On characterization of generalized interval type-2 fuzzy rough sets

This paper presents a systematic study of interval type-2 fuzzy rough sets integrating rough set theory with interval type-2 fuzzy set theory using constructive and axiomatic approaches. From the perspective of a constructive approach, a pair of lower and upper interval type-2 fuzzy rough approximation operators with respect to an interval type-2 fuzzy relation is defined. The basic properties of the interval type-2 fuzzy rough approximation operators are studied. Using cut sets of interval type-2 fuzzy sets, classical representations of interval type-2 fuzzy rough approximation operators are then presented, and the connections between special interval type-2 fuzzy relations and interval type-2 fuzzy rough approximation operators are investigated. Adopting an axiomatic approach, an operator-oriented characterization of interval type-2 fuzzy rough sets is proposed; in other words, interval type-2 fuzzy rough approximation operators are characterized by axioms. Different axiom sets of interval type-2 fuzzy set-theoretic operators guarantee the existence of different types of interval type-2 fuzzy relations that produce the same operators. Finally, the relationship between interval type-2 fuzzy rough sets and interval type-2 fuzzy topological spaces is examined. We obtain sufficient and necessary conditions for the conjecture that an interval type-2 fuzzy interior (closure) operator derived from an interval type-2 fuzzy topological space can associate with an interval type-2 fuzzy reflexive and transitive relation such that the corresponding lower (upper) interval type-2 fuzzy rough approximation operator is the interval type-2 fuzzy interior (closure) operator.