Document Type: Research Paper


Department of Social Sciences, University of Chieti-Pescara, via dei Vestini, 66013, Chieti, Italia


The operations in the set of fuzzy numbers are usually obtained by
the Zadeh extension principle. But these definitions can have some disadvantages
for the applications both by an algebraic point of view and by practical
aspects. In fact the Zadeh multiplication is not distributive with respect to
the addition, the shape of fuzzy numbers is not preserved by multiplication,
the indeterminateness of the sum is too increasing. Then, for the applications
in the Natural and Social Sciences it is important to individuate some suitable
variants of the classical addition and multiplication of fuzzy numbers that have
not the previous disadvantage. Here, some possible alternatives to the Zadeh
operations are studied.


[1] B. Bede and J. Fodor, Product type operations between fuzzy numbers and their applications
in Geology, Acta Polytechnica Hungarica, 3(1) (2006), 123-139.
[2] P. Corsini, Prolegomena of hypergroup theory, Aviani Editore, Tricesimo, 1993.
[3] P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers,
Dordrecht, Hardbound, 2003.
[4] B. De Finetti, Theory of probability, J. Wiley, New York, 1-2 (1974).
[5] D. Dubois and H. Prade, Fuzzy numbers: an overview, in J. C. Bedzek and Ed. Analysis of
Fuzzy Information , CRC-Press, Boca Raton, 2 (1988), 3-39.
[6] S. Hoskova, Binary hyperstructures determined by relational and transformation systems,
Habilitation Thesis, Faculty of Science, University of Ostrava, (2008), 90.
[7] S. Hoskova and J. Chvalina, Discrete transformation hypergroups and transformation hypergroups
with phase tolerance space, Discrete Mathematics, 308(18) (2008), 4133-4143.
[8] G. Klir and B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Prentice Hall, New
Jersey, 1995.
[9] M. Mares, Weak arithmetic on fuzzy numbers, Fuzzy Sets and Systems, 91(2) (1997), 143-
[10] A. Maturo, Grandezze aleatorie fuzzy e loro previsioni per le decisioni in condizione di informazione
parziale, Current Topics in Computer Sciences, Cortellini and Luchian Ed., Panfilus,
Iasi, (2004), 15-24.
[11] A. Maturo, Fuzzy conditional probabilities by the subjective point of view, Advances in Mathematics
of Uncertainty, Tofan Ed., Performantica, Iasi, (2006), 99-108.
[12] A. Maturo, Fuzzy events and their probability assessments, Journal of Discrete Mathematical
Sciences and Cryptography, 3(1-3) (2000), 83-94.
[13] A. Maturo, Alternative fuzzy operations and applications to social sciences, International
Journal of Intelligent Systems, to appear, printed on line by Wiley, 2009.
[14] A. Maturo and A. Ventre, On some extensions of the de Finetti coherent prevision in a fuzzy
ambit, Journal of Basic Science, 4(1) (2008), 95-103.
[15] M. Squillante and A. G. S. Ventre, Consistency for uncertainty measure, International Journal
of Intelligent Systems, 13 (1998), 419-430.
[16] M. Sugeno, Theory of fuzzy integral and its applications, Ph. D. Thesis, Tokyo, 1974.
[17] S. Weber, Decomposable measures and integrals for archimedean t-conorms, J. Math. Anal.
Appl., 101(1) (1984), 114-138.
[18] R. Yager, A characterization of the extension principle, Fuzzy Sets and Systems, 18(3)
(1986), 205-217.
[19] L. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338-353.
[20] L. Zadeh, Probability measures of fuzzy events, J. Math. Anal. Appl., 23 (1968), 421-427.
[21] L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning
I, II, III, Information Sciences, 8 (1975), 199-249 and 301-357, 9 (1975), 43-80.