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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:
 3e+72 Theory of fuzzy sets, etc.