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[46] viXra:1008.0090 [pdf] submitted on 31 Aug 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, Moon Kumar Chetry
Comments: 242 pages
This book introduces several new classes of groupoid, like polynomial groupoids, matrix groupoids, interval groupoids, polynomial interval groupoids, matrix interval groupoids and their neutrosophic analogues.
[45] viXra:1008.0040 [pdf] submitted on 13 Aug 2010
Authors: Kyung Ho Kim, Young Bae Jun, Eun Hwan Roh, Habib Harizavi
Comments: 6 Pages.
We introduce the notion of a Smarandache hyper (∩, ∈)-ideal and Ω-reflexive in hyper K-algebra, and some related properties are given.
[44] viXra:1008.0039 [pdf] submitted on 13 Aug 2010
Authors: AKS Chandra Sekhar Rao.
Comments: 12 Pages.
Two Divisibility Tests for Smarandache semigroups are given . Further, the notion of divisibility of elements in a semigroup is applied to characterize the Smarandache semigroups. Examples are provided for justification.
[43] viXra:1008.0014 [pdf] submitted on 6 Aug 2010
Authors: Marian Dincă
Comments: 2 pages.
In the paper given new proof the inequality using convex function
[42] viXra:1008.0013 [pdf] submitted on 6 Aug 2010
Authors: Marian Dincă
Comments: 3 pages.
In the paper given generalisation inequalities using Lagrange identity.
[41] viXra:1007.0029 [pdf] submitted on 13 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 294 pages
In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra.
[40] viXra:1007.0027 [pdf] submitted on 13 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 410 pages
The systematic study of supermatrices and super linear algebra has been carried out in 2008. These new algebraic structures find their applications in fuzzy models, Leontief economic models and data-storage in computers.
[39] viXra:1007.0014 [pdf] submitted on 13 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K Ilanthenral
Comments: 469 pages
This book for the first time introduces the notion of special set linear algebra and special set fuzzy linear algebra. This is an extension of the book set linear algebra and set fuzzy linear algebra. These algebraic structures basically exploit only the set theoretic property, hence in applications one can include a finite number of elements without affecting the systems property. These new structures are not only the most generalized structures but they can perform multi task simultaneously; hence they would be of immense use to computer scientists.
[38] viXra:1007.0009 [pdf] submitted on 7 Jul 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 404 pages.
This book introduces the concept of neutrosophic bilinear algebras and their generalizations to n-linear algebras, n>2.
[37] viXra:1007.0004 [pdf] submitted on 5 Jul 2010
Authors: A.K.S.Chandra Sekhar Rao
Comments: 4 pages.
The notion of completely regular element of a semigroup is applied to characterize Smarandache Semigroups. Examples are provided for justification.
[36] viXra:1006.0013 [pdf] submitted on 11 Mar 2010
Authors: W.B.Vasantha, Moon K. Chetry
Comments: 9 pages
In this paper we establish the existance of S-idempotents in case of loop rings ZtLn(m) for a special class of loops Ln(m); over the ring of modulo integers Zt for a specific value of t. These loops satisfy the conditions gi2 = 1 for every gi ε Ln(m). We prove ZtLn(m) has an S-idempotent when t is a perfect number or when t is of the form 2ip or 3ip (where p is an odd prime) or in general when t = p1ip2 (p1 and p2 are distinct odd primes). It is important to note that we are able to prove only the existance of a single S-idempotent; however we leave it as an open problem wheather such loop rings have more than one S-idempotent. This paper has three sections. In section one, we give the basic notions about the loops Ln(m) and recall the definition of S-idempotents in rings. In section two, we establish the existance of S-idempotents in the loop ring ZtLn(m). In the final section, we suggest some interesting problems based on our study.
[35] viXra:1005.0110 [pdf] submitted on 11 Mar 2010
Authors: W.B.Vasantha Kandasamy
Comments: 5 pages
In this paper, we study the notion of Smarandache zero divisor in semigroups and rings. We illustrate them with examples and prove some interesting results about them.
[34] viXra:1005.0104 [pdf] submitted on 11 Mar 2010
Authors: Ralf W. Stephan
Comments: 7 pages
Using a personal computer and freely available software, the author factored some members of the Smarandache consecutive sequence and the reverse Smarandache sequence. Nearly complete factorizations are given up to Sm(80) and RSm(80). Both sequences were excessively searched for prime members, with only one prime found up to Sm(840) and RSm(750): RSm(82) = 828180...10987654321.
[33] viXra:1005.0103 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy
Comments: 203 pages
In this book for the first time we introduce the notion of Smarandache neutrosophic algebraic structures. Smarandache algebraic structures had been introduced in a series of 10 books. The study of Smarandache algebraic structures has caused a shift of paradigm in the study of algebraic structures.
