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Keywords:
fuzzy number; semiring; fuzzy product distributivity
Summary:
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.
References:
[1] Allen, P. J.: A fundamental theorem of homomorphisms for semirings. Proc. Amer. Math. Soc. 21 (1969), 412-416. DOI 10.1090/s0002-9939-1969-0237575-4 | MR 0237575
[2] Ban, A. I., Bede, B.: Properties of the cross product of fuzzy numbers. J. Fuzzy Math. 14 (2006), 513-531. MR 2258422
[3] Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Springer-Verlag, Berlin, Heidelberg 2013. MR 3024762 | Zbl 1271.03001
[4] Bede, B., Fodor, J.: Product type operations between fuzzy numbers and their applications in geology. Acta Polytechn. Hungar. 3 (2006), 123-139.
[5] Chou, Ch.-Ch.: 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 | MR 1993230
[6] Coroianu, L.: 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 | MR 3421857
[7] Coroianu, L., Fuller, R.: 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 | MR 3733272
[8] Bica, A. M.: Algebraic structures for fuzzy numbers from categorial point of view. Soft Computing 11 (2007), 1099-1105. DOI 10.1007/s00500-007-0167-x | Zbl 1125.03039
[9] Bica, A. M.: 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 | MR 3035732 | Zbl 1276.92098
[10] Bica, A. M.: 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 | MR 3430181
[11] Delgado, M., Vila, M. A., Voxman, W.: On a canonical representation of fuzzy numbers. Fuzzy Sets Syst. 93 (1998), 125-135. DOI 10.1016/s0165-0114(96)00144-3 | MR 1601513 | Zbl 0916.04004
[12] Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9 (1978), 613-626. DOI 10.1080/00207727808941724 | MR 0491199 | Zbl 0383.94045
[13] Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York 1980. MR 0589341 | Zbl 0444.94049
[14] Atani, R. Ebrahimi, Atani, S. Ebrahimi: Ideal theory in commutative semirings. Bul. Acad. Ştiinţe Repub. Mold. Mat. 57 (2008), 2, 14-23. MR 2435797
[15] Atani, R. Ebrahimi: The ideal theory in quotients of commutative semirings. Glasnik Matematicki 42 (2007), 301-308. DOI 10.3336/gm.42.2.05 | MR 2376908
[16] Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18 (1986), 31-43. DOI 10.1016/0165-0114(86)90026-6 | MR 0825618 | Zbl 0626.26014
[17] Golan, J. S.: Semirings and their Applications. Kluwer Academic Publishers, Dordrecht 1999. DOI 10.1007/978-94-015-9333-5\_21 | MR 1746739
[18] Guerra, M. L., Stefanini, L.: Crisp profile symmetric decomposition of fuzzy numbers. Appl. Math. Sci. 10 (2016), 1373-1389. DOI 10.12988/ams.2016.59598
[19] Hanss, M.: Applied Fuzzy Arithmetic - An Introduction with Engineering Applications. Springer-Verlag, Berlin 2005. DOI 10.1007/b138914
[20] Kolesárová, A., Vivona, D.: Entropy of T-sums and T-products of L-R fuzzy numbers. Kybernetika 37 (2001), 2, 127-145. MR 1839223
[21] Ma, M., Friedman, M., Kandel, A.: A new fuzzy arithmetic. Fuzzy Sets Syst. 108 (1999), 83-90. DOI 10.1016/s0165-0114(97)00310-2 | MR 1714662 | Zbl 0937.03059
[22] Mareš, M.: Multiplication of fuzzy quantities. Kybernetika 28 (1992), 5, 337-356. MR 1197719 | Zbl 0786.04006
[23] Mareš, M.: Brief note on distributivity of triangular fuzzy numbers. Kybernetika 31 (1995), 5, 451-457. MR 1361306 | Zbl 0856.04009
[24] Mareš, M.: Fuzzy zero, algebraic equivalence: yes or no?. Kybernetika 32 (1996), 4, 343-351. MR 1420127 | Zbl 0884.04004
[25] Mareš, M.: Weak arithmetics of fuzzy numbers. Fuzzy Sets Syst. 91 (1997), 143-153. DOI 10.1016/s0165-0114(97)00136-x | MR 1480041
[26] Markov, S.: On quasilinear spaces of convex bodies and intervals. J. Comput. Appl. Math. 162 (2004), 93-112. DOI 10.1016/j.cam.2003.08.016 | MR 2043500
[27] Markov, S.: On directed interval arithmetic and its applications. J. Universal Computer Sci. 7 (1995), 514-526. DOI 10.1007/978-3-642-80350-5\_43 | MR 1403710
[28] Markov, S.: On the algebraic properties of intervals and some applications. Reliable Computing 7 (2001), 113-127. DOI 10.1023/a:1011418014248 | MR 1831373
[29] Mesiar, R., Ribarik, J.: Pan operations structure. Fuzzy Sets Syst. 74 (1995), 365-369. DOI 10.1016/0165-0114(94)00314-w | MR 1351585
[30] Mizumoto, M., Tanaka, K.: The four operations of arithmetic on fuzzy numbers. Systems Comput. Controls 7 (1976), 5, 73-81. MR 0476531
[31] Mizumoto, M., Tanaka, K.: 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. MR 0558721
[32] Mordeson, J. N., Nair, P. S.: Fuzzy Mathematics: An Introduction for Engineers and Scientists. Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, New York 2001. MR 1859718
[33] Nasseri, S. H., Mahdavi-Amiri, N.: 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] Qiu, D., Zhang, W.: Symmetric fuzzy numbers and additive equivalence of fuzzy numbers. Soft Comput. 17 (2013), 1471-1477. DOI 10.1007/s00500-013-1000-3
[35] Schneider, J.: Arithmetic of fuzzy numbers and intervals-a new perspective with examples. arXiv: 1310.5604 [math.GM] (2016).
[36] Stefanini, L., Sorini, L., Guerra, M. L.: 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 | MR 2254174 | Zbl 1109.26024
[37] Stefanini, L., Guerra, M. L.: On fuzzy arithmetic operations: some properties and distributive approximations. Int. J. Appl. Math. 19 (2006), 171-199. MR 2266345
[38] Stupňanová, A.: A probabilistic approach to the arithmetics of fuzzy numbers. Fuzzy Sets Syst. 264 (2015), 64-75. DOI 10.1016/j.fss.2014.08.013 | MR 3303664
[39] Taleshian, A., Rezvani, S.: Multiplication operation on trapezoidal fuzzy numbers. J. Phys. Sci. 15 (2011), 17-26. MR 2881855
[40] Zavadskas, E. K., Antucheviciene, J., Hajiagha, S. H. Razavi, Hashemi, S. Sadat: 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
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