Kybernetika 55 no. 1, 44-62, 2019

Towards the properties of fuzzy multiplication for fuzzy numbers

Alexandru Mihai Bica, Dorina Fechete and Ioan FecheteDOI: 10.14736/kyb-2019-1-0044


In this paper, by using a new representation of fuzzy numbers, namely the ecart-representation, we investigate the possibility to consider such multiplication between fuzzy numbers that is fully distributive. The algebraic and topological properties of the obtained semiring are studied making a comparison with the properties of the existing fuzzy multiplication operations. The properties of the generated fuzzy power are investigated.


fuzzy number, semiring, fuzzy product distributivity




  1. P. J. Allen: A fundamental theorem of homomorphisms for semirings. Proc. Amer. Math. Soc. 21 (1969), 412-416.   DOI:10.1090/s0002-9939-1969-0237575-4
  2. A. I. Ban and B. Bede: Properties of the cross product of fuzzy numbers. J. Fuzzy Math. 14 (2006), 513-531.   CrossRef
  3. B. Bede: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin, Heidelberg 2013.   CrossRef
  4. B. Bede and J. Fodor: Product type operations between fuzzy numbers and their applications in geology. Acta Polytechn. Hungar. 3 (2006), 123-139.   CrossRef
  5. Ch.-Ch. Chou: The canonical representation of multiplication operation on triangular fuzzy numbers. Comput. Math. Appl. 45 (2003), 1601-1610.   DOI:10.1016/s0898-1221(03)00139-1
  6. L. Coroianu: Necessary and sufficient conditions for the equality of the interactive and non-interactive sums of two fuzzy numbers. Fuzzy Sets Syst. 283 (2016), 40-55.   DOI:10.1016/j.fss.2014.10.026
  7. L. Coroianu and R. Fuller: Necessary and sufficient conditions for the equality of interactive and non-interactive extensions of continuous functions. Fuzzy Sets Syst. 331 (2018), 116-130.   DOI:10.1016/j.fss.2017.07.023
  8. A. M. Bica: Algebraic structures for fuzzy numbers from categorial point of view. Soft Computing 11 (2007), 1099-1105.   DOI:10.1007/s00500-007-0167-x
  9. A. M. Bica: One-sided fuzzy numbers and applications to integral equations from epidemiology. Fuzzy Sets Syst. 219 (2013), 27-48.   DOI:10.1016/j.fss.2012.08.002
  10. A. M. Bica: The middle-parametric representation of fuzzy numbers and applications to fuzzy interpolation. Int. J. Approximate Reasoning 68 (2016), 27-44.   DOI:10.1016/j.ijar.2015.10.001
  11. M. Delgado, M. A. Vila and W. Voxman: On a canonical representation of fuzzy numbers. Fuzzy Sets Syst. 93 (1998), 125-135.   DOI:10.1016/s0165-0114(96)00144-3
  12. D. Dubois and H. Prade: Operations on fuzzy numbers. Int. J. Syst. Sci. 9 (1978), 613-626.   DOI:10.1080/00207727808941724
  13. D. Dubois and H. Prade: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York 1980.   CrossRef
  14. R. Ebrahimi Atani and S. Ebrahimi Atani: Ideal theory in commutative semirings. Bul. Acad. Ştiinţe Repub. Mold. Mat. 57 (2008), 2, 14-23.   CrossRef
  15. R. Ebrahimi Atani: The ideal theory in quotients of commutative semirings. Glasnik Matematicki 42 (2007), 301-308.   DOI:10.3336/gm.42.2.05
  16. R. Goetschel and W. Voxman: Elementary fuzzy calculus. Fuzzy Sets Syst. 18 (1986), 31-43.   DOI:10.1016/0165-0114(86)90026-6
  17. J. S. Golan: Semirings and their Applications. Kluwer Academic Publishers, Dordrecht 1999.   DOI:10.1007/978-94-015-9333-5\_21
  18. M. L. Guerra and L. Stefanini: Crisp profile symmetric decomposition of fuzzy numbers. Appl. Math. Sci. 10 (2016), 1373-1389.   DOI:10.12988/ams.2016.59598
