Special Dual like numbers and lattices

248 Pages·03.599 MB·English

by  W. B. Vasantha Kandasamy, Florentin Smarandache

Special Dual like Numbers
and Lattices

W. B. Vasantha Kandasamy
Florentin Smarandache

ZIP PUBLISHING
Ohio
2012
This book can be ordered from:

Zip Publishing
1313 Chesapeake Ave.
Columbus, Ohio 43212, USA
Toll Free: [phone]
E-mail: [email protected]
Website: www.zippublishing.com

Copyright 2012 by Zip Publishing and the Authors

Peer reviewers:
Prof. Ken Nagasaka, Tokyo University A&T, Japan.
Prof. Gabriel Tica, Bailesti College, Bailesti, Jud. Dolj, Romania.
Florentin Popescu, Facultatea de Mecanica, University of Craiova, Romania.
Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA.

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

ISBN-13: 978-1-59973-185-8
EAN: 9781599731858

Printed in the United States of America
2
CONTENTS

Dedication 5

Preface 7

Chapter One
INTRODUCTION 9

Chapter Two
SPECIAL DUAL LIKE NUMBERS 11

Chapter Three
HIGHER DIMENSIONAL SPECIAL DUAL NUMBERS 99

Chapter Four
SPECIAL DUAL LIKE NEUTROSOPHIC NUMBERS 139

3
Chapter Five
MIXED DUAL NUMBERS 165

Chapter Six
APPLICATION OF SPECIAL DUAL LIKE NUMBERS
AND MIXED DUAL NUMBERS 199

Chapter Six
SUGGESTED PROBLEMS 201

FURTHER READING 240

INDEX 243

ABOUT THE AUTHORS 246

4
D
EDICATION



A.P.KOTELNIKOV
5


We dedicate this book to A.P. Kotelnikov. The algebra of
dual numbers has been originally conceived by W.K.
Clifford, but its first applications to mechanics are due to
A.P. Kotelnikov. The original paper of A.P. Kotelnikov,
published in the Annals of Imperial University of Kazan
(1895), is reputed to have
been destroyed during the Russian revolution.


6
PREFACE

In this book the authors introduce a new type of dual numbers
called special dual like numbers.
These numbers are constructed using idempotents in the
place of nilpotents of order two as new element. That is
x = a + bg is a special dual like number where a and b are reals
and g is a new element such that g2 =g. The collection of special
dual like numbers forms a ring. Further lattices are the rich
structures which contributes to special dual like numbers. These
special dual like numbers x = a + bg; when a and b are positive
reals greater than or equal to one we see powers of x diverge on;
and every power of x is also a special dual like number, with
very large a and b. On the other hand if a and b are positive
reals lying in the open interval (0, 1) then we see the higher
powers of x may converge to 0.
Another rich source of idempotents is the Neutrosophic
number I, as I2 = I. We build several types of finite or infinite
rings using these Neutrosophic numbers. We also define the
7
notion of mixed dual numbers using both dual numbers and
special dual like numbers. Neither lattices nor the Neutrosophic
number I can contribute to mixed dual numbers. The two
sources are the linear operators on vector spaces or linear
algebras and the modulo integers Z ; n a suitable composite
n
number, are the ones which contribute to mixed dual numbers.
This book contains seven chapters. Chapter one is
introductory in nature. Special dual like numbers are introduced
in chapter two. Chapter three introduces higher dimensional
special dual like numbers. Special dual like neutrosophic
numbers are introduced in chapter four of this book. Mixed dual
numbers are defined and described in chapter five and the
possible applications are mentioned in chapter six. The last
chapter has suggested over 145 problems.
We thank Dr. K.Kandasamy for proof reading and being
extremely supportive.

W.B.VASANTHA KANDASAMY
FLORENTIN SMARANDACHE

8
Chapter One

INTRODUCTION

In this book the authors for the first time introduce the new
notion of special dual like numbers. Dual numbers were
introduced in 1873 by W.K. Clifford.

We call a number a + bg to be a special dual like number if
a, b  R (or Q or Z or C) and g is a new element such that
n
g2 = g.

We give examples of them.

The natural class of special dual like numbers can also be
got from Z  I = {a + bI | a, b  Z, I2 = I , I the indeterminate}
(Q  I or Z  I or R  I or C  I).
n

Thus introduction of special dual like numbers makes one
identify these neutrosophic rings as special dual like numbers.

Apart from this in this book we use distributive lattices to
build the special dual like numbers.

For S = {a + bg | a, b  R and g  L, L a lattice} paves way
to a special dual like number as g  g = g and g  g = g that is
every element in L is an idempotent under both the operations