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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

03E72 Theory of fuzzy sets, etc.
03B50 Many-valued logic
06B23 Complete lattices, completions
06A06 Partial orders, general
Full Text: DOI
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