Alternative Loop Rings
Alternative Loop Rings

Author: E.G. Goodaire

Publisher: Elsevier

Published: 1996-10-24

Total Pages: 404

ISBN-13: 008052706X

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For the past ten years, alternative loop rings have intrigued mathematicians from a wide cross-section of modern algebra. As a consequence, the theory of alternative loop rings has grown tremendously.One of the main developments is the complete characterization of loops which have an alternative but not associative, loop ring. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups.Another major topic of research is the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group rings is of immense interest.This is the first survey of the theory of alternative loop rings and related issues. Due to the strong interaction between loop rings and certain group rings, many results on group rings have been included, some of which are published for the first time. The authors often provide a new viewpoint and novel, elementary proofs in cases where results are already known.The authors assume only that the reader is familiar with basic ring-theoretic and group-theoretic concepts. They present a work which is very much self-contained. It is thus a valuable reference to the student as well as the research mathematician. An extensive bibliography of references which are either directly relevant to the text or which offer supplementary material of interest, are also included.

Smarandache Non-Associative Rings
Smarandache Non-Associative Rings

Author: W. B. Vasantha Kandasamy

Publisher: Infinite Study

Published: 2002

Total Pages: 151

ISBN-13: 1931233691

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Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B in A which is embedded with a stronger structure S. These types of structures occur in our everyday's life, that's why we study them in this book. Thus, as a particular case: A Non-associative ring is a non-empty set R together with two binary operations '+' and '.' such that (R, +) is an additive abelian group and (R, .) is a groupoid. For all a, b, c in R we have (a + b) . c = a . c + b . c and c . (a + b) = c . a + c . b. A Smarandache non-associative ring is a non-associative ring (R, +, .) which has a proper subset P in R, that is an associative ring (with respect to the same binary operations on R).

Alternative Loop Rings
Alternative Loop Rings

Author: E.G. Goodaire

Publisher: North Holland

Published: 1996-11-07

Total Pages: 386

ISBN-13: 9780444824387

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For the past ten years, alternative loop rings have intrigued mathematicians from a wide cross-section of modern algebra. As a consequence, the theory of alternative loop rings has grown tremendously. One of the main developments is the complete characterization of loops which have an alternative but not associative, loop ring. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups. Another major topic of research is the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group rings is of immense interest. This is the first survey of the theory of alternative loop rings and related issues. Due to the strong interaction between loop rings and certain group rings, many results on group rings have been included, some of which are published for the first time. The authors often provide a new viewpoint and novel, elementary proofs in cases where results are already known. The authors assume only that the reader is familiar with basic ring-theoretic and group-theoretic concepts. They present a work which is very much self-contained. It is thus a valuable reference to the student as well as the research mathematician. An extensive bibliography of references which are either directly relevant to the text or which offer supplementary material of interest, are also included.

Non-Associative Algebra and Its Applications
Non-Associative Algebra and Its Applications

Author: Lev Sabinin

Publisher: CRC Press

Published: 2006-01-13

Total Pages: 553

ISBN-13: 1420003453

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With contributions derived from presentations at an international conference, Non-Associative Algebra and Its Applications explores a wide range of topics focusing on Lie algebras, nonassociative rings and algebras, quasigroups, loops, and related systems as well as applications of nonassociative algebra to geometry, physics, and natural sciences.

NonasSociative Algebra and Its Applications
NonasSociative Algebra and Its Applications

Author: R. Costa

Publisher: CRC Press

Published: 2019-05-20

Total Pages: 488

ISBN-13: 1482270463

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A collection of lectures presented at the Fourth International Conference on Nonassociative Algebra and its Applications, held in Sao Paulo, Brazil. Topics in algebra theory include alternative, Bernstein, Jordan, lie, and Malcev algebras and superalgebras. The volume presents applications to population genetics theory, physics, and more.

Smarandache Near-Rings
Smarandache Near-Rings

Author: W. B. Vasantha Kandasamy

Publisher: Infinite Study

Published: 2002

Total Pages: 201

ISBN-13: 1931233667

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Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B in A which is embedded with a stronger structure S. These types of structures occur in our everyday life, that's why we study them in this book. Thus, as a particular case: A Near-Ring is a non-empty set N together with two binary operations '+' and '.' such that (N, +) is a group (not necessarily abelian), (N, .) is a semigroup. For all a, b, c in N we have (a + b) . c = a . c + b . c. A Near-Field is a non-empty set P together with two binary operations '+' and '.' such that (P, +) is a group (not necessarily abelian), (P \ {0}, .) is a group. For all a, b, c I P we have (a + b) . c = a . c + b . c. A Smarandache Near-ring is a near-ring N which has a proper subset P in N, where P is a near-field (with respect to the same binary operations on N).

Algebra
Algebra

Author: I.B.S. Passi

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 247

ISBN-13: 3034899963

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The Indian National. Science Academy has planned to bring out monographs on special topics with the aim of providing acce~sible surveys/reviews of topics of current research in various fields. Prof. S.K. Malik, FNA, Editor of Publications INSA asked me in October 1997 to edit a volume on algebra in this series. I invited a number of algebraists, several of them working in group rings, and it is with great satisfaction and sincere thanks to the authors that I present here in Algebra: Some Recent Advances the sixteen contributions received in response to my invitations. I.B.S. Passi On Abelian Difference Sets K. r Arasu* and Surinder K. Sehgal 1. Introduction We review some existence and nonexistence results - new and old - on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics. Difference sets are very important in combinatorial design theory and in commu nication engineering while designing sequences with good correlation properties. Our extended bibliography covers a wide variety of papers written in the area of difference sets and related topics.

Smarandache Loops
Smarandache Loops

Author: W. B. Vasantha Kandasamy

Publisher: Infinite Study

Published: 2002

Total Pages: 129

ISBN-13: 1931233632

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Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S.By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A.These types of structures occur in our every day?s life, that?s why we study them in this book.As an example:A non-empty set L is said to form a loop, if on L is defined a binary operation called product, denoted by '?', such that:?For all a, b I L we have a ? b I L (closure property);?There exists an element e I L such that a ? e = e ? a = a for all a I L (e is the identity element of L);?For every ordered pair (a, b) I L ' L there exists a unique pair (x, y) in L such that ax = b and ya = b.Whence:A Smarandache Loop (or S-loop) is a loop L such that a proper subset M of L is a subgroup (with respect to the same induced operation).

Non Associative Algebraic Structures Using Finite Complex Numbers
Non Associative Algebraic Structures Using Finite Complex Numbers

Author: W.B. Vasantha Kandasamy, Florentin Smarandache

Publisher: Infinite Study

Published: 2012

Total Pages: 215

ISBN-13: 159973169X

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The authors have used the concept of finite complex modulo integers to construct non associative algebraic structures like groupoids, loops and quasi-loops.Using these structures we built non associative complex matrix groupoids and complex polynomial groupoids.The authors suggest over 300 problems and some are at the research level.