Sup-t-norm and inf-residuum are a single type of relational equations.

*(English)*Zbl 1259.03065Summary: We show that the sup-t-norm and inf-residuum types of fuzzy relational equations, considered in the literature as two different types, are in fact two particular instances of a single, more general type of equations. We demonstrate that several pairs of corresponding results on the sup-t-norm and inf-residuum types of equations are simple consequences of single results regarding the more general type of equations. We also show that the new type of equations subsumes other types of equations such as equations with constraints on solutions, examples of which are fuzzy relational equations whose solutions are required to be crisp (ordinary) relations.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

##### Keywords:

fuzzy relational equation; inf-residuum product; sup-t-norm product; sup-preserving aggregation structure
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\textit{E. Bartl} and \textit{R. Belohlavek}, Int. J. Gen. Syst. 40, No. 6, 599--609 (2011; Zbl 1259.03065)

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##### References:

[1] | Bandler W., International workshop of fuzzy reasoning theory and applications (1978) |

[2] | DOI: 10.1016/S0020-7373(80)80055-1 · Zbl 0435.68042 |

[3] | Bandler W., Fuzzy sets: theory and applications to policy analysis and information systems pp 341– (1980) |

[4] | Belohlavek R., Fuzzy relational systems: foundations and principles (2002) |

[5] | Belohlavek R., Journal of logic and computation (2010) |

[6] | Belohlavek, R., 2010b. Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences. Fuzzy Sets and Systems (in press) · Zbl 1266.03056 |

[7] | Belohlavek, R. and Vychodil, V. What is a fuzzy concept lattice? Proceedings of the CLA 2005, 3rd international conference on concept lattices and their applications. pp.34–45. Palacky University, Olomouc, Technical University, Ostrava, |

[8] | De Baets B., The handbook of fuzzy set series 1 pp 291– (2000) |

[9] | Di Nola A., Fuzzy relation equations and their applications to knowledge engineering (1989) · Zbl 0694.94025 |

[10] | Goguen J.A., Journal of mathematical analysis and applications pp 145– (1967) · Zbl 0145.24404 |

[11] | Gottwald S., Fuzzy sets and fuzzy logic. Foundations of applications – from a mathematical point of view (1993) · Zbl 0782.94025 |

[12] | Gottwald S., Handbook of granular computing (2002) |

[13] | Hájek P., Metamathematics of fuzzy logic (1998) · Zbl 0937.03030 |

[14] | Klement E.P., Triangular norms (2000) · Zbl 0972.03002 |

[15] | Klir G.J., Fuzzy sets and fuzzy logic. Theory and applications (1995) · Zbl 0915.03001 |

[16] | Krajči S., Logic journal of the IGPL pp 543– (2005) · Zbl 1088.06005 |

[17] | DOI: 10.1016/S0019-9958(76)90446-0 · Zbl 0326.02048 |

[18] | DOI: 10.1016/S0019-9958(65)90241-X · Zbl 0139.24606 |

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.