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

Published: 05 August 2022 Publication History

Abstract

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.

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          Published In

          cover image Fuzzy Sets and Systems
          Fuzzy Sets and Systems  Volume 441, Issue C
          Aug 2022
          378 pages

          Publisher

          Elsevier North-Holland, Inc.

          United States

          Publication History

          Published: 05 August 2022

          Author Tags

          1. Fuzzy implication function
          2. ( S, N )-Implication
          3. Characterization
          4. t-Conorm
          5. t-Norm
          6. Completion

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