Brief note on distributivity of triangular fuzzy numbers.

*(English)*Zbl 0856.04009Summary: The general results summarized by D. Dubois and H. Prade [“Fuzzy numbers: An overview”, in: J. C. Bezdek (ed.), Analysis of fuzzy information, Vol. 1, Math. logic, 3-39 (1987)] and the author [Int. J. Gen. Syst. 20, No. 1, 59-65 (1991; Zbl 0739.90074)] show that fuzzy quantities and more especially fuzzy numbers do not fully preserve some of the classical algebraic properties of addition. The most significant ones are the group property of the opposite elements and one of the distributivity laws. It is shown by the author [loc. cit.] that the concept and properties of the opposite element can be easily formulated if we substitute the crisp equality between fuzzy quantities by a weaker type of relation. It is also shown by the author [loc. cit.] that this method does not influence the problem of distributivity, except a very special sort of fuzzy quantities, as shown by the author [Tatra Mt. Math. Publ. 6, 117-121 (1995; Zbl 0849.04005)]. Here we prove that for the triangular fuzzy numbers and for trapezoidal fuzzy intervals the procedure based on the weaker relation leads to the validity of distributivity.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

##### References:

[1] | D. Dubois, H. Prade: Fuzzy numbers: An overview. Analysis of Fuzzy Information (J. C. Bezdek, CRC Press, Boca Raton 1988, pp. 3-39. |

[2] | M. Kovacs, L.Ii. Tran: Algebraic structure of centered M-fuzzy numbers. Fuzzy Sets and Systems 39 (1991), 1, 91-100. · Zbl 0724.04006 · doi:10.1016/0165-0114(91)90068-2 |

[3] | M. Mareš: Algebra of fuzzy quantities. Internat. J. Gen. Systems 20 (1991), 1, 59-65. · Zbl 0739.90074 · doi:10.1080/03081079108945014 |

[4] | M. Mareš: Equivalentions over fuzzy quantities. Tatra Mountains Mathematical Journal · Zbl 0849.04005 |

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