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