Characterizations and Applications of Fuzzy Implications Generated by a Pair of Generators of T-Norms and the Usual Addition of Real Numbers

Abstract

Motivated by several problems on constructions, algebraic properties, in particular the law of importation (LI) and the flexible ordering property (FOP), and applications in approximate reasoning, the present article defines a new class of fuzzy implications, called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications, by a pair of multiplicative and additive generators of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norms and the usual addition instead of the multiplication or division usually used in the literature. The relations to other generator generated implications are clarified, the intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(S,N)$</tex-math></inline-formula> -implications are discussed in detail and the intersections with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$R$</tex-math></inline-formula> -implications are proved to be conjugates of the Łukasiewicz implication. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norm solutions to the (LI) equation for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications in several cases are characterized in terms of generated t-norms, and two special subclasses of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications are characterized through (LI) and (FOP) on the other hand. A particularly interesting corollary shows that a binary operation on [0,1] enjoying (OP) satisfies (LI) with the Łukasiewicz <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> -norm if and only if it is the Łukasiewicz implication. These results will partially solve or enrich the related open problems pending for fuzzy implications. As applications of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\theta,t)$</tex-math></inline-formula> -generated implications, two fuzzy reasoning methods, called triple I method and parallel hierarchical triple I method, based on (LI) and (FOP) are proposed in order to set a logic foundation for Zadeh’s compositional rule of inference (CRI) and to remedy the dependency of Jayaram’s hierarchical CRI on the order of inputs, respectively.