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