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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:
 [1] C. Drossos: Foundations of fuzzy sets: A nonstandard approach. Fuzzy Sets and Systems · Zbl 0712.03045 · doi:10.1016/0165-0114(90)90027-4 [2] C. Drossos, G. Markakis: Boolean powers and stochastic spaces (submitted 1988). [3] C. Drossos, G. Markakis: Boolean fuzzy sets. Fuzzy Sets and Systems · Zbl 0760.03016 · doi:10.1016/0165-0114(92)90269-A [4] C. Drossos G. Markakis, G. Tzavelas: A non-standard approach to a general theory of random and B-fuzzy sets I. · Zbl 0760.03016 [5] C. Drossos, M. Shakhatreh: Fuzzy Boolean powers and fuzzy probability (in preparation). · Zbl 0795.03093 [6] G. Markakis: A Boolean Generalization of Infinitesimal Analysis with Application to Fuzzy Sets. Ph.D. Thesis in Greek 1990. [7] K. Piasecki: A remark on the definition of fuzzy P/Measures and the Bayes formula. Fuzzy Stes and Systems 27 (1988), 379-383. · Zbl 0655.60004 · doi:10.1016/0165-0114(88)90063-2 [8] A. Sgarro: Fuzziness measures for fuzzy rectangles. Fuzzy Sets and System 34 (1990), 39-45. · Zbl 0683.94019 · doi:10.1016/0165-0114(90)90125-P [9] H. Störmer: A probability approach to fuzzy sets. Beiträge zur Angewandten Math. und Statist., Hanser. Munich 1987, pp. 275-296.
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