Triangular conorms

[donya88]

Triangular conorms

Definition

The dual notion to a triangular norm is a triangular conorm (abbreviation t-conorm, also s-norm), http://pnu-club.com/imported/mising.jpg. Its neutral element is 0 instead of 1, all other conditions remain unchanged:

http://pnu-club.com/imported/mising.jpg (commutativity)
http://pnu-club.com/imported/mising.jpg (associativity)
http://pnu-club.com/imported/mising.jpg (monotonicity)
http://pnu-club.com/imported/mising.jpg (neutral element 0)
Examples of t-conorms


http://pnu-club.com/imported/mising.jpg (maximum or Gödel t-conorm)
http://pnu-club.com/imported/mising.jpg (product t-conorm, probabilistic sum)
http://pnu-club.com/imported/mising.jpg (Lukasiewicz t-conorm, bounded sum))
No t-conorm can attain smaller values than http://pnu-club.com/imported/mising.jpg.
If http://pnu-club.com/imported/mising.jpg is a t-norm, then http://pnu-club.com/imported/mising.jpg is a t-conorm, and vice versa. We obtain a dual pair http://pnu-club.com/imported/mising.jpg of a t-norm and a t-conorm. (Instead of the standard fuzzy negation (http://www.scholarpedia.org/wiki/index.php?title=Fuzzy_negation&action=edit&redlink=1), http://pnu-club.com/imported/mising.jpg, 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 http://pnu-club.com/imported/mising.jpg has a (non-unique) additive generator, which is an increasing bijection http://pnu-club.com/imported/mising.jpg such that
http://pnu-club.com/imported/mising.jpg