A general framework for the characterization of (S,N)-implications with a non-continuous negation based on completions of t-conorms

The characterization of (S,N)-implications when N is a non-continuous negation has remained one the most significant open problems in fuzzy logic for last decades. This paper constitutes first progress this topic. Namely, general family implication functions presented, which central property existence completion binary function defined on certain subregion [0,1]2 to t-conorm. In paper, dual problem finding continuous t-norm studied and solved minimum cancellative function. These results are basis novel axiomatic characterizations case point discontinuity S equal maximum t-conorm or strict