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Any replacements are listed further down
[70] viXra:1202.0019 [pdf] submitted on 2012-02-07 20:30:57
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 342 Pages.
In this book the authors introduce a new type of product on matrices called the natural product Xn. This is an extension of product carried out in the case or row matrices of the same order.
Further, when two column matrices of same order can be added, nothing prevents one from multiplying them.
This sort of multiplication which is natural is defined as natural product Xn on matrices.
We suggest 100 problems and some of them are at the research level.
Category: Algebra
[69] viXra:1201.0098 [pdf] submitted on 2012-01-25 14:05:51
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 213 Pages.
The authors have used the concept of finite complex modulo integers to construct non associative algebraic structures like groupoids, loops and quasi-loops.
Using these structures we built non associative complex matrix groupoids and complex polynomial groupoids.
The authors suggest over 300 problems and some are at the research level.
Category: Algebra
[68] viXra:1201.0066 [pdf] submitted on 2012-01-16 10:44:27
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, D. Datta, H. S. Kushwaha, P. A. Jadhav
Comments: 181 Pages.
In this book the authors introduce and study the properties of natural class of intervals built using
(-∞, ∞) and (∞, -∞). The operations on these matrices with entries from natural class of intervals behave like usual reals. So working with these interval matrices takes the same time as usual matrices. Hence, when applying them to fuzzy finite element methods or finite element methods the determination of solution is simple and time saving.
Category: Algebra
[67] viXra:1111.0078 [pdf] submitted on 22 Nov 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 222 pages
In this book for the first time the authors introduce the notion of real
neutrosophic complex numbers.
Category: Algebra
[66] viXra:1111.0077 [pdf] submitted on 22 Nov 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 299 pages
In this book authors for the first time introduce the notion of
supermatrices of refined labels. Authors prove super row matrix of
refined labels form a group under addition. However super row matrix
of refined labels do not form a group under product; it only forms a
semigroup under multiplication. In this book super column matrix of
refined labels and m x n matrix of refined labels are introduced and
studied.
Category: Algebra
[65] viXra:1111.0076 [pdf] submitted on 22 Nov 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 197 pages
In this book the authors for the first time introduce the notion of
neutrosophic intervals and study the algebraic structures using them.
Concepts like groups and fields using neutrosophic intervals are not
possible. Pure neutrosophic intervals and mixed neutrosophic intervals
are introduced and by the very structure of the interval one can
understand the category to which it belongs.
Category: Algebra
[64] viXra:1110.0038 [pdf] submitted on 12 Oct 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 216 pages
The study of DSm linear algebra of refined labels have been done by
Florentin Smarandache, Jean Dezert, and Xinde Li.
In this book the authors introduce the notion of DSm vector spaces
of refined labels. The reader is requested to refer the paper as the basic
concepts are taken from that paper
Category: Algebra
[63] viXra:1107.0041 [pdf] submitted on 21 Jul 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 289 pages
In this book, super interval matrices using the special type of intervals of the form [0, a] are introduced.
Several algebraic structures like semigroups, groups, semirings, rings, semivector spaces and vector spaces are introduced.
Special fuzzy linear algebras are introduced using the concept of super fuzzy interval matrices.
Category: Algebra
[62] viXra:1106.0060 [pdf] submitted on 27 Jun 2011
Authors: Giuseppe Iurato
Comments: 9 pages.
From physical motivations and from geometrical interpretations of
the Einstein equations, we give a justification of the non-triviality and
non-degeneracy of Einstein bilinear form introduced in [1].
Category: Algebra
[61] viXra:1106.0050 [pdf] submitted on 23 Jun 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 210 pages.
Authors in this book construct interval bistructures using only
interval groups, interval loops, interval semigroups and interval
groupoids.
Category: Algebra
[60] viXra:1106.0019 [pdf] submitted on 11 Jun 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 172 pages
In this book the authors introduce a new class of intervals called the natural class of intervals.
Using these intervals, algebraic structures are defined.
Over 200 problems are given, of which some of them are at the research level.
Category: Algebra
[59] viXra:1105.0029 [pdf] submitted on 20 May 2011
Authors: Giuseppe Iurato
Comments: 11 pages
Motivated by Quantum Mechanics considerations, we expose some
cross product constructions. Furthermore, critical remarks are made
on some basic formal aspects of the Hopf algebra structure.
Category: Algebra
[58] viXra:1105.0009 [pdf] submitted on 6 May 2011
Authors: Elemér E Rosinger
Comments: 18 pages.
It is briefly shown that, due to the growth conditions in their
definition, the Colombeau algebras cannot handle arbitrary Lie groups,
and in particular, cannot allow the formulation, let alone, solution of
Hilbert's Fifth Problem.
Category: Algebra
[57] viXra:1105.0007 [pdf] submitted on 4 May 2011
Authors: Elemér E Rosinger
Comments: 9 pages.
It is briefly shown that, due to the growth conditions in their definition,
the Colombeau algebras cannot handle arbitrary analytic nonlinear
PDEs, and in particular, cannot allow the formulation, let alone,
give the proof of the global Cauchy-Kovalevskaia theorem.
Category: Algebra
[56] viXra:1103.0065 [pdf] submitted on 15 Mar 2011
Authors: Giuseppe Iurato
Comments: 27 pages
In this paper, from algebraic extensions of certain notions
of Functional Analysis, let us deduce some first remarks about
weak nullstellensatz.
Category: Algebra
[55] viXra:1103.0031 [pdf] submitted on 10 Mar 2011
Authors: Marco Ripà
Comments: 3 pages
In this paper we provide an inverse proof of the relation between a particular class of double sums
and tetrahedral numbers. Thus, we present a compact formula to reduce the number of calculations
necessary to solve such a kind of problems. The initial identity is confirmed "a posteriori" using
the formula mentioned above.
