**Previous months:**

2009 - 0902(1) - 0910(1) - 0911(1)

2010 - 1003(17) - 1004(1) - 1005(14) - 1006(1) - 1007(5) - 1008(4) - 1011(3) - 1012(1)

2011 - 1101(2) - 1102(1) - 1103(1) - 1105(2) - 1106(2) - 1107(1) - 1110(1) - 1111(3)

2012 - 1201(2) - 1202(1) - 1203(1) - 1204(1) - 1205(1) - 1206(1) - 1207(2) - 1208(2) - 1209(1) - 1210(4) - 1211(3) - 1212(9)

2013 - 1301(2) - 1302(1) - 1303(2) - 1304(1) - 1305(9) - 1306(17) - 1307(2) - 1309(5) - 1311(2) - 1312(2)

2014 - 1403(1) - 1404(2) - 1406(2) - 1407(2) - 1408(3) - 1409(2) - 1410(1) - 1411(7) - 1412(2)

2015 - 1501(1) - 1503(2)

Any replacements are listed further down

[160] **viXra:1503.0146 [pdf]**
*submitted on 2015-03-18 11:30:24*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 231 Pages.

Here finite real quaternion rings, and finite complex modulo integer quaternion rings, plus neutrosophic finite quaternion rings, and complex neutrosophic quaternion rings for the first time are introduced and analyzed. All these rings behave in a very unique way.

**Category:** Algebra

[159] **viXra:1503.0072 [pdf]**
*submitted on 2015-03-10 19:26:40*

**Authors:** A. A. Salama, Florentin Smarandache

**Comments:** 182 Pages.

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), α-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set and Neutrosophic Continuous Function, etc.
They compute the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree.
The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space.
At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region.

**Category:** Algebra

[158] **viXra:1501.0240 [pdf]**
*submitted on 2015-01-27 20:54:40*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.

**Comments:** 169 Pages.

The distance between row matrices was defined by Hamming. However the distance between column matrices or for that matter and any two matrices of same order are not defined systematically. Here in this book this concept is defined and it can help in the analysis of storage of data, deviation in eigen values and in difference in opinions of experts using some fuzzy models. Some open problems are suggested.

**Category:** Algebra

[157] **viXra:1412.0253 [pdf]**
*submitted on 2014-12-27 09:42:49*

**Authors:** J. S. Markovitch

**Comments:** 5 Pages.

A nonstandard cubic equation is shown to have an unusually economical solution, where this solution incorporates an angle that serves as the equation's discriminant.

**Category:** Algebra

[156] **viXra:1412.0252 [pdf]**
*submitted on 2014-12-27 10:24:17*

**Authors:** J. S. Markovitch

**Comments:** 3 Pages.

A substitution map applied to the simplest algebraic identities is shown to yield second- and third-order equations that share an
interesting property at the minimum 137.036.

**Category:** Algebra

[155] **viXra:1411.0159 [pdf]**
*submitted on 2014-11-15 09:45:29*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 252 Pages.

In this book we introduce several algebraic structures on the special fuzzy interval [0, 1). This study is different from that of the algebraic structures using the interval [0, n) n ≠ 1, as these structures on [0, 1) has no idempotents or zero divisors under ×. Further [0, 1) under product × is only a semigroup. However by defining min(or max) operation in [0, 1); [0, 1) is made into a semigroup.

**Category:** Algebra

[154] **viXra:1411.0158 [pdf]**
*submitted on 2014-11-15 09:47:23*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 218 Pages.

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n.

**Category:** Algebra

[153] **viXra:1411.0154 [pdf]**
*submitted on 2014-11-15 10:03:06*

**Authors:** Marius Coman

**Comments:** 132 Pages.

Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me.

**Category:** Algebra

[152] **viXra:1411.0153 [pdf]**
*submitted on 2014-11-15 10:04:38*

**Authors:** Marius Coman

**Comments:** 84 Pages.

The present collected papers aims to show new applications of Smarandache function in the study of some well known classes of numbers, like prime numbers, Poulet numbers, Carmichael numbers, Sophie Germain primes etc.

**Category:** Algebra

[151] **viXra:1411.0149 [pdf]**
*submitted on 2014-11-15 10:11:33*

**Authors:** Marius Coman

**Comments:** 27 Pages.

It is always difficult to talk about arithmetic, because those who do not know what is about, nor do they understand in few sentences, no matter how inspired these might be, and those who know what is about, do no need to be told what is about.

**Category:** Algebra

[150] **viXra:1411.0148 [pdf]**
*submitted on 2014-11-15 10:12:32*

**Authors:** Marius Coman

**Comments:** 43 Pages.

This collection of papers brings together several articles regarding primes, submitted by the author to the preprint scientific database Vixra.

**Category:** Algebra

[149] **viXra:1411.0147 [pdf]**
*submitted on 2014-11-15 10:13:43*

**Authors:** Marius Coman

**Comments:** 81 Pages.

This collection of articles seeks to expand the knowledge on some well known classes of primes, like for instance Sophie Germain primes, but also to define new classes of primes, like for instance “ACPOW chains of primes”, or classes of integers directly related to primes, like for instance “chameleonic numbers”.

**Category:** Algebra

[148] **viXra:1410.0169 [pdf]**
*submitted on 2014-10-26 14:52:01*

**Authors:** Mumtaz Ali, Florentin Smarandache, Muhammad Shabir

**Comments:** 333 Pages.

In this book, we define several new neutrosophic algebraic structures and their related properties. The main focus of this book is to study the important class of neutrosophic rings such as neutrosophic LA-semigroup ring, neutrosophic loop ring, neutrosophic groupoid ring and so on. We also construct their generalization in each case to study these neutrosophic algebraic structures in a broader sense. The indeterminacy element “I“ gives rise to a bigger algebraic structure than the classical algebraic structures. It mainly classifies the algebraic structures in three categories such as: neutrosophic algebraic structures, strong neutrosophic algebraic structures, and classical algebraic structures respectively. This reveals the fact that a classic algebraic structure is a part of the neutrosophic algebraic structures. This opens a new way for the researcher to think in a broader way to visualize these vast neutrosophic algebraic structures.

**Category:** Algebra

[147] **viXra:1409.0143 [pdf]**
*submitted on 2014-09-17 15:44:59*

**Authors:** Jan S. Xoah

**Comments:** 9 Pages.

The goal of the present paper is to classify subrings. We show that Q = vp . The
groundbreaking work of F. Raman on geometric sets was a major advance. In future work, we plan to
address questions of reducibility as well as regularity.

