Algebra

1003 Submissions

[17] viXra:1003.0231 [pdf] submitted on 7 Mar 2010

Smarandache Special Definite Algebraic Structures

Authors: W. B. Vasantha Kandasamy
Comments: 141 pages

In this book we introduce the notion of Smarandache special definite algebraic structures. We can also call them equivalently as Smarandache definite special algebraic structures. These new structures are defined as those strong algebraic structures which have in them a proper subset which is a weak algebraic structure. For instance, the existence of a semigroup in a group or a semifield in a field or a semiring in a ring. It is interesting to note that these concepts cannot be defined when the algebraic structure has finite cardinality i.e., when the algebraic structure has finite number of elements in it.
Category: Algebra

[16] viXra:1003.0168 [pdf] submitted on 6 Mar 2010

K-Nomial Coefficients

Authors: Florentin Smarandache
Comments: 4 pages

In this article we will widen the concepts of "binomial coefficients" and "trinomial coefficients" to the concept of "k-nomial coefficients", and one obtains some general properties of these. As an application, we will generalize the" triangle of Pascal".
Category: Algebra

[15] viXra:1003.0115 [pdf] submitted on 6 Mar 2010

Special Algebraic Structures

Authors: Florentin Smarandache
Comments: 4 pages

New notions are introduced in algebra in order to better study the congruences in number theory. For example, the <special semigroups> make an important such contribution.
Category: Algebra

[14] viXra:1003.0098 [pdf] submitted on 6 Mar 2010

Applications of Bimatrices to Some Fuzzy and Neutrosophic Models

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 273 pages

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. These concepts are also extended neutrosophically in this book.
Category: Algebra

[13] viXra:1003.0097 [pdf] submitted on 6 Mar 2010

Introduction to Bimatrices

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 181 pages

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other.
Category: Algebra

[12] viXra:1003.0096 [pdf] submitted on 6 Mar 2010

Introduction to Linear Bialgebra

Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 238 pages

The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications.
Category: Algebra

[11] viXra:1003.0079 [pdf] submitted on 7 Mar 2010

Linear Algebra and Smarandache Linear Algebra

Authors: W. B. Vasantha Kandasamy
Comments: 175 pages

While I began researching for this book on linear algebra, I was a little startled. Though, it is an accepted phenomenon, that mathematicians are rarely the ones to react surprised, this serious search left me that way for a variety of reasons. First, several of the linear algebra books that my institute library stocked (and it is a really good library) were old and crumbly and dated as far back as 1913 with the most 'new' books only being the ones published in the 1960s.
Category: Algebra

[10] viXra:1003.0078 [pdf] submitted on 7 Mar 2010

Smarandache Fuzzy Algebra

Authors: W. B. Vasantha Kandasamy
Comments: 455 pages

In 1965, Lofti A. Zadeh introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. It provoked, at first (and as expected), a strong negative reaction from some influential scientists and mathematicians - many of whom turned openly hostile. However, despite the controversy, the subject also attracted the attention of other mathematicians and in the following years, the field grew enormously, finding applications in areas as diverse as washing machines to handwriting recognition. In its trajectory of stupendous growth, it has also come to include the theory of fuzzy algebra and for the past five decades, several researchers have been working on concepts like fuzzy semigroup, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings and so on.
Category: Algebra

[9] viXra:1003.0077 [pdf] submitted on 7 Mar 2010

Bialgebraic Structures and Smarandache Bialgebraic Structures

Authors: W. B. Vasantha Kandasamy
Comments: 272 pages

The study of bialgebraic structures started very recently. Till date there are no books solely dealing with bistructures. The study of bigroups was carried out in 1994-1996. Further research on bigroups and fuzzy bigroups was published in 1998. In the year 1999, bivector spaces was introduced. In 2001, concept of free De Morgan bisemigroups and bisemilattices was studied. It is said by Zoltan Esik that these bialgebraic structures like bigroupoids, bisemigroups, binear rings help in the construction of finite machines or finite automaton and semi automaton. The notion of non-associative bialgebraic structures was first introduced in the year 2002. The concept of bialgebraic structures which we define and study are slightly different from the bistructures using category theory of Girard's classical linear logic. We do not approach the bialgebraic structures using category theory or linear logic.
Category: Algebra

