Fuzzy rough sets are intuitionistic \(L\)-fuzzy sets. (English) Zbl 0924.04001
The concepts of intuitionistic \(L\)-fuzzy sets (IL-FS) [see K. Atanassov and S. Stoeva, “Intuitionistic \(L\)-fuzzy sets”, in: R. Trappl (ed.), Cybernetics and systems research 2, 539-540 (1984; Zbl 0547.03030)], which are extensions of the IFSs [K. T. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets Syst. 20, 87-96 (1986; Zbl 0631.03040)] and of rough sets [Z. Pawlak, “Rough sets and fuzzy sets”, ibid. 17, 99-102 (1995; Zbl 0588.04004)] are described. The author proves the following Theorem: Let \(U\) be a nonempty set, \(\langle L,\leq\rangle\) a complete distributive lattice whose least element and greatest element are denoted by 0 and 1, respectively, with an involutive order-reversing operation \(':L\to L\), and \(B\) be a Boolean subalgebra of the set of all subsets of \(U\). Then any fuzzy rough set in \(X\in B^2\) is an IL-FS in \(U\). This theorem is important for determining the relations between the different extensions of ordinary fuzzy sets.
Reviewer: K.Atanassov (Sofia)
MSC:
| 03E72 | Theory of fuzzy sets, etc. |
References:
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