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

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.