Category: Algebra
[54] viXra:1102.0045 [pdf] submitted on 23 Feb 2011
Authors: Constantin Scheau
Comments:
8 pages.
The multi-space structure has been defined by Fl Smarandache as a union spaces with
some additional conditions hold. The mathematician L. Mao wrote a series of works in which he
introduces the concepts of multi-group, multi-ring, multivector - space etc. In [1] (Smarandache
Multi-Space Theory (I)), at open problems section, he suggests the introduction of a theory of
matrices and applications defined on the multi-linear spaces. This paper will give an example of
a multi-ring structure, introduces the notion of multi-matrix and defines the multi-matrix addition
and multiplication.
Category: Algebra
[53] viXra:1101.0072 [pdf] submitted on 22 Jan 2011
Authors: Jaedoek Kim, Youngmi Kim, Eun Hwan Roh
Comments: 7 pages
We introduce the notion of Smarandache GT-algebras,
and the notion of Smarandache GT-Filters of the Smarandache
GT-algebra related to the Tarski algebra, and related some properties
are investigated.
Category: Algebra
[52] viXra:1101.0063 [pdf] submitted on 21 Jan 2011
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 167 pages
In this book we introduce the notion of interval semigroups
using intervals of the form [0, a], a is real. Several types of
interval semigroups like fuzzy interval semigroups, interval
symmetric semigroups, special symmetric interval semigroups,
interval matrix semigroups and interval polynomial semigroups
are defined and discussed. This book has eight chapters.
The main feature of this book is that we suggest 241
problems in the eighth chapter. In this book the authors have
defined 29 new concepts and illustrates them with 231
examples. Certainly this will find several applications.
The authors deeply acknowledge Dr. Kandasamy for the
proof reading and Meena and Kama for the formatting and
designing of the book.
Category: Algebra
[51] viXra:1012.0028 [pdf] submitted on 12 Dec 2010
Authors: Muhammad Aslam, Saleem Abdullah
Comments: 14 pages
We consider the intuitionistic fuzzi?cation of the concept of several
Γ-ideals in Γ-LA-semigroup S, and investigate some related properties of
such Γ-ideals. We also prove in this paper the set of all intuitionistic fuzzy
left(right) Γ-ideal of S is become LA-semigroup. We prove In Γ-LA band
intuitionistic fuzzy right and left Γ-ideals are coincide..
Category: Algebra
[50] viXra:1011.0038 [pdf] submitted on 17 Nov 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache
Comments: 249 pages
This Interval arithmetic or interval mathematics developed in
1950's and 1960's by mathematicians as an approach to putting
bounds on rounding errors and measurement error in
mathematical computations. However no proper interval
algebraic structures have been defined or studies. In this book
we for the first time introduce several types of interval linear
algebras and study them.
Category: Algebra
[49] viXra:1011.0037 [pdf] submitted on 14 Nov 2010
Authors: Nathaniel S. K. Hellerstein
Comments: 17 pages
This paper redefines the addition of rational numbers, in a way that allows division
by zero. This requires defining a "compensator" on the integers, plus extending
least-common-multiple (LCM) to zero and negative numbers. "Compensated addition"
defines ordinary addition on all ratios, including the 'infinities' n/0, and also
'zeroids' 0/n. The infinities and the zeroids form two 'double ringlets'. The lattice
rationals modulo the zeroids yields the infinities plus the 'wheel numbers'. Due to
the presence of the 'alternator' @ = 0/-1, double-distribution does not apply, but
triple-distribution still does.
Category: Algebra
[48] viXra:1011.0019 [pdf] submitted on 11 Nov 2010
Authors: Nathaniel S. K. Hellerstein
Comments: 33 pages
In this paper I discuss "reduction", a.k.a. "reciprocal addition"; addition conjugated by reciprocal.
I discuss reduction's definition, its laws, its graphs, its geometry, its algebra, its calculus, and its
practical applications. This paper contains a problem set with answer key.
Category: Algebra
[47] viXra:1010.0021 [pdf] submitted on 10 Oct 2010
Authors: A. K. S. Chandra Sekhar Rao
Comments: 12 pages
In this paper we show that a commutative semisimple ring is always a
Smarandache ring. We will also give a necessary and sufficient condition for group
algebra to be a Smarandache ring. Examples are provided for justification.
Category: Algebra
[46] viXra:1008.0090 [pdf] submitted on 31 Aug 2010
Authors: W. B. Vasantha Kandasamy, Florentin Smarandache, Moon K. 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.
Category: Algebra
[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.
Category: Algebra
[44] viXra:1008.0039 [pdf] submitted on 13 Aug 2010
Authors: A.K.S. 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.
Category: Algebra
[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
Category: Algebra
[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.
Category: Algebra
[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.
Category: 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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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".
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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.
Category: Algebra
[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"
Category: Algebra
[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
Category: Algebra
[4] viXra:1105.0029 [pdf] replaced on 24 May 2011
Authors: Giuseppe Iurato
Comments: 15 pages
Motivated by Quantum Mechanics considerations, we expose some
cross product constructions on a groupoid structure. Furthermore,
critical remarks are made on some basic formal aspects of the Hopf
algebra structure.
Category: Algebra
[3] viXra:1103.0065 [pdf] replaced on 26 May 2011
Authors: Giuseppe Iurato
Comments: 27 pages
In this paper, from algebraic extensions of certain notions
of Functional Analysis, let us deduce some first remarks about
weak nullstellensatz.
Category: Algebra
[2] 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
[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.
Category: Algebra