Skip to main content
SpringerLink
Log in

Examples, problems, and results in effect algebras

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

This article discusses various unsolved problems and conjectures that have arisen in the study of effect algebras (orD-posets) during the last few years. We also include some examples, counterexamples, and results that motivate or partially solve these problems. The problems mainly concern sharp and principal elements, the existence of infima in Hilbert space effect algebras, tensor products, and interval algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Beltrametti, E., and Cassinelli, G. (1981).The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Bennett, M. K., and Foulis, D. (n.d.-a). Phi-symmetric effect algebras,Foundations of Physics.

  • Bennett, M. K., and Foulis, D. (n.d.-b). Interval algebras and unsharp quantum logics.

  • Busch, P., Lahti, P., and Mittelstaedt, P. (1991).The Quantum Theory of Measurements, Springer-Verlag, Berlin.

    Google Scholar 

  • 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.

    Google Scholar 

  • Davies, E. B. (1976).Quantum Theory of Open Systems, Academic Press, New York.

    Google Scholar 

  • Dvurečenskij, A. (1995). Tensor products of difference posets,Transactions of the American Mathematical Society,347, 1043–1057.

    MathSciNet  Google Scholar 

  • Dvurečenskij, A., and Pulmannová, S. (1994). Difference posets, effects and quantum measurements,International Journal of Theoretical Physics,33, 819–850.

    MathSciNet  Google Scholar 

  • Foulis, D. (1989). Coupled physical systems,Foundations of Physics,19, 905–922.

    Article  MathSciNet  Google Scholar 

  • Foulis, D., and Bennett, M. K. (1994). Effect algebra and unsharp quantum logics,Foundations of Physics,24, 1331–1352.

    Article  MathSciNet  Google Scholar 

  • Foulis, D., Greechie, R., and Bennett, M. K. (1994). Sums and products of interval algebras,International Journal of Theoretical Physics,33, 2119–2136.

    Article  MathSciNet  Google Scholar 

  • Fuchs, L. (1963).Partially Ordered Algebraic Systems, Pergamon Press, Oxford.

    Google Scholar 

  • Giuntini, R., and Greuling, H. (1989). Toward a formal language for unsharp properties,Foundations of Physics,19, 931–945.

    Article  MathSciNet  Google Scholar 

  • Goodearl, K. (1986).Partially Ordered Abelian Groups, American Mathematical Society, Providence, Rhode Island.

    Google Scholar 

  • Greechie, R. (1971). Orthomodular lattices admitting no states,Journal of Combinatorial Theory,10, 119–132.

    MATH  MathSciNet  Google Scholar 

  • Greechie, R., and Foulis, D. (1995). The transition to effect algebras,International Journal of Theoretical Physics,34, 1–14.

    Article  MathSciNet  Google Scholar 

  • Greechie, R., Foulis, D., and Pulmannová, S. (n.d.). The center of an effect algebras,Order.

  • Gudder, S. (n.d.-a). Chain tensor products and interval effect algebras.

  • Gudder, S. (n.d.-b). Lattice properties of quantum effects.

  • Gudder, S., and Greechie, R. (n.d.). Effect algebra counterexamples,Mathematica Slovaca.

  • Gudder, S., and Moreland, T. (n.d.). Existence of infima for quantum effects.

  • Holevo, A. S. (1982).Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam.

    Google Scholar 

  • Kadison, R. (1951). Order properties of bounded self-adjoint operators,Proceedings of the American Mathematical Society,34, 505–510.

    MathSciNet  Google Scholar 

  • Kôpka, F. (1992).D-posets and fuzzy sets,Tatra Mountain Mathematical Publications,1, 83–87.

    MATH  MathSciNet  Google Scholar 

  • Kôpka, F., and Chovanec, F. (1994).D-posets,Mathematica Slovaca,44, 21–34.

    MathSciNet  Google Scholar 

  • Kraus, K. (1983).States, Effects, and Operations, Springer-Verlag, Berlin.

    Google Scholar 

  • 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.

    Article  ADS  MathSciNet  Google Scholar 

  • Ludwig, G. (1983/1985).Foundations of Quantum Mechanics, Vols. I and II, Springer-Verlag, Berlin.

    Google Scholar 

  • Mackey, G. (1963).The Mathematical Foundations of Quantum Mechanics, Benjamin, New York.

    Google Scholar 

  • Pták, P., and Pulmannová, S. (1991).Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht.

    Google Scholar 

  • Topping, D. (1965). Vector lattices of self-adjoint operators,Transactions of the American Mathematical Society,115, 14–30.

    MATH  MathSciNet  Google Scholar 

  • Varadarajan, V. (1968/1970).Geometry of Quantum Theory, Vols. 1 and 2, Van Nostrand Reinhold, Princeton, New Jersey.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Denver, 80208, Denver, Colorado

    Stanley Gudder

Authors
  1. Stanley Gudder
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gudder, S. Examples, problems, and results in effect algebras. Int J Theor Phys 35, 2365–2376 (1996). https://doi.org/10.1007/BF02302453

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02302453

Keywords

Search

Navigation