Mutually exchangeable fuzzy implications

Recently, Vemuri and Jayaram (2012) have proposed a novel generating method of fuzzy implications, called the ⊛⊛-composition. Further, they have also proposed Mutual Exchangeability (ME), a generalization of the Exchange Principle (EP) to a pair of fuzzy implications and have shown that (ME) plays a central role in the preservation of basic properties, functional equations and families of fuzzy implications w.r.t. the ⊛⊛-composition (Vemuri and Jayaram, 2014). Due to the important role played by (ME), in this work, we investigate the pairs (I,J)(I,J) of fuzzy implications that satisfy (ME). Towards this, we show first that there exist pairs (I,J)(I,J) of fuzzy implications that satisfy (ME) and determine some necessary conditions on such fuzzy implications. Following this, for a given I∈II∈I, we find the set JIJI of all fuzzy implications J   such that the pair (I,J)(I,J) satisfies (ME). Keeping in view the variety of fuzzy implications and the complexity of the functional equation (ME), we restrict our investigations to four important families of fuzzy implications, namely, (S,N)(S,N)-, R-, f- and g  -implications. Further, we discuss a generalization of the Cauchy multiplicative equation, whose solutions help us in obtaining the set JIJI for some families of fuzzy implications.