An Overview of Fuzzy Implication and Its Generalizations

Fuzzy implication operators are vital in the modeling of uncertain reasoning, particularly in approximate reasoning and fuzzy inference systems. The objective of this survey is to provide a structured and comprehensive overview of fuzzy implication, intuitionistic fuzzy implication, and hesitant fuzzy implication. We examine their properties, representations, and characterizations. We discuss a number of findings about fuzzy negations, fuzzy implications, intuitionistic fuzzy implication, and hesitant fuzzy implications, including their characterizations with respect to the identity principle and ordering property, which lead to fundamental results.