[32] viXra:1005.0082 [pdf] submitted on 21 May 2010
Authors: A.K.S. Chandra Sekhar Rao
Comments: 6 pages
It is proved that there are infinitely many infinite Smarandache Groupoids.
[31] viXra:1005.0070 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K Ilanthenral
Comments:
345 pages.
In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book abounds with examples to enable the reader to understand these new concepts easily. Laborious theorems and proofs are avoided to make this book approachable for nonmathematicians. The concepts introduced in this book can be easily put to use by coding theorists, cryptologists, computer scientists, and socio-scientists. Another special feature of this book is the final chapter containing 304 problems. The authors have suggested so many problems to make the students and researchers obtain a better grasp of the subject. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector 6 spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems. When a researcher sets forth to solve them, she/he will certainly gain a deeper understanding of these new notions.
[30] viXra:1005.0069 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy
Comments:
4 pages.
In this paper we study the notion of Smarandache semirings and semifields and obtain some interesting results about them. We show that not every semiring is a Smarandache semiring. We similarly prove that not every semifield is a Smarandache semifield. We give several examples to make the concept lucid. Further, we propose an open problem about the existence of Smarandache semiring S of finite order.
[29] viXra:1005.0065 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy
Comments: 5 pages
In this paper we study the Smarandache pseudo-ideals of a Smarandache ring. We prove every ideal is a Smarandache pseudo-ideal in a Smarandache ring but every Smarandache pseudo-ideal in general is not an ideal. Further we show that every polynomial ring over a field and group rings FG of the group G over any field are Smarandache rings. We pose some interesting problems about them.
[28] viXra:1005.0046 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 231 pages
This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to n-linear algebras of type II. Another use of n-linear algebra (n-vector spaces) of type II is that when this structure is used in coding theory we can have different types of codes built over different finite fields whereas this is not possible in the case of n-vector spaces of type I. Finally in the case of n-vector spaces of type II we can obtain neigen values from distinct fields; hence, the n-characteristic polynomials formed in them are in distinct different fields. An attractive feature of this book is that the authors have suggested 120 problems for the reader to pursue in order to understand this new notion. This book has three chapters. In the first chapter the notion of n-vector spaces of type II are introduced. This chapter gives over 50 theorems. Chapter two introduces the notion of n-inner product vector spaces of type II, n-bilinear forms and n-linear functionals. The final chapter 6 suggests over a hundred problems. It is important that the reader should be well versed with not only linear algebra but also nlinear algebras of type I.
[27] viXra:1005.0045 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 120 pages
With the advent of computers one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure namely n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of nvector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces we can define several types of linear transformations. The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further it is suggested that the concept of nmatrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of 6 these algebraic structures are given in Chapter 3. Chapter four gives some problem to make the subject easily understandable. The authors deeply acknowledge the unflinching support of Dr.K.Kandasamy, Meena and Kama.
[26] viXra:1005.0021 [pdf] submitted on 11 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 154 pages
In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for which the roots can be indeterminates. Further, the notion of neutrosophic matrix rings is defined in this book. Already these neutrosophic matrixes have been applied and used in the neutrosophic models like neutrosophic cognitive maps (NCMs), neutrosophic relational maps (NRMs) and so on.
[25] viXra:1005.0007 [pdf] submitted on 10 Mar 2010
Authors: W. B. Vasantha Kandasamy
Comments: 5 pages
In this paper we study the Smarandache semi-near-ring and nearring, homomorphism, also the Anti-Smarandache semi-near-ring. We obtain some interesting results about them, give many examples, and pose some problems. We also define Smarandache semi-near-ring homomorphism.
[24] viXra:1005.0005 [pdf] submitted on 10 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 149 pages
Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit the factor of indeterminacy. It also plays a significant role, and utilizes the concept of neutrosophic graphs. Thus neutrosophic graphs and neutrosophic bipartite graphs plays the role of representing the neutrosophic models. Thus to construct the neutrosophic graphs one needs some of the neutrosophic algebraic structures viz. neutrosophic fields, neutrosophic vector spaces and neutrosophic matrices. So we for the first time introduce and study these concepts. As our analysis in this book is application of neutrosophic algebraic structure we found it deem fit to first introduce and study neutrosophic graphs and their applications to neutrosophic models.
