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... AMERICAN RESEARCH PRESS REHOBOTH 2002 A 1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Depart... ...1 L 15 (8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...(8) {e} A 15 B 1 B 5 1 Smarandache Loops W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...exas 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can... ...s 76019, USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be... ...USA. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be download... ....O., Phillips J.D., Robinson D. A., Solarin A. R. T., Tim Hsu, Wright C.R.B. and by others who have worked on Moufang loops and other loops like Bol ... ... a new class of loops and loop rings, Ph.D. thesis IIT(Madras), guided by Vasantha.W.B., (1994). 57. Smith.J.D.H, Commutative Moufang loops: the f...

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... 1 Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... Bialgebraic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...aic Structures and Smarandache Bialgebraic Structures W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Nigeria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...geria. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...n of time. I also thank my daughters Meena and Kama and my husband Dr. Kandasamy without whose combined help this book would have been impossible... ...inary operation, called the product and denoted by ' ' such that i. a, b ∈ G implies a b ∈ G. ii. a, b, c ∈ G implies (a b) c = a (b c). i... ...a new class of loops and loop rings, Ph.D. thesis IIT (Madras), guided by Vasantha. W.B., (1994). 49. SMARANDACHE, Florentin, Special Algebraic St...

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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 Tec... ...1 Smarandache Semirings, Semifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techno... ...andache Semirings, Semifields, and Semivector spaces W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Mad... ...ity, India. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ...a. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...rs, Vol. III, Abaddaba, Oradea, 78-81, (2000). 5. Vasantha Kandasamy, W. B. Smarandache Semirings and Semifields, Smaranda... ...III, Abaddaba, Oradea, 78-81, (2000). 5. Vasantha Kandasamy, W. B. Smarandache Semirings and Semifields, Smarandache Notio...

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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... ... American Research Press Rehoboth 2002 1 W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology ... ...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 ... ...ge, China. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be do... ... said to define an equivalence relation on A if (a, a) ∈ S for all a ∈ A (a, b) ∈ S implies (b, a) ∈ S (a, b) ∈ S and (b, c) ∈ S implies that (a,... ... of Pure and Applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 2. W.B.Vasantha Kandasamy, Smarandache cosets, Smarandache Notions Journal,... ...and Applied Sciences, Delhi, Vol. 17E, No. 1, 119-121, 1998. 2. W.B.Vasantha Kandasamy, Smarandache cosets, Smarandache Notions Journal, American...

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...d group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ... group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...group loop 1 SMARANDACHE FUZZY ALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology Madras... ... Indian Institute of Technology Madras Chennai – 600 036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv ... ...tion by: Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...n by: Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ... DEFINITION 1.2.8: Let µ be a fuzzy subgroup of a group G. Then for any a, b ∈ G a fuzzy middle coset a µ b of the group G is defined by (a µ b) (x) ... ...ps, Ph.D. Thesis, IIT (Madras), June 1998. Research Guide: W. B. Vasantha Kandasamy. 90. MILIC, Svetozar, and TEPAVCEVIC, Andreja, On P-fuzzy corres...

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... REHOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...HOBOTH, NM 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...M 2002 1 Smarandache Non-associative rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...e, Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books c... ...Nigeria. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can ... ... Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downlo... ...port and encouragement made me write this book. I thank my husband Dr. Kandasamy and my daughters Meena and Kama, who tirelessly worked for very ... ...α and β in such a way that a. addition is commutative α + β = β + α . b. addition is associative that is α + ( β + γ) = ( α + β) + γ. c. There i... ...rings, Proc. London Math. Soc., 46, 231-248 (1940). 33. IQBAL UNNISA and VASANTHA KANDASAMY, W. B., Supermodular lattices, J. Madras Univ., 44, 58-...

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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... ... a 1 a 4 a 15 a 12 a 0 1 Smarandache Rings W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ...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... ... Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be download... ... following axioms are satisfied. 1. The set G is non-empty. 2. If a, b, c ∈ G then a(bc) = (ab) c. 3. There are exists in G an element e such t... ...semigroups is very recent one, done by F. Smarandache, R. Padilla and W.B. Vasantha Kandasamy [73, 60, 154, 156], we felt it is appropriate that the ... ... is very recent one, done by F. Smarandache, R. Padilla and W.B. Vasantha Kandasamy [73, 60, 154, 156], we felt it is appropriate that the notion of ...

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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... ... 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 Technology M... ...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... ..., Spain. Copyright 2002 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be down... ...l be used in this book. As concerned with notations, the familiar symbols a > b, a ≥ b, |a|, a / b, b / a occur with their usual meaning. DEFINIT... ...man, Introduction to number theory, Wiley Eastern Limited, (1989). 3. W.B. Vasantha Kandasamy, Smarandache Groupoids, http://www.gallup.unm.edu/~... ...oduction to number theory, Wiley Eastern Limited, (1989). 3. W.B. Vasantha Kandasamy, Smarandache Groupoids, http://www.gallup.unm.edu/~smarandac...

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...4 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ... 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Te... ... FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology,... ...matics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...alia. Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, Univers... ...r Science, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama K... ...cience, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kand... ...niversity of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. C... ... and Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kandasamy. Many books can be downloaded from: http://www.gallup.u...

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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... ...TROSOPHIC ANALYSIS OF PERIYAR’S VIEWS ON UNTOUCHABILITY W. B. Vasantha Kandasamy e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv... ....iitm.ac.in/~wbv Florentin Smarandache e-mail: smarand@unm.edu K. Kandasamy e-mail: dr.k.kandasamy@gamil.com Translation of... ...ranslation of the speeches and writings of Periyar from Tamil by Meena Kandasamy HEXIS Phoenix, Arizona 2005 2 This book can be... ...Russia. Prof. G. Tica, M. Viteazu College, Bailesti, jud. Dolj, Romania. Dr. B.S.Kirangi, University of Mysore, Mysore, Karnataka, India ... ...ity of Mysore, Mysore, Karnataka, India Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K. Kandasamy ... ... of Mysore, Mysore, Karnataka, India Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K. Kandasamy ... ...ed his entire life to working for the rights of the subjugated people. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.KANDASAMY 26-11-2005 ...

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...VICE 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...E 1 Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ...uzzy Cognitive Maps and Neutrosophic Cognitive Maps W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madra... ... Indian Institute of Technology, Madras Chennai – 600036, India E-mail: vasantha@iitm.ac.in Florentin Smarandache Department of Mathematics... ... Finance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...nance, Griffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave.... ...iffith University, Australia. Copyright 2003 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix... ...port and encouragement towards the writing of this book. We also thank Dr.Kandasamy, who diligently proofread four rough drafts of this book, Kama wh... ...the book was intact and Meena who collected the existing literature. W.B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One BASIC CO...

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