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...Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it ...

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... 5 Preface Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equation... ...a, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in the construction of biau... ...e reader to supply proofs for our main motivation is introduce several new concepts and the proofs which are left for the reader is more a matter o...

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...gebraic structures viz. neutrosophic fields, neutrosophic vector spaces and neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic algebraic structure we found it deem fit to first introduce and study neutrosophic graphs and their applications to neutrosophic models....

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...d neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic al... ...is devoted for this purpose. Chapter Three introduces many new neutrosophic concepts in graphs and applies it to the case of neutrosophic cognitive ... ...at is when we are not in a position to associate a relation between any two concepts then we denote it as an indeterminacy or when a concept cannot ... ...proceed on to give the definition of factor of a graph for more about these concepts please refer [23]. DEFINITION 2.5.7: A factor of a graph G is... ...of a FCM by an example. DEFINITION 2.8.5:An FCM is a directed graph with concepts like policies, events etc as nodes and causalities as edges. It... ...s nodes and causalities as edges. It represents causal relationship between concepts. The problem studied in this case is for a fixed source S,... ...ining the cognitive structure of the relationship, we give notations to the concepts involved in the analysis as below. a b c δ α β γ .5 .5 ... ...signed directed graph, we obtain a connection matrix E, since the number of concepts used here are eight, the connection matrix is a 8 × 8 matrix. ... ...). DEFINITION 2.8.6: A FRM is a directed graph or a map from D to R with concepts like policies or events etc as nodes and causalities as edges. ...

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...which can be undoubtedly characterized as interesting mathematics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as a substructure. Several researchers around the world working on Smarandache notions have systematically carried out this study. This is the first book ...

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... capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as... ...in order to make an organized study possible, we have also included all the concepts about semirings, semifields and semivector spaces; since, to the... ...e best of our knowledge we do not have books, which solely deals with these concepts. This book introduces several new concepts about Smarandache se... ...his book consists of seven chapters. The first chapter introduces the basic concepts which are very essential to make the book self-contained. The se... ...major parts of this book contains a complete systematic introduction of all concepts together with a sequential analysis of these concepts. Examples... ...phical list for supplementary reading, since referring to and knowing these concepts will equip and enrich the researcher's knowledge. The book also... ...e semirings and semifields. Overall in this book we have totally defined 65 concepts related to the Smarandache notions in semirings and its general... ...HAPTER ONE PRELIMINARY NOTIONS This chapter gives some basic notions and concepts used in this book to make this book self-contained. The serious... ...ctor spaces. The purpose of this book is two-fold, firstly to introduce the concepts of semirings, semifields and semivector spaces (which we will s...

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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 AUTHOR...

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...ound in existing literature the author has taken utmost efforts to see that the concepts are completely understood by illustrating with examples and... ...n chapters. The first chapter on preliminaries gives some important notions and concepts, which are used in this book. Chapters 2 and 3 gives most o... ...concepts, which are used in this book. Chapters 2 and 3 gives most of the basic concepts on group theory and results in group theory which have been... ...ntroduce some Smarandache notions in groups. The introduction of Smarandache concepts in groups and Smarandache semigroups in a way makes a mathe... ...ffort is taken to see that the chapters two and three give most of the relevant concepts in group theory for which we have studied the Smarandache no... ...nter-examples and problems. The prominent feature of the book is - all abstract concepts are illustrated by examples. As to the best of the autho... ...ncepts are illustrated by examples. As to the best of the author's knowledge concepts and results about Smarandache semigroups and Smarandache no... ...of Smarandache semigroups as given by Raul (1998) and introduce in this section concepts like Smarandache commutative semigroup, Smarandache weakly ... ...ial definitions were made about S-semigroup, their properties studied and those concepts illustrated with examples. So now, we proceed to introduce ...

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... basic properties or classical results that the standard textbooks contain. I feel that such a venture, which has consolidated several ring theoretic concepts, has made the current book a unique one from the angle of research. One of the major highlights of this book is by creating the Smarandache analogue of the various ring theoretic concepts we have succeeded in definin...

