Topological and lattice structures of LL-fuzzy rough sets determined by lower and upper sets

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