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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... ...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... ... ISBN: 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS Preface 5 1... ... 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS Preface 5 1. Gene... ... property). 2. a, b, c ∈ G implies a • (b • c) = (a • b) • c (associative law). 3. There exist an element e ∈ G such that a • e = e • a = a for all ... ...tional condition (L 4 ) x ∩ x = x, x ∪ x = x. (L 1 ) is the commutative law (L 2 ) is the associative law (L 3 ) is the absorption law and (L 4... ...of the classical Cayley's theorem for groups to S-loops. Cayley's theorem states "Every group is isomorphic to a subgroup of S n for some appropriat...

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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... ...ity, India. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... ISBN: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed in the United States of America C... ...N: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed in the United States of America CONTENTS... ...ure). 2. For all a, b, c ∈ S we have (a ∗ b) ∗ c = a ∗ (b ∗ c) (associative law), We denote by (S, ∗) the semigroup. DEFINITION 1.1.2: If in a se... ...ure property) 2. a, b, c ∈ G implies a • (b • c) = (a • b) • c (associative law) 3. There exists an element e ∈ G such that a • e = e • a = a for al... ...uence simple finite semiring. Using the classical theorem of Cayley which states, "Every group is isomorphic to a subgroup of A(S) for some approp...

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...7 a 11 a 9 a 1 a 4 a 15 a 12 a 0 1 Smarandache Rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...a 11 a 9 a 1 a 4 a 15 a 12 a 0 1 Smarandache Rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...sam, India. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can... ..., India. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be... ... ISBN: 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 ... ... 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. P... ...correcting this book. I dedicate this book to be my beloved mother-in-law Mrs. Salagramam Alamelu Ammal, whose only son, an activist-writer and ... ...e concept of lattices as we have a well known result in ring theory which states that “the set of all two sided ideals of a ring form a modular latt... ... ∪ (x ∩ x)) = x lead to x ∩ x = x and x ∪ x = x. L 1 is the commutative law, L 2 is the associative law, L 3 is the absorption law, and L 4 is ...

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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... ... 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 ... ...931233-71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface Chapter One: INTRODUC... ...71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface Chapter One: INTRODUCTORY CO... ...3 ∈ N; (n 1 + n 2 ) n 3 = n 1 n 3 + n 2 n 3 (right distributive law). This near-ring will be termed as right near-ring. If n 1 (n 2 ... ...ents of N are left cancelable we say that N fulfils the left cancellation law. Similarly one defines right cancelable. N is an integral domain if N ... ... ∪ A 2 and a function δ : Z × (A 1 ∪ A 2 ) → Z, Z is called the set of states and A = A 1 ∪ A 2 the input alphabet and δ the next state function...

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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... ... 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... ... ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 PART ONE 1. ... ... 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 PART ONE 1. SOME RE... ... a –1 )(x) = µ (a –1 xa) for every x ∈ G. By using earlier theorems which states o(aµ a –1 ) = o(µ ) for any a ∈ G. For the sake of simplicity an... ...t (right) cancelable we say that N fulfills the left (right) cancellation law. N is a bi-integral domain if both N 1 and N 2 has no zero divisors. ... ...erly contained in 〈S 〉; as in 〈S 〉 we may or may not have the associative law to be true. Remark: Even (ab)c ≠ a(bc) in general for all a, b, c ∈ ...

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... American Research Press Rehoboth 2002 A 1 × A 2 Z 2 Z 1 B 1 × B 2 A 1 A 2 B 1 B 2 4 0 1 2 3 1 W. B. Vasantha ... ... Z 2 Z 1 B 1 × B 2 A 1 A 2 B 1 B 2 4 0 1 2 3 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tech... ...ipiona 11550, Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books... ...ona 11550, Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books ca... ... ISBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 Pre... ...SBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 Preface 5... ...osure). 2. For all a, b, c ∈ S we have a  (b  c) = (a  b)  c (associative law). (S, ) is a semigroup. DEFINITION: If in a semigroup (S, ) w... ...s properly contained in <S>; as in <S> we may or may not have the associative law to be true. Remark: Even (ab) c ≠ a (bc) in general for all a, b... ... non - empty sets Z and A and a function δ: Z × A → Z, Z is called the set of states, A the input alphabet and δ the "next state function" of Y. ...