**Category:** Algebra

[146] **viXra:1409.0054 [pdf]**
*submitted on 2014-09-08 08:51:46*

**Authors:** Mumtaz Ali, Florentin Smarandache, Muhammad Shabir

**Comments:** 288 Pages.

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.

**Category:** Algebra

[145] **viXra:1408.0182 [pdf]**
*submitted on 2014-08-27 06:03:34*

**Authors:** Andrzej Okninski

**Comments:** 4 Pages.

In the present work we make a small step towards inding explicit formula for polynomial roots. xplicit formulae for simple real roots, based on the Frobenius companion matrix, are derived. After some modifications the method can be used to compute multiple or complex roots.

**Category:** Algebra

[144] **viXra:1408.0124 [pdf]**
*submitted on 2014-08-18 12:19:02*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 206 Pages.

Here for the first time we introduce the semi open square using modulo integers. This square under addition modulo ‘n’ is a group, and however under product ‘x’ this semi open square is only a semigroup as under ‘x’ the square has infinite number of zero divisors.
We define the new type of rings called pseudo rings.
Several open problems are suggested.

**Category:** Algebra

[143] **viXra:1408.0118 [pdf]**
*submitted on 2014-08-18 12:16:59*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 235 Pages.

In this book the authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers were introduced by the authors in 2011. Now, using this finite complex modulo integer intervals several algebraic structures are built. Open problems are suggested.

**Category:** Algebra

[142] **viXra:1407.0189 [pdf]**
*submitted on 2014-07-25 00:44:10*

**Authors:** Ouannas Moussa

**Comments:** 03 Pages.

Basing on my study, "Distribution of primes"viXra:1407.0167 [pdf] submitted on 2014-07-21 I've shown that we can determine the distribution of prime numbers while remaining within the set of natural numbers; However, I present in this paper the numbers which are susceptible to be twins

**Category:** Algebra

[141] **viXra:1407.0167 [pdf]**
*submitted on 2014-07-21 16:17:56*

**Authors:** Ouannas Moussa

**Comments:** 08 Pages.

In this paper I present the distribution of prime
numbers which was treated in many
researches by studying the function of Riemann; because it has a remarkable property; its non
trivial zeros are prime numbers; but in this work I will show that we can find the distribution
of prime numbers on remaining in natural numbers only.

**Category:** Algebra

[140] **viXra:1406.0183 [pdf]**
*submitted on 2014-06-30 08:46:28*

**Authors:** Richard J. Mathar

**Comments:** 182 Pages. 131 Figures and 129 Tables.

The cycle graphs for all finite abstract groups up to order 37 are plotted, except the trivial ring graphs of the cyclic groups themselves.

**Category:** Algebra

[139] **viXra:1406.0098 [pdf]**
*submitted on 2014-06-15 18:28:07*

**Authors:** Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

**Comments:** 264 Pages.

Study of soft sets was first proposed by Molodtsov in 1999 to deal with uncertainty in a non-parametric manner. The researchers did not pay attention to soft set theory at that time but now the soft set theory has been developed in many areas of mathematics.
Algebraic structures using soft set theory are very rapidly developed. In this book we developed soft neutrosophic algebraic structures by using soft sets and neutrosophic algebraic structures. In this book we study soft neutrosophic groups, soft neutrosophic semigroups, soft neutrosophic loops, soft neutrosophic LA-semigroups, and their generalizations respectively.

**Category:** Algebra

[138] **viXra:1404.0120 [pdf]**
*submitted on 2014-04-14 11:09:26*

**Authors:** W. B. vasantha Kandasamy, Florentin Smarandache

**Comments:** 221 Pages.

In this book authors build algebraic structures on fuzzy unit semi-open square UF = {(a,b), with a, b in [0, 1)} and on neutrosophic unit semi-open square UN = {a+bI, with a, b in [0, 1)}.
As distributive laws are not true, we are not in a position to develop several properties of rings, semirigs and linear algebras. Seven open conjectures are proposed.

**Category:** Algebra

[137] **viXra:1404.0119 [pdf]**
*submitted on 2014-04-14 11:10:37*

**Authors:** W. B. vasantha Kandasamy, Florentin Smarandache

**Comments:** 178 Pages.

Study of algebraic structures built using [0, n) looks to be one of interesting and innovative research. Here we define two types of groupoids using [0, n), both of them are of infinite order. It is an open conjecture to find whether this new class of groupoids satisfy any of the special identities like Moufang identity or Bol identity and so on.

**Category:** Algebra

[136] **viXra:1403.0958 [pdf]**
*submitted on 2014-03-28 09:56:58*

**Authors:** Nikolay Dementev

**Comments:** 7 Pages.

The report suggests an approach to extend a concept of determinant to the systems of any order.

**Category:** Algebra

[135] **viXra:1312.0213 [pdf]**
*submitted on 2013-12-26 19:15:38*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 216 Pages.

The algebraic structures built using [0, n) interval are new and innovative. They happen to have different properties. The interval [0, n) can be realized as the real algebraic closure of the modulo ring Zn. The algebraic behavior of [0, n) is different from the ring Zn.

**Category:** Algebra

[134] **viXra:1312.0212 [pdf]**
*submitted on 2013-12-26 19:16:40*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 211 Pages.

The concept of non associative topological space is very new and interesting. In this book we have built non associative topological spaces using subsets of non associative algebraic structures like loops, groupoids, non associative rings and non associative semirings. We also find conditions on these non associative subset topological spaces to satisfy special identities like Bol, Moufang, right alternative, etc. The new notion will find several applications.

**Category:** Algebra

[133] **viXra:1311.0040 [pdf]**
*submitted on 2013-11-06 00:52:42*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 241 Pages.

Special type of subset topological spaces introduced by the authors pave way for the topological spaces, which basically inherit the algebraic structures from which the subsets are taken. This study is new and happens to be a mixture of algebra and topology.

**Category:** Algebra

[132] **viXra:1311.0039 [pdf]**
*submitted on 2013-11-06 00:54:17*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 254 Pages.

The authors use the subset semigroups over the semifields to build the semilinear algebras of both finite order and infinite order. The concept of subset linear independence and subset linear dependence which leads to the dimension and basis of subset semilinear algebras is analyzed in this book.

