[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.