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This is Smarandache's Russian translation from Romanian on Algebra.

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...Ion GOlAN Ralsa GRIGOR Vasile MARIN Aorentm SMARANDACHE P I . f • • 1 Algebra In exerc;lii 1; probleme pentru lic... ...Ion GOlAN Ralsa GRIGOR Vasile MARIN Aorentm SMARANDACHE P I . f • • 1 Algebra In exerc;lii 1; probleme pentru liceu • , II /(11- "., ,., • ...

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...Prezenta lucrare contine exercitii §i probleme de algebra, grupate pe capitole, pentru clasele superioare de licee §i §coli medii de cultura generala. Scopul ei este pregatirea matematica a elevilor din liceele de toate categoriile §i va fi utila in lucrul de sine statator...

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...Ion GOlAN Ralsa GRIGOR Vasile MARIN Aorentm SMARANDACHE P I . f • • 1 Algebra In exerc;lii 1; probleme pentru lic... ...Ion GOlAN Ralsa GRIGOR Vasile MARIN Aorentm SMARANDACHE P I . f • • 1 Algebra In exerc;lii 1; probleme pentru liceu • , II /(11- "., ,., • ... ...ca • ~ " '21. 1 { ,.f.- II J ;.\ Ion GOlAN Raisa GRIGOR Vasile MARIN Florentin SMARANDACHE Algebra in exerciJii Ii probleme pentru liceu Mul... ... '21. 1 { ,.f.- II J ;.\ Ion GOlAN Raisa GRIGOR Vasile MARIN Florentin SMARANDACHE Algebra in exerciJii Ii probleme pentru liceu Mulfimi, oper... ...f.- II J ;.\ Ion GOlAN Raisa GRIGOR Vasile MARIN Florentin SMARANDACHE Algebra in exerciJii Ii probleme pentru liceu Mulfimi, operafii cu mulfi... ... Ot~iniiu: bd. Mircea cel Batrin, Dr. 9, sectoml Ciocana. Tel.: 34 64 61. ALGEBRA IN EXEROTII $1 PROBLEME PENTRU LICEU (Multimi, operatii cu multimi... ...lemente de combinatorica.) Autori: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache. Coperta: Vitalie Coroban Prepress: Centrul de Mate... ... combinatorica.) Autori: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache. Coperta: Vitalie Coroban Prepress: Centrul de Matematica Apl... ...tematica Aplicata ~i Informatica © Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache, 2000, pentru prezenta editie. Aceasta editie a apa...

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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 eco...

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...In this section we give the definition of linear algebra and just state the few important theorems like Cayley Hamilton theorem, Cyclic Decomposition Theorem, Generalized Cayley Hamilton Theorem and give some properties about linear algebra....

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...6, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico... ...e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, ... ...sore, Karnataka, India. Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral Layout by Kama Kandasamy. ... ...ataka, India. Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral Layout by Kama Kandasamy. Cover des... ...andasamy. Many books can be downloaded from: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1931233-97-7 ... ...NTENTS Preface 5 Chapter One INTRODUCTION TO LINEAR ALGEBRA AND S- LINEAR ALGEBRA 1.1 Basic properties of linear algebra ... ... 1.1 Basic properties of linear algebra 7 1.2 Introduction to s-linear algebra 15 1.3 Some aapplications of S-linear algebra 30 Chapter... ...PHIC LINEAR BIALGEBRA AND ITS APPLICATION 4.1 Some basic neutrosophic algebraic structures 131 4.2 Smarandache neutrosophic linear bialgebr... ...ent in every manner made this book possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 7 Chapter One I...

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...ices are used. Matrices find their applications in several real models. In fact it is not an exaggeration 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, bigroup...

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...36, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexic... ... e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup,... ...ity, Atlanta, GA 30302, USA Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral Layout by Kama Kandasamy.... ...ta, GA 30302, USA Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral Layout by Kama Kandasamy. Cover de... ...an be downloaded from this E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1931233-95-0 HEXIS Pu... ...UCTION TO NEUTROSOPHIC BIMATRICES 3.1. Introduction to some neutrosophic algebraic structures 139 3.2. Neutrosophic bimatrices 148 3.3. Neutro... ...In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other... ...r bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in the constructio... ... and Kama Kandasamy for formatting this book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 7 Chapter One BIMATRI...