**Category:** Algebra

[131] **viXra:1309.0120 [pdf]**
*submitted on 2013-09-17 10:05:33*

**Authors:** Arsham Borumand Saeid

**Comments:** 63 Pages.

There are three types of Smarandache Algebraic Structures:
1.A Smarandache Strong Structure on a set S means a structure on S that has a proper subset P with a stronger structure.
2.A Smarandache Weak Structure on a set S means a structure on S that has a proper subset P with a weaker structure.
3.A Smarandache Strong-Weak Structure on a set S means a structure on S that has two proper subsets: P with a stronger structure, and Q with a weaker structure.
By proper subset of a set S, one understands a subset P of S, different from the empty set, from the original set S, and from the idempotent elements if any.
Having two structures {u} and {v} defined by the same operations, one says that structure {u} is stronger than structure {v}, i.e. {u} > {v}, if the operations of {u} satisfy more axioms than the operations of {v}.
Each one of the first two structure types is then generalized from a 2-level (the sets P ⊂ S and their corresponding strong structure {w1}>{w0}, respectively their weak structure {w1}<{w0}) to an n-level (the sets Pn-1 ⊂ Pn-2 ⊂ … ⊂ P2 ⊂ P1 ⊂ S and their corresponding strong structure {wn-1} > {wn-2} > … > {w2} > {w1} > {w0}, or respectively their weak structure {wn-1} < {wn-2} < … < {w2} < {w1} < {w0}). Similarly for the third structure type, whose generalization is a combination of the previous two structures at the n-level.
A Smarandache Weak BE-Algebra X is a BE-algebra in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a CI-algebra.
And a Smarandache Strong CI-Algebra X is a CI-algebra X in which there exists a proper subset Q such that 1 belongs to Q, |Q| ≥ 2, and Q is a BE-algebra.
The book elaborates a recollection of the BE/CI-algebras, then introduces these last two particular structures and studies their properties.

**Category:** Algebra

[130] **viXra:1309.0107 [pdf]**
*submitted on 2013-09-17 09:01:32*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 260 Pages.

The authors have constructed subset semirings using rings of both finite and infinite order. Thus, using finite rings we construct infinite number of finite semirings, both commutative as well as non-commutative, which is the main advantage of using this algebraic structure. For finite distribute lattices alone contribute for finite semirings.

**Category:** Algebra

[129] **viXra:1309.0027 [pdf]**
*submitted on 2013-09-05 21:38:31*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 246 Pages.

When the subsets of a loop are taken they also form only a subset groupoid and not a subset loop.
Thus the concept of subset interval loop is not there, and they only form a subset interval groupoid.
Subset matrix interval groupoid S using the loops Ln(m) has no S-Cauchy elements.

**Category:** Algebra

[128] **viXra:1309.0026 [pdf]**
*submitted on 2013-09-05 21:41:06*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 207 Pages.

In this book the authors introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this topic are still rare. Some open problems are suggested in this book.

**Category:** Algebra

[127] **viXra:1309.0019 [pdf]**
*submitted on 2013-09-04 21:27:50*

**Authors:** Eckhard Hitzer, Roxana Bujack, Gerik Scheuermann

**Comments:** 5 Pages. Proc. of the Third SICE Symposium on Computational Intelligence, August 30, 2013, Osaka University, Osaka, pp. 91-95.

Exactly 125 years ago G. Peano introduced the modern concept of vectors
in his 1888 book "Geometric Calculus - According to the Ausdehnungslehre (Theory of
Extension) of H. Grassmann". Unknown to Peano, the young British mathematician
W. K. Clifford (1846-1879) in his 1878 work "Applications of Grassmann's Extensive
Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern
concept of geometric algebras, including the measurement of lengths (areas and volumes)
and angles (between arbitrary subspaces). This leads currently to new ideal methods for
vector field computations in geometric algebra, of which several recent exemplary results
will be introduced.

**Category:** Algebra

[126] **viXra:1307.0017 [pdf]**
*submitted on 2013-07-03 09:19:40*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 149 Pages.

The authors in this book construct a large class of finite subset groupoids as well as a large class of infinite subset groupoids. Here the conditions under which these subset groupoids satisfy special identities like Bol Identity, Moufang Identity, right alternative identity and so on are found.
In fact, it is an open problem to find subset groupoids to satisfy special identities, even if the groupoid over which they are defined do not satisfy the special identities.

**Category:** Algebra

[125] **viXra:1307.0016 [pdf]**
*submitted on 2013-07-03 09:21:52*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 267 Pages.

In this book the authors introduce the new notions of subset polynomial semirings and subset matrix semirings. Solving subset polynomial equations is an interesting exercise.
Open problems about the solution set of subset polynomials are proposed.

**Category:** Algebra

[124] **viXra:1306.0234 [pdf]**
*submitted on 2013-06-29 19:34:04*

**Authors:** John Frederick Sweeney

**Comments:** 6 Pages.

In math, especially within the Lie Algebra groups, there exist a small group of "exceptional isomorphisms" or "accidental isomorphisms." Following the lead of Swiss psychologist Carl Jung, the author does not accept the existence of "exceptional" or "accidental;" instead, these are merely phenomena which heretofore have not been explained satisfactorily by mathematical theory. However, articulation of the Qi Men Dun Jia Cosmic Board Model in a recent and forthcoming paper has helped to explain several of the exceptional isomorphisms.

**Category:** Algebra

[123] **viXra:1306.0178 [pdf]**
*submitted on 2013-06-21 04:43:19*

**Authors:** Eckhard Hitzer

**Comments:** 8 Pages. 3 figures, 1 table. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Axioms for Geometric Algebra R_{p,q} - Definitions using quadratic form, basic multiplication rules. Grade r subspaces, geometric algebra R_2, complex numbers, reflections and rotations, 2-dim. point groups, geometric algebra R_3 and quaternions.

**Category:** Algebra

[122] **viXra:1306.0177 [pdf]**
*submitted on 2013-06-21 04:46:56*

**Authors:** Eckhard Hitzer

**Comments:** 3 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition, Adjoint and inverse linear mappings, References.

**Category:** Algebra

[121] **viXra:1306.0176 [pdf]**
*submitted on 2013-06-21 04:49:53*

**Authors:** Eckhard Hitzer

**Comments:** 2 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

We describe Gram-Schmidt orthogonalization in W.K. Clifford's geometric algebra.

**Category:** Algebra

[120] **viXra:1306.0175 [pdf]**
*submitted on 2013-06-21 04:54:51*

**Authors:** Eckhard Hitzer

**Comments:** 11 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The aim of this work is to show how the geometric product of multivectors is defined in
general, extending the basic geometric product of vectors given by Clifford. An alternative
definition of Clifford geometric algebra, that guarantees existence as quotient algebra of
the tensor algebra was given by Chevalley in 1954.[2] We further treat the scalar product, the outer product, the cross product in three dimensions, linear dependence and independence, as well as right- and left contractions.
!"#$%

**Category:** Algebra

[119] **viXra:1306.0174 [pdf]**
*submitted on 2013-06-21 04:59:20*

**Authors:** Eckhard Hitzer

**Comments:** 6 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

Definition of geometric algebra with quadratic form - examples of quadratic forms and associated geometric algebras - geometric algebras with degenerate quadratic forms - new interpretation of the geometric algebra of the Minkowski plane - generalizing to the geometric mother algebra with p=q=n.

**Category:** Algebra

[118] **viXra:1306.0173 [pdf]**
*submitted on 2013-06-21 05:03:08*

**Authors:** Eckhard Hitzer

**Comments:** 4 Pages. Support Website for the Linear Algebra Lectures of the University of the Air Japan 2004-2008.

The previous Japanese emperor is said to have asked this question. Today many students and scientists still ask it, but the traditional canon of mathematics at school and university needs to be widened for the answer.

**Category:** Algebra

[117] **viXra:1306.0150 [pdf]**
*submitted on 2013-06-19 03:03:40*

**Authors:** Eckhard Hitzer, Christian Perwass

**Comments:** 2 Pages. 1 figure, 1 table. Bulletin of the Society for Science on Form, 21(1), pp. 55,56 (2006).

This paper focuses on the symmetries of crystal space lattices. All two dimensional
(2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described
by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric
multiplication of these vectors completely generates all symmetries, including reflections,
rotations, inversions, rotary-reflections and rotary-inversions. We then extend this treatment to 3D
space groups by including translations, glide reflections and screw rotations. We focus on the
monoclinic case as an example. A software demonstration shows the spacegroup visualizer.
**Keywords:** Crystal lattice, space group, geometric algebra, OpenGL, interactive software.

**Category:** Algebra

[116] **viXra:1306.0125 [pdf]**
*submitted on 2013-06-17 02:20:49*

**Authors:** Eckhard Hitzer, Luca Redaelli

**Comments:** 22 Pages. 27 figures. Advances in Applied Clifford Algebras, 13(2), pp. 157-181 (2003). DOI: 10.1007/s00006-003-0013-x .

Conventional illustrations of the rich elementary relations and physical applications of geometric algebra are helpful,
but restricted in communicating full generality and time dependence. The main restrictions are one special perspective
in each graph and the static character of such illustrations. Several attempts have been made to overcome such
restrictions. But up till now very little animated and fully interactive, free, instant access, online material is available.
This report presents therefore a set of over 90 newly developed (freely online accessible [1]) JAVA
applets. These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle
relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically different
approaches to the representation of conics; and even more ways to describe ellipses. Next Clifford's famous circle chain
theorem is illustrated. Finally geometric applications important for crystallography and structural mechanics give a
glimpse of the vast potential for applied mathematics. The interactive geometry software Cinderella [2]
was used for creating these applets. The interactive features of many of the applets invite the user to freely explore
by a few mouse clicks as many different special cases and perspectives as he likes. This is of great help in "visualizing"
geometry encoded by geometric algebra.

**Category:** Algebra

[115] **viXra:1306.0113 [pdf]**
*submitted on 2013-06-17 04:25:16*

**Authors:** Eckhard Hitzer

**Comments:** 21 Pages. in T. M. Karade (ed.), Proc. of the Nat. Symp. on Math. Sc., 1-5 March, 2001, Nagpur, India, Einst. Foundation Int. 1, pp. 1-26 (2001).

This paper first reviews how anti-symmetric matrices in two dimensions yield
imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to
rotations by means of the Cayley transformation. Then the necessary tools from real
geometric algebra are introduced and a real geometric interpretation is given to the
eigenvalues and eigenvectors. The latter are seen to be two component eigenspinors
which can be further reduced to underlying vector duplets. The eigenvalues are
interpreted as rotors, which rotate the underlying vector duplets. The second part of
this paper extends and generalizes the treatment to three dimensions. The final part
shows how all entities and relations can be obtained in a constructive way, purely
assuming the geometric algebras of 2-space and 3-space.

**Category:** Algebra

[114] **viXra:1306.0112 [pdf]**
*submitted on 2013-06-17 04:30:00*

**Authors:** Eckhard Hitzer

**Comments:** 16 Pages. Mem. Fac. Eng. Fukui Univ. 49(2), pp. 283-298 (2001).

This paper briefly reviews the conventional method of obtaining the canonical form of an antisymmetric (skew-symmetric, alternating) matrix. Conventionally a vector space over the complex field has to be introduced. After a short introduction to the universal mathematical "language" Geometric Calculus, its fundamentals, i.e. its "grammar" Geometric Algebra (Clifford Algebra) is explained. This lays the groundwork for its real geometric and coordinate free application in order to obtain the canonical form of an antisymmetric matrix in terms of a bivector, which is isomorphic to the conventional canonical form. Then concrete applications to two, three and four dimensional antisymmetric square matrices follow. Because of the physical importance of the Minkowski metric, the canonical form of an antisymmetric matrix with respect to the Minkowski metric is derived as well. A final application to electromagnetic fields concludes the work.
Keywords: Geometric Calculus, Geometric Algebra, Clifford Algebra, antisymmetric (alternating, skewsymmetric) matrix, Real Geometry

**Category:** Algebra

[113] **viXra:1306.0038 [pdf]**
*submitted on 2013-06-06 10:12:21*

**Authors:** H. J. Siamwalla, A.S.Muktibodh

**Comments:** 7 Pages.

In this paper we prove some results towards classifying Smarandache groupoids
which are in Z*(n) and not in Z(n) when n is even and n is odd.

**Category:** Algebra

[112] **viXra:1306.0036 [pdf]**
*submitted on 2013-06-06 10:15:44*

**Authors:** Arun S. Muktibodh

**Comments:** 7 Pages.

In [4] we have introduced Smarandache quasigroups which are Smarandache
non-associative structures. A quasigroup is a groupoid whose composition table is a Latin
square. There are squares in the Latin squares which seem to be of importance to study the
structure of Latin Squares. We consider the particular type of squares properly contained in
the Latin squares which themselves contain a Latin square. Such Latin squares are termed
as Smarandache Mukti-Squares or SMS. Extension of some SMS to Latin squares is also
considered.

