Extending the notion of dimension of a hyperring, we introduce the concept of height of a prime hyperideal of a hyperring, similarly as in ring theory, and we present some basic properties and relations with the dimension notion. In the second part of the article we illustrate some results concerning the height of prime hyperideals in Noetherian/Artinian hyperrings.

Neutrosophic $\mathcal{N}$ -structures with applications in $BCK/BCI$ -algebras is discussed. The notions of a neutrosophic $\mathcal{N}$ -subalgebra and a (closed) neutrosophic $\mathcal{N}$ -ideal in a $BCK/BCI$ -algebra are introduced, and several related properties are investigated. Characterizations of a neutrosophic $\mathcal{N}$ -subalgebra and a neutrosophic $\mathcal{N}$ -ideal are considered, and relations between a neutrosophic $\mathcal{N}$ -subalgebra and a neutrosophic $\mathcal{N}$ -ideal are stated. Conditions for a neutrosophic $\mathcal{N}$ -ideal to be a closed neutrosophic $\mathcal{N}$ -ideal are provided.

The notion of a neutrosophic quadruple $BCK/BCI$-number is considered, and a neutrosophic quadruple $BCK/BCI$-algebra, which consists of neutrosophic quadruple $BCK/BCI$-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple $BCK$-algebra and a closed ideal in a neutrosophic quadruple $BCI$-algebra are studied. Given subsets A and B of a $BCK/BCI$-algebra, the set $NQ(A,B)$, which consists of neutrosophic quadruple $BCK/BCI$-numbers with a condition, is established. Conditions for the set $NQ(A,B)$ to be a (positive implicative) ideal of a neutrosophic quadruple $BCK$-algebra are provided, and conditions for the set $NQ(A,B)$ to be a (closed) ideal of a neutrosophic quadruple $BCI$-algebra are given. An example to show that the set $\{\tilde{0}\}$ is not a positive implicative ideal in a neutrosophic quadruple $BCK$-algebra is provided, and conditions for the set $\{\tilde{0}\}$ to be a positive implicative ideal in a neutrosophic quadruple $BCK$-algebra are then discussed.

The concept of hesitant union (⋓) is introduced. Using this, the notions of ⋓-hesitant fuzzy subalgebras and ⋓-hesitant fuzzy ideals are introduced and several properties are investigated. Relations between ∪-hesitant fuzzy subalgebras (resp. ideals) and ⋓-hesitant fuzzy subalgebras (resp. ideals) are considered. Relations between a ⋓-hesitant fuzzy subalgebra/ideal and its lower level set are investigated. Characterizations of ∪-hesitant fuzzy subalgebras/ideals are discussed. Given a subalgebra (resp., ideal) A of a BCK / BCI -algebra X , a ⋓-hesitant fuzzy subalgebra (resp., ideal) is established.

Inspired by the concepts of projective and injective modules in classical algebraic structure theory, in this paper we initiate the study of the chains of hypermodules over a Krasner hyperring R, endowing first the set $Ho{m}_R^n(M,N)$ of all normal homomorphisms between two R-hypermodules M and N with a structure of R-hypermodule. Then, our study focuses on the concepts of normal injectivity and projectivity of hypermodules over a Krasner hyperring R, characterizing them by the mean of chains of R-hypermodules.