Using one axiom to characterize rough set and fuzzy rough set approximations

Axiomatic approaches are important for understanding the concepts of rough set theory. The properties of the approximation operators constructed in rough set theory are determined by a binary relation. A linkage, in the form of a necessary and sufficient condition, between the constructive approach and an axiomatic system is established in this paper. Various classes of rough set algebras are characterized by different sets of axioms. With no restriction on the cardinality of the universal set, we use only one axiom to describe the approximation generated by the reflexive, symmetric, and transitive relations, respectively. In addition, we characterize lower and upper approximations of the Pawlak rough set with only one axiom. We also study a similar problem in the context of fuzzy sets.