**Category:** Algebra

[111] **viXra:1306.0035 [pdf]**
*submitted on 2013-06-06 10:17:09*

**Authors:** Arun S. Muktibodh

**Comments:** 6 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smarandache non-
associative structures. W.B.Kandasamy [2] has studied groupoid ring and loop ring. We have de¯ned
Smarandache quasigroup rings which are again non-associative structures having two binary operations.
Substructures of quasigroup rings are also studied.

**Category:** Algebra

[110] **viXra:1306.0034 [pdf]**
*submitted on 2013-06-06 10:18:29*

**Authors:** Arun S. Muktibodh

**Comments:** 7 Pages.

In this paper, we have introduced Smarandache quasigroups which are Smaran-
dache non-associative structures. W.B.Kandasamy [2] has studied Smarandache groupoids
and Smarandache semigroups etc. Substructure of Smarandache quasigroups are also studied.

**Category:** Algebra

[109] **viXra:1306.0033 [pdf]**
*submitted on 2013-06-06 10:20:01*

**Authors:** Arun S. Muktibodh

**Comments:** 3 Pages.

G. Pilz [1] has dened near-rings and semi-near-rings. In this paper we introduce
the concepts of quasi-near ring and semiquasi-near ring. We have also dened Smarandache
semiquasi-near-ring. Some examples are constructed. We have posed some open problems.

**Category:** Algebra

[108] **viXra:1306.0029 [pdf]**
*submitted on 2013-06-06 10:29:40*

**Authors:** A. S. Muktibodh, V. M. Wagh

**Comments:** 3 Pages.

In this paper we have constructed two chains of semifields. All the semifields in the chains are
Smarandache semifields. Every member of the chain is an extension semifield of Ordered equilateral
Integral triangles with Zero triangle such that it is a semivector space over R^I_e\delta.

**Category:** Algebra

[107] **viXra:1306.0019 [pdf]**
*submitted on 2013-06-05 00:29:16*

**Authors:** S. Kalimuthu

**Comments:** 4 Pages. NA

The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850."Matrix" is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced. The orgins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of the use of matrix methods to solve simultaneous equations. Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps! The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory. In this work, by applying the fundamental concepts of matrices, the author attempts to study the fifth Euclidean postulate problem and Godel’s incompleteness theorems.

**Category:** Algebra

[106] **viXra:1305.0183 [pdf]**
*submitted on 2013-05-30 00:59:35*

**Authors:** S.Kalimuthu

**Comments:** 6 Pages. This is a new attempt in set theory 7 geometry

The sum of the interior angles of a number triangles were transformed into linear algebraic equations. The analysis of these equations without assuming the fifth Euclidean postulate established the following theorem: There exists a triangle whose interior angle sum is equal to two right angles

**Category:** Algebra

[105] **viXra:1305.0182 [pdf]**
*submitted on 2013-05-30 01:05:03*

**Authors:** S.Kalimuthu

**Comments:** 4 Pages. This is an entirely new approach in algebra & geometry

By the application of abstract algebra the famous unsolved classical problems trisection of an angle,squaring the circle , duplicating the cube and drawing a regular septagon were shown not possible to solve.. On the other hand,in this work, the application of classical algebra proposed a proposition for the origin of a new field of geometry.

**Category:** Algebra

[104] **viXra:1305.0181 [pdf]**
*submitted on 2013-05-30 01:11:48*

**Authors:** S.Kalimuthu

**Comments:** 3 Pages. Quo Vadis

By applying the basic operations of number theory and set theory to geometry, a new challenging finding has been obtained

**Category:** Algebra

[103] **viXra:1305.0152 [pdf]**
*submitted on 2013-05-24 09:24:45*

**Authors:** S. Kalimuthu

**Comments:** 8 Pages. This is an extra-ordinary math. article.

After establishing the fundamental physics prizes, Yuri Milner said: “Unlike the Nobel in physics, the Fundamental Physics Prize can be awarded to scientists whose ideas have not yet been verified by experiments, which often occurs decades later. Sometimes a radical new idea “really deserves recognition right away because it expands our understanding of at least what is possible.”. Keeping this mind the author formulated two spherical geometrical theorems which may applied for the studies and probes of fundamental particles, quantum gravity, gravitational waves ,dark matter and dark energy.

**Category:** Algebra

[102] **viXra:1305.0151 [pdf]**
*submitted on 2013-05-24 09:27:04*

**Authors:** S. Kalimuthu

**Comments:** 4 Pages. This is an extra-ordinary math. article.

Miles Mathis has shown the following equivalent propositions to Gödel’s incompleteness theorems: [ http://milesmathis.com/godel.html ]
Theorem 1: In any logical system one can construct statements that are neither true nor false (mathematical variations of the liar’s paradox).
Theorem 2: Therefore no consistent system can be used to prove its own consistency. No proof can be proof of itself. In this work , we attempt to prove the first theorem mention above.
Key words: Gödel’s incompleteness theorems ; Miles Mathis’s proposition

**Category:** Algebra

[101] **viXra:1305.0150 [pdf]**
*submitted on 2013-05-24 09:29:12*

**Authors:** S. Kalimuthu

**Comments:** 2 Pages. This is an extra-ordinary math. article.

The classical Euclidean geometry is widely used in all fields of science, engineering and other art fields. The hyperbolic and elliptic non Euclidean geometries are applied in theoretical physics and cosmology. In this short paper, we have come across an interesting non geometrical phenomenon.

**Category:** Algebra

[100] **viXra:1305.0058 [pdf]**
*submitted on 2013-05-08 14:47:25*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 197 Pages.

Study of algebraic structures using subsets was started by George Boole. After the invention of Boolean algebra, subsets were not used in any algebraic structures.
In this book we develop algebraic structures using subsets of a set, or of a group, or semigroup, or a ring and we get algebraic structures.
Using ring or semiring we get only subset semirings.
BY the method we get an infinite number of non-commutative semirings of finite order. We build subset semivector spaces, and describe and develop interesting properties over them.

**Category:** Algebra

[99] **viXra:1305.0057 [pdf]**
*submitted on 2013-05-08 14:50:08*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 167 Pages.

Use of fuzzy neutrosophic models in the study of social problem or technical problem has become a demand for the day due to the involvement of indeterminacy. For always one is not in the position to say true or false, or assign a fuzzy value to any problem. At times, on is not in the position to commit to any of the three statements. In that situation, neutrosophic lmodels play a vital role.
In this book we introduce several fuzzy neutrosophic models to study social problems.

