Table Of Content

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: [email protected] 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