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...graphs plays the role of representing the neutrosophic models. Thus to construct the neutrosophic graphs one needs some of the neutrosophic algebraic 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 str...

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... 1 Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Models ... ... 5 Chapter One INTRODUCTION TO SOME NEUTROSOPHIC ALGEBRAIC STRUCTURES 1.1 Neutrosophic Fields 7 1.2 Neutrosophic Vector spaces ... ...E AUTHORS 149 5 Preface Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has p... ...truct the neutrosophic graphs one needs some of the neutrosophic algebraic structures viz. neutrosophic fields, neutrosophic vector spaces and neut... ...epts. As our analysis in this book is application of neutrosophic algebraic structure we found it deem fit to first introduce and study neutrosophic...

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.... Chapter five just introduces the concept of neutrosophic groupoids and neutrosophic Ngroupoids. Sixth chapter innovatively gives mixed neutrosophic structures and their duals. The final chapter gives problems for the interested reader to solve. Our main motivation is to attract more researchers towards algebra and its various applications....

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...1.1 Groups, N-group and their basic Properties It is a well-known fact that groups are the only algebraic structures with a single binary operation that is mathematically so perfect that an introduction of a richer structure within it is impossible. Now we proceed on to define a group. DEFINITION 1.1.1: A non empty set of elemen...

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...p and their basic Properties 7 1.2 Semigroups and N-semigroups 11 1.3 Loops and N-loops 12 1.4 Groupoids and N-groupoids 25 1.5 Mixed N-algebraic Structures 32 Chapter Two NEUTROSOPHIC GROUPS AND NEUTROSOPHIC N-GROUPS 2.1 Neutrosophic Groups and their Properties 40 2.2 Neutrosophic Bigroups and their Properties 52 2.3 Neutrosophic N-groups and their Properties 68...

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...tion and desire for writing this book, is the direct appreciation and attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibit properties of both a group and a semigroup simultaneously. This book is a piece of work on Smarandache semigroups and assumes the reader to have a good background o...

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...attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibit properti... ...eorem, and Sylow's theorem. Since Smarandache semigroups are the overlap of two structures, we are able to see how the mixture of a group and a semig... ...ade do Porto, Portugal, 9-11 July 1998. 2. R. Padilla, Smarandache Algebraic Structures, Smarandache Notions Journal, USA, Vol.9, No. 1-2, 36-38,... ...al, USA, Vol.9, No. 1-2, 36-38, (1998). 3. R. Padilla. Smarandache Algebraic Structures, Bulletin of Pure and Applied Sciences, Delhi, Vol. 17 E,... ...allup.unm.edu/~smarandache/ALG-S-TXT.TXT 4. F. Smarandache, Special Algebraic Structures, in Collected Papers, Vol. III, Ab...

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...ssical linear algebra theorems. In case of n-vector spaces we can define several types of linear transformations. The applications of these algebraic structures are given in Chapter 3. Chapter four gives some problem to make the subject easily understandable....

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...ration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup b...

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...EUTROSOPHIC BIMATRICES 3.1. Introduction to some neutrosophic algebraic structures 139 3.2. Neutrosophic bimatrices 148 3.3. Neutrosophic bive... ...ces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache... ... simplify the writing or printing of A B . (2) It exhibits some particular structure of A B which is of interest. (3) It simplifies computation. ... ...ophic matrices and neutrosophic vector spaces and the related neutrosophic structures used to define it. This Chapter has three sections. The first... ...s three sections. The first section introduces some neutrosophic algebraic structures and neutrosophic matrices and vector spaces. In section two ...

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...c interval bivector spaces are introduced and their properties are described. In the third chapter we introduce the notion of neutrosophic interval n-structures or neutrosophic interval n-structures. Over 60 examples are given and various types of n-structures are studied. Possible applications of these new structures are given in chapter four. The final chapter suggests o...

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... Neutrosophic Interval Bisemirings 85 2.3 Neutrosophic Interval Bivector Spaces and their Generalization 93 Chapter Three NEUTROSOPHIC n- INTERVAL STRUCTURES (NEUTROSOPHIC INTERVAL n-STRUCTURES) 127 Chapter Four APPLICATIONS OF NEUTROSOPHIC INTERVAL ALGEBRAIC STRUCTURES 159 Chapter Five SUGGESTED PROBLEMS 161 FURTHER READING 187 INDEX 189 ABOUT THE AUTHORS 19...