**Category:** Algebra

[98] **viXra:1305.0045 [pdf]**
*submitted on 2013-05-07 21:51:26*

**Authors:** H. J. Siamwalla, A.S.Muktibodh

**Comments:** 15 Pages.

In this paper, we defined the Smarandache seminormal subgroupoids. We have proved
some results for finding the Smarandache seminormal subgroupoids in Z(n) when n is even
and n is odd.

**Category:** Algebra

[97] **viXra:1304.0021 [pdf]**
*submitted on 2013-04-04 12:05:34*

**Authors:** Vasantha Kandasamy, Florentin Smarandache

**Comments:** 166 Pages.

This book brings out how sets in algebraic structures can be used to construct the most generalized algebraic structures, like set linear algebra / vector space, set ideals in groups and rings and semigroups, and topological set vector spaces. This sort of study is innovative and will find applications in data handling.

**Category:** Algebra

[96] **viXra:1303.0016 [pdf]**
*submitted on 2013-03-03 12:42:55*

**Authors:** Nasir ermain

**Comments:** 2 Pages.

Nasir Germain newest problem

**Category:** Algebra

[95] **viXra:1303.0014 [pdf]**
*submitted on 2013-03-03 11:09:03*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, Indra Venkatbabu

**Comments:** 227 Pages.

In this book authors have given the innovative ways in which they have used matrices, starting from ATD, RTD & CETD matrices which can be applied to any collected data depending on time.
A new type of matrices, called fuzzy linguistic matrices, which play a major role in the construction of fuzzy linguistic models, is defined.
Then, the super matrices are used in the construction of super linear algebra and super fuzzy models.
Further, interval matrices using natural class of intervals is introduced. The ingenious way of defining DSm matrices of refined labels is an interesting feature. Finally the construction of n-matrices and its use in the building of n-codes is described.

**Category:** Algebra

[94] **viXra:1302.0088 [pdf]**
*submitted on 2013-02-13 11:27:51*

**Authors:** Joerg Siegler

**Comments:** 39 Pages.

This paper about indefinite summation describes a natural approach to discrete calculus. Two natural operators for discrete difference and summation are defined. They preserve symmetry and show a duality in contrast to the classical operators. Several summation and differentiation algorithms will be presented.

**Category:** Algebra

[93] **viXra:1301.0175 [pdf]**
*submitted on 2013-01-28 04:43:38*

**Authors:** Martin Erik Horn

**Comments:** 25 Pages. 6 Figures.

Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric Algebra it is helpful to restate Geometric Algebra in the
mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. ** Special emphasis will be given to the definition of some wedge products which fifit better to this algebra of (3 x 3) matrices than the usual Geometric Algebra wedge product. ** And as S3 permutation symmetry is flavour symmetry a unifified flavour picture of Geometric Algebra will emerge.

**Category:** Algebra

[92] **viXra:1301.0049 [pdf]**
*submitted on 2013-01-10 09:26:05*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 152 Pages.

In this book, the authors introduce the notion of quasi set topological vector subspaces. The
advantage of such study is that given a vector subspace we can have only one topological
space associated with the collection of all subspaces. However, we can have several
quasi set topological vector subspaces of a given vector space. Further, we have defined
topological spaces for set vector spaces, semigroup vector spaces and group vector spaces.

**Category:** Algebra

[91] **viXra:1212.0157 [pdf]**
*submitted on 2012-12-27 16:40:31*

**Authors:** Colin Naturman

**Comments:** 20 Pages.

Certain topological countability properties are generalized to interior algebras and basic reaults concerning these properies are investigated. The preservation of these properties under the formation of principal quotients and under a new construction called a join of interior algebras, is investigated.

**Category:** Algebra

[90] **viXra:1212.0151 [pdf]**
*submitted on 2012-12-27 03:05:04*

**Authors:** Colin Naturman

**Comments:** 26 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 139

Interior algebras are Boolean algebras enriched with an interior operator and corresponding closure operator. Alternative descriptions of interior algebras in terms of generalized topologies in Boolean algebras and neighbourhood functions on Boolean algebras are found. The topological concepts of convergence and accumulation of systems and nets are generalized to interior algebras. Relationships between different forms of convergence and accunulation are found.

**Category:** Algebra

[89] **viXra:1212.0148 [pdf]**
*submitted on 2012-12-26 04:38:39*

**Authors:** Colin Naturman

**Comments:** 17 Pages. Results published in "Naturman, C.A., 1991, Interior Algebras and Topology, PhD Thesis, University of Cape Town Department of Mathematics" and this paper presented as "Research Reports, Department of Mathematics. The University of Cape Town, Volume 144

The intervals in an interior algebra A can be turned into interior algebras called interval algebras. Generalizations of homomorphisms, called topomorphisms, are introduced and certain quotient structures of A in the category of interior algebras and topomorphisms (the principal quotients) are shown to be (up to isomorphism) precisely the interval algebras of A.

**Category:** Algebra

[88] **viXra:1212.0146 [pdf]**
*submitted on 2012-12-25 11:02:30*

**Authors:** Peter Bruyns, Colin Naturman, Henry Rose

**Comments:** 17 Pages.

The amalgamation class Amal (N) of a lattice variety generated by a pentagon is considered. It is shown that Amal (N) is closed under reduced products and therefore is an elementary class determined by Horn sentences. The above result is based on a new characterization of Amal (N). The lattice varieties whose amalgamation classes contain Amal (N) as a subclass are considered.

**Category:** Algebra

[87] **viXra:1212.0138 [pdf]**
*submitted on 2012-12-23 08:11:59*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 26 Pages.

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.

**Category:** Algebra

[86] **viXra:1212.0133 [pdf]**
*submitted on 2012-12-21 12:55:50*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 25 Pages. Published in Journal of the Korean Mathematical Society, 30(1), 1993, pp.1–23 content is free for download but PDFs distributed by the publisher are missing the diagrams and/or abstract and errata.

An interior algebra is a Boolean algebra enriched with an interior operator. Congruences on interior algebras are investigated. Simple, subdirectly irreducible, finitely subdirectly irreducible and directly indecomposable interior algebras are characterized and the classes of these are shown to be finitely axiomatizable elementary classes. Quotients by open elements, dissectable and openly decomposable interior algebras are investigated. Basic results concerning interior algebras and their connection to topology are discussed.

