On the T-power Inverse Invariance Property on Fuzzy Implication Functions

Among the great bunch of additional properties which fuzzy implication functions may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication function when both the antecedent and the consequent are modified using the same quantifier modelled using powers of t-norms. In this paper, a related property called T-power inverse invariance property is introduced. This property ensures the invariance of the truth value of the fuzzy implication function when the consequent is modified using the quantifier inverse to the one used to modify the antecedent. The characterization of those binary mappings I : [0, 1]
²
→ [0, 1] fulfilling this property for continuous Archimedean t-norms is presented as well as the particular characterization result for fuzzy implication functions.