Abstract
Aggregation is a mathematical process consisting in the fusion of a set of values into a unique one and representing them in some sense. Aggregation functions have demonstrated to be very important in many problems related to the fusion of information. This has resulted in the extended use of these functions not only to combine a family of numbers but also a family of certain mathematical structures such as metrics or norms, in the classical context, or indistinguishability operators or fuzzy metrics in the fuzzy context. In this paper, we study and characterize the functions through which we can obtain a single weak fuzzy (quasi-)norm from an arbitrary family of weak fuzzy (quasi-)norms in two different senses: when each weak fuzzy (quasi-)norm is defined on a possibly different vector space or when all of them are defined on the same vector space. We will show that, contrary to the crisp case, weak fuzzy (quasi-)norm aggregation functions are equivalent to fuzzy (quasi-)metric aggregation functions.
Highlights
An aggregation procedure amounts to a method for merging a family of structures of the same type into the only structure of this type
As the main objective of this paper is to study this problem in the fuzzy context, we recall the known results for crisp norms
Aggregation of Weak Fuzzy (Quasi-)Norms In Section 2 we have summarized the known results about the aggregation ofmetrics and asymmetric norms on products and on sets
Summary
An aggregation procedure amounts to a method for merging a family of structures of the same type into the only structure of this type. F is a quasi-metric aggregation function on products; f −1 (0) = {0} and f preserves asymmetric triangular triplets; f −1 (0) = {0}, f is subadditive and isotone. The definition of a (weak) fuzzy (quasi-)normed space (V, N, ∗) given above requires the continuity of the triangular norm ∗.
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