[23] viXra:1005.0004 [pdf] submitted on 10 Mar 2010
Authors: W. B. Vasantha Kandasamy
Comments: 13 pages
In this paper we introduce the concept of Smarandache non-associative rings, which we shortly denote as SNA-rings as derived from the general definition of a Smarandache Structure (i.e., a set A embedded with a week structure W such that a proper subset B in A is embedded with a stronger structure S). Till date the concept of SNA-rings are not studied or introduced in the Smarandache algebraic literature. The only non-associative structures found in Smarandache algebraic notions so far are Smarandache groupoids and Smarandache loops introduced in 2001 and 2002. But they are algebraic structures with only a single binary operation defined on them that is nonassociative. But SNA-rings are non-associative structures on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from the right and left. Further to understand the concept of SNA-rings one should be well versed with the concept of group rings, semigroup rings, loop rings and groupoid rings. The notion of groupoid rings is new and has been introduced in this paper. This concept of groupoid rings can alone provide examples of SNA-rings without unit since all other rings happens to be either associative or nonassociative rings with unit. We define SNA subrings, SNA ideals, SNA Moufang rings, SNA Bol rings, SNA commutative rings, SNA non-commutative rings and SNA alternative rings. Examples are given of each of these structures and some open problems are suggested at the end.
[22] viXra:1005.0002 [pdf] submitted on 1 May 2010
Authors: Rajesh Singh, Mukesh Kumar, Florentin Smarandache
Comments: 14 pages
In this paper we have proposed an almost unbiased estimator using known value of some population parameter(s). Various existing estimators are shown particular members of the proposed estimator. Under simple random sampling without replacement (SRSWOR) scheme the expressions for bias and mean square error (MSE) are derived. The study is extended to the two phase sampling. Empirical study is carried out to demonstrate the superiority of the proposed estimator.
[21] viXra:1004.0084 [pdf] submitted on 9 Mar 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, K. Ilanthenral
Comments: 288 pages.
In this book we introduce mainly three new classes of linear algebras; neutrosophic group linear algebras, neutrosophic semigroup linear algebras and neutrosophic set linear algebras. The authors also define the fuzzy analogue of these three structures.
[20] viXra:1003.0231 [pdf] submitted on 7 Mar 2010
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.
[19] viXra:1003.0168 [pdf] submitted on 6 Mar 2010
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".
[18] viXra:1003.0115 [pdf] submitted on 6 Mar 2010
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.
[17] viXra:1003.0098 [pdf] submitted on 6 Mar 2010
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.
[16] viXra:1003.0097 [pdf] submitted on 6 Mar 2010
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.
[15] viXra:1003.0096 [pdf] submitted on 6 Mar 2010
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.
[14] viXra:1003.0079 [pdf] submitted on 7 Mar 2010
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.
[13] viXra:1003.0078 [pdf] submitted on 7 Mar 2010
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.
[12] viXra:1003.0077 [pdf] submitted on 7 Mar 2010
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.
[11] viXra:1003.0076 [pdf] submitted on 7 Mar 2010
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.
[10] viXra:1003.0075 [pdf] submitted on 7 Mar 2010
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.
[9] viXra:1003.0074 [pdf] submitted on 7 Mar 2010
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.
[8] viXra:1003.0073 [pdf] submitted on 7 Mar 2010
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.
[7] viXra:1003.0072 [pdf] submitted on 7 Mar 2010
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.
[6] viXra:1003.0071 [pdf] submitted on 7 Mar 2010
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.
[5] viXra:1003.0070 [pdf] submitted on 7 Mar 2010
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.
[4] viXra:1003.0066 [pdf] submitted on 5 Mar 2010
Authors: Ion Goian, Raisa Grigor, Vasile Marin, Florentin Smarandache
Comments: 119 pages, In Romanian language.
Theory and problems on algebraic structures.
[3] viXra:0911.0034 [pdf] submitted on 13 Nov 2009
Authors: Por Kujonai
Comments: 82 pages, In Spanish
A continuación, pretendo relacionar varios conceptos como modulo, opuestos (o signos), aritmética, el cuarto nivel de hypernumeros de Musean, politopos, especialmente el triangulo, matrices y determinantes, complejos, raices, ..., ya que de esta sopa de conceptos nace mi trabajo, aunque a un nivel mas profundo nace por darle un sentido matemático simple al concepto de opuesto, especialmente a una aritmética de 3 signos, y lo demás fue saliendo a medida de que avanzaba en esto, mientras iba adquiriendo sentido y fuerza.
[2] viXra:0910.0026 [pdf] submitted on 16 Oct 2009
Authors: Hideyuki Ohtsuka
Comments: 2 Pages
In this paper, we show a geometry approach to the expansion of (1 + x + x2 + ... + xn)3. This proof is a "Proof Without Words"
[1] viXra:0902.0006 [pdf] submitted on 14 Feb 2009
Authors: Jaiyeola Temitope Gbolahan
Comments: recovered from sciprint.org
A Study Of New Concepts In Smarandache Quasigroups And Loops
[1] viXra:1003.0066 [pdf] replaced on 6 Mar 2010
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.