Skip to main content
SpringerLink
Log in

Factorization of Observables

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

Abstract

Categorical approach to probability leads to better understanding of basic notions and constructions in generalized (fuzzy, operational, quantum) probability, where observables—dual notions to generalized random variables (statistical maps)—play a major role. First, to avoid inconsistencies, we introduce three categories \(\mathbb {L}\), \(\mathbb {S}\), and \(\mathbb {P}\), the objects and morphisms of which correspond to basic notions of fuzzy probability theory and operational probability theory, and describe their relationships. To illustrate the advantages of categorical approach, we show that two categorical constructions involving observables (related to the representation of generalized random variables via products, or smearing of sharp observables, respectively) can be described as factorizing a morphism into composition of two morphisms having desired properties. We close with a remark concerning products.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Adámek, J.: Theory of mathematical structures. Reidel, Dordrecht (1983)

    MATH  Google Scholar 

  2. Bugajski, S.: Statistical maps I. Basic properties. Math. Slovaca 51, 321–342 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Bugajski, S.: Statistical maps II. Operational random variables. Math. Slovaca 51, 343–361 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Chovanec, F., Kôpka, F.: D-posets. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures: Structures, Quantum, pp. 367–428. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  5. Coecke, B.: Introducing categories to the practicing physicist. In: Sica, G. (ed.) What is Category Theory, pp. 45–74. Polimetrica, Monza (2006)

    Google Scholar 

  6. Dvurecenskij, A.: Smearing of observables and spectral measures on quantum structures. Found. Phys. 43, 210–224 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Dvurecenskij, A., Pulmannová, S.: New trends in quantum structures. Kluwer Academic Publ. and Ister Science, Dordrecht and Bratislava (2000)

  8. Foulis, J.D.: Algebraic measure theory. Atti Sem. Mat. Fis. Univ. Modena 48, 435–461 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Fric, R.: Remarks on statistical maps and fuzzy (operational) random variables. Tatra Mt. Math. Publ. 30, 21–34 (2005)

    MATH  MathSciNet  Google Scholar 

  10. Fric, R.: Extension of domains of states. Soft Comput. 13, 63–70 (2009)

    Article  MATH  Google Scholar 

  11. Fric, R.: On D-posets of fuzzy sets. Math. Slovaca 64, 545–554 (2014)

    MATH  MathSciNet  Google Scholar 

  12. Fric, R.: Statistical maps: a categorical approach. Math. Slovaca 57, 41–57 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fric, R., Papčo, M.: On probability domains. Internat. J. Theoret. Phys. 49, 3092–3100 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Fric, R., Papco, M.: A categorical approach to probability. Stud. Logica 94, 215–230 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fric, R., Papco, M.: Fuzzification of crisp domains. Kybernetika 46, 1009–1024 (2010)

    MATH  MathSciNet  Google Scholar 

  16. Fric, R., Papco, M.: On probability domains II. Internat. J. Theoret. Phys. 50, 3778–3786 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Fric, R., Papco, M.: On probability domains III. Internat. J. Theoret. Phys. 54, 4237–4246 (2015)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Fric, R., Papco, M.: Upgrading probability via fractions of events. Commun. Math. 24, 29–41 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fric, R., Papco, M.: On probability domains IV (Submitted)

  20. Greechie, R.J., Foulis, D.J.: The transition to effect algebras. Internat. J. Theoret. Phys. 34, 1369–1382 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Gudder, S.: Fuzzy probability theory. Demonstratio Math. 31, 235–254 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Gudder, S.: Observables and statistical maps. Found. Phys. 29, 877–897 (1999)

    Article  MathSciNet  Google Scholar 

  23. Jencová, A., Pulmannová, S., Vinceková, E.: Observables on σ-MV-algebras and σ-lattice effect algebras. Kybernetika 47, 541–559 (2011)

    MATH  MathSciNet  Google Scholar 

  24. Mesiar, R.: Fuzzy sets and probability theory. Tatra Mt. Math. Publ. 1, 105–123 (1992)

    ADS  MATH  MathSciNet  Google Scholar 

  25. Navara, M.: Probability theory of fuzzy events. In: Montseny, E., Sobrevilla, P. (eds.) Fourth Conference of the European Society for Fuzzy Logic and Technology and 11 Rencontres Francophones sur la Logique Floue et ses Applications, pp. 325–329. Universitat Politecnica de Catalunya, Barcelona, Spain (2005)

    Google Scholar 

  26. Papco, M.: On effect algebras. Soft Comput. 12, 373–379 (2008)

    Article  MATH  Google Scholar 

  27. Papco, M.: Fuzzification of probabilistic objects. In: 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013), pp. 67–71 (2013) doi:10.2991/eusat.2013.10

  28. Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23, 421–27 (1968)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by VEGA 1/0097/16 and VEGA 2/0031/15.

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01, Košice, Slovak Republic

    Peter Eliaš & Roman Frič

Authors
  1. Peter Eliaš
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Frič
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Eliaš.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eliaš, P., Frič, R. Factorization of Observables. Int J Theor Phys 56, 4073–4083 (2017). https://doi.org/10.1007/s10773-017-3436-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-017-3436-1

Keywords

Search

Navigation