International Journal of Theoretical Physics

, Volume 53, Issue 8, pp 2855–2866 | Cite as

Representable Effect Algebras and Observables

  • Anatolij Dvurečenskij


We introduce a class of monotone σ-complete effect algebras, called representable, which are σ-homomorphic images of a class of monotone σ-complete effect algebras of functions taking values in the interval [0, 1] and with effect algebra operations defined by points. We exhibit different types of compatibilities and show their connection to representability. Finally, we study observables and show situations when information of an observable on all intervals of the form (−∞, t) gives full information about the observable.


Effect algebra Compatibility Strong-compatible Internal compatibility Monotone σ-completeness Homogeneous algebra Observable Block 



The author is very indebted to anonymous referees for their careful reading and suggestions which helped to improve the readability of the paper.


  1. 1.
    Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Loomis-Sikorski representation of monotone σ-complete effect algebras. Fuzzy Sets Syst. 157, 683–690 (2006)CrossRefMATHGoogle Scholar
  2. 2.
    Catlin, D.: Spectral theory in quantum logics. Inter. J. Theor. Phys. 1, 285–297 (1968)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chang, C.C.: Algebraic analysis of many-valued logics. Trans. Am. Math. Soc. 88, 467–490 (1958)CrossRefMATHGoogle Scholar
  4. 4.
    Dvurečenskij, A.: Gleason’s Theorem and Its Applications, p 325+xv. Kluwer, Dordrecht/Boston/London (1993)CrossRefMATHGoogle Scholar
  5. 5.
    Dvurečenskij, A.: Loomis–Sikorski theorem for σ-complete MV-algebras and ℓ-groups. J. Aust. Math. Soc. Ser. A 68, 261–277 (2000)CrossRefMATHGoogle Scholar
  6. 6.
    Dvurečenskij, A.: On effect algebras which can be covered by MV-algebras. Inter. J. Theor. Phys. 41, 221–229 (2002)CrossRefMATHGoogle Scholar
  7. 7.
    Dvurečenskij, A., Kuková, M.: Observables on quantum structures. Inf. Sci. 262, 215–222 (2014).  10.1016/j.ins.2013.09.014 CrossRefGoogle Scholar
  8. 8.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures, p 541 + xvi. Kluwer, Dordrecht, Ister Science, Bratislava (2000)CrossRefMATHGoogle Scholar
  9. 9.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)ADSCrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys Monographs No. 20. American Mathematics Society, Providence, Rhode Island (1986)Google Scholar
  11. 11.
    Halmos, P.R.: Measure Theory. Springer, Berlin (1974)MATHGoogle Scholar
  12. 12.
    Jenča, G.: Blocks of homogeneous effect algebras. Bull. Aust. Math. Soc. 64, 81–98 (2001)CrossRefMATHGoogle Scholar
  13. 13.
    Mundici, D.: Tensor products and the Loomis–Sikorski theorem for MV-algebras, . Adv. Appl. Math. 22, 227–248 (1999)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Niederle, J., Paseka, J.: Homogeneous orthocomplete effect algebras are covered by MV-algebras. Fuzzy Sets Syst. 210, 89–101 (2013)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Pulmannová, S.: Compatibility and decomposition of effects. J. Math. Phys. 43, 2817–2830 (2002)ADSCrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Pulmannová, S.: Blocks in homogeneous effect algebras and MV-algebras. Math. Slovaca 53, 525–539 (2003)MathSciNetMATHGoogle Scholar
  17. 17.
    Riečanová, Z.: A generalization of blocks for lattice effect algebras. Inter. J. Theoret. Phys. 39, 231–237 (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Sikorski, R.: Boolean Algebras. Springer, Berlin, Heidelberg, New York (1964)MATHGoogle Scholar
  19. 19.
    Varadarajan, V.S.: Geometry of Quantum Theory, Vol. 1. van Nostrand, Princeton, New Jersey (1968)CrossRefGoogle Scholar
  20. 20.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Department of Algebra and GeometryPalacký UniversityOlomoucCzech Republic

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