Axiomatic characterizations of \(L\)-valued rough sets using a single axiom. (English) Zbl 1535.03259
Summary: Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory. There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach. Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering \(L\) being a GL-quantale, we will develop the theory of \(L\)-valued rough sets with an \(L\)-set as the basic universe of defining \(L\)-valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets, we will present the axiomatic characterizations of \(L\)-valued upper and lower rough approximation operators on an \(L\)-set with respect to reflexive, symmetric, transitive \(L\)-valued relations on an \(L\)-set as well as their compositions. Choosing an \(L\)-set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.
MSC:
| 03E72 | Theory of fuzzy sets, etc. |
| 68T37 | Reasoning under uncertainty in the context of artificial intelligence |
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