Axiomatic approaches to three types of L-valued rough sets

In this paper, considering L being a GL-quantale, we further develop the theory of L-valued rough sets with an L-set as the basic universe of defining L-valued rough approximation operators. Choosing an L-set as the universe can break the rules of adopting Zadeh’s fuzzy sets as the universe. We first introduce three types of L-valued relations on L-sets, namely, inverse serial, mediate, Euclidean, and then characterize them by L-valued rough sets. Adopting the idea of single axiomatic characterizations of L-valued rough sets, we present the axiomatical characterizations of L-valued upper and lower rough approximation operators on an L-set concerning these new L-valued relations by fuzzy unions and fuzzy intersections. Moreover, in the framework of category, we introduce the concepts of L-valued Alexander co-topological spaces and L-valued closure spaces on L-sets by fuzzy unions and fuzzy intersections, then prove they are category isomorphic. By fuzzy unions, we obtain a simplified axiomatic system of the L-valued closure spaces, which highlights the advantages of fuzzy unions. Finally, we obtain the category of L-valued Alexander co-topological spaces and their continuous mappings is isomorphic to the category of L-valued preordered approximation spaces and their order-preserving mappings.