For a complete residuated lattice \({\mathcal {L}}\) , the aim of this paper is to develop a general framework of \({\mathcal {L}}\) -valued rough sets. A pair of approximation operators derived from \({\mathcal {L}}\) -valued neighborhood systems ( \({\mathcal {L}}\) -VNSs for short) are constructed and discussed. It is proved that these operators include, as special cases, \({\mathcal {L}}\) -valued relation (and Zhao’s \({\mathcal {L}}\) -fuzzy generalized neighborhood system)-based approximation operators. Furthermore, special cases of \({\mathcal {L}}\) -VNS and their corresponding approximation operators are discussed and characterized. Based on such \({\mathcal {L}}\) -valued approximation operators ( \({\mathcal {L}}\) -VApprXOs for short), we define a method that allows us to measure the “quality" of \({\mathcal {L}}\) -valued approximations of \({\mathcal {L}}\) -subsets. Such measures are interpreted in terms of lattice-valued fuzzy topologies on \({\mathcal {L}}\) -sets. Finally, the relationships between \({\mathcal {L}}\) -VApprXOs based on \({\mathcal {L}}\) -VNS and many valued topologies on \({\mathcal {L}}\) -sets are studied.