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...vorbindu-va despre el. Ovidiu Florentin a vazut pentru prima data lumina zilei/ noaptea (de 2 spre 3 octombrie 1980) în Craiova, din creierul meu. Dar el a fost conceput mult timp, dupa lungi cautari în suflet. Nu are mama. Taica-su facându-l din flori......

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...t ă de: MIRCEA MONU Tehnoredactare computerizat ă: MIRELA BOLDEANU 3 FLORENTIN SMARANDACHE LEGI DE COMPOZI ŢIE INTERN Ă ... ...RCEA MONU Tehnoredactare computerizat ă: MIRELA BOLDEANU 3 FLORENTIN SMARANDACHE LEGI DE COMPOZI ŢIE INTERN Ă ... ...Coperta: „Silly computer design III” (Desen prostesc de computer), de Florentin Smarandache, din cartea „Outer-Art”, vol. II, Editura „Co... ...Silly computer design III” (Desen prostesc de computer), de Florentin Smarandache, din cartea „Outer-Art”, vol. II, Editura „Conphys”, Râ... ...re importan ţă, am s ă tac despre mine, vorbindu-v ă despre el. Ovidiu Florentin a v ăzut pentru prima dat ă lumina zilei/ noaptea (de 2 spre 3 ... ... aser ţiune; şi-n matematic ă (în special analiza matematic ă, geometria, algebra) exist ă atâtea spa ţii abstracte, imaginate de omul de ştiin ţă, ... ...ţă, care au devenit utile, foarte aplicate în practic ă: amintesc numai de algebra lui Boole, care p ărea atât de departe de realitate, iar peste 10... ... odat ă/ ca nici o alt ă dat ă - şi poate va mai fi. 19 Florentin Smarandache În continuare, urm ări ţi din crea ţ... ...TE MÂINE. Şi-a smuls limba cu creieri cu tot pe coala de scris. Algebra îi d ă pumni în cap, în fa ţa personalit ăţii cifrelor are sup...

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...2 3 FLORENTIN SMARANDACHE FRATE CU MERIDIANELE SI PARALEL... ... 3 FLORENTIN SMARANDACHE FRATE CU MERIDIANELE SI PARALELELE ... ... Descrierea CIP a Bibliotecii Na ţionale a României SMARANDACHE, FLORENTIN Frate cu meridianele şi paralelele / Florentin Sma... ... Descrierea CIP a Bibliotecii Na ţionale a României SMARANDACHE, FLORENTIN Frate cu meridianele şi paralelele / Florentin Smarandache – ... ...şi... Dennis Hathance, pasionat de matematic ă (inele neasociative – din algebra abstract ă) şi... vioar ă, m ă ia la un concert simfonic: Verdi (U... ...am terminat doctoratul ( şi nici nu prea am chef de el!)... „Func ţia Smarandache” – în ţeleas ă, „Matematici paradoxiste” – nu! Am cinat la ... ...studen ţi premia ţi la matematic ă. Dennis râdea când a v ăzut meniul: – Florentin, comand ă Escargot Florentine Parmezan (parmezan florentin de me... ... Carlos Castro Perelman (de la Universitatea din Atlanta), cu lucr ări în Algebra Clifford şi Extinderi Supraluminale ale Relativit ăţii Generale, d... ...s, (1811-1832) celebrul matematician francez, autor al Teoriei Galois din Algebra Abstract ă, mort la 21 de ani în duel pentru c ă era anti-monarh, ...

