Single axiomatic characterization of a hesitant fuzzy generalization of rough approximation operators

Hesitant fuzzy set is a natural generalization of the classical fuzzy set. A hesitant fuzzy set on a universe of discourse is in terms of a function that when applied to the universe returns a finite subset of [0, 1]. Since the axiomatic method of approximation operator is of great significance in the research of the mathematical structure of rough set theory, it is a fundamental problem in axiomatic method to find the minimum set of abstract axioms. This paper first introduces the basic concepts, properties and related operations of hesitant fuzzy set, hesitant fuzzy rough set and hesitant fuzzy rough approximation operator. Secondly, by defining inner product, outer product and by exploring their related properties, the single axiomatization problem of the classical hesitant fuzzy rough approximation operator is solved. Furthermore, we study the single axiomatization of hesitant fuzzy rough approximation operators derived from serial, reflexive, symmetric and transitive hesitant fuzzy relations, respectively. Finally, we compare and analyze the advantages and disadvantages of hesitant fuzzy set, fuzzy rough set and hesitant fuzzy rough set through some cases.