Quasigroups and loops

introduction

Author: Hala O. Pflugfelder

Publisher: Heldermann

ISBN: N.A

Category: Quasigroups

Page: 147

View: 5057

Posted in Quasigroups

Quasigroups and loops

theory and applications

Author: Orin Chein,Hala O. Pflugfelder,Jonathan D. H. Smith

Publisher: N.A

ISBN: N.A

Category: Loops (Group theory)

Page: 568

View: 9336

Posted in Loops (Group theory)

Smooth Quasigroups and Loops

Author: L. Sabinin

Publisher: Springer Science & Business Media

ISBN: 9401144915

Category: Mathematics

Page: 249

View: 2380

During the last twenty-five years quite remarkable relations between nonas sociative algebra and differential geometry have been discovered in our work. Such exotic structures of algebra as quasigroups and loops were obtained from purely geometric structures such as affinely connected spaces. The notion ofodule was introduced as a fundamental algebraic invariant of differential geometry. For any space with an affine connection loopuscular, odular and geoodular structures (partial smooth algebras of a special kind) were introduced and studied. As it happened, the natural geoodular structure of an affinely connected space al lows us to reconstruct this space in a unique way. Moreover, any smooth ab stractly given geoodular structure generates in a unique manner an affinely con nected space with the natural geoodular structure isomorphic to the initial one. The above said means that any affinely connected (in particular, Riemannian) space can be treated as a purely algebraic structure equipped with smoothness. Numerous habitual geometric properties may be expressed in the language of geoodular structures by means of algebraic identities, etc.. Our treatment has led us to the purely algebraic concept of affinely connected (in particular, Riemannian) spaces; for example, one can consider a discrete, or, even, finite space with affine connection (in the form ofgeoodular structure) which can be used in the old problem of discrete space-time in relativity, essential for the quantum space-time theory.
Posted in Mathematics

A Study of New Concepts in Smarandache Quasigroups and Loops

Author: Jaiyeola Temitope Gbolahan

Publisher: Infinite Study

ISBN: 1599730839

Category: Loops (Group theory)

Page: 127

View: 5943

This monograph is a compilation of results on some new Smarandache concepts in Smarandache;groupoids, quasigroups and loops, and it pin points the inter-relationships and connections between andamong the various Smarandache concepts and notions that have been developed. This monograph isstructured into six chapters. The first chapter is an introduction to the theory quasigroups and loops withmuch attention paid to those quasigroup and loop concepts whose Smarandache versions are to bestudied in the other chapters. In chapter two, the holomorphic structures of Smarandache loops ofBol-Moufang type and Smarandache loops of non-Bol-Moufang type are studied. In the third chapter,the notion of parastrophy is introduced into Smarandache quasigroups and studied. Chapter four studiesthe universality of some Smarandache loops of Bol-Moufang type. In chapter five, the notion ofSmarandache isotopism is introduced and studied in Smarandache quasigroups and loops. In chaptersix, by introducing Smarandache special mappings in Smarandache groupoids, the SmarandacheBryant-Schneider group of a Smarandache loop is developed.
Posted in Loops (Group theory)

An Introduction to Quasigroups and Their Representations

Author: Jonathan D. H. Smith

Publisher: CRC Press

ISBN: 9781420010633

Category: Mathematics

Page: 352

View: 5326

Collecting results scattered throughout the literature into one source, An Introduction to Quasigroups and Their Representations shows how representation theories for groups are capable of extending to general quasigroups and illustrates the added depth and richness that result from this extension. To fully understand representation theory, the first three chapters provide a foundation in the theory of quasigroups and loops, covering special classes, the combinatorial multiplication group, universal stabilizers, and quasigroup analogues of abelian groups. Subsequent chapters deal with the three main branches of representation theory-permutation representations of quasigroups, combinatorial character theory, and quasigroup module theory. Each chapter includes exercises and examples to demonstrate how the theories discussed relate to practical applications. The book concludes with appendices that summarize some essential topics from category theory, universal algebra, and coalgebras. Long overshadowed by general group theory, quasigroups have become increasingly important in combinatorics, cryptography, algebra, and physics. Covering key research problems, An Introduction to Quasigroups and Their Representations proves that you can apply group representation theories to quasigroups as well.
Posted in Mathematics

An Introduction to Quasigroups and Their Representations

Author: Jonathan D. H. Smith

Publisher: CRC Press

ISBN: 9781420010633

Category: Mathematics

Page: 352

View: 1214

Collecting results scattered throughout the literature into one source, An Introduction to Quasigroups and Their Representations shows how representation theories for groups are capable of extending to general quasigroups and illustrates the added depth and richness that result from this extension. To fully understand representation theory, the first three chapters provide a foundation in the theory of quasigroups and loops, covering special classes, the combinatorial multiplication group, universal stabilizers, and quasigroup analogues of abelian groups. Subsequent chapters deal with the three main branches of representation theory-permutation representations of quasigroups, combinatorial character theory, and quasigroup module theory. Each chapter includes exercises and examples to demonstrate how the theories discussed relate to practical applications. The book concludes with appendices that summarize some essential topics from category theory, universal algebra, and coalgebras. Long overshadowed by general group theory, quasigroups have become increasingly important in combinatorics, cryptography, algebra, and physics. Covering key research problems, An Introduction to Quasigroups and Their Representations proves that you can apply group representation theories to quasigroups as well.
Posted in Mathematics

