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Generalized CF1F2-integrals: From Choquet-like aggregation to ordered directionally monotone functions

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    0531646 - ÚTIA 2021 RIV NL eng J - Journal Article
    Dimuro, G. P. - Lucca, G. - Bedregal, B. - Mesiar, Radko - Sanz, A. - Ling, S.-T. - Bustince, H.
    Generalized CF1F2-integrals: From Choquet-like aggregation to ordered directionally monotone functions.
    Fuzzy Sets and Systems. Roč. 378, č. 1 (2020), s. 44-67. ISSN 0165-0114
    Institutional support: RVO:67985556
    Keywords : Uninorm * Fuzzy Implication * Distributivity
    Subject RIV: BA - General Mathematics
    OBOR OECD: Applied mathematics
    Impact factor: 3.305, year: 2019

    This paper introduces the theoretical framework for a generalization of CF1F2-integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by gCF1F2-integrals, is based on the so-called pseudo pre-aggregation function pairs (F1,F2), which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the gCF1F2-integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of gCF1F2-integrals. We study several properties of gCF1F2-integrals, considering different constraints for the functions F1 and F2, and state under which conditions gCF1F2-integrals present or not averaging behaviors. Several examples of gCF1F2-integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation functions.
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