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Keywords:
probability; fuzzy sets; MV-algebra; IF events
Summary:
It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations – multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.
References:
[1] Atanassov K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica–Verlag, New York 1999 MR 1718470 | Zbl 0939.03057
[2] Cignoli L. O., D’Ottaviano M. L., Mundici D.: Algebraic Foundations of Many–valued Reasoning. Kluwer, Dordrecht 2000 Zbl 0937.06009
[3] Nola A. Di, Dvurečenskij A., Hyčko, M., Manara C.: Entropy of effect algebras with the Riesz decomposition property I: Basic properties. Kybernetika 41 (2005), 143–160 MR 2138765
[4] Nola A. Di, Dvurečenskij A., Hyčko, M., Manara C.: Entropy of effect algebras with the Riesz decomposition property II: MV-algebras. Kybernetika 41 (2005), 161–176 MR 2138766
[5] Dvurečenskij A., Pulmannová S.: New Trends in Quantum Structures. Kluwer, Dordrecht 2000 MR 1861369
[6] Gerstenkorn T., Manko J.: Probabilities of intuitionistic fuzzy events. In: Issues in Intelligent Systems: Paradigms (O. Hryniewicz et al., eds.). EXIT, Warszawa, pp. 63–58
[7] Grzegorzewski P., Mrowka E.: Probability of intuitionistic fuzzy events. In: Soft Methods in Probability, Statistics and Data Analysis (P. Grzegorzewski et al., eds.). Physica–Verlag, New York 2002, pp. 105–115 MR 1987681
[8] Halmos P. R.: Measure Theory. Van Nostrand, New York 1950 MR 0033869 | Zbl 0283.28001
[9] Kolmogorov A. N.: Foundations of the Theory of Probability. Chelsea Press, New York 1950 (German original appeared in 1933) MR 0032961
[10] Lendelová K.: Measure Theory on Multivalued Logics and its Applications. Ph.D. Thesis. M. Bel University, Banská Bystrica 2005
[11] Lendelová K.: Probability on L-posets. In: Proc. Fourth Conference of the European Society for Fuzzy Logic and Technology and 11 Rencontres Francophones sur la Logique Floue et ses Applications (EUSFLAT-LFA 2005 Joint Conference), Technical University of Catalonia, Barcelona, pp. 320–324
[12] Lendelová K.: A note on invariant observables. Internat. J. Theoret. Physics 45 (2006), 915–923 MR 2245606 | Zbl 1105.81008
[13] Lendelová K.: Central Limit Theorem for L-posets. J. Electr. Engrg. 12/S (2005), 56, 7–9 Zbl 1096.60015
[14] Montagna F.: An algebraic approach to propositional fuzzy logic. J. Logic. Lang. Inf. 9 (2000), 91–124 MR 1749775 | Zbl 0942.06006
[15] Neumann J. von: Grundlagen der Quantenmechanik. Berlin 1932
[16] Riečan B.: A new approach to some notions of statistical quantum mechanics. BUSEFAL 36 (1988), 4–6
[17] Riečan B.: On the product MV-algebras. Tatra Mt. Math. Publ. 16 (1999), 143–149 MR 1725292 | Zbl 0951.06013
[18] Riečan B.: Representation of probabilities on IFS events. In: Advances in Soft Computing, Soft Methodology and Random Information Systems (M. Lopez–Diaz et al., eds.) Springer–Verlag, Berlin 2004, pp. 243–246 MR 2118103 | Zbl 1061.03058
[19] Riečan B.: Free products of probability MV-algebras. Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 173–186 MR 1910785 | Zbl 1072.06007
[20] Riečan B.: The conjugacy of probability MV-$\sigma $-algebras with the unit interval. Atti Sem. Mat. Fis. Univ. Modena 52 (2004), 241–248 MR 2152490 | Zbl 1115.28018
[21] Riečan B.: Kolmogorov–Sinaj entropy on MV-algebras. Internat. J. Theoret. Physics 44 (2005), 1041–1052 MR 2199519 | Zbl 1119.81302
[22] Riečan B.: On the probability on IF-sets and MV-algebras. Notes on IFS 11 (2005), 6, 21–25
[23] Riečan B.: On the probability and random variables on IF events. In: Applied Artificial Intelligence (Proc. 7th FLINS Conf. Genova, Da Ruan et al., eds.), World Scientific 2006, pp. 138–145
[24] Riečan B., Jurečková M.: On invariant observables and the individual ergodic theorem. Internat. J. Theoret. Physics 44 (2005), 1587–1597 MR 2198158
[25] Riečan B., Mundici D.: Probability on MV-algebras. In: Handbook on Measure Theory (E. Pap, ed.), Elsevier, Amsterdam 2002 MR 1954631 | Zbl 1017.28002
[26] Riečan B., Neubrunn T.: Integral, Measure, and Ordering. Kluwer, Dordrecht 1997 MR 1489521 | Zbl 0916.28001
[27] Schmidt E., Kacprzyk J.: Probability of intuitionistic fuzzy events and their applications in decision making. In: Proc. EUSFLAT’99, Palma de Mallorca 1999, pp. 457–460
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