This study is based on the structure of pseudo hyper GR-algebra, analgebra that is partially related on some class of hyper BCI-algebras.This allows us to create a new structure and how this two algebrasare related to each other. A pseudo hyper GR-algebra involves twohyper operations and a set of axioms that come in pairs or a combina-tion of both making it interesting like some algebras established. Thispaper focuses on some properties of pseudo hyper GR-algebras and itsideals. Moreover, pseudo hyper GR-ideals were dened and classied todetermine their relationship to each other.

The implicative hyper GR-ideals, the fuzzy implicative hyper GR-ideals of type 1 and the fuzzy implicative hyper GR-ideals of type 2 are introduced, and several properties are investigated. Characterizations of fuzzy implicative hyper GR-ideals of type 1 are established using level subsets of fuzzy sets.

Let $P(H)$ be the power set of $H$. Consider $\ds P^*(H)=P(H)\setminus\{\phi\}$. A hyperoperation on a nonempty set $H$ is a function $\ds\circledast:H\times H\rightarrow P^*(H).$ A set $H$ endowed with a family $\Gamma$ of hyperoperations is called ahyperstructure. Hyperstructures have many applications to several sectors of both pure and applied sciences. One of the well developed hyperstructures is the hyper BCI-algebra. Recently, by following this hyperstructure, a new hyperstructure was created by Indangan et al., named as hyper GR-algebras. In this paper, fuzzy set and intuitionistic fuzzy set are applied to hyper GR-algebra. Particularly, the fuzzy hyper GR-ideal of type 1 and the intuitionistic fuzzy hyper GR-ideal are introduced, and a relationship between them are obtained. Moreover, some of their characterizations are established by the use of their level subsets.

We characterize the dimension of Lie algebras of white noise operators containing the quantum white noise derivatives of the conservation operator. We establish isomorphisms to filiform Lie algebras, Engel-type algebras, and solvable Lie algebras with Heisenberg nilradical and Abelian nilradical. A new class of solvable Lie algebras is proposed, those having an Engel-type algebra as nilradical. This arises in white noise analysis as a 2n+3-dimensional Lie algebra containing the identity operator, annihilation operators, creation operators (Heisenberg algebra), number operator, and Gross Laplacian.

In this paper, the notions of SVN hyper UP-algebra and SVNS hyper UP-algebra are introduced, and some of their structural properties are investigated. Moreover, the Cartesian product of SVNS hyper UP-algebra is discussed and proved to be a SVNS hyper UP- algebra. Finally, the homomorphic image and preimage of SVNS hyper UP-algebra under SVNS functions are studied and showed also to be SVNS hyper UP-algebra.