Zbl 0219.06007
[16] Greechie, R., and Foulis, D. (1995). The transition to effect algebras,International Journal of Theoretical Physics,34, 1–14. · Zbl 0839.03049
[17] Greechie, R., Foulis, D., and Pulmannová, S. (n.d.). The center of an effect algebras,Order. · Zbl 0846.03031
[18] Gudder, S. (n.d.-a). Chain tensor products and interval effect algebras. · Zbl 0884.03060
[19] Gudder, S. (n.d.-b). Lattice properties of quantum effects. · Zbl 0879.47045
[20] Gudder, S., and Greechie, R. (n.d.). Effect algebra counterexamples,Mathematica Slovaca.
[21] Gudder, S., and Moreland, T. (n.d.). Existence of infima for quantum effects. · Zbl 0964.46010
[22] Holevo, A. S. (1982).Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam. · Zbl 0497.46053
[23] Kadison, R. (1951). Order properties of bounded self-adjoint operators,Proceedings of the American Mathematical Society,34, 505–510. · Zbl 0043.11501
[24] Kôpka, F. (1992).D-posets and fuzzy sets,Tatra Mountain Mathematical Publications,1, 83–87. · Zbl 0797.04011
[25] Kôpka, F., and Chovanec, F. (1994).D-posets,Mathematica Slovaca,44, 21–34. · Zbl 0789.03048
[26] Kraus, K. (1983).States, Effects, and Operations, Springer-Verlag, Berlin. · Zbl 0545.46049
[27] Lahti, P., and Maczynski, M. (1995). On the order structure of the set of effects in quantum mechanics,Journal of Mathematical Physics,36, 1673–1680. · Zbl 0829.46060
[28] Ludwig, G. (1983/1985).Foundations of Quantum Mechanics, Vols. I and II, Springer-Verlag, Berlin.
[29] Mackey, G. (1963).The Mathematical Foundations of Quantum Mechanics, Benjamin, New York. · Zbl 0114.44002
[30] Pták, P., and Pulmannová, S. (1991).Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht. · Zbl 0743.03039
[31] Topping, D. (1965). Vector lattices of self-adjoint operators,Transactions of the American Mathematical Society,115, 14–30. · Zbl 0137.10301
[32] Varadarajan, V. (1968/1970).Geometry of Quantum Theory, Vols. 1 and 2, Van Nostrand Reinhold, Princeton, New Jersey. · Zbl 0155.56802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.