Journal of Neutrosophic and Fuzzy Systems

Recent time data representation and visualization is considered as one of the major issues. It become more crucial when the data sets exists beyond the non-euclidean geometry and its hybridization also. There are several examples given by non-euclidean geometry by Lobachevsky, Bolyai, Riemannian which contains failure of Euclid postulates V and II, respectively. The problem arises when none of the Euclid Postulates exists. It might possible that the data sets contains unknown or co-ordinate free geometry. In this case the data can be explored based on a defined vector space rather than available co-ordinate geometry. It require human Turiyam consciousness to explore these types of unknown, undefined, co-ordinate free data. To understand this problem current paper explores the Turiyam geometry and its basic for exploring the unknown or undefined data with an example.

The objective of this paper is to find all formulas that describe the 3-cyclic refined neutrosophic real solutions of the equation 𝑋𝑛=1 which are called 3-cyclic refined real roots of unity. Also, we classify the algebraic group represented by these solutions as a direct product of some familiar finite abelian groups. On the other hand, we illustrate many examples to clarify the validity of our work.

Let I be a right (left) ideal of a ring R, then R/I is a right (left) generalized m – flat module (GmF – module) if and only if for each a Î I , there exists b Î I and a fixed positive integer m such that . In this paper, we study the characterization and properties this class of flat modules, and we give the relation between this class and generalized m- flat modules and m– regular rings, reduced rings, reversible rings and uniform rings.

This paper is dedicated to define and study for the first time the concept of two-fold neutrosophic group by combining two-fold algebras with the classical concept of neutrosophic group. We study the elementary properties of this novel concept by many related theorems and illustrated examples.

This paper is concerned with the study of the classification of 3-cyclic refined real vector spaces by semi-module isomorphisms, where we use this algebraic technique to find the algebraic relationship between real 3-cyclic refined vector spaces and classical vector spaces. Also, we define the inner products on these spaces and prove many related inequalities.  

This paper is dedicated to finding all invertible elements (algebraic units) in 14 different finite neutrosophic rings modulo integers 𝑍𝑛(𝐼) for all 3≤n≤16. We arrange the elements of the group of units of each ring in a table called the units table of the ring.