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.

的表征 $(秒,N)$-implications当ñ是不连续的否定一直在模糊逻辑中最显著开放的问题，为过去的几十年之一。本文构成了这一主题的第一份进展。即，该家族的模糊蕴涵功能的通用特征被呈现，其中，所述中央属性是基于的特定子区域定义的二元函数的完成的存在${[0,1]}^2$到叔余模。在本文中，找到一个二进制函数的完成的对偶问题上的子区域限定${[0,1]}^2$到连续t-模被研究和解决的最小和一个消功能。这些结果是对的新颖公理表征的基础$(秒,N)$- 当N有一个不连续点并且S等于某个子区域中的最大 t-conorm 时的含义${[0,1]}^2$ 或严格叔余模。