Investigations of fuzzy implications satisfying generalized hypothetical syllogism

Generalized hypothetical syllogism (GHS) is one of the fuzzy inference rules that plays a central role in the selection of suitable fuzzy implications for a specific task. Because of the important role played by the classical hypothetical syllogism, in this work, we attempt to investigate the fuzzy implications that do satisfy (GHS). Due to the variety of fuzzy implications and the complexity of the functional equation, we restrict our investigations of fuzzy implications only for those which come from some well known families of fuzzy implications, viz., (S,N)-, R-, QL-, and Yager's families of f-, g-implications. Since there do not exist many fuzzy implications satisfying (GHS) from these families of fuzzy implications, we propose some classes of fuzzy implications and show that every element from these classes satisfies (GHS). Finally, we examine the preservation of (GHS) by some important generating methods of fuzzy implications that exist in the literature.