Type-2 fuzzy rough sets based on extended t-norms

•A type-2 fuzzy rough set based on extended t-norms is investigated from constructive and axiomatic approaches.•An approximation algorithm of type-2 fuzzy rough sets is listed based on α-cut sets.•Weak type-2 fuzzy topologies are proposed to investigate the topological properties of type-2 fuzzy rough sets.•An interval-valued type-2 fuzzy rough set is proposed and investigated.

This paper is devoted to investigate type-2 fuzzy rough sets based on extended t-norms with respect to type-2 fuzzy relations with convex normal fuzzy truth values. Type-2 fuzzy rough sets are proposed on two finite universes of discourse and characterized from both constructive and axiomatic approaches. In the constructive approach, different classes of type-2 fuzzy rough sets are investigated. An approximate algorithm of type-2 fuzzy rough sets based on α-cut sets is given with numerical examples. In the axiomatic approach, the axiomatic characterizations of type-2 fuzzy rough approximation operators are studied. Moreover, the topological properties of type-2 fuzzy rough sets are discussed. In the end, this paper introduces interval-valued type-2 fuzzy rough sets and discusses their properties.