MOD Pseudo Linear Algebras

2015·2.8 MB·English

by  W. B. Vasantha Kandasamy

Pseudo
MOD
Linear Algebras

W. B. Vasantha Kandasamy
Ilanthenral K
Florentin Smarandache

2015
This book can be ordered from:

EuropaNova ASBL
Clos du Parnasse, 3E
1000, Bruxelles
Belgium
E-mail: [email protected]
URL: http://www.europanova.be/

Copyright 2014 by EuropaNova ASBL and the Authors

Peer reviewers:

Dr. Stefan Vladutescu, University of Craiova, Romania.
Dr. Octavian Cira, Aurel Vlaicu University of Arad, Romania.
Prof. Gabriel Tica, Bailesti College, Bailesti, Jud. Dolj, Romania.

Many books can be downloaded from the following
Digital Library of Science:
http://www.gallup.unm.edu/eBooks-otherformats.htm

ISBN-13: 978-1-59973-369-2
EAN: 9781599733692

Printed in the United States of America
2
CONTENTS

Preface 5

Chapter One
INTRODUCTION
7

Chapter Two
REAL MOD MATRICES AND THEIR PROPERTIES 39

Chapter Three
ALGEBRAIC STRUCTURES ON MOD SUBSETS
OF MOD PLANES 93

3
Chapter Four
MULTIDIMENSIONAL MOD PSEUDO
LINEAR ALGEBRAS 209

FURTHER READING
261

INDEX 265

ABOUT THE AUTHORS 269

4
PREFACE

In this book authors for the first time elaborately study the
notion of MOD vector spaces and MOD pseudo linear algebras.
This study is new, innovative and leaves several open
conjectures. In the first place as distributive law is not true we
can define only MOD pseudo linear algebras.
Secondly most of the classical theorems true in case of
linear algebras are not true in case of MOD pseudo linear
algebras. Finding even eigen values and eigen vectors happens
to be a challenging problem. Further the notion of
multidimensional MOD pseudo linear algebras are defined using
the notion of MOD mixed matrices.
These function only under the natural product × as the
n
usual product × cannot be even defined on these mixed MOD
matrices.
5
Several innovative and interesting results are given in this
book. Many open problems are proposed.
We wish to acknowledge Dr. K Kandasamy for his
sustained support and encouragement in the writing of this
book.

W.B.VASANTHA KANDASAMY
ILANTHENRAL K
FLORENTIN SMARANDACHE

6
Chapter One

INTRODUCTION

In this chapter we introduce the notion of MOD interval,
MOD plane and MOD transformation. For more about these
notions [26-30].

Throughout this book (–∞, ∞) is the real line, Z the ring of
m
modulo integers m and R the real plane. C the complex plane.
(–∞i, ∞i) the imaginary line, R(g) (g2 = 0) the dual number
plane, R(h) (h2 = h) the special dual like number plane. R(k); (k2
= –k) the special quasi dual number plane, C(Z ) the finite
m
complex modulo integer i2 = m – 1.
F

(–∞g, ∞g), (g2 = 0) the dual number line. (–∞h, ∞h) (h2 =
h); the special dual like number line. (–∞k, ∞k) the special quasi
dual number line. k2 = –k. 〈R ∪ I〉 is the real neutrosophic plane.
(–∞I, ∞I) is the neutrosophic number line.

〈R ∪ I〉 = {a + bI | a, b ∈ R, I2 = I},

R(g) = {a + bg | g2 = 0, a, b ∈ R},

R(h) = {a + bh | h2 = h, a, b ∈ R},
8 MOD Pseudo Linear Algebras

C = {a + bi | a, b ∈ R, i2 = –1} and

R(k) = {a + bk | a, b ∈ R, k2 = –k}

denotes the neutrosophic plane, dual number plane, special dual
like number plane, complex plane and special quasi dual
number plane respectively.

(–∞, ∞) is the real line,

(–∞g, ∞g) = {mg | m ∈ R, g2 = 0},

(–∞I, ∞I) = {mI | m ∈ R, I2 = 0},

(–∞i, ∞i) = {mi | m ∈ R, i2 = –1},

(–∞h, ∞h) = {mh | m ∈ R, h2 = h} and

(–∞k, ∞k) = {mk | m ∈ R, k2 = –k2}

are the real line, dual line, neutrosophic line, complex number
line, special dual like number line and special quasi dual
number line respectively.

We can now proceed onto define the new notion of MOD
transformation. For this to occur we recall the definition of MOD
intervals of 6 types and the MOD planes of 6 types [26-30].

Let [0, m); 2 ≤ m < ∞ be the interval. We define this
interval as MOD real interval.

[0, 7), [0, 12), [0, 148), [0, 19) and so on [0, m); 2 ≤ m < ∞
are defined as MOD real intervals.

[0, 1) is defined as the fuzzy MOD interval or MOD fuzzy
interval [26-30].

We have infinite number of real MOD intervals but only one
MOD fuzzy interval.
Introduction 9

[0, 43) is the MOD real interval.

[0, m) = {a | a ∈ [0, m), that is a cannot take the value m}.

[0, 5) = {a | 0 ≤ a < 5; that is a cannot take the value 5}.

[0, 24) = {a | 0 ≤ a < 24; that is a can take 23.99…9 but
c annot take a = 24}.

The advantage of using MOD real intervals is there exists
infinitely many MOD real intervals but however one and only

one real interval (–∞, ∞).

Next we define the complex interval (–∞i, ∞i). We see we
have one and only one complex interval (–∞i, ∞i).

However [0, m)i ; 2 ≤ m < ∞ is the MOD complex interval
F
where i2 = m – 1.
F

We have infinite number of MOD complex intervals for each
of this i2 = m – 1, 2 ≤ m < ∞.
F

[0, 3)i is the MOD complex interval; i2 = 2 and so on.
F F

[0, 42)i is the MOD complex interval; i2= 41 and so on.
F F

[0, 14)i = {ai | i2 = 13; 0 ≤ ai < 14i }.
F F F F F

[0, 215)i = {ai | 0 ≤ ai < 215i , i2= 24}
F F F F F

are the MOD complex interval.

Next we proceed onto describe MOD neutrosophic intervals
(–∞I, ∞I); I2 = I is defined as the neutrosophic interval.