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Generalized and extended fuzzy sets with applications. (English) Zbl 0649.94028
In the paper different generalizations of fuzzy sets are proposed and studied. Fuzzy set \(\bar x\) defined in a finite space is described in terms of ordered pairs \(\bar x=(\#/x)\) where # stands for a number between 0 and 1 while x denotes a certain element of the universe of discourse. Two generalizations of fuzzy sets refer to (i) membership fuzzification (g-fuzzification), \(({\#}\bar{\;}/x),\) or equivalently ((#/#)/x) which lead to type-2 sets, and (ii) support fuzzification (s-fuzzification) \(({\#}/\bar x)\) (i.e. \(({\#}/({\#}/x))\) generating level-2 fuzzy sets. Next classes of generalized fuzzy sets are generated in a recursive manner which essence is to substitute any number # by \(\overline{\#}\) and/or to replace x by the fuzzy set \(\bar x\).
A reduction problem of generalized fuzzy sets is formulated and put into consideration. Three applicational examples dealing with decision-making, hierarchical analysis, expert systems within which generalized fuzzy sets arise are also provided.
Reviewer: W.Pedrycz

MSC:
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
03E72 Theory of fuzzy sets, etc.
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