[14] **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.

**Category:** Algebra

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

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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*

**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