Triangular norms. basic notions and properties

Abstract

This chapter discusses the basic definitions concerning triangular norms and conorms. It describes the most important algebraic and analytical properties that a triangular norm (t-norm) may have. The construction of triangular norms by means of additive and multiplicative generators and via ordinal sums is detailed, and some other construction methods are also illustrated. General construction methods are based on additive and multiplicative generators and also on ordinal sums. Some constructions leading to non-continuous t-norms and a presentation of some distinguished families of t-norms are also illustrated in the chapter. Because t-norms are just functions from the unit square into the unit interval, the comparison of t-norms is done in the usual point wise way. Clearly, each t-norm is a t-subnorm, but not vice versa. For example, the zero function is a t-subnorm but not a t-norm. There is a close relationship between the existence of non-trivial idempotent elements and ordinal sums. Finally, the chapter describes the representation theorems of continuous Archimedean triangular norms (via continuous additive or multiplicative generators) and of continuous triangular norms (as ordinal sums of continuous Archimedean triangular norms). © 2005 Elsevier B.V. All rights reserved.