An analysis of the invariance property with respect to powers of nilpotent t-norms on fuzzy implication functions

Abstract

The invariance property with respect to powers of continuous t-norms is an additional property of fuzzy implication functions which is particularly interesting in the area of approximate reasoning. In this paper, we provide the characterization of all the binary functions which are invariant with respect to the powers of a nilpotent t-norm and, from the restriction of this result to the definition of fuzzy implication function, we introduce the family of nilpotent T-power invariant implications. We study when the members of this family do satisfy other additional properties apart from the invariance. Moreover, the intersection between the family of nilpotent T-power invariant implications and several families of fuzzy implication functions is investigated.