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...I feel that such a venture, which has consolidated several ring theoretic concepts, has made the current book a unique one from the angle of researc... ...ook is by creating the Smarandache analogue of the various ring theoretic concepts we have succeeded in defining around 243 Smarandache concepts. ... ...ally makes the study easier and also gives way to greater number of newer concepts. Also localization of properties automatically comes when Smarand... ...e study of this book. In section one we introduce certain group theory concepts to make the reader understand the notions of Smarandache semigrou... ... in building rings viz. semigroup rings. Section three aims to give basic concepts in lattices. The final section on Smarandache semigroups gives th... ...he main motivation to make the book self- contained; atleast for the basic concepts. We give examples and ask the reader to solve the problems at the... ...and Smarandache notions about rings; not only for comparison of these two concepts, but to make them build more Smarandache structures. DEFINIT... ...is section we introduce the concept of semigroups mainly to study the two concepts; Smarandache semigroups and semigroup rings. Several types of semi... ...ero divisors and regular elements, we just recall the definition of these concepts and illustrate them with examples. All properties and results rel...

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...This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly any n-dimensional space of refined labels...

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...the National Cheng Kung University, Taiwan. These newsletters give an overall picture of the research carried out and the recent advancements and new concepts in the field. Conferences devoted solely to near-rings are held once every two years. There are about half a dozen books on near-rings apart from the conference proceedings. Above all there is a online searchable dat...

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...ly invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. These concepts are also extended neutrosophically in this book....

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...rica 3 CONTENTS Preface 5 Chapter One BASIC CONCEPTS ON BIMATRICES 1.1 Definition and Basic Operation on Bimatr... ... bigraphs, trigraphs and n-graphs and describe some of its properties. These concepts are also extended neutrosophically in this book. Using th... ...his book is organized into four chapters. The first chapter recalls the basic concepts of bimatrices and neutrosophic bimatrices. Second chapter int... ...LORENTIN SMARANDACHE K.ILANTHENRAL 7 Chapter One BASIC CONCEPTS ON BIMATRICES In this chapter we just briefly recall ... ...ysis of a model when we have some common attributes given by experts i.e. the concepts / attributes are not totally disjoint but have certain concep... ...m when the FCM model is to analyze two sets of attributes and in that pair of concepts / attributes only when a point or a edge or some points and s... ...1 and G 2 respectively. A directed bigraph is called the dibigraph. All concepts incident into, incident out, outneighbour and inneighbour are... ...onnection matrices. 72 DEFINITION 3.1.1: An FCM is a directed graph with concepts like policies, events etc. as nodes and causalities as edges.... ...as nodes and causalities as edges. It represents causal relationship between concepts. We illustrate this by the example 3.1.1: Example 3.1.1:...

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...Dedication 5 Preface 7 Chapter One INTRODUCTION TO BASIC CONCEPTS 9 1.1 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 S...

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...Preface 5 Chapter One INTRODUCTION TO SOME BASIC CONCEPTS 7 1.1 Properties of Rooted Trees 7 1.2 Basic Concepts 9 Chapter Two GROUPS AS GRAPHS 17 Chapter Three IDENTITY GRAPHS OF SOME ALGEBRAIC STRUCTURES 89 3.1 Identity Graphs of Semigroups 89 3.2 Special Identity ...

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...ill be helpful in working independently. Also work can be used for extra school work because the reader will find in it important formulas, theorems, concepts, and basic definitions which are not always included in school textbooks....

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...hic bigroup linear algebras, neutrosophic semigroup (bisemigroup) linear algebras and neutrosophic biset bivector spaces. The fuzzy analogue of these concepts are given in chapter six. An interesting feature of this book is it contains nearly 424 examples of these new notions. The final chapter suggests over 160 problems which is another interesting feature of this book....

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... neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophic group can have substructures which are pseudo neutrosophic groups which is evident. ...

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...Preface 5 Chapter One BASIC CONCEPTS 7 1.1 Neutrosophic Interval Bisemigroups 8 1.2 Neutrosophic Interval Bigroups 32 1.3 Neutrosophic Biinterval Groupoids 41 1.4 Neutrosophic Interval Biloops 57 Chapter Two NEUTROSOPHIC INTERVAL BIRINGS AND NEU...

