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

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
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