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...inite using complex groupoids and complex loops. This chapter gives over 120 examples and thirty theorems. Forth chapter introduces nonassociative structures using complex modulo integer groupoids 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 f...

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...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 new class of loops. All these non-associative linear ...

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...ic rings, S-semigroup neutrosophic rings which can be realized as a type of extension of group rings or generalization of group rings. Study of these structures will throw light on the research on the algebraic structure of group rings. Chapter four is entirely devoted to the problems on this new topic, which is an added attraction to researchers. A salient feature of this...

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... is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo 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 ...

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... is a group. Pseudo neutrosophic subgroups can be found as a substructure of neutrosophic groups. Thus a pseudo 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 group...

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...to 5 chapters. The first chapter introduces real neutrosophic complex numbers. Chapter two introduces the notion of finite complex numbers; algebraic structures like groups, rings etc are defined using them. Matrices and polynomials are constructed using these finite complex numbers. Chapter three introduces the notion of neutrosophic complex modulo integers. Algebraic str...

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...MPLEX NUMBERS 89 Chapter Three NEUTROSOPHIC COMPLEX MODULO INTEGERS 125 Chapter Four APPLICATIONS OF COMPLEX NEUTROSOPHIC NUMBERS AND ALGEBRAIC STRUCTURES USING THEM 175 Chapter Five SUGGESTED PROBLEMS 177 FURTHER READING 215 INDEX 217 ABOUT THE AUTHORS 220...

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...pout in primary school one of the causes in both cases may be due to high fee structure in primary school. We read the resultant bivector given by I... ...r repulsion. But it is forgotten in all these situations we may have molecule structures which need not attract or repel but remain without action o... ...phs and n = 4 we say the graphs as quad graphs. The main motivation for these structures are they have application in fuzzy and neutrosophic real mo... ...o introduce the indeterminacy in FCMs, and we choose to call this generalized structure as Neutrosophic Cognitive Maps (NCMs). In our view this will... ... on them. Example 4.6.2: Since the very concept of women’s empowerment and community mobilization in the context of AIDS epidemic is an unsupervi... ...e bigraph. 234 The attributes related with the women’s empowerment and community mobilization are taken as W 1 – Gender balance W 2 –... ...0, 1999. http://law.vanderbilt.edu/lawreview/vol525/adams.pdf 3. Albert, A. Structure of Algebras, Colloq. Pub., 24, Amer. Math. Soc., 1939. 4. ...

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...cal theorems for neutrosophic algebras are proved with appropriate modifications. Chapter four indicates the probable applications of these algebraic structures. The final chapter suggests about 80 innovative problems for the reader to solve....

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... 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 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bi...

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...ONCEPTS 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 Graphs of Loops 129 3.3 The Identity Graph of a Finite Commutative Ring with Unit 134 Chapter Four SUGGESTED PROBLEMS 157 FURTHER READING 162 IND...

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...s book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super m...

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...s to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter suggests several research level problems. Fifth chapter indicates the applications of dual numbers. The forth chapter introduces the concept of i...

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...have to define natural operations existing on reals to the intervals what changes should be made in the definition of intervals. Here we redefine the structure of intervals to adopt or extend to the operations on reals to these new class of intervals....

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...ral different locations simultaneously, it can become overloaded and “see” structure that does not really exist. 71 F. Smarandache defined several A... ...nce inside blocks of zeros, he created two images that appear to be spiral structures that expand from top to bottom. The first was created using the ... ...stereograms, while there is some difference in the appearance, the general structure is still the same. However, we can use it to ask a more general q... ...74 27 32.71052 28 44.11692 29 51.44522 30 42.61618 116 FIGURE 3 From the structure of the graph, it appears that the sum is generally increasing in ...

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...The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic mode...

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...BIALGEBRA AND ITS APPLICATION 4.1 Some basic neutrosophic algebraic structures 131 4.2 Smarandache neutrosophic linear bialgebra 170 4.3 ... ... 229 ABOUT THE AUTHORS 238 5 Preface The algebraic structure, linear algebra happens to be one of the subjects which yields ... ...ls and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebrai... ...sophic linear bialgebra and also give the applications of these algebraic structures. It is important to mention here it is a matter of simple ex... ...f bigroups. The fourth chapter introduces several neutrosophic algebraic structures since they help in defining the new concept of neutrosophic li...