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... Chief Editor Dmitri Rabounski rabounski@yahoo.com Associate Editors Prof. Florentin Smarandache smarand@unm.edu Dr. Larissa Borissova lborissova@yaho... ...tor Dmitri Rabounski rabounski@yahoo.com Associate Editors Prof. Florentin Smarandache smarand@unm.edu Dr. Larissa Borissova lborissova@yahoo.com Step... ...downloaded free from the E-library of Science: http://www.gallup.unm.edu/ smarandache ISSN: 1555-5534 (print) ISSN: 1555-5615 (online) Standard Addre... ...New Method to Measure the Speed of Gravitation. . . . . . . . . . . . 3 F. Smarandache Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes. . . 7 F.... ... as follows b σ g αβ ∇ α ∇ β F σ =0, h i σ g αβ ∇ α ∇ β F σ =0. After some algebra we obtain the chr.inv.-d’Alembert equations for the feld of the gra... ... IN PHYSICS Volume 1 Quantum Quasi-Paradoxes and Quantum Sorites Paradoxes Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 C... ... A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 C... ... modifed Newtonian dynamics induced by the Yang’s Noncommutative Spacetime algebra involving a lower and upper scale in Nature. 1 Introduction In rece... ... include a theory with all topologies as well. It turned out that Clifford algebras contained the appropriate algebro-geometric features to im- plemen...

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...0036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mex... ...a e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallu... ..., Alaska, USA. Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K.Ilanthenral Layout by Kama Kandas... ...SA. Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K.Ilanthenral Layout by Kama Kandasamy. Cover... ...wnloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1931233-98-5 Stand... ...rt Kosko, the father of Fuzzy Cognitive Maps. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.ILANTHENRAL 7 Chapter One ... ...nd their extensions to these models. 1. Abraham, R. Linear and Multilinear Algebra, W. A. Benjamin Inc., 1966. 2. Adams, E.S., and D.A. Farber. ... ...p://law.vanderbilt.edu/lawreview/vol525/adams.pdf 3. Albert, A. Structure of Algebras, Colloq. Pub., 24, Amer. Math. Soc., 1939. 4. Allen, J., S.... ...in/32/3271/A327183.php 11. Birkhoff, G., and MacLane, S. A Survey of Modern Algebra, Macmillan Publ. Company, 1977. 12. Birkhoff, G. On the struc...

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... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC... ... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATION... ...600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New M... ...dia e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gal... ...Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by... ...rnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kan... ...lem of unique solvability of a fuzzy relation equation in an arbitrary max-min algebra. By a max-min fuzzy algebra B we mean any linearly ordered ... ...and the relation X ∈ B (n, p) is unknown. An important special case of max-min algebra is Godel algebra, in which the underlying set is the closed ... ...is the closed unit interval with natural ordering of real numbers. This max-min algebra will be denoted by B G . An implication operator in B G is ...

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...allup 1-3 December 2001 (second printed edition) FLORENTIN SMARANDACHE editor PROCEEDINGS OF THE FIRST INTERNATIO... ... 1-3 December 2001 (second printed edition) FLORENTIN SMARANDACHE editor PROCEEDINGS OF THE FIRST INTERNATIONAL CONFE... ...co Contributed papers were sent, by November 30, 2001, to the organizer: Florentin Smarandache, University of New Mexico, 200 College Road, Ga... ...ntributed papers were sent, by November 30, 2001, to the organizer: Florentin Smarandache, University of New Mexico, 200 College Road, Gallup, NM 8... ...Neutrosophic Conference). E-mail: smarand@unm.edu, http://www.gallup.unm.edu/~smarandache/FirstNeutConf.htm. A selection of them is being published... ...ophic Logic, Neutrosophic Set, and Neutrosophic Probability and Statistics) Florentin Smarandache Department of Mathematics University of New Me... ...versity of Texas at Arlington, Vol. XIX, 143-154, 1999. [42] Stoyanova, D., Algebraic structures of intuitionistic fuzzy sets, Third Sci. Sessio... ...ork of A. Kolmogorov in 1933, the axiomatic of the probability theory based on algebras and measure theory has been definitely adopted. We remind now ... .... (54) The last constraint in (54) can also be written as (55) Now with basic algebraic manipulation, the optimal global bpa we are searching for, c...

