Bustince Sola, Humberto
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Bustince Sola
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Humberto
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Estadística, Informática y Matemáticas
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ISC. Institute of Smart Cities
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Publication Open Access From fuzzy sets to interval-valued and Atanassov intuitionistic fuzzy sets: a unified view of different axiomatic measures(IEEE, 2019) Couso, Inés; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y MatemáticasWe examine a broad collection of axiomatic definitions from various and diverse contexts, within the domain of fuzzy sets. We evaluate their respective extensions to the case of interval-valued fuzzy sets and intuitionistic fuzzy sets, from a purely formal point of view. We conclude that a large number of such extensions follow similar formal procedures This fact allows us to formulate a general procedure which encompasses all the reviewed extensions as particular cases of it. The new general formulation allows us to identify three different procedures to derive the corresponding extension to the field of interval-valued fuzzy sets or to the field of intuitionistic fuzzy sets from a specific real-valued measure in the context of fuzzy sets. These three processes agglutinate a multitude of particular constructions found in the literature.Publication Open Access Three categories of set-valued generalisations from fuzzy sets to interval-valued and Atanassov intuitionistic fuzzy sets(IEEE, 2018) Couso, Inés; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y MatemáticasMany different notions included in the fuzzy set literature can be expressed in terms of functionals defined over collections of tuples of fuzzy sets. During the last decades, different authors have independently generalised those definitions to more general contexts, like interval-valued fuzzy sets and Atanassov intuitionistic fuzzy sets. These generalised versions can be introduced either through a list of axioms or in a constructive manner. We can divide them into two further categories: setvalued and point-valued generalized functions. Here we deal with constructive set-valued generalisations. We review a long list of functions, sometimes defined in quite different contexts and we show that we can group all of them into three main different categories, each of them satisfying a specific formulation. We respectively call them the set-valued extension, the max-min extension and the max-min-varied extension. We conclude that the set-valued extension admits a disjunctive interpretation, while the max-min extension can be interpreted under an ontic perspective. Finally, the max-min varied extension provides a kind of compromise between both approaches.