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On Probability Domains II

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Abstract

We continue our studies of the foundation of probability theory using elementary category theory. We propose a classification scheme of probability domains in terms of cogenerators and their algebraic and topological properties and use the scheme to describe the transition from classical to fuzzy probability. We show that Łukasiewicz tribes form a category of natural probability domains in which σ-fields of sets are “minimal” and measurable [0,1]-valued functions are “maximal” objects. The maximal objects form an epireflective subcategory in which both the classical and fuzzy probability can be modelled. This leads to a better understanding of the transition.

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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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  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: Quantum Structures, pp. 367–428. Elsevier, Amsterdam (2007)

    Chapter  Google Scholar 

  5. Chovanec, F., Drobná, E., Kôpka, F., Nánasiová, O.: Conditional states and independence in D-posets. Soft Comput. 14, 1027–1034 (2010)

    Article  MATH  Google Scholar 

  6. Chovanec, F., Frič, R.: States as morphisms. Int. J. Theor. Phys. 49, 3050–3060 (2010)

    Article  MATH  Google Scholar 

  7. Di Nola, A., Navara, M.: A characterization of σ-complete MV-algebras with enough states. Colloq. Math. 103, 121–130 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  9. Frič, R.: Łukasiewicz tribes are absolutely sequentially closed bold algebras. Czechoslovak Math. J. 52, 861–874 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Frič, R.: Simplex-valued probability. Math. Slovaca 60, 607–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Frič, R., Papčo, M.: On probability domains. Int. J. Theor. Phys. 49, 3092–3100 (2010)

    Article  MATH  Google Scholar 

  14. Frič, R., Papčo, M.: A categorical approach to probability. Stud. Log. 94, 215–230 (2010)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Kroupa, T.: Conditional probability on MV-algebras. Fuzzy Sets Syst. 149, 369–381 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Loève, M.: Probability theory. Nostrand, Princeton (1963)

    MATH  Google Scholar 

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

    MathSciNet  ADS  MATH  Google Scholar 

  22. Navara, M.: Probability and conditional probability on tribes of fuzzy sets. In: Dubois, D., Klement, E.P., Mesiar, R. (eds.) Fuzzy sets, Probability, and Statistics–Gaps and Bridges. Johannes Kepler University, Linz, Austria, pp. 84–88 (2007)

    Google Scholar 

  23. Navara, M.: Tribes revisited. In: Bodenhofer, U., De Baets, B., Klement, E.P., Saminger-Platz, S. (eds.) 30th Linz Seminar on Fuzzy Set Theory: The Legacy of 30 Seminars—Where Do We Stand and Where Do We Go? Johannes Kepler University, Linz, Austria, pp. 81–84 (2009)

    Google Scholar 

  24. Papčo, M.: On measurable spaces and measurable maps. Tatra Mt. Math. Publ. 28, 125–140 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Papčo, M.: On fuzzy random variables: examples and generalizations. Tatra Mt. Math. Publ. 30, 175–185 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Rényi, A.: Foundations of Probability. Holden-Day, San Francisco (1970)

    MATH  Google Scholar 

  28. Riečan, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, Vol. II. North-Holland, Amsterdam, pp. 869–910 (2002)

    Google Scholar 

  29. Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer Academic, Dordrecht (1997

    MATH  Google Scholar 

  30. Rodabaugh, S.E.: Categorical foundations of variable-basis fuzzy topology. In: Höhle, U., Rodabaugh, S.E. (eds.) Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, vol. 3, pp. 273–388. Kluwer Academic, Dordrecht (1999)

    Chapter  Google Scholar 

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Authors and Affiliations

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

    Roman Frič

  2. Department of Mathematics, Faculty of Education, Catholic University in Ružomberok, Hrabovská cesta 1, 034 01, Ružomberok, Slovak Republic

    Roman Frič & Martin Papčo

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  1. Roman Frič
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  2. Martin Papčo
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Correspondence to Roman Frič.

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Frič, R., Papčo, M. On Probability Domains II. Int J Theor Phys 50, 3778–3786 (2011). https://doi.org/10.1007/s10773-011-0855-2

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  • DOI: https://doi.org/10.1007/s10773-011-0855-2

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