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New Classes of Neutrosophic Linear Algebras

By Smarandache, Florentin

Book Id:WPLBN0002828482 Format Type:PDF eBook: File Size:6.55 MB Reproduction Date:8/2/2013

Smarandache, B. F., & Vasantha Kandasamy, W. B. (2013). New Classes of Neutrosophic Linear Algebras. Retrieved from http://gutenberg.us/

Description
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.

Summary
In this book we introduce mainly three new classes of linear algebras; neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras. The authors also define the fuzzy analogue of these three structures.

Excerpt
Now we proceed onto define the notion of neutrosophic subgroup of a neutrosophic group.
DEFINITION 1.2: Let N(G) = (GuI) be a neutrosophic group generated by G and I. A proper subset P(G) is said to be a neutrosophic subgroup if P(G) is a neutrosophic group i.e. P(G) must contain a (sub) group.
Example 1.3: Let N(Z2) = Z2 I be a neutrosophic group under addition. N(Z2) = {0, 1, I, 1 + I}. Now we see {0, I} is a group under + in fact a neutrosophic group {0, 1 + I} is a group under ‘+’ but we call {0, I} or {0, 1 + I} only as pseudo neutrosophic groups for they do not have a proper subset which is a group. So {0, I} and {0, 1 + I} will be only called as pseudo neutrosophic groups (subgroups).
We can thus define a pseudo neutrosophic group as a neutrosophic group, which does not contain a proper subset which is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophic group can have substructures which are pseudo neutrosophic groups which is evident.

Table of Contents
Preface 5
Chapter One
INTRODUCTION 7
Chapter Two
SET NEUTROSOPHIC LINEAR ALGEBRA 13
2.1 Type of Neutrosophic Sets 13
2.2 Set Neutrosophic Vector Space 16
2.3 Neutrosophic Neutrosophic Integer Set Vector Spaces 40
2.4 Mixed Set Neutrosophic Rational Vector Spaces and their Properties 52
Chapter Three
NEUTROSOPHIC SEMIGROUP LINEAR ALGEBRA 91
3.1 Neutrosophic Semigroup Linear Algebras 91
3.2 Neutrosophic Group Linear Algebras 113
Chapter Four
NEUTROSOPHIC FUZZY SET LINEAR ALGEBRA 135
Chapter Five
NEUTROSOPHIC SET BIVECTOR SPACES 155
Chapter Six
NEUTROSOPHIC FUZZY GROUP BILINEAR ALGEBRA 219
Chapter Seven
SUGGESTED PROBLEMS 247
FURTHER READING 277
INDEX 280
ABOUT THE AUTHORS 286