Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic.

*(English)*Zbl 0638.03051Using the language of many-valued logic (with [0,1] as set of truth values and max, min as well as an additional conjunction described by a left continuous t-norm T as main connectives), the author introduces in a rather algebraic style, fuzzy sets and fuzzy partial orderings with respect to a many-valued identity predicate which is playing a basic role. In this way the approach becomes analogous to an earlier one of the reviewer [Fuzzy Sets Syst. 2, 125-151 (1979; Zbl 0408.03042)]; a novelty yet is that here idempotent elements of [0,1] w.r.t. T play an essential part. On that basis the author gives a successful fuzzification of the completion of a partially ordered set by Dedekind cuts. An interesting special case is reached with \(T(x,y)=\max (0,x+y-1)\) where the fuzzy completion of the rationals produces the usual probability measures supporting the point of view that they represent a preferrable version of fuzzy real numbers.

Reviewer: S.Gottwald

##### MSC:

03E72 | Theory of fuzzy sets, etc. |

03B50 | Many-valued logic |

06B23 | Complete lattices, completions |

06A06 | Partial orders, general |

##### Keywords:

completions of fuzzy partial orderings; fuzzy sets; many-valued identity predicate; Dedekind cuts; probability measures; fuzzy real numbers
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##### References:

[1] | Birkhoff, G., Lattice theory, AMS colloquium publications, () |

[2] | Cerruti, U.; Höhle, U., Categorical foundations of probabilistic microgeometry, () |

[3] | Cerruti, U.; Höhle, U., An approach to uncertainty using algebras over a monoidal closed category, Suppl. rend. circ. mat. Palermo ser. II, 12, 47-63, (1986) · Zbl 0633.18002 |

[4] | Dedekind, R., Was sind und was sollen die zahlen, 1888, (), English translation by · JFM 36.0087.04 |

[5] | Dubois, D.; Prade, H., Fuzzy real algebra, some results, Fuzzy sets and systems, 2, 327-348, (1979) · Zbl 0412.03035 |

[6] | Goldblatt, R., Topoi, the categorical analysis of logic, (1979), North-Holland Amsterdam · Zbl 0434.03050 |

[7] | Höhle, U., Representation theorems for L-fuzzy quantities, Fuzzy sets and systems, 5, 83-107, (1981) · Zbl 0448.03041 |

[8] | Höhle, U., The category CM-SET, an algebraic approach to uncertainty, (1986), Preprint |

[9] | MacNeille, H., Partially ordered sets, Trans. amer. math. soc., 42, 416-460, (1937) · JFM 63.0833.04 |

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

[11] | Wong, C.K., Fuzzy points and local properties, J. math. anal. appl., 46, 316-328, (1974) · Zbl 0278.54004 |

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