Lifting negations and implications on bounded subposets of a complete lattice

Abstract

The main purpose of this paper is to apply the generic extension methods of uplifting and downlifting associative operations in our previous work for fuzzy negations and implications. We also characterize when the NG-uplift (resp. GH-uplift) of a fuzzy negation N (resp. fuzzy implication) on a bounded subposet of a complete lattice is a fuzzy negation (resp. fuzzy implication) on the entire complete lattice. The NG-uplift (resp. GH-uplift) of a fuzzy negation N (resp. fuzzy implication) involves the use of a set-valued mapping G (resp. two set-valued mappings G and H) between that complete lattice and the subposet. Moreover, we discuss about which properties of fuzzy implications are preserved through the use of this extension method. Furthermore, we discuss the relationship between these two extensions of fuzzy negations and implications, and investigate the extensions of R-implications and (S,N)-implications. Finally, we formulate the dual results on the NG-downlift (resp. GH-downlift) of a fuzzy negation N (resp. fuzzy implication).