(S,N)- and R-implications: A state-of-the-art survey

In this work we give a state-of-the-art review of two of the most established classes of fuzzy implications, viz., (S,N)- and R-implications. Firstly, we discuss their properties, characterizations and representations. Many new results concerning fuzzy negations and (S,N)-implications, notably their characterizations with respect to the identity principle and ordering property, are presented, which give rise to some representation results. Finally, using the presented facts, an almost complete characterization of the intersections that exist among some subfamilies of (S,N)- and R-implications are obtained.