**Category:** Algebra

[85] **viXra:1212.0131 [pdf]**
*submitted on 2012-12-21 03:39:27*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 6 Pages. Published in Ordered Set and Lattices, 11, 1995 pp. 39-44, publisher does not provide offprints

An interior algebra is a Boolean algebra enriched with an interior operator. Given an interior algebra there is a natural way of forming interior algebras from its principal ideals. Basic results concerning these ideal algebras, Stone spaces of ideal algebras and preservation properties of ideal algebras are investigated.

**Category:** Algebra

[84] **viXra:1212.0124 [pdf]**
*submitted on 2012-12-20 04:53:25*

**Authors:** Deepak Ponvel Chermakani

**Comments:** 6 Pages, 6 Theorems, 5 Figures

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real-variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.

**Category:** Algebra

[83] **viXra:1212.0018 [pdf]**
*submitted on 2012-12-03 12:30:39*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 200 Pages.

The authors introduce the concept of neutrosophic super matrices and the new notion of quasi super matrices. This new notion of quasi super matrices contains the class of super matrices. The larger class contains more partitions of the usual simple matrices. Studies in this direction are interesting and find more applications in fuzzy models. The authors also suggest in this book some open problems.

**Category:** Algebra

[82] **viXra:1211.0044 [pdf]**
*submitted on 2012-11-08 12:33:29*

**Authors:** Iqbal Unnisa, W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 132 Pages.

In this book the authors introduce a new class of lattices called Supermodular Lattices, which is an equational class of lattices lying between the equational class of distributive lattices and modular lattices.
Several other new properties related with these lattices are introduced, described and developed.

**Category:** Algebra

[81] **viXra:1211.0038 [pdf]**
*submitted on 2012-11-07 13:25:27*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 114 Pages.

In this book the authors use set ideal of rings (or semigroups) to build topological spaces. These spaces are dependent on the set over which the set ideals are defined.
It is left as an open problem whether this newly constructed topological space of finite order increases the existing number of finite topological spaces.

**Category:** Algebra

[80] **viXra:1211.0029 [pdf]**
*submitted on 2012-11-06 06:11:24*

**Authors:** J. S. Markovitch

**Comments:** 2 Pages.

A rewriting system applied to the simplest algebraic identities is shown to yield second- and third-order equations that share a property associated with 137.036.

**Category:** Algebra

[79] **viXra:1210.0142 [pdf]**
*submitted on 2012-10-25 16:03:56*

**Authors:** Martin Erik Horn

**Comments:** 14 Pages. Poster presentation at AGACSE 2012 in La Rochelle

Quarks are described mathematically by (3 x 3) matrices. To include these quarkonian mathematical structures into Geometric algebra it is helpful to restate Geometric algebra in the mathematical language of (3 x 3) matrices. It will be shown in this paper how (3 x 3) permutation matrices can be interpreted as unit vectors. And as S3 permutation symmetry is flavour symmetry a unified flavour picture of Geometric algebra will emerge.

**Category:** Algebra

[78] **viXra:1210.0086 [pdf]**
*submitted on 2012-10-17 11:01:47*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 193 Pages.

A new notion of special quasi dual numbers is introduced.
If a+bg is the special quasi dual number with a, b reals, g the new element is such that g^2 = - g.
The rich source of getting new elements of the form g^2 = - g is from Z_n, the ring of modulo integers.
For the first time we construct non associative structures using them.
We have proposed some research problems.

**Category:** Algebra

[77] **viXra:1210.0085 [pdf]**
*submitted on 2012-10-17 11:06:59*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 246 Pages.

In this book we define x = a+bg to be a special dual like number, where a, b are reals and g is a new element such that g^2 = g.
The new element which is idempotent can be got from Z_n or from lattices or from linear operators.
Mixed dual numbers are constructed using dual numbers and special dual like numbers.
Neutrosophic numbers are a natural source of special dual like numbers, since they have the form a+bI, where I = indeterminate and I^2 = I.

**Category:** Algebra

[76] **viXra:1210.0084 [pdf]**
*submitted on 2012-10-17 11:08:46*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 153 Pages.

The zero divisor graph of semigroups of finite modulo integers n under product is studied and characterized. If n is a non-prime, the zero divisor graph is not a tree.
We introduce the new notion of tree covering a pseudo lattice. When n is an even integer of the form 2p, p a prime, then the modulo integer zero divisor graph is a tree-covering pseudo lattice.

**Category:** Algebra

[75] **viXra:1209.0014 [pdf]**
*submitted on 2012-09-05 00:31:28*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 159 Pages.

The concept of dual numbers was introduced in 1873 by W.K.Clifford. In this book the authors build higher dimensional dual numbers, interval dual numbers and impose some algebraic structures on them. The S-vector space of dual numbers built over a Smarandache dual ring can have eigen values and eigen vectors to be dual numbers.
Complex modulo integer dual numbers and neutrosophic dual numbers are also introduced. The notion of fuzzy dual numbers can play a vital role in fuzzy models. Some research level problems are also proposed here.

**Category:** Algebra

[74] **viXra:1208.0200 [pdf]**
*submitted on 2012-08-20 03:38:01*

**Authors:** Pierre-Yves Gaillard

**Comments:** 2 Pages.

We give a mild generalization of Zariski's Lemma.

**Category:** Algebra

[73] **viXra:1208.0020 [pdf]**
*submitted on 2012-08-07 13:16:43*

**Authors:** Arsham Borumand Saeid

**Comments:** Pages.

In this paper, we introduce the notions of Smarandache weak
BE-algebra, Q-Smarandache filters and Q-Smarandache ideals. We show
that a nonempty subset F of a BE-algebra X is a Q-Smarandache filter
if and only if A(x, y) is included in or equal to F, which A(x, y) is a Q-Smarandache upper set.
The relationship between these notions are stated and proved.

**Category:** Algebra

[72] **viXra:1207.0076 [pdf]**
*submitted on 2012-07-19 20:57:11*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 150 Pages.

In this book we explore the possibility of extending the natural operations on reals to intervals and matrices. The extension to intervals makes us define a natural class of intervals in which we accept [a, b], a > b. Further, we introduce a complex modulo integer in Zn (n, a positive integer) and denote it by i_F with i_F^2 = n-1.

**Category:** Algebra

[71] **viXra:1207.0051 [pdf]**
*submitted on 2012-07-13 00:51:32*

**Authors:** Pierre-Yves Gaillard

**Comments:** 2 Pages.

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory for finite degree extensions.

