Solutions to the functional equation I(x, y) = I(x, I(x, y)) for a continuous D-operation
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. (C) 2010 Elsevier Inc. All rights reserved.
d-operations iterative boolean-like laws t-norms t-conorms strong negations d-implications