On the distributivity of fuzzy implications and the weighted S-implications

The distributivity properties of fuzzy implication functions play an important role in fuzzy research. Making use of the solution of the autodistributivity functional equations, we give the necessary and sufficient conditions of two types of the distributivity property of fuzzy implications. It is essentially based on the weighted disjunctive operator, and new weighted and mean (S,N)-implication functions are introduced. Then using the pliant operators – where all the operators have a common generator function – we give a parametric form of the implication, namely the neutral value-dependent implication, and we show that some weakened properties of the fuzzy weighted implications are valid such as the identity principle, the ordering principle, the T-conditionality, the strong negation principle, etc. With this new concept, we use the fixed point of the negation as a threshold. We give a specific form of the new implication.