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...N 1-879585-76-6 American Research Press Rehoboth 1999, 2000, 2003 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEU... ...-76-6 American Research Press Rehoboth 1999, 2000, 2003 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, ... ...script from the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philo... ...m the mathematician, experimental writer and innovative painter Florentin Smarandache, especially because the treated subject was of philosophy - re... ...ables, odds, paradoxes in creations. It was set up and led by the writer Florentin Smarandache since 1980's, who said: "The goal is to enlargement ... ...s, paradoxes in creations. It was set up and led by the writer Florentin Smarandache since 1980's, who said: "The goal is to enlargement of the art... ...paradoxist intuition is a high level of awareness; - postmodernist; - an algebraic, physical and chemical philosophy; - consistent with its inconsi... ...rough the objects and see the essence. What could be a philosophical algebra? But a philosophical vector space? And how should we introduce a p... ..., Õ), where NL is the set of neutrosophic logical propositions, is not an algebra. Nor ( c(. 0, 1 + 0), 3, 4, C), where c(. 0, 1 + 0) is the set o...

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... Chief Editor Dmitri Rabounski rabounski@yahoo.com Associate Editors Prof. Florentin Smarandache smarand@unm.edu Dr. Larissa Borissova lborissova@yaho... ...tor Dmitri Rabounski rabounski@yahoo.com Associate Editors Prof. Florentin Smarandache smarand@unm.edu Dr. Larissa Borissova lborissova@yahoo.com Step... ...downloaded free from the E-library of Science: http://www.gallup.unm.edu/ smarandache ISSN: 1555-5534 (print) ISSN: 1555-5615 (online) Standard Addre... ...s. . . . . . . . . . . . . . . . . . . . 92 D. Rabounski, L. Borissova, F. Smarandache Quantum Causality Threshold in the General Theory of Relativity... ...00 =F 00 =0, as for any anti- symmetric tensor of the 2nd rank, after some algebra we obtain the other components of the feld tensor F αβ F 0i = 1 c 2... ...ccount that the Lorentz condition for the feld b α is D=0 (48), after some algebra we obtain the chr.inv.-d’Alembert equations for the time density fe... ...onsiderations, the theory of gravitational waves is directly linked to the algebraical classi- fcation of spaces given by Petrov [37] ( Petrov classif... ...d in the General Theory of Relativity Dmitri Rabounski, Larissa Borissova, Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 C... ... Experiments, More Types of Unmatter, and a Quantum Chromodynamics Formula Florentin Smarandache Dept. of Mathematics, University of New Mexico, 200 C...

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... 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 linear tr...

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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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...This paper contains exercises and problems of algebra and grouped chapters, the upper classes of secondary schools and coli culture media general. Its purpose is to prepare students from high schools mathematics categories and will be helpful in working independently. Al...

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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 new class...

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...e 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 16...

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...make use of the literature in case the reader is not thorough with the 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 t...

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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 Strong Neutrosophic Bivector Spaces 32 2....

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...This 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 fiv...

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...t defined on column matrices, m * n (m ≠ n) matrices. This extension is the same in case of row matrices. We make use of the notion of semigroups and Smarandache semigroups. Also the notion of semirings, Smarandache semirings, semi vector spaces and semifields are used....

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...ures 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...

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...ps, N-group 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 Pro...

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...THEOREM 2.1: Let G = {C(Zn), *, (t, u); t, u ∈ Zn} be a complex modulo integer groupoid. If H ⊆ G is such that H is a Smarandache modulo integer subgroupoid, then G is a Smarandache complex modulo integer groupoid. But every subgroupoid of G need not be a Smarandache complex modulo interger subgroupoid even if G is a Smarandache groupoid. ...

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...We want to find efficient algebraic methods to improve the realiability of the transmission of messages. In this chapter we give only simple coding and decoding algorithms which can be easily understood by a beginner. Binary symmetric channel is an il...

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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 ...

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...roups or neutrosophic biinterval semigroups. We derive results pertaining to them. The new notion of quasi bisubsemigroups and ideals are introduced. Smarandache interval neutrosophic bisemigroups are also introduced and analysed. Also notions like neutrosophic interval bigroups and their substructures are studied in section two of this chapter. Neutrosophic interval bigro...

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... 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 195 ...

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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 group line...

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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 Chapt...

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...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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...2 Lukasiewicz Multi-Valued-logic: History and Introduction to Multi- Valued Algebra 2.1 Introduction to trivalent logic and plurivalent logic We all have heard of typical binary logic, Yes or No. Or in a famous phrase by Shakespeare: “To be or not to be.” In the same way all computer hardwares ...

