The solution sets of infinite fuzzy relational equations with sup-conjunctor composition on complete distributive lattices.

*(English)*Zbl 1183.03056Summary: This paper deals with sup-conjunctor composition fuzzy relational equations in infinite domains and on complete distributive lattices. When its right-hand side is a continuous join-irreducible element or has an irredundant continuous join-decomposition, a necessary and sufficient condition describing an attainable solution (resp. an unattainable solution) is formulated and some properties of the attainable solution (resp. the unattainable solution) are shown. Further, the structure of solution sets is investigated.

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

##### Keywords:

complete distributive lattice; conjunctor; relational equation; continuous join-irreducible element; irredundant continuous join-decomposition; solution set
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\textit{X.-p. Wang} and \textit{C. Xia}, Fuzzy Sets Syst. 160, No. 20, 2989--3006 (2009; Zbl 1183.03056)

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##### References:

[1] | G. Birkhoff, Lattice Theory, revised ed., Vol. XXV, American Mathematical Society Colloquium, Providence, RI, 1948. |

[2] | De Baets, B., An order-theoretic approach to solving sup-\(\mathcal{T}\) equations, (), 67-87 · Zbl 0874.04005 |

[3] | De Baets, B.; Mesiar, R., Triangular norms on product lattices, Fuzzy sets and systems, 104, 61-75, (1999) · Zbl 0935.03060 |

[4] | B. De Baets, Analytical solution methods for fuzzy relation equations, in: D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Vol. 1, Kluwer Academic Publishers, Dordrecht, 2000, pp. 291-340. · Zbl 0970.03044 |

[5] | Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy relation equations and their applications to knowledge engineering, (1989), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0694.94025 |

[6] | Fodor, J.C.; Keresztfalvi, T., Nonstandard conjunctions and implications in fuzzy logic, Internat. J. approx. reason., 12, 69-84, (1995) · Zbl 0815.03017 |

[7] | Han, S.C.; Li, H.X.; Wang, J.Y., Resolution of finite fuzzy relation equations based on strong pseudo-t-norms, Appl. math. lett., 19, 752-757, (2006) · Zbl 1121.03075 |

[8] | L. Noskova, I. Perfilieva, System of fuzzy relation equations with sup*-composition in semi-linear spaces: minimal solutions, in: Proc. FUZZ-IEEE Conf. on Fuzzy Systems, July 23-26, 2007, London, pp. 1520-1525. |

[9] | K. Peeva, Y. Kyosev, Fuzzy relational calculus-theory, applications and software (with CD-ROM), Series Advances in Fuzzy Systemsâ€”Applications and Theory, Vol. 22, World Scientific, Singapore, 2004. · Zbl 1083.03048 |

[10] | Perfilieva, I.; Gottwald, S., Fuzzy function as a solution to a system of fuzzy relation equations, Int. J. gen. syst., 32, 361-372, (2003) · Zbl 1059.03060 |

[11] | Qu, X.B.; Wang, X.P., Some properties of infinite fuzzy relational equations on complete Brouwerian lattices, Fuzzy sets and systems, 158, 1327-1339, (2007) · Zbl 1120.03041 |

[12] | Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048 |

[13] | Szasz, G., Introduction to lattice theory, (1963), Academic Press New York · Zbl 0126.03703 |

[14] | Wang, X.P., Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy sets and systems, 120, 409-414, (2001) · Zbl 0981.03055 |

[15] | Wang, X.P., Infinite fuzzy relational equations in a complete Brouwerian lattice, Indian J. pure appl. math., 33, 87-95, (2002) · Zbl 1002.03531 |

[16] | Wang, X.P., Conditions under which a fuzzy relational equation has minimal solution in a complete Brouwerian lattice, Adv. math., 31, 220-228, (2002), (in Chinese) · Zbl 1264.03113 |

[17] | Wang, X.P., Infinite fuzzy relational equations on a complete Brouwerian lattice, Fuzzy sets and system, 138, 657-666, (2003) · Zbl 1075.03026 |

[18] | Wang, X.P.; Xiong, Q.Q., The solution set of a fuzzy relational equation with sup-conjunctor composition in a complete lattice, Fuzzy sets and systems, 153, 249-260, (2005) · Zbl 1073.03539 |

[19] | Wang, X.P.; Qu, X.B., Continuous join-irreducible elements and their applications to describing the solution set of fuzzy relational equations, Acta math. sinica (chin. ser.), 49, 1171-1180, (2006), (in Chinese) · Zbl 1120.03042 |

[20] | Wang, Z.D.; Yu, Y.D., Direct product decomposition of pseudo-t-norms and implications, J. China univ. sci. tech., 31, 657-662, (2001), (in Chinese) · Zbl 1042.03022 |

[21] | Zhang, K.L.; Li, D.H.; Song, L.X., On finite relation equations with sup-conjunctor composition over a complete lattice, Fuzzy sets and systems, 160, 119-128, (2009) · Zbl 1183.03060 |

[22] | Zhao, C.K., On matrix equations in a class of complete and completely distributive lattices, Fuzzy sets and systems, 22, 303-320, (1987) · Zbl 0621.06006 |

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