Characterization of some families of fuzzy inplication functions

Abstract

[eng] Fuzzy implication functions are the logical operators which generalize to the fuzzy logic framework the binary implication in classical logic. Over the last years, several families of fuzzy implication functions with different properties have been proposed. Among them, the so-called (h,e)-implications were introduced as a generalization of h-implications in order to provide a family which satisfies the left-neutrality principle with respect to a parameter e 2 (0,1). In this report, we deal with an open problem on fuzzy implication functions, which is the axiomatic characterization of (h,e)-implications. First, we overview the basic concepts of the theory of fuzzy connectives and study the properties of the pseudo-inverse of a monotone function. Then, we recall a representation theorem for (h,e)-implications which describes the structure of the operator in terms of two families of fuzzy implication functions called ( f ,e) and (g ,e)-implications. These two families can be interpreted as particular cases of the so-called ( f , g ) and (g , f )-implications, which are two families of fuzzy implication functions which generalize the well-known Yager’s f and g -generated implications. Thus, these two more general families are deeply studied retrieving particular results for the ( f ,e) and (g ,e)-implications. Next, thanks to the introduction of two new properties closely related to the law of importation, an axiomatic characterization of ( f ,e) and (g ,e)-implications is presented. Finally, applying these characterizations and recalling the representation theorem, an axiomatic characterization of (h,e)-implications in terms of its own properties is provided.