# Triangular norms. basic notions and properties

Erich Peter Klement, Radko Mesiar, E. Pap

Research output: Chapter in Book/Report/Conference proceedingChapter

24 Citations (Scopus)

### 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).

Original language English Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms Elsevier 17-60 44 9780444518149 https://doi.org/10.1016/B978-044451814-9/50002-1 Published - 2005

### ASJC Scopus subject areas

• Computer Science(all)

### Cite this

Klement, E. P., Mesiar, R., & Pap, E. (2005). Triangular norms. basic notions and properties. In Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms (pp. 17-60). Elsevier. https://doi.org/10.1016/B978-044451814-9/50002-1

Triangular norms. basic notions and properties. / Klement, Erich Peter; Mesiar, Radko; Pap, E.

Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier, 2005. p. 17-60.

Research output: Chapter in Book/Report/Conference proceedingChapter

Klement, EP, Mesiar, R & Pap, E 2005, Triangular norms. basic notions and properties. in Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier, pp. 17-60. https://doi.org/10.1016/B978-044451814-9/50002-1
Klement EP, Mesiar R, Pap E. Triangular norms. basic notions and properties. In Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier. 2005. p. 17-60 https://doi.org/10.1016/B978-044451814-9/50002-1
Klement, Erich Peter ; Mesiar, Radko ; Pap, E. / Triangular norms. basic notions and properties. Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier, 2005. pp. 17-60
title = "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).",
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