Solutions to the functional equation image 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 image. 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 image, Information Sciences, 177 (2007) 2954–2970), this work provides a complete characterization of the equation for the usual implication operations currently known.