As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in
-algebra is considered. The notions of
-external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of
-external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in
-algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed.