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Operation properties and \(\delta \)-equalities of complex fuzzy sets. (English) Zbl 1196.03077

Complex fuzzy sets are characterized by a membership function taking values in the unit complex circle, and thus they are special \(L\)-fuzzy sets, as introduced by Goguen. These fuzzy sets can be equivalently characterized by membership functions with values in the half-open square \([0,1]\times ]0,1]\). The paper is devoted to the study of operations on complex fuzzy sets and of \(\delta\)-equalities of complex fuzzy sets. All introduced results are premature once we look on complex numbers as ordered pairs of reals (this is, in fact, the approach applied in the paper and all operations are defined on single coordinates). Then this is enough to apply well-known results from fuzzy set theory. For example, complex fuzzy union, introduced in Definition 2.3., is simply coordinatewise maximum (i.e., in both coordinates it corresponds to the classical fuzzy union introduced by Zadeh). Similarly, the distance function \(d\), introduced in 3.1., is the maximum of classical Chebychev norms applied to both coordinates, and thus all results in Section 3 follow from the standard properties of Chebychev norms.
Thus, this paper is superfluously long, the majority of the proofs can be shortened or even omitted. Moreover, the terminology introduced by the authors is confusing compared with the standard one (e.g., t-norm is an abbreviation for a triangular norm, while in the paper these are two different notions).
On the other hand, the idea of complex fuzzy sets certainly has its potential, similarly as the idea of complex numbers is more than pairs of reals. Therefore, this paper can be seen as an opening to a deeper study of complex fuzzy sets theory, expanding the original idea of D. Ramot, M. Milo, M. Friedman and A. Kandel [IEEE Trans. Fuzzy Syst. 10, No. 2, 171–186 (2002)].

MSC:

03E72 Theory of fuzzy sets, etc.
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