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... American Research Press Rehoboth 2002 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ... American Research Press Rehoboth 2002 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Te... ...achers College, China. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many boo... ...ers College, China. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books ... ... ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America 3 ... ... ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Prefac... ...τ o δ) that is the composition of mappings in general satisfies the associative law. Further if σ, τ : S → S is one to one then σ o τ and τ o σ ar... ...  b ∈ G (closed) 2. a, b, c ∈ G implies a  (b  c) = (a  b)  c (associative law) 3. There exists an element e ∈ G such that a  e = e  a = a fo... ...Z(G), the center of G, then N(a) ≠ G. Recall, too, that the class equation of G states that ∑ = )) a ( N ( o ) G ( o ) G ( o where the sum runs ove...

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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 ... ...-931233-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. BAS... ...-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. BASIC CONC... ... (closure axiom) 2. a. (b.c) = (a.b).c for all a, b, c ∈ L. (associative law) (S, .) is called the semigroup. The semigroup in general may not ha... ...b) . c for some a, b, c ∈ R. 3. a. (b + c) = a.b + a.c (The distributive law are true) for all a, b, c, ∈ R. Let R be a non-assoiciative ring. R ...

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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... ...tics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...ova, Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Ka... ...ISBN: 1931233-96-9 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface ... ...1931233-96-9 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 ... ... lacking more in India and especially in villages of Tamil Nadu than in other states. We see that they are the passive victims for no cause of their... ...and differential competitive. I. The deterministic signal Hebbian learning law correlates local neuronal signals. XY ij ij i i j j m m S( x) S... ...0, inhibitory if m ij < 0. 17 II. The deterministic competitive learning law modulates the signal-synaptic differences S i (x i ) – m ij wit...

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...UZZY AND NEUTROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iit... ...Y AND NEUTROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iitm.a... ...Russia. Prof. G. Tica, M. Viteazu College, Bailesti, jud. Dolj, Romania. Dr. B.S.Kirangi, University of Mysore, Mysore, Karnataka, India ... ... ISBN: 1931233-00-4 Standard Address Number: 297-5092 Printed in the United States of America 3 PERIYAR, THE PROPHET ... ... 1931233-00-4 Standard Address Number: 297-5092 Printed in the United States of America 3 PERIYAR, THE PROPHET OF THE... ... hidden pattern of the situation. As we have a very well known theory, which states that the strength of the data depends on, the number of experts... .... Combined conflicting opinions tend to cancel out and assisted by the strong law of large numbers, a consensus emerges as the sample opinion appro... ...s, in case of a model, add two or more FCMs to produce a new FCM. The strong law of large numbers ensures in some sense that, knowledge reliability... ..., we can know the problems and sufferings of the other castes so that we can unitedly help them come out of their problems. Except the selfish lazy...

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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... ... Finance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ... ISBN: 1-931233-76-4 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Dedication 4 Preface 5 Cha... ... 1-931233-76-4 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Dedication 4 Preface 5 Chapter On... ...dden pattern of the situation. As we have a very well known theory, which states that the strength of the data depends on, the number of experts' opi... ...ombined conflicting opinions tend to cancel out and assisted by the strong law of large numbers, a consensus emerges as the sample opinion approximat... ...in case of a model, add two or more FCMs to produce a new FCM. The strong law of large numbers ensures in some sense that knowledge reliability incr... ... election. Now each party has its own opinion and when they join up as an united front we will see the result. So, let us take the opinion of 4 expe...

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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... ...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... ...SBN: 1-931233-86-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS D e d i c a t i o n ... ...-931233-86-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS D e d i c a t i o n 6... ...system the fundamental method is to analyze its recursive relation of the fuzzy states. M. Kurano et al [47] gave 114 the existence and the uni... ... 2 ) given in Figure 2.1.3. 173 The linear equation is based on Ohm’s law, the drop in voltage V across a resistor R is given by the linea... ...in voltage V across a resistor R is given by the linear relation V = iR (Ohm’s law). The hydrodynamic analogue to the mean velocity v for laminar f...