Triangular norms – basic properties and representation theorems. (English) Zbl 1010.03046

Klement, Erich Peter; Mesiar, Radko; Pap, Endre

Novák, Vilém (ed.) et al., Discovering the world with fuzzy logic. Heidelberg: Physica-Verlag. Stud. Fuzziness Soft Comput. 57, 63-81 (2000).

This is a review of the basic results on triangular norms (t-norms). The definition of a t-norm and its dual conorm is followed by examples of the most important t-norms and classes of t-norms (Frank, Yager). The mutual relations of continuous, strictly monotone, Archimedean and nilpotent norms are explained together with the respective proofs. The construction of new t-norms by ordinal sums is explained as well as their construction by additive and multiplicative generators. Basic statements on the connection between properties of generators and those of corresponding generated t-norms are proved.
For the entire collection see [Zbl 0979.00015].

Reviewer: Vladimír Janiš (Banská Bystrica)

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

tiangular norm; ordinal sum; additive generator; multiplicative generator; survey