$L$-valued rough approximation oper

$L$-valued rough approximation operators are important notions in $L$-valued rough sets. Two approaches for the development of this theory of $L$-valued rough sets have been reviewed, i.e., the constructive way and the axiomatic way. Single axioms to characterize $L$-valued rough approximation operators have become very popular for many years. In this paper, considering $L$ being a GL-quantale or an MV-algebra, i.e., a GL-quantale with a pseudo complementary involution, we further investigates single axiomatic characterizations of $L$-valued rough approximation operators, where fuzzy sets are called $L$-sets that can be seen a farther generalization of Zadeh' fuzzy sets. Firstly, we characterize $L$-valued upper rough approximation operators by single axioms. Then, we give the axiomatic definition of $L$-valued upper rough approximation operators. Further, we use single axioms to describe $L$-valued upper rough approximation operators with various kinds of $L$-valued relations, such as reflexive, symmetric, transitive as well as their compositions in a GL-quantale or an MV-algebra. Finally, similar to the research of $L$-valued upper rough approximations operators, we discuss the related properties of the $L$-valued lower rough approximation operators in an MV-algebra.