Characterization of generalized (h,e)-implications based on the characterization of (f,e) and (g,e)-implications

The open problem of the characterization of generalized (h,e)-implications is studied and totally solved. It is retrieved that the structure of generalized (h,e)-implications can be described in terms of two families which are generalizations of the well-known Yager’s implications, called (f,e) and (g,e)-implications. Thus, to obtain the characterization of generalized (h,e)-implications first, an axiomatic characterization of these two families is presented, and then these two results are applied to obtain an axiomatic characterization of generalized (h,e)-implications in terms of their own properties. The characterizations presented rely on two new additional properties of fuzzy implication functions which are modifications of the law of importation.