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...Foreword 6 1 Introduction: Paradoxes, Lukasiewicz, Multi-Valued logic 7 2 Lukasiewicz Multi-Valued Logic: History and Introduction to Multi-Valued Algebra 10 2.1. Introduction to trivalent logic and plurivalent logic 10 2.2. History of Lukasiewicz and Multi-Valued Logic 12 2.3. Introduction to Multi-Valued Algebra, Chang’s Notation 15 2.4. Linkage between Multi-Valu...

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...Abstract We explore Yang’s Noncommutative space-time algebra (involving two length scales) within the context of QM defined in Noncommutative spacetimes and the holographic area-coordinates algebra in Clifford spaces. Casimir invariant wave equations corresponding to Noncommuta...

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...This second book devoted on advances and applications of Dezert-Smarandache Theory (DSmT) for information fusion collects recent papers from different researchers working in engineering and mathematics. Part 1 of this book presents the current state-of-the-art on theoretical investigation...

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...Preamble iii Prefaces v Part I Advances on DSmT 1 Chapter 1 Proportional Conflict Redistribution Rules for Information Fusion 3 by Florentin Smarandache and Jean Dezert 1.1 Introduction . . . . . . . . . . . 3 1.2 The principal rules of combination . . . . . . . 6 1.2.1 Notion of total and partial conflicting masses . . . . . 6 1.2.2 The conjunctive ...

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...This book has four chapters. Chapter one is introductory in nature. The reader is expected to have a good background of algebra and graph theory in order to derive maximum understanding of this research. The second chapter represents groups as graphs. The main feature of this chapter is that it contains 93 examples with diagrams and 18 theorem...

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...ME 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 Graphs of Loops 129 3.3 The Identity Graph of a Finite Commutative Ring with Unit 134 Chapter Four SUGGESTED PROBLEMS 157 FURTHER READIN...

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...Part I Advances on DSmT 1 Chapter 1 An introduction to DSmT 3 by J. Dezert and F. Smarandache 1.1 Introduction . . . . . . . . . . 4 1.2 Foundations of DSmT . . . . . . . . . 4 1.2.1 The power set, hyper-power set and super-power set . . 6 1.2.2 Notion of free and hybrid DSm models . . . . 18 1.2.3 Ge...

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...perations on Fuzzy relations . . . . . . . . . . . . 43 3.3 Neutrosophic Relations . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Generalized Algebra on Neutrosophic Relations . . . . . . . . . . . . . . . 47 3.5 An Application . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 An In_nite-Valued Tuple Relational Calculus . . . . . . . . . . . . . . . 54 3....

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...This book contains 23 papers, most of which were written by participants to the fifth International Conference on Number Theory and Smarandache Notions held in Shangluo University, China, in March, 2009. In this Conference, several professors gave a talk on Smarandache Notions and many participants lectured on them both extensively and intensively. All th...

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...J. Wang : An equation related to the Smarandache power function 1 X. Lu and J. Hu : On the F.Smarandache 3n-digital sequence 5 B. Cheng : An equation involving the Smarandache double factorial function and Euler function 8 A. A. K. Majumdar : S-perfect and c...

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.... . . . . . . . . . . . . . . . . . 81 10.2 Basic fuzzy logic BL8 . . . . . . . . . . . . . . . . . . . . . . . . 84 10.3 Non-Archimedean valued BL-algebras . . . . . . . . . . . . . . . 85 10.4 Non-Archimedean valued predicate logical language . . . . . . . . 87 10.5 Non-Archimedean valued basic fuzzy propositional logic BL1 . . 89 11 Neutrosophic sets. . . . . . . ...

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...SUPER INTERVAL SEMILINEAR ALGEBRAS In this chapter we for the first time introduce the notion of semilinear algebra of super interval matrices over semifields of type I (super semilinear algebra of type I) and semilinear algebra of super interval mat...

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... INTERVAL SUPERMATRICES 9 Chapter Three SEMIRINGS AND SEMIVECTOR SPACES USING SUPER INTERVAL MATRICES 83 Chapter Four 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...

