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**Convex combinations in terms of triangular norms: A characterization of idempotent, bisymmetrical and self-dual compensatory operators.**
*(English)*
Zbl 0928.03063

Summary: It is observed that most of the well-known compensatory operators are defined in terms of convex transformations of the values of triangular norms and conorms or other operators. In this paper, the characterization and the applicability of parametrized operators constructed again in terms of triangular norms and conorms, but as transformations of their arguments, are studied. It turns out that these operators are closely related to the ordinary convex combination. Moreover, when idempotency is required, these operators can be characterized algebraically as bisymmetrical and self-dual operators that satisfy the cancellation law.

### MSC:

03E72 | Theory of fuzzy sets, etc. |

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

### Keywords:

bisymmetry; compensatory operators; triangular norms; convex combination; idempotency; self-dual operators; cancellation law
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\textit{B. Moser} et al., Fuzzy Sets Syst. 104, No. 1, 97--108 (1999; Zbl 0928.03063)

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### References:

[1] | Aczél, J., Lectures on functional equations and their applications, (1966), Academic Press New York · Zbl 0139.09301 |

[2] | Alsina, C., On a family of connectives for fuzzy sets, Fuzzy sets and systems, 16, 231-235, (1985) · Zbl 0603.39005 |

[3] | Fodor, J., On bisymmetric operations on the unit interval, (), 277-280 |

[4] | Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (1994), Kluwer Academic Publishers Dordrecht · Zbl 0827.90002 |

[5] | Klement, E.P.; Mesiar, R.; Pap, E., On the relationship of associative compensatory operators to triangular norms and conorms, Int. J. uncertainty, fuzziness and knowledge-based systems, 10, 282-293, (1996) |

[6] | E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, in preparation. · Zbl 0972.03002 |

[7] | Mayor, G., On a family of quasi arithmetic means, Aequationes mathematicae, 48, 137-142, (1994) · Zbl 0809.39011 |

[8] | Mayor, G.; Torrens, J., On some classes of idempotent operators, Int. J. uncertainty, fuzziness knowledge-based systems, 5, 401-410, (1997) · Zbl 1232.03042 |

[9] | Mizumoto, M., Pictorial representations of fuzzy connectives. part 2: cases of compensatory and self-dual operators, Fuzzy sets and systems, 32, 245-252, (1989) |

[10] | Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), North-Holland Amsterdam · Zbl 0546.60010 |

[11] | Turksen, I.B., Interval-valued fuzzy sets and ‘compensatory AND’, Fuzzy sets and systems, 51, 295-307, (1992) |

[12] | Zimmermann, H.-J., Fuzzy set theory and its applications, (1991), Kluwer Boston · Zbl 0719.04002 |

[13] | Zimmermann, H.-J.; Zysno, P., Latent connectives in human decision making, Fuzzy sets and systems, 4, 37-51, (1980) · Zbl 0435.90009 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.