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This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with multiple arities (extended means).
We introduce a method to construct piecewise linear binary aggregation functions on the unit interval, based on a triangulation of the unit square with one additional vertex. We derive conditions under which such piecewise linear aggregation functions possess additional interesting properties, such as idempotence, symmetry, Lipschitz continuity and 2-monotonicity. This construction method can also...
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