Skip to main content
SpringerLink
Log in

States on Pseudo Effect Algebras and Integrals

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.

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

  1. Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Springer, Berlin (1971)

    MATH  Google Scholar 

  2. Alfsen, E.M., Schultz, F.W.: State Spaces of Operator Algebras. Birkhäuser, Boston/Basel/Berlin (2001)

    Book  MATH  Google Scholar 

  3. Baudot, R.: Non-commutative logic programming language NoClog. In: Symposium LICS, Santa Barbara, pp. 3–9 (2000) (Short Presentation)

    Google Scholar 

  4. Bennett, M.K., Foulis, D.J.: Phi-symmetric effect algebras. Found. Phys. 25, 1699–1722 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  5. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. Springer, New York/Heidelberg/Berlin (1979)

    MATH  Google Scholar 

  6. de Finetti, B.: Theory of Probability, vol. 1. Wiley, Chichester (1974)

    MATH  Google Scholar 

  7. Dvurečenskij, A.: States on pseudo MV-algebras. Stud. Log. 68, 301–327 (2001)

    Article  MATH  Google Scholar 

  8. Dvurečenskij, A.: Pseudo MV-algebras are intervals in ℓ-groups. J. Aust. Math. Soc. A 72, 427–445 (2002)

    Article  MATH  Google Scholar 

  9. Dvurečenskij, A.: Every state on interval effect algebra is integral. J. Math. Phys. 51, 083508–12 (2010). doi:10.1063/1.3467463

    Google Scholar 

  10. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic/Ister Science, Dordrecht/Bratislava (2000)

    MATH  Google Scholar 

  11. Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras. I. Basic properties. Int. J. Theor. Phys. 40, 685–701 (2001)

    Article  MATH  Google Scholar 

  12. Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras. II. Group representation. Int. J. Theor. Phys. 40, 703–726 (2001)

    Article  MATH  Google Scholar 

  13. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  14. Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford/London/Paris (1963)

    MATH  Google Scholar 

  15. Georgescu, G., Iorgulescu, A.: Pseudo-MV algebras. Mult. Valued Log. 6, 95–135 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Glass, A.M.W.: Partially Ordered Groups. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  17. Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs, vol. 20. Am. Math. Soc., Providence, Rhode Island (1986)

    MATH  Google Scholar 

  18. Kolmogorov, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin (1933)

    Google Scholar 

  19. Kroupa, T.: Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst. 157, 2771–2782 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Panti, G.: Invariant measures in free MV-algebras. Commun. Algebra 36, 2849–2861 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Phelps, R.R.: Lectures on Choquet’s Theorem. Van Nostrand, Princeton (1965)

    Google Scholar 

  22. Ravindran, K.: On a structure theory of effect algebras. Ph.D. thesis, Kansas State Univ., Manhattan, Kansas (1996)

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

Authors
  1. Anatolij Dvurečenskij
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatolij Dvurečenskij.

Additional information

The author thanks for the support by Center of Excellence SAS—Quantum Technologies—, ERDF OP R&D Projects CE QUTE ITMS 26240120009 and meta-QUTE ITMS 26240120022, the grant VEGA No. 2/0032/09 SAV.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dvurečenskij, A. States on Pseudo Effect Algebras and Integrals. Found Phys 41, 1143–1162 (2011). https://doi.org/10.1007/s10701-011-9537-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-011-9537-4

Keywords

Search

Navigation