Non-Associative Linear Algebras

By Smarandache, Florentin

Book Id: WPLBN0002828488
Format Type: PDF eBook:
File Size: 3.66 MB
Reproduction Date: 8/2/2013

Title:	Non-Associative Linear Algebras
Author:	Smarandache, Florentin
Volume:
Language:	English
Subject:	Non Fiction, Education, Algebra
Collections:	Authors Community, Mathematics
Historic Publication Date:	2013
Publisher:	World Public Library

Member Page:	Florentin Smarandache
Citation APA MLA Chicago Smarandache, B. F., & Vasantha Kandasamy, W. B. (2013). Non-Associative Linear Algebras. Retrieved from http://www.self.gutenberg.org/

Description
This book has six chapters. First chapter is introductory in nature. The new concept of non-associative semi-linear algebras is introduced in chapter two. This structure is built using groupoids over semi-fields. Third chapter introduces the notion of non-associative linear algebras. These algebraic structures are built using the new class of loops. All these non-associative linear algebras are defined over the prime characteristic field Zp, p a prime. However if we take polynomial non associative, linear algebras over Zp, p a prime; they are of infinite dimension over Zp. We in chapter four introduce the notion of groupoid vector spaces of finite and infinite order and their generalizations. Only when this study becomes popular and familiar among researchers several applications will be found. The final chapter suggests around 215 problems some of which are at research level.

Summary
In this book authors for the first time introduce the notion of non associative vector spaces and non associative linear algebras over a field. We construct non associative space using loops and groupoids over fields.

Excerpt
Ln(m) is a loop of order n + 1 that is Ln(m) is always of even order greater than or equal to 6. For more about these loops please refer [37]. All properties discussed in case of groupoids can be done for the case of loops. However the resultant product may not be a loop in general. For the concept of semifield refer [41]. We will be using these loops and groupoids to build linear algebra and semilinear algebras which are non associative.

Table of Contents
Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two NON ASSOCIATIVE SEMILINEAR ALGEBRAS 13 Chapter Three NON ASSOCIATIVE LINEAR ALGEBRAS 83 Chapter Four GROUPOID VECTOR SPACES 111 Chapter Five APPLICATION OF NON ASSOCIATIVE VECTOR SPACES / LINEAR ALGEBRAS 161 Chapter Six SUGGESTED PROBLEMS 163 FURTHER READING 225 INDEX 229 ABOUT THE AUTHORS 231