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

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Iterative boolean-like laws in fuzzy logic have been studied 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,Ix,y)). 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,Ix,y), for all (x,y) in [0, 1].


Original languageEnglish
Pages (from-to)2954-2970
JournalInformation Sciences
Issue number14
Publication statusPublished - Jul 2007


  • Iterative boolean-like law, Fuzzy implications, S-implications, R-implications, QL-implications

ID: 221805