On fuzzy \(n\)-cell numbers and \(n\)-dimension fuzzy vectors
The concepts of fuzzy \(n\)-cell numbers \(u\in{L(E^n)}\) and \(n\)-dimensional fuzzy vectors \((u_1,u_2,\ldots,u_n)\) \(\in(E)^n\) are introduced by two different ideas. Fuzzy \(n\)-cell numbers are a kind of special \(n\)-dimensional fuzzy numbers, whose sets of cuts are all hyperrectangles and it is introduced by the convenience to use. \(n\)-dimensional fuzzy vectors are the Cartesian products of \(n\) one-dimensional fuzzy numbers and it is natural to introduce. The authors show that fuzzy \(n\)-cell numbers and \(n\)-dimensional fuzzy vectors can be represented by each other (Theorem 3.1) and obtain the representations of joint member function \(u(x_1,x_2,\dots,x_n)\) and edge membership functions \(u_i(x_i)\) \((i=1,2,\ldots,n)\) by each other (Theorem 3.2). Then, the authors get the relations between \(L(E^n)\) and \((E)^n\) for the rules of operations, the metrics, the supremum (infimum), the derivative and \((k)\)integral (Theorems 3.3--3.6).
scientific article; zbMATH DE number 5130385