Boolean fuzzy sets and possibility measures.

*(English)*Zbl 1096.94538B-possibility and B-necessity measures are defined as the qualitative analogues of the well known \([0, 1]\)-measures. Their connection to B-fuzzy sets (Boolean fuzzy sets) is established. It is also proved that B-fuzzy sets define B-possibility and B-necessity measures by their membership function. We also introduce the possibility of a B-fuzzy set and prove that it is the extension of the possibility measure defined on crisp sets. This extension can be defined in two ways which are proved to be equivalent.

##### MSC:

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

03E72 | Theory of fuzzy sets, etc. |

PDF
BibTeX
XML
Cite

\textit{G. Markakis}, Fuzzy Sets Syst. 110, No. 2, 279--285 (2000; Zbl 1096.94538)

Full Text:
DOI

##### References:

[1] | Banon, G., Distinction between several subsets of fuzzy measures, Fuzzy sets and systems, 5, 291-305, (1981) · Zbl 0449.60001 |

[2] | Dempster, A.P., Upper and lower probabilities induced by a multivalued mapping, Ann. math. statist., 38, 235-339, (1967) · Zbl 0168.17501 |

[3] | Drossos, C.A., Foundations of fuzzy sets: a nonstandard approach, Fuzzy sets and systems, 37, 287-307, (1990) · Zbl 0712.03045 |

[4] | Drossos, C.A.; Markakis, G., Boolean fuzzy sets, Fuzzy sets and systems, 46, 81-95, (1992) · Zbl 0760.03016 |

[5] | Drossos, C.A.; Markakis, G., Boolean powers and stochastic spaces, Math. Slovaca, 1994, (1944) |

[6] | Drossos, C.A.; Markakis, G.; Shakhatreh, M., A non-standard approach to fuzzy set theory, Suppl. kybernetika, 28, 41-44, (1992) · Zbl 0861.03041 |

[7] | Drossos, C.A.; Theodoropoulos, P.L., B-fuzzy probabilities, Fuzzy sets and systems, 78, 355-369, (1996) · Zbl 0868.03023 |

[8] | Dubois, D.; Prade, H., Possibility theory, (1988), Plenum Press New York · Zbl 0645.68108 |

[9] | Mansfield, R., The theory of Boolean ultrapowers, Ann. math. logic, 2, 297-323, (1971) · Zbl 0216.29401 |

[10] | Markakis, G., The Boolean generalization of infinitesimal analysis with applications to fuzzy sets, ph.D. dissertation (in Greek), (1990), University of Patras Greece |

[11] | Shafer, G., A mathematical theory of evidence, (1976), Princeton Univ. Press Princeton, NJ · Zbl 0359.62002 |

[12] | Sugeno, M., Theory of fuzzy integrals and its applications, ph.D. dissertation, (1974), Tokyo Institute of Technology |

[13] | Wang, Z.; Klir, G.I., Fuzzy measure theory, (1992), Plenum Press New York |

[14] | Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606 |

[15] | Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002 |

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.