The $S$-Power Invariance Property and the Mixed Invariance Property for Fuzzy Implication Functions

The invariance property based on triangular norms is one of many fuzzy implication function properties. It ensures that the value of the conditional is the same whenever the same linguistic modifier is applied to the antecedent and the consequent. In one of the previous FUZZ-IEEE conferences, we presented some results connected with the invariance of fuzzy implications concerning triangular norms. The main goal of this article is to extend this property and investigate the invariance property based on the powers of triangular conorms. Important characterizations are presented in addition to introduced definitions. Moreover, the mixed invariance property with respect to both
$T$
-powers and
$S$
-powers and their inverses is considered.