Triangular conorms

Definition

The dual notion to a triangular norm is a triangular conorm (abbreviation t-conorm, also s-norm), . Its neutral element is 0 instead of 1, all other conditions remain unchanged:
  • (commutativity)
  • (associativity)
  • (monotonicity)
  • (neutral element 0)
Examples of t-conorms

  • (maximum or Gödel t-conorm)
  • (product t-conorm, probabilistic sum)
  • (Lukasiewicz t-conorm, bounded sum))
No t-conorm can attain smaller values than .
If is a t-norm, then is a t-conorm, and vice versa. We obtain a dual pair of a t-norm and a t-conorm. (Instead of the standard fuzzy negation, , another strong fuzzy negation can be used in the duality formula.)
The classification and representations of t-conorms are dual to those of t-norms. Each continuous Archimedean t-conorm has a (non-unique) additive generator, which is an increasing bijection such that