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...A collection of papers concerning Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, experimental geometries, algebraic structures, neutrosophic probability, set, and logic, etc. is published this year....

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... This paper explains this action by mathematical logic on mathematical systems generalized to Smarandache systems, or such systems with combinatorial structures, i.e., combinatorial systems, and shows how to extend the homomorphism theorem in abstract algebra to multi-systems or combinatorial systems. All works in this paper are motivated by a combinatorial speculation of ...

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...verify or refute the propositions described herein. It is of our hope that this volume could open a new chapter in our knowledge on the formation and structure of Astrophysical systems. ...

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...1. Why is the WORLD a Combinatorial One? The multiplicity of the WORLD results in modern sciences overlap and hybrid, also implies its combinatorial structure. To see more clear, we present two meaningful proverbs following. Proverb 1. Ames Room An Ames room is a distorted room constructed so that from the front it appears to be an ordinary cubic-shaped room, with a bac...

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...A structure theorem of right C-rpp semigroups1 Abstract A new method of construction for right C-rpp semigroups is given by using a right cross product of a right regular band and a strong semilattice of left cancellative mono...

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...s 4 K. Liu : On mean values of an arithmetic function 12 L. Wang and Y. Liu : On an infinite series related to Hexagon-numbers 16 X. Ren, etc. : A structure theorem of right C-rpp semigroups 21 A. R. Gilani and B. N. Waphare : Fuzzy extension in BCI-algebras 26 A. S. Shabani : The Pell's equation x2 ¡ Dy2 = §a 33 L. Cheng : On the mean value of the Smarandache LCM fu...

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...ion 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 Neutrosophic Graphs 22 1.6 On Neutrosophic Cognitive Maps with Examples 28 1.7 Definition and Illustration of Fuzzy Relational Maps (FRMs) 33 1.8 Introduction to Fuzzy Associativ...

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...nd X. Yan : Global attractivity of a recursive sequence 15 J. Earls : Smarandache reversed slightly excessive numbers 22 X. Ren and H. Zhou : A new structure of super R¤-unipotent semigroups 24 J. Li : An equation involving the Smarandache-type function 31 P. Zhang : An equation involving the function ±k(n) 35 Y. B. Jun : Smarandache fantastic ideals of Smarandache BC...

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... mean value of the Smarandache ceil function 42 Y. Wang : Some identities involving the Smarandache ceil function 45 J. Yan, X. Ren and S. Ma : The Structure of principal lters on po-semigroups 50 Y. Lu : F. Smarandache additive k-th power complements 55 M. Le : The Smarandache reverse auto correlated sequences of natural numbers 58 M. Karama : Smarandache partitions ...

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...60 8.5.1 Many-body mechanics . . . . . . . . . . . . . . . . . .460 8.5.2 Cosmology . . . . . . . . . . . . . . . . . . . . . . 462 8.5.3 Physical structure . . . . . . . . . . . . . . . . . . . . . . 465 8.5.4 Economical field . . . . . . . . . . . . . . . . . . . . . . . 466 8.5.5 Engineering field . . . . . . . . . . . . . . . . . . . . . . . 467 References . . . ...

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...Philosophical Lexicon, Econompress, Minsk-Moscow, Belarus-Russia, 2008.................520 60. Neutrosophic Transdisciplinarity (Multi-Space & Multi-Structure), by Florentin Smarandache, Arhivele Statului, Filiala Vâlcea, Rm. Vâlcea, 1969; presented at _coala de Var_ - Interna ional_, Interdisciplinar_ _i Academic_, Romanian Academy, Bucharest, 6-10 July 2009................

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...Philosophical Lexicon, Econompress, Minsk-Moscow, Belarus-Russia, 2008.................520 60. Neutrosophic Transdisciplinarity (Multi-Space & Multi-Structure), by Florentin Smarandache, Arhivele Statului, Filiala Vâlcea, Rm. Vâlcea, 1969; presented at _coala de Var_ - Interna ional_, Interdisciplinar_ _i Academic_, Romanian Academy, Bucharest, 6-10 July 2009................