On fuzzy implications: An axiomatic approach

Authors:
Sebastia Massanet;Gaspar Mayor;Radko Mesiar;Joan Torrens
Affiliations:
Department of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa, Km. 7.5, E-07122 Palma de Mallorca, Spain;Department of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa, Km. 7.5, E-07122 Palma de Mallorca, Spain;Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia and Institute for Research and Applications of Fuzzy Modelli ...;Department of Mathematics and Computer Science, University of the Balearic Islands, Ctra. de Valldemossa, Km. 7.5, E-07122 Palma de Mallorca, Spain
Venue:
International Journal of Approximate Reasoning
Year:
2013

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Abstract

Fuzzy operations acting on entire fuzzy sets with the stress on fuzzy implications are discussed and studied. In the case of binary operations, the input fuzzy sets are fuzzy subsets of possibly different universal spaces X and Y, and the output fuzzy set is a fuzzy subset of the Cartesian product XxY. The standard approach to fuzzy operations is based on functions acting on [0,1], and then these fuzzy operations are called functionally expressible. We give a characterization of functionally expressible fuzzy implications (and other fuzzy operations), and include several examples of fuzzy operations which are not functionally expressible.