$l$-fuzzy approximation space

We consider properties of L-fuzzy relations and L-normal operators for a residuated lattice L in detail and show that the class RL(U) of all L-fuzzy relations on U and the class NL(U) of all L-normal operators are residuated lattices and they are isomorphic as lattices. Moreover, we prove that for any L-normal operators F, it is reflexive (or transitive) if and only if the L-fuzzy relation RF induced by F is reflexive (or transitive) respectively.

Galois connection in category theory play an important role inestablish the relationships between different spatial structures. Inthis paper, we prove that there exist many interesting Galoisconnections between the category of Alexandroff $L$-fuzzytopological spaces, the category of reflexive $L$-fuzzyapproximation spaces and the category of Alexandroff $L$-fuzzyinterior (closure) spaces. This indicates that there is a closeconnection between the three structures.