  19. M. Hanss: Applied Fuzzy Arithmetic - An Introduction with Engineering Applications. Springer-Verlag, Berlin 2005.   DOI:10.1007/b138914
  20. A. Kolesárová and D. Vivona: Entropy of T-sums and T-products of L-R fuzzy numbers. Kybernetika 37 (2001), 2, 127-145.   CrossRef
  21. M. Ma, M. Friedman and A. Kandel: A new fuzzy arithmetic. Fuzzy Sets Syst. 108 (1999), 83-90.   DOI:10.1016/s0165-0114(97)00310-2
  22. M. Mareš: Multiplication of fuzzy quantities. Kybernetika 28 (1992), 5, 337-356.   CrossRef
  23. M. Mareš: Brief note on distributivity of triangular fuzzy numbers. Kybernetika 31 (1995), 5, 451-457.   CrossRef
  24. M. Mareš: Fuzzy zero, algebraic equivalence: yes or no? Kybernetika 32 (1996), 4, 343-351.   CrossRef
  25. M. Mareš: Weak arithmetics of fuzzy numbers. Fuzzy Sets Syst. 91 (1997), 143-153.   DOI:10.1016/s0165-0114(97)00136-x
  26. S. Markov: On quasilinear spaces of convex bodies and intervals. J. Comput. Appl. Math. 162 (2004), 93-112.   DOI:10.1016/
  27. S. Markov: On directed interval arithmetic and its applications. J. Universal Computer Sci. 7 (1995), 514-526.   DOI:10.1007/978-3-642-80350-5\_43
  28. S. Markov: On the algebraic properties of intervals and some applications. Reliable Computing 7 (2001), 113-127.   DOI:10.1023/a:1011418014248
  29. R. Mesiar and J. Ribarik: Pan operations structure. Fuzzy Sets Syst. 74 (1995), 365-369.   DOI:10.1016/0165-0114(94)00314-w
  30. M. Mizumoto and K. Tanaka: The four operations of arithmetic on fuzzy numbers. Systems Comput. Controls 7 (1976), 5, 73-81.   CrossRef
  31. M. Mizumoto and K. Tanaka: Some properties of fuzzy numbers. In: Advances in Fuzzy Set Theory and Applications (M. H. Gupta, R. K. Ragade, and R. R. Yager, eds.), North-Holland, Amsterdam, 1979, pp. 156-164.   CrossRef
  32. J. N. Mordeson and P. S. Nair: Fuzzy Mathematics: An Introduction for Engineers and Scientists. Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, New York 2001.   CrossRef
  33. S. H. Nasseri and N. Mahdavi-Amiri: Some duality results on linear programming problems with symmetric fuzzy numbers. Fuzzy Inf. Eng. 1 (2009), 1, 59-66.   DOI:10.1007/s12543-009-0004-2
  34. D. Qiu and W. Zhang: Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft Comput. 17 (2013), 1471-1477.   DOI:10.1007/s00500-013-1000-3
  35. J. Schneider: Arithmetic of fuzzy numbers and intervals-a new perspective with examples. arXiv: 1310.5604 [math.GM] (2016).   CrossRef
  36. L. Stefanini, L. Sorini and M. L. Guerra: Parametric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets Syst. 157 (2006), 2423-2455.   DOI:10.1016/j.fss.2006.02.002
  37. L. Stefanini and M. L. Guerra: On fuzzy arithmetic operations: some properties and distributive approximations. Int. J. Appl. Math. 19 (2006), 171-199.   CrossRef
  38. A. Stupňanová: A probabilistic approach to the arithmetics of fuzzy numbers. Fuzzy Sets Syst. 264 (2015), 64-75.   DOI:10.1016/j.fss.2014.08.013
  39. A. Taleshian and S. Rezvani: Multiplication operation on trapezoidal fuzzy numbers. J. Phys. Sci. 15 (2011), 17-26.   CrossRef
  40. E. K. Zavadskas, J. Antucheviciene, S. H. Razavi Hajiagha and S. Sadat Hashemi: Extension of weighted aggregated sum product assessment with interval-valued intuitionistic fuzzy numbers (WASPAS-IVIF). Appl. Soft Comput. 24 (2014), 1013-1021.   DOI:10.1016/j.asoc.2014.08.031