### 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 |
---|---|

Title of host publication | Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms |

Publisher | Elsevier |

Pages | 17-60 |

Number of pages | 44 |

ISBN (Print) | 9780444518149 |

DOIs | |

Publication status | Published - 2005 |

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*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.

Research output: Chapter

*Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms.*Elsevier, pp. 17-60. https://doi.org/10.1016/B978-044451814-9/50002-1

}

TY - CHAP

T1 - Triangular norms. basic notions and properties

AU - Klement, Erich Peter

AU - Mesiar, Radko

AU - Pap, E.

PY - 2005

Y1 - 2005

N2 - 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).

AB - 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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U2 - 10.1016/B978-044451814-9/50002-1

DO - 10.1016/B978-044451814-9/50002-1

M3 - Chapter

AN - SCOPUS:79958808681

SN - 9780444518149

SP - 17

EP - 60

BT - Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms

PB - Elsevier

ER -