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...36, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mex... ... e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallu... ...Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by ... ...ofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kanda... ... can be downloaded from this E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1931233-96-9 ... ...e world who are bravely battling the AIDS epidemic. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One INTRODUCTION ... ...asamy and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In t... ...e has guided eight Ph.D. scholars in the different fields of non- associative algebras, algebraic coding theory, transportation theory, fuzzy groups... ...d a Book Series, consisting of ten research books on the topic of Smarandache Algebraic Structures which were published by the American Research Pr...

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... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC... ... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATION... ...600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New M... ...dia e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gal... ...Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by... ...rnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kan... ...lem of unique solvability of a fuzzy relation equation in an arbitrary max-min algebra. By a max-min fuzzy algebra B we mean any linearly ordered ... ...and the relation X ∈ B (n, p) is unknown. An important special case of max-min algebra is Godel algebra, in which the underlying set is the closed ... ...is the closed unit interval with natural ordering of real numbers. This max-min algebra will be denoted by B G . An implication operator in B G is ...

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...bipartite 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 alg...

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... 1 Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Mo... ...36, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexic... ... e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup,... ...re, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Mee... ...aka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kandasa... ...Kandasamy. Many books can be downloaded from: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1-931233-87-X ... ... f a c e 5 Chapter One INTRODUCTION TO SOME NEUTROSOPHIC ALGEBRAIC STRUCTURES 1.1 Neutrosophic Fields 7 1.2 Neutrosophic Vect... ... ABOUT THE AUTHORS 149 5 Preface Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic th... ...standing of Periyar's ideology and approach. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One Introduction to So...

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...origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. This was introduced by one of the authors, Florentin Smarandache. A few of the mentionable characteristics of this mode of thinking are [90-94]: It proposes new philosophical theses, principles, laws, methods, formulas and movements; it reveals that the world is full ...

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...W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY COGNITIVE MAPS AND NEUTROSOPHIC COGNITIV... ...W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY COGNITIVE MAPS AND NEUTROSOPHIC COGNITIVE MAPS ... ...ology, Madras Chennai – 600036, India E-mail: vasantha@iitm.ac.in Florentin Smarandache Department of Mathematics University of New Mexico ... ...ras Chennai – 600036, India E-mail: vasantha@iitm.ac.in Florentin Smarandache Department of Mathematics University of New Mexico Gallup, N... ...ersity, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix, AZ 85020,... ...stralia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix, AZ 85020, USA ... ...eutrosophic structures In this section we define some new neutrosophic algebraic structures like neutrosophic fields, neutrosophic spaces and neu... ...asantha Kandasamy, W.B., N.R. Neelakantan and S.R. Kannan. Replacement of Algebraic Linear Equations by Fuzzy Relation Equations in Chemical Enginee... ...y and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding / communication theory. In the p...

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...W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE ANALYSIS OF SOCIAL ASPECTS OF MIGRANT LABOURERS... ...W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE ANALYSIS OF SOCIAL ASPECTS OF MIGRANT LABOURERS LIVING W... ..., India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico ... ...-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, N... ...ad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix, AZ 85020 ... ... Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix, AZ 85020 USA L... ... are Maximum Infected, Using Fuzzy Matrices In this section we give an algebraic approach to the problem of studying the psychological aspects an... ...y and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In the p... ...s guided eight Ph.D. scholars in the different fields of non- associative algebras, algebraic coding theory, transportation theory, fuzzy groups, an...

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...a Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache e-mail: smarand@unm.edu K. Kandasamy e-... ...y e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smarandache e-mail: smarand@unm.edu K. Kandasamy e-mail: dr.... ...Karnataka, India Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K. Kandasamy Cover Design and Lay... ...India Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K. Kandasamy Cover Design and Layout by Kama... ...wnloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1931233-00-4 S... ...elational Maps 18 1.4 An Introduction to Neutrosophy and some Neutrosophic algebraic structures 22 1.5 Neutrosophic Cognitive Maps 27 1.6 Neut... ...to working for the rights of the subjugated people. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.KANDASAMY 26-11-2005 9 Chapter On... ... E 1 +…+ E p . 1.4 An Introduction to Neutrosophy and some Neutrosophic algebraic structures In this section we introduce the notion of neut... ...asamy and daughters Meena and Kama. Her current interests include Smarandache algebraic structures, fuzzy theory, coding/ communication theory. In t...