### Abstract

Given two t-norms T_{1} and T_{2}, it is quite often difficult or even impossible to check directly whether T_{1}≤ T_{2}. In Schweizer and Sklar (1983), a necessary and sufficient condition is given for continuous Archimedean t-norms to be comparable. We simplify this result, proving a sufficient condition which involves only one argument of a one-place function rather than two arguments. This result is applied to show the monotonicity of some well-known classes of t-norms. Next, we generalize the result of Schweizer and Sklar to the case of all continuous t-norms and present also a sufficient condition, which is again easier to check.

Original language | English |
---|---|

Pages (from-to) | 189-195 |

Number of pages | 7 |

Journal | Fuzzy Sets and Systems |

Volume | 86 |

Issue number | 2 |

Publication status | Published - 1997 |

### Fingerprint

### Keywords

- Additive generator
- Comparison of t-norms
- Ordinal sum
- t-norm

### ASJC Scopus subject areas

- Artificial Intelligence
- Computer Science Applications
- Computer Vision and Pattern Recognition
- Information Systems and Management
- Statistics, Probability and Uncertainty
- Electrical and Electronic Engineering
- Statistics and Probability

### Cite this

*Fuzzy Sets and Systems*,

*86*(2), 189-195.

**A characterization of the ordering of continuous t-norms.** / Klement, Erich Peter; Mesiar, Radko; Pap, E.

Research output: Contribution to journal › Article

*Fuzzy Sets and Systems*, vol. 86, no. 2, pp. 189-195.

}

TY - JOUR

T1 - A characterization of the ordering of continuous t-norms

AU - Klement, Erich Peter

AU - Mesiar, Radko

AU - Pap, E.

PY - 1997

Y1 - 1997

N2 - Given two t-norms T1 and T2, it is quite often difficult or even impossible to check directly whether T1≤ T2. In Schweizer and Sklar (1983), a necessary and sufficient condition is given for continuous Archimedean t-norms to be comparable. We simplify this result, proving a sufficient condition which involves only one argument of a one-place function rather than two arguments. This result is applied to show the monotonicity of some well-known classes of t-norms. Next, we generalize the result of Schweizer and Sklar to the case of all continuous t-norms and present also a sufficient condition, which is again easier to check.

AB - Given two t-norms T1 and T2, it is quite often difficult or even impossible to check directly whether T1≤ T2. In Schweizer and Sklar (1983), a necessary and sufficient condition is given for continuous Archimedean t-norms to be comparable. We simplify this result, proving a sufficient condition which involves only one argument of a one-place function rather than two arguments. This result is applied to show the monotonicity of some well-known classes of t-norms. Next, we generalize the result of Schweizer and Sklar to the case of all continuous t-norms and present also a sufficient condition, which is again easier to check.

KW - Additive generator

KW - Comparison of t-norms

KW - Ordinal sum

KW - t-norm

UR - http://www.scopus.com/inward/record.url?scp=0001497254&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001497254&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001497254

VL - 86

SP - 189

EP - 195

JO - Fuzzy Sets and Systems

JF - Fuzzy Sets and Systems

SN - 0165-0114

IS - 2

ER -