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... AMERICAN RESEARCH PRESS REHOBOTH 2002 A 1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Depart... ...1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...(8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...exas 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can... ...s 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be... ...USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be download... ...NITION 1.3.3: A partial order relation ≤ on A is called a total order (or linear order) if for each a, b ∈ A either a ≤ b or b ≤ a. (A, ≤) is then c... ...NITION 1.3.7: A partial order relation ≤ on A is called a total order (or linear order) if for each a, b ∈ A either a ≤ b or b ≤ a. (A, ≤) is then ca...

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...e ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity...

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... REHOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...HOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...M 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...e, Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ... 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algeb... ...sociative rings boils down to the study of properties of vector spaces or linear algebras over fields. But study of non-associative algebras usin...

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... concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category theory or linear logic....

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... 1 Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...aic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Nigeria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...geria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...fferent from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category... ...c. We do not approach the bialgebraic structures using category theory or linear logic. We can broadly classify the study under four heads : ...

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...d group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology Madras... ... Indian Institute of Technology Madras Chennai – 600 036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv ... ...tion by: Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ... Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...x ( , , ) x ( L where 0 < t i < A(x i ) for i = 1, 2, …, k is said to be linearly independent in A if and only if n 1 t t k k t 1 0 ) x ( n ) x ( k... ...let C and D be fuzzy subfields of F such that D, C ⊇ δ K . If C and D are linearly disjoint over δ K then we write CD = C ⊗ D where CD is the compo...

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... american research press rehoboth 2002 { φ} {a} {b} {c} {d} {a,b,c} {a,b,c,d} {a,b} {a,c} {a,d} {b,c} {b,d} {d,c} {a... ... {a,b,c} {a,b,c,d} {a,b} {a,c} {a,d} {b,c} {b,d} {d,c} {a,b,d} {a,d,c} {b,d,c} 1 Smarandache Semirings, Semifields, and Semivector spaces... ...1 Smarandache Semirings, Semifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techno... ...andache Semirings, Semifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Mad... ...ity, India. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ...a. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...r spaces with examples 103 6.2 S-subsemivector spaces 104 6.3 Smarandache linear transformation 106 6.4 S-anti semivector spaces 109 7. Res... ...nd properties about vector spaces. DEFINITION 1.4.1: A vector space (or a linear space) consists of the following 1. a field F of scalars. 2. a...

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...DS With Specific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of T... ... With Specific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tech... ...pecific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, M... ...tics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...l: smarand@gallup.unm.edu Translation of the Tamil interviews by Meena Kandasamy H E X I S Phoenix, Arizona 2005 2 This... ...ova, Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Ka... ...l function S j (y j ). However the competitive learning law can be reduced to linear competitive learning law ij m = S j (y j ) (x – m j ), ...

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...4 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ... 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Te... ... FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology,... ...matics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...alia. Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, Univers... ...niversity of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. C... ... resolution based on Archimedean triangular norms 40 1.6 Solving non-linear optimization problem with FRE constraints 48 1.7 Method... ...8 4 1.19 A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal sol...

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... Structures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ...ructures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techno... ...and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Mad... ...cs Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smara... .... Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, University ... ...rsity of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover... ...rosophic vector space over the neutrosophic field K(I ). By basis we mean a linearly independent set which spans the neutrosophic space. Now we ill... ...developed the theory of trees, in order to solve the system of simultaneous linear equations, which give the current in each branch and each circuit...

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...VICE 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...E 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...uzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Indian Institute of Technology, Madras Chennai – 600036, India E-mail: vasantha@iitm.ac.in Florentin Smarandache Department of Mathematics... ... Finance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...iffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix... ...ors C repeatedly through the FCM connection matrix E, thresholding or non-linearly transforming the result after each pass. Independent of the FCMs ... ...by Hagiwara [41]. The features of the E-FCMs are: a. Weights having non-linear membership functions. b. Conditional weights. c. Time-delay weight...

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...GNITIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...TIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ... With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarand... ...and@gallup.unm.edu Translation of the Tamil interviews by Meena Kandasamy XIQUAN Phoenix 2004 The picture on the cover is... ... University of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...ors C repeatedly through the FCM connection matrix E, thresholding or non-linearly transforming the result after each pass. Independent of the FCMs ... ... networks consists of many simple computing elements generally simple non linear summing junctions connected together by connections of varying stre...

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...This book has six chapters. First chapter is introductory in nature. The new concept of non-associative semi-linear algebras is introduced in chapter two. This structure is built using groupoids over semi-fields. Third chapter introduces the notion of non-associative linear algebras. These algebraic structures are built using the ne...

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...ever the resultant product may not be a loop in general. For the concept of semifield refer [41]. We will be using these loops and groupoids to build linear algebra and semilinear algebras which are non associative. ...

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two NON ASSOCIATIVE SEMILINEAR ALGEBRAS 13 Chapter Three NON ASSOCIATIVE LINEAR ALGEBRAS 83 Chapter Four GROUPOID VECTOR SPACES 111 Chapter Five APPLICATION OF NON ASSOCIATIVE VECTOR SPACES / LINEAR ALGEBRAS 161 Chapter Six SUGGESTED PROBLEMS 163 FURTHER READING 225 INDEX 229 ABOUT THE ...

