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Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic. (English) Zbl 0638.03051
Using 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
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[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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