Algebra

1005 Submissions

[14] viXra:1005.0110 [pdf] submitted on 11 Mar 2010

Smarandache Zero Divisors

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.
Category: Algebra

[13] viXra:1005.0104 [pdf] replaced on 25 Aug 2011

Factors and Primes in Two Smarandache Sequences

Authors: Ralf W. Stephan
Comments: 10 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.
Category: Algebra

[12] viXra:1005.0103 [pdf] submitted on 11 Mar 2010

Smarandache Neutrosophic Algebraic Structures

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.
Category: Algebra

[11] viXra:1005.0082 [pdf] submitted on 21 May 2010

Infinite Smarandache Groupoids

Authors: A.K.S. Chandra Sekhar Rao
Comments: 6 pages

It is proved that there are infinitely many infinite Smarandache Groupoids.
Category: Algebra

[10] viXra:1005.0070 [pdf] submitted on 11 Mar 2010

Set Linear Algebra and Set Fuzzy Linear Algebra

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.
Category: Algebra

[9] viXra:1005.0069 [pdf] submitted on 11 Mar 2010

Smarandache Semirings and Semifields

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.
Category: Algebra

[8] viXra:1005.0065 [pdf] submitted on 11 Mar 2010

Smarandache Pseudo-Ideals

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.
Category: Algebra

[7] viXra:1005.0046 [pdf] submitted on 11 Mar 2010

N-Linear Algebra of Type II

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.
Category: Algebra

[6] viXra:1005.0045 [pdf] submitted on 11 Mar 2010

N-Linear Algebra of Type I and Its Applications

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.
Category: Algebra

[5] viXra:1005.0021 [pdf] submitted on 11 Mar 2010

Neutrosophic Rings

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.
Category: Algebra

[4] viXra:1005.0007 [pdf] submitted on 10 Mar 2010

Smarandache Near-Rings and Their Generalizations

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.
Category: Algebra

[3] viXra:1005.0005 [pdf] submitted on 10 Mar 2010

Basic Neutrosophic Algebraic Structures and Their Application to Fuzzy and Neutrosophic Models

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.
Category: Algebra

[2] viXra:1005.0004 [pdf] submitted on 10 Mar 2010

Smarandache Non-Associative (SNA-) rings

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.
Category: Algebra

[1] viXra:1005.0002 [pdf] submitted on 1 May 2010

Almost Unbiased Estimator for Estimating Population Mean Using Known Value of Some Population Parameter(s)

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.
Category: Algebra