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...poids and quasi loops. This new notion is well illustrated by 140 examples. These can find applications only in due course of time, when these new concepts become familiar. The final chapter suggests over 300 problems some of which are research problems....

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...In this chapter we only indicate as reference of those the concepts we are using in this book. However the interested reader should refer them for a complete understanding of this book. In this book we define the notion of natural product in matrices so that we have a nice natural pr...

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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 / LIN...

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... neutrosophic group though has a group structure is not a neutrosophic group and a neutrosophic group cannot be a pseudo neutrosophic group. Both the concepts are different. Now we see a neutrosophic group can have substructures which are pseudo neutrosophic groups which is evident from the following example....

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...ted States of America 3 CONTENTS Preface 5 1. BASIC CONCEPTS 1.1 Basics of vector space and bilinear forms 7 1.2 Smarandach... ...algebras. Further as the study of these two algebras heavily exploits the concepts of vector spaces and matrix theory we do not venture to approach ... ...ly introductory but also devotes an entire section to introduce the basic concepts in Smarandache vector spaces. Chapter two studies about Smarandac... ... and analysed. Chapter four is devoted to the introduction of several new concepts in Lie algebras and Smarandache Lie algebras. Chapter five define... ...continues and requires our intensive struggle. 7 Chapter 1 BASIC CONCEPTS This chapter on basic concepts has three sections. First sect... ... and skew symmetric forms. As we expect the reader to be well versed with concepts in algebra we do not recall the definition of fields or rings. ... ...ties of Smarandache K-vectorial spaces here we have developed some of the concepts, which will be very useful in the study of linear structure and L... ... is a basis of V. 10 Proof: Obvious by the very definition of these concepts. DEFINITION 1.2.3: Let V be a finite dimensional vector space o... ...a S-basis of V. Proof: Easily follows from the very definition of these concepts. THEOREM 1.2.4: Let V be a vector space over the field K. If A ...

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...a and for the past five decades, several researchers have been working on concepts like fuzzy semigroup, fuzzy groups, fuzzy rings, fuzzy ideals, fu... ...mirings and fuzzy near-rings though there are several papers on these two concepts. This book has seven chapters, which are divided into two part... ...ivided into twelve sections, we deal with eleven distinct fuzzy algebraic concepts and in the concluding section list the miscellaneous properties o... ...the miscellaneous properties of fuzzy algebra. The eleven fuzzy algebraic concepts which we analyze are fuzzy sets, fuzzy subgroups, fuzzy sub- bigro... ...d in these sections are extensive and we have succeeded in presenting new concepts defined by several researchers. In the second chapter we introduc... ...tions. In these six chapters, we have succeeded in introducing around 664 concepts related to Smarandache fuzzy algebra. The reader is expected to b... ...bilities of applications, all that I have done here is to put forth those concepts that clearly have relevant applications. When I informed my inter... ...ed upon them. He was the first political leader in India to fight for the concepts of Self-Respect and Social Justice; and in terms of social reform... ...aic operations, or to be more precise we are not interested in discussing concepts topologically or analytically like continuity, connected, increas...

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...ons and some important definitions used in this book. In Chapter 2 almost all concepts, most of them new have been introduced to groupoids in genera... ... In this chapter, we give some basic notion and preliminary concepts used in this book so as to make this book self contained. The... ... to the introduction to groupoids and the study of several properties and new concepts in them. We make this explicit by illustrative examples. Apar... ...entities on SGs and study them. We have given many examples of SG to make the concepts defined about them explicit. 4.1 Smarandache Groupoids ... ...res in Smarandache Groupoids In this section we extend the research to the concepts of Smarandache subgroupoids, Smarandache left (right) ideal, ...

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...ms. Further, this book gives lots of references so that the interested reader can make use of them. This book has two chapters. In Chapter One, basic concepts about fuzzy matrices are introduced. Basic notions of matrices are given in section one in order to make the book self contained. Section two gives the properties of fuzzy matrices. Since the data need to be transfor...

