On fuzzy implications: An axiomatic approach

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 X×Y. 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.