Topological and lattice structures of \(\mathcal L\)-fuzzy rough sets determined by lower and upper sets. (English) Zbl 1293.03025

Ma, Zhen Ming; Hu, Bao Qing

Inf. Sci. 218, 194-204 (2013).

Summary: This paper builds the topological and lattice structures of \(\mathcal L\)-fuzzy rough sets by introducing lower and upper sets. In particular, it is shown that when the \(\mathcal L\)-relation is reflexive, the upper (resp. lower) set is equivalent to the lower (resp. upper) \(\mathcal L\)-fuzzy approximation set. Then by the upper (resp. lower) set, it is indicated that an \(\mathcal L\)-preorder is the equivalence condition under which the set of all the lower (resp. upper) \(\mathcal L\)-fuzzy approximation sets and the Alexandrov \(\mathcal L\)-topology are identical. However, associating with an \(\mathcal L\)-preorder, the equivalence condition that \(\mathcal L\)-interior (resp. closure) operator accords with the lower (resp. upper) \(\mathcal L\)-fuzzy approximation operator is investigated. At last, it is proven that the set of all the lower (resp. upper) \(\mathcal L\)-fuzzy approximation sets forms a complete lattice when the \(\mathcal L\)-relation is reflexive.

MSC:

03E72	Theory of fuzzy sets, etc.
54A40	Fuzzy topology

Keywords:

fuzzy rough set; lower (upper) set; fuzzy topology; residuated lattice