... 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...

... 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...

... 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...

... 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...

...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...

... 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-...

...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 ...

... 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...

...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... ...pecific Reference To Rural Tamil Nadu in India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, M... ...tics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin S... ...l: smarand@gallup.unm.edu Translation of the Tamil interviews by Meena Kandasamy H E X I S Phoenix, Arizona 2005 2 This... ...ova, Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Ka... ..., Bulgarian Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kanda... ...an Academy of Sciences, Sofia, Bulgaria. Copyright 2005 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Co... ...ok to women around the world who are bravely battling the AIDS epidemic. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 7 Chapter One ...

... Structures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ...ructures and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techno... ...and Their Application to Fuzzy and Neutrosophic Models W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Mad... ...cs Indian Institute of Technology, Madras Chennai – 600036, India e-mail: vasantha@iitm.ac.in web: http://mat.iitm.ac.in/~wbv Florentin Smara... .... Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, University ... ...ience, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kanda... ...ce, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasam... ...rsity of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama Kandasamy. Cover... ... Florentin Smarandache Layout by Kama Kandasamy. Cover design by Meena Kandasamy. Many books can be downloaded from: http://www.gallup.unm.e...

...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...

...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 ...

...GNITIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Techn... ...TIVE MAPS With Specific Reference To Rural Tamilnadu In India W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technolo... ... With Specific Reference To Rural Tamilnadu In India 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 web: http://mat.iitm.ac.in/~wbv Florentin Smarand... ...and@gallup.unm.edu Translation of the Tamil interviews by Meena Kandasamy XIQUAN Phoenix 2004 The picture on the cover is... ... University of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley A... ...niversity of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave.... ... of Hyderabad, Andhra Pradesh, India Copyright 2004 by W. B. Vasantha Kandasamy, Florentin Smarandache, and Xiquan, 510 E. Townley Ave., Phoenix... ... men and women living with HIV/AIDS in the Third- World countries. W.B. Vasantha Kandasamy Florentin Smarandache Chapter One INTRO...

...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...