MOD Rectangular Natural Neutrosophic Numbers

2018·3.7 MB·

by  W. B. Vasantha Kandasamy

Mod Rectangular Natural
Neutrosophic Numbers

W. B. Vasantha Kandasamy
Ilanthenral K
Florentin Smarandache

2018
Copyright 2018 by EuropaNova ASBL and the Authors

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

Peer reviewers:

Professor Paul P. Wang, Ph D, Department of Electrical & Computer
Engineering, Pratt School of Engineering, Duke University, Durham, NC
27708, USA
Dr.S.Osman, Menofia University, Shebin Elkom, Egypt.
Prof. Xingsen Li, School of Management, Ningbo Institute of Technology,
Zhejiang University, Ningbo 315100, P. R. China

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

ISBN-13: 978-1-59973-539-9
EAN: 9781599735399

Printed in the United States of America
2
CONTENTS

Preface 4

Chapter One
INTRODUCTION 7

Chapter Two
RECTANGULAR MOD PLANES 9

Chapter Three
ALGEBRAIC STRUCTURES USING
RECTANGULAR MOD PLANES 45

3
Chapter Four
SEMIGROUPS ON MOD RECTANGULAR
NATURAL NEUTROSOPHIC NUMBERS
AND THEIR PROPERTIES 127

FURTHER READING 253

INDEX 259

ABOUT THE AUTHORS 261

4
PREFACE

In this book authors introduce the new notion of MOD
rectangular planes. The functions on them behave very
differently when compared to MOD planes (square). These
are different from the usual MOD planes. Algebraic
structures on these MOD rectangular planes are defined and
developed.
However we have built only MOD interval natural
neutrosophic products of the form I[0, m)  I[0, n) where
m ≠ n and 2 ≤ m, n < . These can be called as planes as
one can accommodate the mod natural neutrosophic
numbers in these planes. Further MOD rectangular natural
neutrosophic numbers ZI ZI ; m ≠ n; 2 ≤ m, n <  are
n m
also constructed and algebraic structures on them are
defined and described. They happen to be of finite
cardinality. On these MOD rectangular numbers,
5
semigroups with respect to + and  (or  ) are defined and
0
described. They happen to be of finite MOD rectangular
natural neutrosophic sets. MOD matrix subsets are
constructed and under + (or  or  ) these collections yield
n
only semigroups.
On similar lines MOD rectangular natural neutrosophic
subset coefficient polynomials are defined and under + and
 or  they happen to be only semigroups. Study in this
0
direction yields nice algebraic structures under a single
binary operator + or  (or  )
0
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 book authors introduce the new notion of MOD
rectangular planes [0, n) [0, m) where m n; 2  m, n < .
This concept is analogously defined to MOD rectangular modulo
integers as Zn  Zm, MOD rectangular natural neutrosophic
modulo integers ZI ZI and MOD rectangular natural
n m
neutrosophic interval set I[0, n)  I[0, m). However these are
not planes but we choose to call them as MOD rectangular
numbers. Further we cannot define MOD rectangular complex
plane or dual plane or neutrosophic plane for we need m = n.
For more about MOD structures please refer [31-41].
Here we give how the usual functions behave in the case
of MOD rectangular plane [0, n)  [0, m); (m  n). Then we
proceed onto define algebraic structures on Z  Z, ZI  ZI ,
n , n m
I[0, n)  I[0, m) and [0, n)  [0, m); m  n. We see Z Z and
n m
[0, n)  [0, m) are groups under + modulo pair operation. For if
(a, b) and (c, d)  [0, n)  [0, m) then (a, b) + (c, d) = ((a +c)
(mod n), (b + d) (mod m)).
8 MOD Rectangular Natural Neutrosophic Numbers

But ZI ZI and I[0, n)  I[0, m) are only semigroups
n m

under +. So we are in a position to give semigroups of finite

and infinite order which are only semigroups under +. When
we extend this concept to matrices and polynomials we get
matrices such that A + A = A and polynomials p(x) + p(x) =
p(x) respectively.
Except for this structure we would not be in a position to
get all these. ZI  ZI and Z  Z under product are finite order
n m n m
semigroups which enjoy several special features. Likewise
I[0, n)  I[0, m) and [0, n)  [0, m) are infinite order semigroups
which are also only infinite order semigroups under product.
All these give examples of non-abstract semigroups of
either finite or infinite order under + or . Finally we see these
semigroups have special and distinct features. Certainly these
finite structures can find applications in several important
problems related to automaton theory.
The authors also mention about matrix and polynomial
MOD rectangular natural neutrosophic semigroups under + and
 (or ). They are always commutative monoids. In case of
0
square matrices we can have the usual product  and get the
infinite order non-commutative monoids if [0, n)  [0,m) or
I[0, n)  I[0, m) is used and finite non commutative monoids in
case of Z  Z or ZI  ZI .
n m n m
Several innovative results are obtained. For more about
neutrosophic algebraic structures, refer [2-12].
Chapter Two

RECTANGULAR MOD PLANES

In this chapter we introduce the new notion of rectangular
MOD planes. These are distinctly different from the existing real
planes. Further these planes will serve a better purpose for one
need not go for different types of scales only for rectangular
planes. In fact rectangular MOD planes are infinitely many.
When both the x and y coordinates are real we call them
as real MOD rectangular planes or MOD rectangular real planes.
We will first illustrate this by some examples.
Example 2.1. Let us consider the two MOD intervals [0, m) and
[0, p); p  m, 2  p, m < .
Now R (m, p) = [0, m)  [0, p) = {(a, b) / a  [0, m) and
n
b  [0, p)}.
This Rn(m, p) is defined as the MOD rectangular plane,
which is described by the figure given in the next page.