Distributivity of residual implications over conjunctive and disjunctive uninorms

This paper deals with implications defined from uninorms U by residuation through the expressionI(x,y)=sup{z∈[0,1]∣U(x,z)⩽y}.I(x,y)=sup{z∈[0,1]∣U(x,z)⩽y}.The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. A special case is considered when the uninorm that defines the implication is a t-norm. The obtained results show many new solutions generalizing those obtained in previous works, where the implications are derived from t-norms and when they are strong implications derived from disjunctive uninorms.