New results of (U,N)-implications satisfying I(r,I(s,t))=I(I(r,s),I(r,t))

Abstract

Generalized Frege's law has been extensively explored by numerous scholars in the field of fuzzy mathematics, particularly within the framework of fuzzy logic. This study aims to further investigate the $(U,N)$-implications that satisfy this law and presents a multitude of novel findings. First, to efficiently determine the satisfiability of the generalized Frege's law for any $(U,N)$-implication, two new necessary conditions have been introduced that are simple and practical: for the fuzzy negation N, it must be noncontinuous, and its values in the interval $[0,e]$ should remain the constant 1. Next, the necessary and sufficient conditions for any $(U,N)$-implication to satisfy the generalized Frege's law are provided. Several complete characterizations are described depending on the position of α in $[e,1]$. To be more specific, the full characterization is achieved when $\alpha =e$ ($\alpha =1$) and a disjunctive uninorm with a continuous underlying t-norm (t-conorm). The necessary and sufficient conditions are presented when $\alpha \in ]e,1[$ and U is a locally internal and disjunctive uninorm.