Fuzzy Implications: Classification and a New Class

Abstract

One of the most important and interesting topics in fuzzy mathematics is the study of fuzzy connectives and in particular fuzzy implications. Fuzzy implications are supposed to have at least some fundamental properties in common with the classical binary implication. Besides these fundamental properties there are many additional potential properties for fuzzy implications, among which eight are widely used in the literature. Fuzzy implications satisfying different subsets of these eight properties have been constructed and some interrelationships between these eight properties have been established. This paper aims to lay bare all the interrelationships between the eight additional properties. Where needed suitable counterexamples are provided. In our search for these counterexamples we discovered a new class of fuzzy implications that is completely determined by a fuzzy negation. For this new class we examine the conditions under which the eight properties are satisfied and we obtain the intersection with the class of strong and residual fuzzy implications.