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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 Z  Z , MOD rectangular natural neutrosophic n m 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 R (m, p) is defined as the MOD rectangular plane, n which is described by the figure given in the next page.