[Paper Review] New classes of Neutrosophic Linear Algebras

Kandasamy, W. B. Vasantha, Florentín Smarandache+1 more|arXiv (Cornell University)|Mar 9, 2010

Fuzzy and Soft Set Theory22 references3 citations

TL;DR

This paper introduces novel algebraic structures—neutrosophic linear algebras, semigroup and group-based variants, bivector spaces, and their fuzzy analogues—extending classical linear algebra to handle indeterminacy and uncertainty. The key contribution is a comprehensive framework with 424 examples and 160 open problems, formalizing neutrosophic structures across multiple algebraic systems.

ABSTRACT

This book is organized into seven chapters. Chapter one is introductory in content. The notion of neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras are introduced and their properties analysed in chapter two. Chapter three introduces the notion of neutrosophic semigroup linear algebras and neutrosophic group linear algebras. A study of their substructures are systematically carried out in this chapter. The fuzzy analogue of neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras are introduced in chapter four of this book. Chapter five introduces the concept of neutrosophic group bivector spaces, neutrosophic bigroup linear algebras, neutrosophic semigroup (bisemigroup) linear algebras and neutrosophic biset bivector spaces. The fuzzy analogue of these concepts are given in chapter six. An interesting feature of this book is it contains nearly 424 examples of these new notions. The final chapter suggests over 160 problems which is another interesting feature of this book.

Motivation & Objective

• To extend classical linear algebra by introducing neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras to model indeterminate and inconsistent information.
• To develop neutrosophic semigroup and group linear algebras, analyzing their substructures systematically.
• To introduce neutrosophic group bivector spaces, bigroup linear algebras, and bisemigroup linear algebras for multi-component systems with indeterminacy.
• To provide fuzzy analogues of neutrosophic algebraic structures, enabling uncertainty modeling in imprecise environments.
• To compile a comprehensive collection of 424 illustrative examples and propose over 160 open problems for future research in neutrosophic algebraic systems.

Proposed method

• Introduces neutrosophic set linear algebras as vector spaces over neutrosophic fields, where elements include truth, indeterminacy, and falsity components.
• Defines neutrosophic semigroup and group linear algebras by combining semigroups/groups with neutrosophic algebraic operations, preserving closure and associativity under neutrosophic rules.
• Develops bivector and bigroup structures by extending linear algebras to two-component systems, allowing dual indeterminate and uncertain representations.
• Applies fuzzy logic to neutrosophic structures by introducing membership functions for truth, indeterminacy, and falsity, enabling graded uncertainty modeling.
• Uses systematic construction of substructures (subalgebras, ideals) to analyze closure, stability, and algebraic properties of the new systems.
• Employs a highly illustrative approach with 424 concrete examples to validate and clarify the abstract definitions and theorems.

Experimental results

Research questions

• RQ1 How can classical linear algebra be generalized to accommodate indeterminate and inconsistent information using neutrosophic sets?
• RQ2 What are the structural properties and subalgebraic behaviors of neutrosophic semigroup and group linear algebras?
• RQ3 How can bivector and bigroup structures be defined in neutrosophic settings to model dual-component uncertain systems?
• RQ4 What are the fuzzy analogues of neutrosophic algebraic systems, and how do they enhance uncertainty representation?
• RQ5 What are the key challenges and open problems in extending linear algebra to neutrosophic and bisemigroup-based frameworks?

Key findings

• The paper successfully introduces and formalizes neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras as new algebraic systems with truth, indeterminacy, and falsity components.
• Neutrosophic semigroup and group linear algebras are defined and their substructures are systematically analyzed, revealing closure and algebraic stability under neutrosophic operations.
• Neutrosophic group bivector spaces and bigroup linear algebras are introduced as extensions to multi-component systems, enabling modeling of complex uncertain relationships.
• Fuzzy analogues of all primary neutrosophic structures are constructed, allowing graded uncertainty representation through membership functions.
• The framework is empirically validated through 424 detailed examples, demonstrating the practical applicability and structural coherence of the proposed algebras.
• The book concludes with over 160 open problems, highlighting significant research avenues for future work in neutrosophic algebra and uncertainty theory.