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Aggregation operators. (English) Zbl 0960.03045
Herceg, D. (ed.) et al., PRIM ’96. Proceedings of the XI conference on applied mathematics, Budva, Yugoslavia, June 3-6, 1996. Novi Sad: Univ. of Novi Sad, Fac. of Science, Inst. of Math. 193-211 (1997).
The paper is an overview of recent results on the aggregation operators. An aggregation operator \({\mathbf A}\) is a non-decreasing mapping \([{\mathbf A}: \bigcup_{n \in {\mathbf N}} [0,1]^{n} \rightarrow [0,1] ]\) fulfilling the following conditions: (i) \(0 \leq x_{i} \leq y_{i} \leq 1, i=1, \cdots, n\) imply \({\mathbf A}(x_{1}, \cdots, x_{n}) \leq {\mathbf A}(y_{1}, \cdots, y_{n});\) (ii) \({\mathbf A}(x) =x\) for all \(x \in [0,1];\) (iii) \({\mathbf A}(0, \cdots,0) = 0\) and \({\mathbf A}(1, \cdots,1) =1.\) Property (i) is the monotonicity and properties (ii) and (iii) are the boundary conditions. Each aggregation operator \({\mathbf A}\) can be represented as a system \(({\mathbf A}_{n})_{n \in {\mathbf N}}\) of \(n\)-ary operators \({\mathbf A}_{n}, n \in {\mathbf N},\) on the unit interval, where \({\mathbf A} _{1}\) is the identity operator on \([0,1]\) and each \({\mathbf A}_{n}, n \geq 2,\) is non-decreasing and \({\mathbf A}_{n}(0, \cdots,0) = 0, {\mathbf A}_{n}(1, \cdots,1) = 1.\) Depending on the field of application, several additional properties can be required and/or examined, such as commutativity, associativity, continuity, idempotency, compensation, cancellativity, etc. Three main approches are presented: aggregation operators based on the Choquet integral and similar integrals, aggregation operators based on triangular norms and conorms and aggregation operators based on additive generators.
For the entire collection see [Zbl 0872.00028].

MSC:
03E72 Theory of fuzzy sets, etc.
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