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**Fibred triangular norms.**
*(English)*
Zbl 0946.26017

Summary: We deal with the formula \(g[T(f(x), f(y))]\) where \(f\), \(g\) are functions mapping the unit interval onto itself and \(T\) is a triangular norm. We are interested in the conditions under which this formula yields a triangular norm. The case, when \(f\) is continuous is investgated in detail. As a result, a new method which generates new triangular norms from an arbitrary triangular norm is developed. These new triangular norms are the so-called ‘fibred triangular norms’ which are closely related to the homomorphism of semigroups. It is well known that continuous Archimedean triangular norms can be generated with additive generator functions. We investigate a possible generalization of the additive generator function and characterize the class of t-norms which can be generated with them. This class turns out to be the class of fibred continuous Archimedean t-norms. Finally, some examples are given for the case when \(f\) is not continuous.

### Keywords:

Łukasiewicz t-norm; triangular norms; homomorphism; semigroups; additive generator function; t-norms; fibred continuous Archimedean t-norms
Full Text:
DOI

### References:

[1] | B. Demant, Deformationen von t-Normen, ihre Symmetrien und Symmetriebrechungen, preprint.; B. Demant, Deformationen von t-Normen, ihre Symmetrien und Symmetriebrechungen, preprint. · Zbl 0817.04003 |

[2] | Drossos, C.; Navara, M., Generalized t-conorms and closure operators, Abstracts EUFIT, 96 (1996) |

[3] | Fodor, J. C., Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems, 69, 141-156 (1995) · Zbl 0845.03007 |

[4] | S. Jenei, New family of triangular norms via contrapositive symmetrization of residuated implications, submitted.; S. Jenei, New family of triangular norms via contrapositive symmetrization of residuated implications, submitted. · Zbl 0941.03059 |

[5] | Ling, C.-H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401 |

[6] | Mesiar, R., On some constructions of new triangular norms, Math. Soft Comput., 2, 39-45 (1995) · Zbl 0837.47056 |

[7] | Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203 |

[8] | Schweizer, B.; Sklar, A., Associative functions and abstract semigroups, Publ. Math. Debrecen, 10, 69-81 (1963) · Zbl 0119.14001 |

[9] | Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010 |

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