On the characterization of fuzzy implications satisfying the laws of left or right contraposition

The law of contraposition is one of the most well-known tautologies in classical logic. In addition, there are two other notions of contrapositions: the law of right contraposition and the law of left contraposition. In fuzzy logic, these tautologies are modeled by a fuzzy implication and a fuzzy negation. The characterization of fuzzy implications satisfying the laws of right or left contraposition with respect to an arbitrary fuzzy negation is a problem that has gained attention recently. This article aims to propose characterizations of such implications, presenting necessary and sufficient conditions under which a fuzzy implication satisfies these laws with respect to an arbitrary fuzzy negation.