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Roughness based on fuzzy ideals. (English) Zbl 1112.03049

Summary: The theory of rough sets, proposed by Pawlak, and the theory of fuzzy sets, proposed by Zadeh, are complementary generalizations of classical set theory. Many sets are naturally endowed with two binary operations: addition and multiplication. One concept which does this is a ring. This paper concerns a relationship between rough sets, fuzzy sets and ring theory. It is a continuation of ideas presented by N. Kuroki and P. P. Wang [“The lower and upper approximations in a fuzzy group”, Inf. Sci. 90, 203–220 (1996; Zbl 0878.20050)]. We consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal. In fact, we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring. Some characterizations of the above approximations are made and some examples are presented.

MSC:

03E70 Nonclassical and second-order set theories
03E72 Theory of fuzzy sets, etc.
16D25 Ideals in associative algebras
16Y99 Generalizations

Citations:

Zbl 0878.20050
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