On (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices

Rough sets were introduced by Pawlak as a formal tool for dealing with imprecision and uncertainty in data analysis. After that time, varieties of fuzzy generalizations of rough approximations have been investigated in the literature. In particular, generalized fuzzy rough sets determined by a triangular norm have been considered. On the other hand, binary operations & and ⊙ on complete residuated and co-residuated lattice L can be regarded as two more extensive operations than left continuous triangular norms and right continuous triangular conorms, respectively. Therefore, as a further generalization of the notion of rough sets, this paper is devoted to proposing (⊙,&)-fuzzy rough sets based on residuated and co-residuated lattices from both constructive and axiomatic approaches. In the constructive approach, we define a pair of lower and upper L-fuzzy rough approximation operators determined by ⊙ and &, respectively. Meanwhile, various classes of (⊙,&)-fuzzy rough sets are investigated. In the axiomatic approach, the axiomatic characterizations of different (⊙,&)-fuzzy rough approximation operators are studied. Furthermore, the topological properties of (⊙,&)-fuzzy rough sets are discussed, especially from the categorical point of view. At the end of this paper, we give a brief introduction to a new model of fuzzy rough sets and discuss some properties of it.