**Category:** Algebra

[70] **viXra:1206.0009 [pdf]**
*submitted on 2012-06-02 04:13:28*

**Authors:** Ren Shiquan

**Comments:** 24 Pages. this is the bachelor thesis of Mr. Ren Shiquan in 2010.

We give a review on the classification of complex semi-simple Lie algebras, according to our study process.

**Category:** Algebra

[69] **viXra:1204.0005 [pdf]**
*submitted on 2012-04-02 14:11:51*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 231 Pages.

In this book we introduce the notion of Non Associative Linear Algebras. We mainly use the concepts of loops and groupoids to build these structures. We have also introduced the concept of Non Associative Semi-Linear Algebras.
In the future, non-associative linear algebras will find applications in mathematical models that do not in general need to be associative.

**Category:** Algebra

[68] **viXra:1203.0008 [pdf]**
*submitted on 2012-03-02 14:01:07*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, K. Amal

**Comments:** 193 Pages.

In this book the authors introduce the notion of fuzzy linguistic topological spaces. These linguistic topological spaces enjoy many properties depending on the problem and the linguistic variables associated with them.
These structures find applications in mathematical modeling and fuzzy linguistic neural networks.

**Category:** Algebra

[67] **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

[66] **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

[65] **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

[64] **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

[63] **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

[62] **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

[61] **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

[60] **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

[59] **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

[58] **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

[57] **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

[56] **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

[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
Z_{t}L_{n}(m) for a special class of loops L_{n}(m); over the ring of modulo integers
Z_{t} for a specific value of t. These loops satisfy the conditions g_{i}^{2} = 1 for every
g_{i} ε L_{n}(m). We prove Z_{t}L_{n}(m) has an S-idempotent when t is a perfect number
or when t is of the form 2^{i}p or 3^{i}p (where p is an odd prime) or in general when
t = p_{1}^{i}p_{2} (p_{1} and p_{2} 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 L_{n}(m) and recall the definition of S-idempotents in rings. In section
two, we establish the existance of S-idempotents in the loop ring Z_{t}L_{n}(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 + x^{2} + ... + x^{n})^{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

[12] **viXra:1412.0253 [pdf]**
*replaced on 2015-01-12 08:33:03*

**Authors:** J. S. Markovitch

**Comments:** 5 Pages.

A nonstandard cubic equation is shown to have an unusually economical solution, where this solution incorporates an angle that serves as the equation's discriminant.

**Category:** Algebra

[11] **viXra:1409.0054 [pdf]**
*replaced on 2014-09-08 12:19:39*

**Authors:** Mumtaz Ali, Florentin Smarandache, Muhammad Shabir

**Comments:** 288 Pages.

In this book, the authors define several new types of soft neutrosophic algebraic structures over neutrosophic algebraic structures and we study their generalizations. These soft neutrosophic algebraic structures are basically parameterized collections of neutrosophic sub-algebraic structures of the neutrosophic algebraic structure. An important feature of this book is that the authors introduce the soft neutrosophic group ring, soft neutrosophic semigroup ring with its generalization, and soft mixed neutrosophic N-algebraic structure over neutrosophic group ring, then the neutrosophic semigroup ring and mixed neutrosophic N-algebraic structure respectively.

**Category:** Algebra

[10] **viXra:1407.0189 [pdf]**
*replaced on 2014-09-02 17:35:11*

**Authors:** Ouannas Moussa

**Comments:** 03 Pages.

Basing on my study, "Distribution of primes"viXra:1407.0167 [pdf] submitted on 2014-07-21 I've shown that we can determine the distribution of prime numbers while remaining within the set of natural numbers; However, I present in this paper the numbers which are susceptible to be twins

**Category:** Algebra

[9] **viXra:1406.0183 [pdf]**
*replaced on 2015-01-09 15:56:12*

**Authors:** Richard J. Mathar

**Comments:** 182 Pages. Corrected figure caption of Figure 50.

The cycle graphs for all finite abstract groups up to order 37 are plotted, except the trivial ring graphs of the cyclic groups themselves.

**Category:** Algebra

[8] **viXra:1404.0466 [pdf]**
*replaced on 2014-06-21 19:29:49*

**Authors:** Hajime Mashima

**Comments:** 8 Pages.

During the 360 years of Fermat's last theorem is to be proved, this proposition
was the presence appear full-length novel in "The Lord of the Rings", such as
the "One Ring". And finally in 1994, it was proved completely by Andrew
Wiles. However interesting proof is Fermat has been is still unknown. This will
be assumed in the category of algebra probably.

**Category:** Algebra

[7] **viXra:1403.0958 [pdf]**
*replaced on 2014-03-29 00:04:16*

**Authors:** Nikolay Dementev

**Comments:** 7 Pages.

The report suggests an approach to extend a concept of determinant to the systems of any order.

**Category:** Algebra

[6] **viXra:1312.0213 [pdf]**
*replaced on 2013-12-28 19:47:58*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache

**Comments:** 216 Pages.

The algebraic structures built using [0, n) interval are new and innovative. They happen to have different properties. The interval [0, n) can be realized as the real algebraic closure of the modulo ring Zn. The algebraic behavior of [0, n) is different from the ring Zn.

**Category:** Algebra

[5] **viXra:1212.0124 [pdf]**
*replaced on 2012-12-21 04:13:24*

**Authors:** Deepak Ponvel Chermakani

**Comments:** There are 6 Pages, 6 Theorems, 7 Figures. I also made a small correction that in Theorem-1, the correct word is "NP-Hard" and not "NP-Complete".

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.

**Category:** Algebra

[4] **viXra:1211.0029 [pdf]**
*replaced on 2012-11-27 13:25:20*

**Authors:** J. S. Markovitch

**Comments:** 2 Pages.

A rewriting system applied to the simplest algebraic identities is shown to yield second- and third-degree equations that share a property associated with the constant 137.036, which is a minimal case.

**Category:** Algebra

[3] **viXra:1205.0093 [pdf]**
*replaced on 2012-06-08 19:15:06*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, A. Praveen Prakash

**Comments:** 167 Pages.

The authors in this book have analyzed the socio-economic and
psychological problems faced by People with Disabilities
(PWDs) and their families. The study was made by collecting
data using both fuzzy linguistic questionnaire / by interviews in
case they are not literates from 2,15,811 lakhs people. This data
was collected using the five Non Government Organizations
(NGOs) from northern Tamil Nadu.

**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