Some remarks on the solutions to the functional equation I(x, y) = I(x, I(x, y)) for D-operations

Abstract

This paper is devoted to the iterative functional equation I(x,y) = I(x,I(x,y)) for all x, y ∈ [0, 1] where I denotes a fuzzy implication. This equation, that comes from a tautology in crisp logic, is revised from the results obtained in A. Xie and F. Qin (2010) Information Sciences, doi:10.1016/j.ins.2010.01.023, clarifying which kinds of continuous t-norms and t-conorms can be used in order to generate D-operations satisfying the mentioned equation. © Springer-Verlag Berlin Heidelberg 2010.