Extension of the concept of mappings using fuzzy sets
The paper gives a new method to extend the extension principle. It is well known that the classical extension principle means that the set- value mapping \(f: 2^ X\to 2^ Y\), \(A\mapsto f(A)= \{f(x): x\in A\}\), is induced by a mapping \(f: X\to Y\). It is interesting that the authors of this paper have found the following fact: for any \(y\in f(A)\), there may be a point \(x\in X= A\) such that \(f(x)= y\), which gives us a new point of view. By doing so, the authors introduce a generalized extension principle -- the \(\mu\)-extension principle, which includes the classical extension principle as well as Zadeh's extension as special cases. For different \(\mu\) there are different extensions. So it has many applications, especially in decision making under fuzzy environment. And the authors give a comparison of the different extensions. Based on this extension, I think that there will be a lot of work to do.
scientific article; zbMATH DE number 496162