Solutions to the functional equation I(x,y)=I(x,I(x,y)) for a continuous D-operation

Authors:
Aifang Xie;Feng Qin
Affiliations:
College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nanchang, PR China;College of Mathematics and Information Science, Jiangxi Normal University, 330022 Nanchang, PR China
Venue:
Information Sciences: an International Journal
Year:
2010

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Abstract

We present a study, in a fuzzy logic framework, of the fuzzy implication operations involved in the iterative Boolean-like law I(x,I(x,y))=I(x,y). And then, a full characterization is given of this functional equation based on a Dishkant operation (or D-operation for short), where the D-operation is generated by a continuous t-norm T, a continuous t-conorm S, and a strong negation N. Together with the work by Shi et al. (Y. Shi, D. Ruan, E.E. Kerre, On the characterization of fuzzy implications satisfying I(x,y)=I(x,I(x,y)), Information Sciences, 177 (2007) 2954-2970), this work provides a complete characterization of the equation for the usual implication operations currently known.