On interval type-2 rough fuzzy sets

In this paper, we present a general framework for the study of interval type-2 rough fuzzy sets by using both constructive and axiomatic approaches. First, several concepts and properties of interval type-2 fuzzy sets are introduced. Then, a pair of lower and upper interval type-2 rough fuzzy approximation operators with respect to a crisp binary relation is proposed. Classical representations of the interval type-2 rough fuzzy approximation operators are then constructed, and the connections between the special binary relations and the interval type-2 rough fuzzy approximation operators are investigated. Furthermore, an operator-oriented characterization of interval type-2 rough fuzzy sets is proposed; that is, interval type-2 rough fuzzy approximation operators are characterized by axioms. Different axiom sets of interval type-2 fuzzy set-theoretic operators guarantee the existence of different types of crisp binary relations, which produce the same operators. Furthermore, the relationship between interval type-2 rough fuzzy sets and interval type-2 fuzzy topological spaces is obtained. The sufficient and necessary condition for the conjecture that an interval type-2 fuzzy interior (closure) operator derived from an interval type-2 fuzzy topological space can be associated with a reflexive and transitive binary relation such that the corresponding lower (upper) interval type-2 rough fuzzy approximation operator is the interval type-2 fuzzy interior (closure) operator is examined. Finally, we provide a practical application to illustrate the usefulness of the interval type-2 rough fuzzy sets model.