As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in $BCK/BCI$ -algebra is considered. The notions of $\alpha$ -internal, $\beta$ -internal, $\alpha$ -external and $\beta$ -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of $\alpha$ -internal and $\alpha$ -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in $BCK/BCI$ -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.

The notion of a neutrosophic positive implicative $\mathcal{N}$ -ideal in $BCK$ -algebras is introduced, and several properties are investigated. Relations between a neutrosophic $\mathcal{N}$ -ideal and a neutrosophic positive implicative $\mathcal{N}$ -ideal are discussed. Characterizations of a neutrosophic positive implicative $\mathcal{N}$ -ideal are considered. Conditions for a neutrosophic $\mathcal{N}$ -ideal to be a neutrosophic positive implicative $\mathcal{N}$ -ideal are provided. An extension property of a neutrosophic positive implicative $\mathcal{N}$ -ideal based on the negative indeterminacy membership function is discussed.

The notion of a neutrosophic commutative N -ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N-ideal and a neutrosophic commutative N-ideal are discussed. Characterizations of a neutrosophic commutative N-ideal are considered.

The notions of hyperfuzzy ideals in $BCK/BCI$ -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy ideals are provided.

The notion of a hesitant fuzzy semigroup with a frontier is introduced, and several properties are investigated. Characterizations of a hesitant fuzzy semigroup with a frontier are considered, and a condition for a special set to be a subsemigroup is provided. The hesitant union and hesitant intersection of two hesitant fuzzy semigroups with a frontier are dealt with, and the hesitant fuzzy pre-image and hesitant fuzzy image of a hesitant fuzzy semigroup with a frontier under the homomorphism are discussed.