Characterization of a Class of Fuzzy Implications Satisfying the Law of Importation With Respect to Uninorms With Continuous Underlying Operators

Abstract

The law of importation, given by the equality <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(x\wedge y) \longrightarrow z\equiv (x\longrightarrow (y\longrightarrow z))$</tex-math></inline-formula> , is a tautology in classical logic and has been proved to be widely used in approximate reasoning and image processing. Some open problems of fuzzy implication dealing with the law of importation were suggested on 8th International Conference on Fuzzy Set Theory and Applications (FSTA 2006). In this article, we partially solve one open problem associated with this property. Specifically, we mainly devote ourselves to solving the general form of the law of importation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I(U(x,y),z)=I(x,I(y,z))$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$I$</tex-math></inline-formula> is a fuzzy implication and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$U$</tex-math></inline-formula> is a conjunctive uninorm with a continuous underlying t-norm and a continuous underlying t-conorm. Along this study, given a fixed uninorm with continuous underlying operators, all fuzzy implications that satisfy the law of importation with respect to this uninorm, and having an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> -section that is a continuous negation, are characterized.