On interval fuzzy negations

There exists infinitely many ways to extend the classical propositional connectives to the set [0, 1] such that the behavior in its extremes is as in the classical logic. Still, it is a consensus that it is not sufficient, demanding that these extensions also preserve some minimal logical properties of the classical connectives. Thus, the notions of t-norms, t-conorms, fuzzy negations, and fuzzy implications were introduced. In previous works, the authors generalize these notions to the set U = {[a, b]/0 ≤ a ≤ b ≤ 1} and provided canonical constructions to obtain, for example, an interval t-norm which is the best interval representation of a t-norm. In this paper, we considered the notion of interval fuzzy negation and generalized, in a natural way, several notions related with fuzzy negations, such as equilibrium point and negation-preserving automorphism, and we show that the main properties of these notions are preserved for the proposed interval generalizations.