Tsallis entropy of fuzzy σ-algebras

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran



The Shannon entropy and the logical entropy of fuzzy σ-algebras are well-known instances of entropy. In this paper, we introduce and study the Tsallis entropy of order α of fuzzy σ−algebras on F −probability measure spaces, where α ∈ (0, 1)∪(1, ∞). Moreover, we study the conditional version of this entropy and examine its basic properties.


[1] P.A. Alemany and D.H. Zanette, Fractal random walks from a variational formalism for Tsallis entropies, Phys. Rev. E 49 (1994), 956–958.
[2] L. Alizadeh and M. Ebrahimi, Renyi and Tsallis and Shannon entropies on D−posets, J. Intell. Fuzzy Syst. 34 (2018), 2771–2781.
[3] M.P. Almeida, Generalized entropies from first principles, Phys. A: Statist. Mech. Appl. 300 (2001) 424–432.
[4] L. Borland, Long-range memory and nonextensivity in financial markets, Europhys. News 36 (2005), 228–231.
[5] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions, J. Math. Anal. Appl. 176 (1993), 359–373.
[6] A. Ebrahimzadeh and J. Jamalzadeh, Conditional logical entropy of fuzzy σ-algebras, J. Intell. Fuzzy Syst. 33 (2017), 1019–1026.
[7] R. Hanel and S. Thurner, Generalized Boltzmann factors and the maximum entropy principle: Entropies for complex systems, Phys. A: Statist. Mech. Appl. 380 (2007), 109–114.
[8] J. Havrda and F. Charvat, Quantification methods of classification processes: Concept of structural alpha-entropy, Kybernetika 3 (1967), 30–35.
[9] H. Huang, H. Xie, and Z. Wang, The analysis of VF and VT with wavelet-based Tsallis information measure, Phys. Lett. A 336 (2005), 180–187.
[10] G. Kaniadakis, Statistical mechanics in the context of special relativity, Phys. Rev. E 66 (2002), 56–125.
[11] B.N. Karayiannis, Fuzzy partition entropies and entropy constrained fuzzy clustering algorithms, J. Intell. Fuzzy Syst. 5 (1997), no. 2, 103–111.
[12] M. Khare, Fuzzy σ-algebras and conditional entropy, Fuzzy Sets Syst. 102 (1999), 287–292.
[13] M. Khare, Sufficient families and entropy of inverse limit, Math. Slovaca 49 (1999), 443–452.
[14] E.-P. Klement, Fuzzy σ-algebras and fuzzy measurable functions, Fuzzy Sets Syst. 4 (1980), 83–93.
[15] A.N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue space, Doklady Russian Acad. Sci. 119 (1958), no. 5, 861–864.
[16] V. Kumar, Kapur’s and Tsalli’s Entropies: A Communication System Perspective, LAP LAMBERT Academic Publishing, Saarbrucken, Germany, 2015.
[17] D. Markechova, The entropy of fuzzy dynamical systems and generators, Fuzzy Sets Syst. 48 (1992), 351–363.
[18] D. Markechova, Entropy of complete fuzzy partitions, Math. Slovaca 43 (1993), 1–10.
[19] J. Naudts, Deformed exponentials and logarithms in generalized thermostatistics, Phys. A: Statist. Mech. Appl. 316 (2002), 323–334.
[20] Nonextensive Statistical Mechanics and Thermodynamics, Available online: http://tsallis.cat.cbpf.br/biblio.htm (accessed on 16 November 2018)
[21] K. Piasecki, Fuzzy partitions of sets, Busefal 25 (1986), 52–60.
[22] D.G. Perez, L. Zunino, M.T Martın, D.G. Perez, L. Zunino, M.T. Martın, M. Garavaglia, A. Plastino, and O.A. Rosso, Model-free stochastic processes studied with q-wavelet-based informational tools, Phys. Lett. A 364 (2007), 259–266.
[23] A. Ramirez-Reyes, A.R. Hernandez-Montoya, G. Herrera-Corral, and I. Domınguez-Jimenez, Determining the entropic index q of Tsallis entropy in images through redundancy, Entropy 18 (2016), 299.
[24] O.A. Rosso, M.T. Mart´ın, and A. Plastino, Brain electrical activity analysis using wavelet-based informational tools (II): Tsallis non-extensivity and complexity measures, Phys. A: Statist. Mech. Appl. 320 (2003), 497–511.
[25] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 379–423.
[26] Y.G. Sinai, On the notion of entropy of dynamical system, Doklady Russian Acad. Sci. 124 (1959), no. 3, 768–771.
[27] P. Srivastava and M. Khare, Conditional entropy and Rokhlin metric, Math Slovaca 49 (1999), 433–441.
[28] P. Srivastava and M. Khare, Y.-K. Srivastava, m-Equivalence, entropy and F-dynamical systems, Fuzzy Sets Syst. 121 (2001), 275–283.
[29] S. Tong, A. Bezerianos, J. Paul, Y. Zhu, and N. Thakor, Nonextensive entropy measure of EEG following brain injury from cardiac arrest, Phys. A: Statist. Mech. Appl. 305 (2002), 619–628.
[30] C. Tsallis, Possible generalization of Boltzmann–Gibbs statistics, J. Statist. Phys. 52 (1988), 479–487.
[31] C. Tsallis, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World, New York, NY, USA, Springer, 2009.
[32] C. Tsallis, Generalized entropy-based criterion for consistent testing, Phys. Rev. E 58 (1998), 1442–1445.
[24] C. Tsallis, Nonextensive thermostatistics and fractals, Fractals 3 (1995), 541–547.
[34] C. Tsallis, C. Anteneodo, L. Borland, and R. Osorio, Nonextensive Statistical mechanics and economics, Phys. A: Statist. Mech. Appl. 324 (2003), 89–100

Articles in Press, Corrected Proof
Available Online from 06 February 2024
  • Receive Date: 14 July 2022
  • Revise Date: 22 February 2023
  • Accept Date: 19 October 2023