Interval valued versions of T-conorms, fuzzy negations and fuzzy implications

There exists infinitely many way to extend the classical propositional connectives to the set [0, 1] such that the behavior in their extremes are as in the classical logic. Still, is a consensus that it is not sufficient, demanding that these extensions also preserves some logical properties of the classical. connectives. Thus, were introduced the notions of t-norms, t-conorms, fuzzy negations, and fuzzy implications. In a previous work, the authors generalize the t-norm notion to the set I = {[a, b] : 0 <= a <= b <= 1}, named interval t-norms, and provided canonical constructions to obtain an interval t-norm which is the best interval representation of the t-norm. In this paper, we generalize the notions of t-conorm, fuzzy negation and fuzzy implication to the set I and provide canonical constructions to obtain their best interval representations. We will also provide a way to obtain: an interval fuzzy t-conorm from an interval t-norm and an interval fuzzy negation, an interval fuzzy implication from an interval t-norm, and an interval fuzzy negation from an interval fuzzy implication. We also prove several properties for this constructions.