Non-Associative Algebra and Its Applications

Author: Lev Sabinin,Larissa Sbitneva,Ivan Shestakov

Publisher: CRC Press

ISBN: 9781420003451

Category: Mathematics

Page: 552

View: 6454

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. This book covers material such as Jordan superalgebras, nonassociative deformations, nonassociative generalization of Hopf algebras, the structure of free algebras, derivations of Lie algebras, and the identities of Albert algebra. It also includes applications of smooth quasigroups and loops to differential geometry and relativity.
Posted in Mathematics

Loops in Group Theory and Lie Theory

Author: Péter Nagy,Karl Strambach

Publisher: Walter de Gruyter

ISBN: 3110900580

Category: Mathematics

Page: 370

View: 1095

In this book the theory of binary systems is considered as a part of group theory and, in particular, within the framework of Lie groups. The novelty is the consequent treatment of topological and differentiable loops as topological and differentiable sections in Lie groups. The interplay of methods and tools from group theory, differential geometry and topology, symmetric spaces, topological geometry, and the theory of foliations is what gives a special flavour to the results presented in this book. It is the first monograph devoted to the study of global loops. So far books on differentiable loops deal with local loops only. This theory can only be used partially for the theory of global loops since non-associative local structures have, in general, no global forms. The text is addressed to researchers in non-associative algebra and foundations of geometry. It should prove enlightening to a broad range of readers, including mathematicians working in group theory, the theory of Lie groups, in differential and topological geometry, and in algebraic groups. The authors have produced a text that is suitable not only for a graduate course, but also for selfstudy in the subjectby interested graduate students. Moreover, the material presented can be used for lectures and seminars in algebra, topological algebra and geometry.
Posted in Mathematics

Proceedings of the Sixth International Conference on Number Theory and Smarandache Notions

Author: Wenpeng Zhang

Publisher: Infinite Study

ISBN: 1599731274

Category: Number theory

Page: 140

View: 642

This Book is devoted to the proceedings of the Sixth International Conferenceon Number Theory and Smarandache Notions held in Tianshui during April 24-25,2010. The organizers were Prof. Zhang Wenpeng and Prof. Wangsheng He from Tianshui Normal University. The conference was supported by Tianshui Normal University and there were more than 100 participants.
Posted in Number theory

Alternative Loop Rings

Author: E.G. Goodaire,E. Jespers,C. Polcino Milies

Publisher: Elsevier

ISBN: 9780080527062

Category: Mathematics

Page: 386

View: 5932

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.
Posted in Mathematics

Soft Neutrosophic Algebraic Structures and Their Generalization, Vol. 2

Author: Mumtaz Ali,Florentin Smarandache,Muhammad Shabir

Publisher: Infinite Study

ISBN: 1599733064

Category:

Page: N.A

View: 562

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.
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Scientia Magna, Vol. 2, No. 1, 2006

international book series

Author: Zhang Wenpeng,W. B. Vasantha Kandasamy

Publisher: Infinite Study

ISBN: 159973009X

Category:

Page: N.A

View: 9257

Collection of papers from various scientists dealing with Smarandache Notions in science. Papers on holomorphic study of the Smarandache concept in loops, some arithmetical properties of primitive numbers of power p, Smarandache quasigroups, the mean value of the Smarandache simple divisor function, and other similar topics. Contributors: Z. Xu, Y. Shao, X. Zhao, X. Pan, T. Kim, C. Adiga, J. Han, Q. Yang, and many others.
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Elements of Quasigroup Theory with Applications in Coding and Cryptology

Author: Victor Shcherbacov,J. D. Phillips

Publisher: Chapman & Hall/CRC

ISBN: 9781498721554

Category:

Page: 576

View: 9101

This book provides an introduction to quasigroup theory along with new structural results on some of the quasigroup classes. Many results are presented with some of them from mathematicians of the former USSR. These included results have not been published before in the western mathematical literature. In addition, many of the achievements obtained with regard to applications of quasigroups in coding theory and cryptology are described.
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Generalized Lie Theory in Mathematics, Physics and Beyond

Author: Sergei D. Silvestrov,Eugen Paal,Viktor Abramov,Alexander Stolin

Publisher: Springer Science & Business Media

ISBN: 3540853324

Category: Mathematics

Page: 306

View: 4150

This book explores the cutting edge of the fundamental role of generalizations of Lie theory and related non-commutative and non-associative structures in mathematics and physics.
Posted in Mathematics

Scientia Magna, Vol. 4, No. 1, 2008

international book series

Author: Zhang Wenpeng

Publisher: Infinite Study

ISBN: 1599730618

Category:

Page: N.A

View: 9427

Proceedings of the Fourth International Conference on Number Theory and Smarandache Problems.
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Latin Squares

New Developments in the Theory and Applications

Author: József Dénes,A. Donald Keedwell

Publisher: Elsevier

ISBN: 9780080867861

Category: Mathematics

Page: 452

View: 8319

In 1974 the editors of the present volume published a well-received book entitled ``Latin Squares and their Applications''. It included a list of 73 unsolved problems of which about 20 have been completely solved in the intervening period and about 10 more have been partially solved. The present work comprises six contributed chapters and also six further chapters written by the editors themselves. As well as discussing the advances which have been made in the subject matter of most of the chapters of the earlier book, this new book contains one chapter which deals with a subject (r-orthogonal latin squares) which did not exist when the earlier book was written. The success of the former book is shown by the two or three hundred published papers which deal with questions raised by it.
Posted in Mathematics