[8] viXra:1003.0076 [pdf] submitted on 7 Mar 2010

Smarandache Non-Associative Rings

Authors: W. B. Vasantha Kandasamy
Comments: 201 pages

An associative ring is just realized or built using reals or complex; finite or infinite by defining two binary operations on it. But on the contrary when we want to define or study or even introduce a non-associative ring we need two separate algebraic structures say a commutative ring with 1 (or a field) together with a loop or a groupoid or a vector space or a linear algebra. The two non-associative well-known algebras viz. Lie algebras and Jordan algebras are mainly built using a vector space over a field satisfying special identities called the Jacobi identity and Jordan identity respectively. Study of these algebras started as early as 1940s. Hence the study of non-associative algebras or even non-associative rings boils down to the study of properties of vector spaces or linear algebras over fields.
Category: Algebra

[7] viXra:1003.0075 [pdf] submitted on 7 Mar 2010

Smarandache Near-Rings

Authors: W. B. Vasantha Kandasamy
Comments: 201 pages

Near-rings are one of the generalized structures of rings. The study and research on near-rings is very systematic and continuous. Near-ring newsletters containing complete and updated bibliography on the subject are published periodically by a team of mathematicians (Editors: Yuen Fong, Alan Oswald, Gunter Pilz and K. C. Smith) with financial assistance from the National Cheng Kung University, Taiwan. These newsletters give an overall picture of the research carried out and the recent advancements and new concepts in the field. Conferences devoted solely to near-rings are held once every two years. There are about half a dozen books on near-rings apart from the conference proceedings. Above all there is a online searchable database and bibliography on near-rings. As a result the author feels it is very essential to have a book on Smarandache near-rings where the Smarandache analogues of the near-ring concepts are developed. The reader is expected to have a good background both in algebra and in near-rings; for, several results are to be proved by the reader as an exercise.
Category: Algebra

[6] viXra:1003.0074 [pdf] submitted on 7 Mar 2010

Smarandache Rings

Authors: W. B. Vasantha Kandasamy
Comments: 222 pages

Over the past 25 years, I have been immersed in research in Algebra and more particularly in ring theory. I embarked on writing this book on Smarandache rings (Srings) specially to motivate both ring theorists and Smarandache algebraists to develop and study several important and innovative properties about S-rings.
Category: Algebra

[5] viXra:1003.0073 [pdf] submitted on 7 Mar 2010

Smarandache Loops

Authors: W. B. Vasantha Kandasamy
Comments: 129 pages

The theory of loops (groups without associativity), though researched by several mathematicians has not found a sound expression, for books, be it research level or otherwise, solely dealing with the properties of loops are absent. This is in marked contrast with group theory where books are abundantly available for all levels: as graduate texts and as advanced research books.
Category: Algebra

[4] viXra:1003.0072 [pdf] submitted on 7 Mar 2010

Smarandache Semirings, Semifields, and Semivector Spaces

Authors: W. B. Vasantha Kandasamy
Comments: 122 pages

Smarandache notions, which can be undoubtedly characterized as interesting mathematics, has the capacity of being utilized to analyse, study and introduce, naturally, the concepts of several structures by means of extension or identification as a substructure. Several researchers around the world working on Smarandache notions have systematically carried out this study. This is the first book on the Smarandache algebraic structures that have two binary operations.
Category: Algebra

[3] viXra:1003.0071 [pdf] submitted on 7 Mar 2010

Groupoids and Smarandache Groupoids

Authors: W. B. Vasantha Kandasamy
Comments: 115 pages

The study of Smarandache Algebraic Structure was initiated in the year 1998 by Raul Padilla following a paper written by Florentin Smarandache called "Special Algebraic Structures". In his research, Padilla treated the Smarandache algebraic structures mainly with associative binary operation. Since then the subject has been pursued by a growing number of researchers and now it would be better if one gets a coherent account of the basic and main results in these algebraic structures. This book aims to give a systematic development of the basic non-associative algebraic structures viz. Smarandache groupoids. Smarandache groupoids exhibits simultaneously the properties of a semigroup and a groupoid. Such a combined study of an associative and a non associative structure has not been so far carried out. Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache such types of studies would have been completely absent in the mathematical world.
Category: Algebra

[2] viXra:1003.0070 [pdf] submitted on 7 Mar 2010

Smarandache Semigroups

Authors: W. B. Vasantha Kandasamy
Comments: 95 pages

The main motivation and desire for writing this book, is the direct appreciation and attraction towards the Smarandache notions in general and Smarandache algebraic structures in particular. The Smarandache semigroups exhibit properties of both a group and a semigroup simultaneously. This book is a piece of work on Smarandache semigroups and assumes the reader to have a good background on group theory; we give some recollection about groups and some of its properties just for quick reference.
Category: Algebra

[1] viXra:1003.0066 [pdf] replaced on 6 Mar 2010

Theory and Problems on Algebraic Structures.

Authors: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache
Comments: 119 pages, v1 in Romanian language, v2 in Russian language.

Theory and problems on algebraic structures.
Category: Algebra