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The concept of rough set was originally proposed by Pawlak [1]. This theory has been developed significantly due to its importance for the study of intelligent systems with insufficient and incomplete information. In rough sets introduced by Pawlak, the key role is played by equivalence relations. In literature [2–4], several generalizations of rough sets have been made by replacing the equivalence relation by an arbitrary relation. Further, Dubois and Prade [5] introduced the concept of fuzzy rough set, in which fuzzy relations play a key role instead of crisp relations. Recently, the combinations of fuzzy sets and rough sets were investigated with different fuzzy logic operations and binary fuzzy relation in [6–14], where fuzzy implications play an important role in the extensions of fuzzy rough sets. In a recent study, Li and Yue established more general model of fuzzy rough sets based on L-valued relation. They investigated it from both constructive and axiomatic approaches, where L is a GL-quantale. In this model of fuzzy rough sets, the key components are a universal set A (as an L-set), an Lvalued equivalence relation on A and an L-subset of A (cf., [15], for details).

On Relationship between L-valued Approximation Spaces and L-valued Transformation Systems