an:00975730
Zbl 0861.03041
Drossos, Costas A.; Markakis, George; Shakhatreh, M.
A nonstandard approach to fuzzy set theory
EN
Kybernetika 28, Suppl., 41-44 (1992).
00012642
1992
j
03E72 03H05
possibility distribution; nonstandard approach to fuzzy sets; Boolean generalization of Infinitesimal Analysis; fuzzy partitions; Boolean probability algebra
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 \(f=p\circ \pi\).
Zbl 0760.03016; Zbl 0712.03045; Zbl 0789.03038