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...This book is organized into seven chapters. Chapter one is introductory in content. The notion of neutrosophic set linear algebras and neutrosophic neutrosophic set linear algebras are introduced and their properties analysed in chapter two. Chapter three introduces the notion of neutrosophic semigroup linear algebras and neutrosophic gro...

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...Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET NEUTROSOPHIC LINEAR ALGEBRA 13 2.1 Type of Neutrosophic Sets 13 2.2 Set Neutrosophic Vector Space 16 2.3 Neutrosophic Neutrosophic Integer Set Vector Spaces 40 2.4 Mixed Set Neutrosophic Rational Vector Spaces and their Properties 52 ...

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...ntroduced. This chapter gives over 50 theorems. Chapter two introduces the notion of n-inner product vector spaces of type II, n-bilinear forms and n-linear functionals. The final chapter suggests over a hundred problems. It is important that the reader should be well versed with not only linear algebra but also n-linear algebras of type I....

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...This book has four chapters. The first chapter just introduces n-group which is essential for the definition of n-vector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces we can define several types of li...

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... spaces or n-vector spaces of type I. Several interesting properties about them are derived in this chapter. DEFINITION 2.1: A n-vector space or a n-linear space of type I (n 2)...

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two n-VECTOR SPACES OF TYPE I AND THEIR PROPERTIES 13 Chapter Three APPLICATIONS OF n-LINEAR ALGEBRA OF TYPE I 81 Chapter Four SUGGESTED PROBLEMS 103 FURTHER READING 111 INDEX 116 ABOUT THE AUTHORS 120...

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...super FAMs etc. In chapter five we define interval matrix and matrix interval using the natural class of intervals, that is intervals of the form [a, b] where a b or a and b not comparable. In the final chapter we have defined the new notion of bimatrices and n-matrices. Using these new concepts we have constructed linear algebra of type I and linear algebra of type II. U...

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...e basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modific...

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...1 Introduction to Bilinear Algebras and their Generalizations 9 1.2 Introduction to Neutrosophic Algebraic Structures 11 Chapter Two NEUTROSOPHIC LINEAR ALGEBRA 15 2.1 Neutrosophic Bivector Spaces 15 2.2 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bivector Space 198 Chapter Three NEUTROSOP...

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...easily understood by a beginner. Binary symmetric channel is an illustration of a model for a transmission channel. Now we will proceed onto define a linear code algebraically....

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two ALGEBRAIC LINEAR CODES AND THEIR PROPERTIES 29 Chapter Three ERASURE DECODING OF MAXIMUM RANK DISTANCE CODES 49 3.1 Introduction 49 3.2 Maximum Rank Distance Codes 51 3.3 Erasure Decoding of MRD Codes 54 Chapter Four MRD CODES ...

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... in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research...

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... chemical industries using fuzzy neutral networks. Chapter five gives the method of minimization of waste gas flow in chemical industries using fuzzy linear programming. The final chapter suggests when in these studies indeterminancy is an attribute or concept involved, the notion of neutrosophic methods can be adopted. The authors feel that the reader should be well verse...

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...ur SUPER INTERVAL SEMILINEAR ALGEBRAS 137 Chapter Five SUPER FUZZY INTERVAL MATRICES 203 5.1 Super Fuzzy Interval Matrices 203 5.2 Special Fuzzy Linear Algebras Using Super Fuzzy Interval Matrices 246 Chapter Six APPLICATION OF SUPER INTERVAL MATRICES AND SET LINEAR ALGEBRAS BUILT USING SUPER INTERVAL MATRICES 255 Chapter Seven SUGGESTED PROBLEMS 257 FURT...

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...In this book authors for the first time elaborately study the notion of MOD vector spaces and MOD pseudo linear algebras. This study is new, innovative and leaves several open conjectures. In the first place as distributive law is not true one can define only MOD pseudo linear algebras. Secondly most of the classical theorems tr...

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...In this book we introduce some special type of linear algebras called pseudo special linear algebras using the interval [0, n). These new types of special pseudo interval linear algebras has several interesting properties. Special pseudo interval linear algebras are built...

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...In this book authors build algebraic structures on fuzzy unit semi open square UF = {(a, b) | a, b ε [0, 1)} and on the fuzzy neutrosophic unit semi open square UN = {a + bI | a, b ε [0, 1)}. This study is new and we define, develop and describe several interesting and innovative theories about them. We cannot bu...

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...so we can have only pseudo ring as the distributive law is not true, in general in C([0, n)). Finally the concepts of pseudo vector spaces and pseudo linear algebras are introduced. At every stage the neutrosophic analogue is also defined, developed and described. There are several open problems suggested....

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...peration this semi real open square is a semiring. We define the new type of ring call pseudo ring. Finally we define S-vector spaces and S-pseudo linear algebras using them. This concept will in due course of time find lots of applications. Several open problems are suggested for interested researchers....