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**Rough approximate operators: Axiomatic rough set theory.**
*(English)*
Zbl 0818.03028

Ziarko, Wojciech P. (ed.), Rough sets, fuzzy sets and knowledge discovery. Proceedings of the international workshop, RSKD ’93, Banff, Alberta, Canada, 12-15 October 1993. London: Springer-Verlag (in collab. with the British Computer Society). Workshops in Computing. 256-260 (1994).

Summary: In rough set theory, the upper and lower approximations are defined in terms of equivalence relation. In this paper, the reverse problem is considered. Let \(H\) and \(L\) be two abstract operators acting on the power set of \(U\), the universe of discourse. If the two operators satisfy six axioms, then there is an equivalence relation defined on \(U\) such that \(H(X)\) and \(L(X)\) are precisely the upper and lower approximations. The six axioms are adopted from the axioms of Kuratowski’s closure operator. The proof is an easy application of point set topology. Similar results (five axioms) are also obtained for neighborhood systems (a generalized rough set theory) which are based on FrĂ©chet (V) spaces. The results can be viewed as a beginning of an axiomatic rough set theory.

For the entire collection see [Zbl 0812.00038].

For the entire collection see [Zbl 0812.00038].

### MSC:

03E70 | Nonclassical and second-order set theories |

### Keywords:

approximations; abstract operators; universe of discourse; equivalence relation; neighborhood systems; axiomatic rough set theory
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\textit{T. Y. Lin} and \textit{Q. Liu}, in: Rough sets, fuzzy sets and knowledge discovery. Proceedings of the international workshop, RSKD '93, Banff, Alberta, Canada, 12-15 October 1993. London: Springer-Verlag (in collab. with the British Computer Society). 256--260 (1994; Zbl 0818.03028)