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...Preface 4 Dedication 6 Chapter One BASIC MATRIX THEORY AND FUZZY MATRIX THEORY 7 1.1 Basic Matrix Theory 7 1.2 Basic Concepts on Fuzzy Matrices 33 1.3 Basic Concepts on Graphs 47 Chapter Two DESCRIPTION OF SIMPLE FUZZY MODELS AND THEIR APPLICATIONS TO REAL WORLD PROBLEMS 71 2.1 Description of Simple Fuzzy Matrix Model 72 2.2 Definitio...

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...gebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic structures....

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...TRODUCTORY COCEPTS OF BASIC BISTRUCTURES AND S-BISTRUCTURES 2.1 Basic concepts of bigroups and bivector spaces 37 2.2 Introduction of S-bigr... ...uction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialg... ...e study of these properties in case of complex fields. Here we recall the concepts of linear functionals, adjoint, unitary operators and normal ope... ...n case of different models. 37 Chapter Two INTRODUCTORY CONCEPTS ON BASIC BISTRUCTURES AND S-BISTRUCTURES In this chapt... ...RES AND S-BISTRUCTURES In this chapter we recall some of the basic concepts on bistructures and S-bistructures used in this book to make the... ...finition in section 2 some basic notions of S-bistructure. 2.1 Basic concepts of bigroups and bivector spaces This section is devoted to the... ...n we have at least some of the classical theorems and some more classical concepts to be true in them. DEFINITION 2.1.1: Let (G, +, ) be a bigrou... ...strong linear 57 bialgebras are defined over bifields. Thus these three concepts are not fully related. It is important to mention here that anal... ...is point is very vital for in several situations certain relations between concepts may certainly remain an indeterminate. So this section will pur...

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...e 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. Using these matrices we can build n-eigen values and n-eigen vectors....

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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...

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...t as a “method of solving problems by honest confession of one’s ignorance”. Using this definition, Boole introduces, in a conversational manner, the concepts of logic and algebra, illustrating these concepts with stories and anecdotes, often from biblical sources. At times, her discussion seems somewhat mystical, speaking of the imagination and angels as messengers which ...

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...simple and an effective one as it can analyse the data by directed graphs and connection matrices. DEFINITION 1.2.1: An FCM is a directed graph with concepts like policies, events etc. as nodes and causalities as edges. It represents causal relationship between concepts. Example 1.2.1: In Tamil Nadu (a southern state in India) in the last decade several new engineering c...

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...Dedication 5 Preface 6 Chapter One BASIC CONCEPTS 1.1 Definition of Interval Matrices and Examples 8 1.2 Definition of Fuzzy Cognitive Maps 9 1.3 An Introduction to Neutrosophy 13 1.4 Some Basic Neutrosophic Structures 16 1.5 Some Basic Notions about Neutrosoph...

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... WITH ILLUSTRATIONS 1.1 Fuzzy matrices and their applications 9 1.2 Definition and illustration of Fuzzy Relational Maps (FRMs) 16 1.3 Some basic concepts of BAM with illustration 20 1.3.1 Some basic concepts of BAM 22 1.3.2 Use of BAM Model to study the cause of vulnerability to HIV/AIDS and factors for migration 28 1.4 Introduction to Fuzzy Associative Memories (F...

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...; therefore he introduced new terms ‘contingency’ and ‘possibility’ into his modal logic [5]. And then American logician Lewis first formulated these concepts of logical modality. ...

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...loop. For more on loops and their properties, readers should check [16], [3], [5], [8], [9] and [19]. In her book, she introduced over 75 Smarandache concepts on loops. In her ¯rst paper [20], she introduced Smarandache : left(right) alternative loops, Bol loops, Moufang loops, and Bruck loops. But in this paper, Smarandache : inverse property loops (IPL), weak inverse pro...

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... 11.4.2 Settings and other model information . . . . 360 11.4.3 Results . . . . . . . . . 361 11.4.4 Discussion . . . . . . . . . 367 11.5 Latest concepts . . . . . . . . . 368 11.6 Conclusion . . . . . . . . . . 370 11.7 References . . . . . . . . . . . 370 Chapter 12 Utilizing classifier conflict for sensor management and user interaction 371 by W.L. van Norden ...