Neutrosophic LA-Semigroup Rings

Deleted Journal, 2024

Since F. Smarandache introduced the "neutrosophic" in [1,2] "neutrosophic" studied in many sciences, such as algebra by introducing neutrosophic group which played a basic role in studying many neutrosophic algebraic structures as a neutrosophic ring, neutrosophic module, and neutrosophic vector space. In recent years, Agboola et al. studied the concept of " neutrosophic group " and " neutrosophic ring " in [3, 4], They also, In 2015, introduced the concept of " refined neutrosophic algebraic structures " [5], and presented " refined neutrosophic groups". Also, Adeleke et al. in [6, 7] in 2020, studied several refined concepts such as "refined neutrosophic rings", "refined neutrosophic ideals" and "refined neutrosophic homomorphisms" in detail. Also, many researchers study applications of " neutrosophic group " in topology, such as R. Al-Hamido, in 2021, [8] studied " neutrosophic bi-topological groups ", and investigated its basic properties. Before this study, Q. Imran, et al. in [9] studied several types of "neutrosophic topological groups" and introduced their basic properties. Also, Sumathi et al. in [10] introduced the concept of "neutrosophic topological groups". Also, P. K. Singh, et al. in [11] defined the concept of a "symbolic Turiyam ring "as an application of the "Turiyam symbolic set" [12, 13] and as a "new generalization" of "neutrosophic rings".

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures ina broader sense. The indeterminacy element “ I ” give rise to a more bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three catagories such as: neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical algebraic structures respectively. This reveals the fact that a classical algebraic structure is a part of the neutrosophic algebraic structures. This open a new way for the researchers to think in a broader way to visualize these vast neutrosophic algebraic structures.

In this paper we define neutrosophic bi-LAsemigroup and neutrosophic N-LA-semigroup. Infact this paper is an extension of our previous paper neutrosophic left almost semigroup shortly neutrosophic LAsemigroup. We also extend the neutrosophic ideal to neutrosophic biideal and neutrosophic N-ideal. We also find some new type of neutrosophic ideal which is related to the strong or pure part of neutrosophy. We have given sufficient amount of examples to illustrate the theory of neutrosophic bi-LA-semigroup, neutrosophic N-LAsemigroup and display many properties of them this paper.

2006

Neutrosophic components (NC) under addition and product form different algebraic structures over different intervals. In this paper authors for the first time define the usual product and sum operations on NC. Here four different NC are defined using the four different intervals: (0, 1), [0, 1), (0, 1] and [0, 1]. In the neutrosophic components we assume the truth value or the false value or the indeterminate value to be from the intervals (0, 1) or [0, 1) or (0, 1] or [0, 1]. All the operations defined on these neutrosophic components on the four intervals are symmetric. In all the four cases the NC collection happens to be a semigroup under product. All of them are torsion free semigroups or weakly torsion free semigroups. The NC defined on the interval [0, 1) happens to be a group under addition modulo 1. Further it is proved the NC defined on the interval [0, 1) is an infinite commutative ring under addition modulo 1 and usual product with infinite number of zero divisors and the ring has no unit element. We define multiset NC semigroup using the four intervals.