MOD Planes: A New Dimension to Modulo Theory

2015·5.6 MB·English

by  W. B. Vasantha Kandasamy

MOD Planes:
A New Dimension
to Modulo Theory

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: [e-mail]
URL: http://www.europanova.be/

Copyright 2014 by EuropaNova ASBL and the Authors

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. Valeri Kroumov, Dept. of Electrical and Electronic Engineering, Faculty
of Engineering, Okayama University of Science, 1-1, Ridai-cho, Kita-ku,
Okayama

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

ISBN-13: 978-1-59973-363-0
EAN: 9781599733630

Printed in the United States of America
2
CONTENTS

Preface 5

Chapter One

INTRODUCTION
7

Chapter Two

THE MOD REAL PLANES 9

Chapter Three

MOD FUZZY REAL PLANE 73
3
Chapter Four

MOD NEUTROSOPHIC PLANES AND MOD FUZZY
NEUTOSOPHIC PLANES 103

Chapter Five

THE MOD COMPLEX PLANES 147

Chapter Six

MOD DUAL NUMBER PLANES 181

FURTHER READING
215

INDEX 218

ABOUT THE AUTHORS 221

4
PREFACE

In this book for the first time authors study mod planes
using modulo intervals [0, m); 2 ≤ m ≤ ∞. These planes unlike
the real plane have only one quadrant so the study is carried out
in a compact space but infinite in dimension. We have given
seven mod planes viz real mod planes (mod real plane) finite
complex mod plane, neutrosophic mod plane, fuzzy mod plane,
(or mod fuzzy plane), mod dual number plane, mod special dual
like number plane and mod special quasi dual number plane.
These mod planes unlike real plane or complex plane or
neutrosophic plane or dual number plane or special dual like
number plane or special quasi dual number plane are infinite in
numbers.
Further for the first time we give a plane structure to the
fuzzy product set [0, 1) × [0, 1); where 1 is not included; this is
defined as the mod fuzzy plane. Several properties are derived.
5
This study is new, interesting and innovative. Many open
problems are proposed. Authors are sure these mod planes will
give a new paradigm in mathematics.

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 define the new notion of several types
of MOD planes, using reals, complex numbers, neutrosophic
numbers, dual numbers and so on. Such study is very new and
in this study we show we have a MOD transformation from R to
the MOD real plane and so on.

We give the references needed for this study.

In the first place we call intervals of the form [0, m) where
m ∈ N to be the MOD interval or small interval. The term MOD
interval is used mainly to signify the small interval [0, m) that
can represent (–∞, ∞). If m = 1 we call [0, 1) to be the MOD
fuzzy interval.

We define R (n) = {(a, b) | a, b ∈ [0, m)} to be the MOD real
n
plane.

If m = 1 we call R (1) as the MOD fuzzy plane.
n

The definitions are made and basic structures are given and
these MOD planes extended to MOD polynomials and so on. For
study of intervals [0, m) refer [20-1].

For neutrosophic numbers please refer [3, 4].
8 MOD Planes

For the concept of finite complex numbers refer [15]. The

notion of dual number can be had from [16].

The new notion of special quasi dual numbers refer [18].

The concept of special dual like numbers and their
properties refer [17]. For decimal polynomials [5-10].

We have only mentioned about the books of reference.
Several open problems are suggested in this book for the
researchers.

It is important that the authors name these structures as MOD
intervals or MOD planes mainly because they are small when
compared to the real line or real plane. The term MOD is used
mainly to show it is derived using MOD operation.
Chapter Two

THE MOD REAL PLANES

In this chapter authors for the first time introduce a new
notion called “MOD real plane”. We call this newly defined
plane as MOD real plane mainly because it is a very small plane
in size but enjoys certain features which is very special and very
different from the general real plane.

Recall the authors have defined the notion of semi open
squares of modulo integers [23]. Several algebraic properties
about them were studied [23]. Here we take [0, m) the semi
open interval, 1 ≤ m < ∞ and find [0, m) × [0, m); this
represents a semi open square which we choose to call as the
MOD real plane.

This is given the following representation.

Figure 2.1