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A nonstandard approach to fuzzy set theory. (English) Zbl 0861.03041
Summary: The nonstandard approach to fuzzy sets [the first author, Fuzzy Sets Syst. 37, No. 3, 287-307 (1990; Zbl 0712.03045)] is based on a Boolean generalization of Infinitesimal Analysis [see, e.g., the first two authors, Math. Slovaka 44, No. 1, 1-19 (1994; Zbl 0789.03038)]. This paper, gives a short review of this approach, describes some applications to mathematical structures and indicates the way for an extension using fuzzy partitions. In addition, we prove that the theory is general, since for any ordinary fuzzy set \(f:X\to [0,1]\) there exists a unique Boolean probability algebra \((\mathbb{B},p)\) and a \(\mathbb{B}\)-possibility distribution \(\pi:X\to\mathbb{B}\), such that \(f=p\circ \pi\).

MSC:
03E72 Theory of fuzzy sets, etc.
03H05 Nonstandard models in mathematics
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References:
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