On the characterizations of fuzzy implications satisfying I(x,y)=I(x,I(x,y))I(x,y)=I(x,I(x,y))

Iterative boolean-like laws in fuzzy logic have been studied by Alsina and Trillas [C. Alsina, E. Trillas, On iterative boolean-like laws of fuzzy sets, in: Proc. 4th Conf. Fuzzy Logic and Technology, Barcelona, Spain, 2005, pp. 389–394] for functional equations with boolean background in which only fuzzy conjunctions, fuzzy disjunctions and fuzzy negations are contained. In this paper we study an iterative boolean-like law with fuzzy implications, more precisely we derive characterizations of some classes of fuzzy implications satisfying I(x,y)=I(x,I(x,y))I(x,y)=I(x,I(x,y)), for all (x,y)∈[0,1]2(x,y)∈[0,1]2. Our discussion mainly focuses on the three important classes of implications: S-implications, R-implications and QL-implications. We prove the sufficient and necessary conditions for an S-implication generated by any t-conorm and any fuzzy negation, an R-implication generated by a left-continuous t-norm, a QL-implication generated by a continuous t-conorm, a continuous t-norm and a strong fuzzy negation to satisfy I(x,y)=I(x,I(x,y))I(x,y)=I(x,I(x,y)), for all (x,y)∈[0,1]2(x,y)∈[0,1]2.