Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented.

In an attempt to generalize and create alternatives for classical algebraic structures, Smarandache in 2019 introduced the idea of treating binary operations on sets as partially well (totally outer) defined with axioms that are partially true (totally false) and even indeterminate for some elements of any given algebraic structure. The structures formed with these types of operations called Neutro(Anti) Operations are called Neutro(Anti) Structures and are named according to the axioms that are neutro(anti)-sophicated. In this chapter, the authors consider some properties of neutrosophic quadruple numbers with examples. In particular, they introduce and study the concept of NeutroQuadrupleRings and their substructures. Some basic definitions and a few important results are presented. It is shown that the intersection and union of any two NeutroQuadrupleSubrings of a particular class does not necessarily belong to the same class. Also, they give a necessary and sufficient condition for a neutrosophic quadruple ring (NQ(X), +, ·) to be a NeutroQuadrupleRing.

The process through which AntiAlgebraicStructures can be generated from any given classical structure is called AntiSophication. In this chapter, the authors introduce the concept of AntiHyperStructure and present some of its basic properties. It has been shown in our previous works that there are 19 types/classes of AntiHyperGroups. The chapter is devoted to the study of a particular class of these 19 classes, called a [2, 3] AntiHyperGroup, that is, a class in which associativity and reproductive axioms are totally false. Several examples and properties of [2, 3] AntiHyperGroup are presented. It is shown that the union of [2, 3] AntiSubhypergroups is a [2, 3] AntiHyperGroup and their intersection is not a [2, 3] AntiSubhypergroup. Also, it is shown that the Cartesian product of a [2, 3] AntiHyperGroup with any other class of AntiHyperGroup is not in general a [2, 3] AntiHyperGroup but its product with a classical Hypergroup/[2, 3]NeutroHyperGroup is a [2, 3] AntiHyperGroup. Further, it is shown that a [2, 3] AntiHyperGroup factored by a [2, 3] AntiSubhypergroup is a [2, 3] AntiHyperGroup.