Number Theory Commons^™

Soft Neutrosophic Algebraic Structures And Their Generalization - Vol. 1, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduced the notions of soft neutrosophic algebraic structures. These soft neutrosophic algebraic structures are basically defined over the neutrosophic algebraic structures which means a parameterized collection of subsets of the neutrosophic algebraic structure. For instance, the existence of a soft neutrosophic group over a neutrosophic group or a soft neutrosophic semigroup over a neutrosophic semigroup, or a soft neutrosophic field over a neutrosophic field, or a soft neutrosophic LA-semigroup over a neutrosophic LAsemigroup, or a soft neutosophic loop over a neutrosophic loop. It is interesting to note that these notions are defined over finite and …

New Research On Neutrosophic Algebraic Structures, Florentin Smarandache, Mumtaz Ali, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

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 more 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 …

Solving Diophantine Equations, Florentin Smarandache, Octavian Cira

Branch Mathematics and Statistics Faculty and Staff Publications

In recent times, we witnessed an explosion of Number Theory problems that are solved using mathematical software and powerful computers. The observation that the number of transistors packed on integrated circuits doubles every two years made by Gordon E. Moore in 1965 is still accurate to this day. With ever increasing computing power more and more mathematical problems can be tacked using brute force. At the same time the advances in mathematical software made tools like Maple, Mathematica, Matlab or Mathcad widely available and easy to use for the vast majority of the mathematical research community. This tools don’t only …

Subset Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of constructing non associative algebraic structures using subsets of a groupoid. Thus subset groupoids are constructed using groupoids or loops. Even if we use subsets of loops still the algebraic structure we get with it is only a groupoid. However we can get a proper subset of it to be a subset loop which will be isomorphic with the loop which was used in the construction of the subset groupoid. To the best of the authors’ knowledge this is the first time non associative algebraic structures are constructed using subsets. We get …

Intuitionistic Neutrosophic Soft Set, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper we study the concept of intuitionistic neutrosophic set of Bhowmik and Pal. We have introduced this concept in soft sets and defined intuitionistic neutrosophic soft set. Some definitions and operations have been introduced on intuitionistic neutrosophic soft set. Some properties of this concept have been established.

More On Intuitionistic Neutrosophic Soft Sets, Said Broumi, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Intuitionistic Neutrosophic Soft Set theory proposed by S. Broumi and F. Samarandache [28], has been regarded as an effective mathematical tool to deal with uncertainties. In this paper new operations on intuitionistic neutrosophic soft sets have been introduced . Some results relating to the properties of these operations have been established. Moreover ,we illustrate their interconnections between each other.

Subset Non Associative Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of non associative topological space is new and innovative. In general topological spaces are defined as union and intersection of subsets of a set X. In this book authors for the first time define non associative topological spaces using subsets of groupoids or subsets of loops or subsets of groupoid rings or subsets of loop rings. This study leads to several interesting results in this direction.

Over hundred problems on non associative topological spaces using of subsets of loops or groupoids is suggested at the end of chapter two. Also conditions for these non associative subset topological spaces …

Subset Polynomial Semirings And Subset Matrix Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of subset polynomial semirings and subset matrix semirings. The study of algebraic structures using subsets were recently carried out by the authors. Here we define the notion of subset row matrices, subset column matrices and subset m × n matrices. Study of this kind is developed in chapter two of this book. If we use subsets of a set X; say P(X), the power set of the set X....

Hence if P(X) is replaced by a group or a semigroup we get the subset matrix to be only a subset matrix semigroup. If …

Subset Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of groupoids is meager and we have recently introduced the new notion of subset groupoids and have studied them. It is interesting to keep on record that interval groupoids have been studied by us in 2010. Further when the subsets of a loop are taken they also form only a subset groupoid and not a subset loop. Thus we do not have the concept of subset interval loop they only form a subset interval groupoid. Special elements like subset interval zero divisors, subset interval idempotents and subset interval units are studied. Concept of subset interval groupoid homomorphism is …

Algebraic Structures Using Subsets, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a …

Algebraic Structures Using [0,N), Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

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. We see [0, n) under both max and min operation is a semiring. [0, n) under + and × is not in general a ring. We define S = {[0, n), +, ×} to be a pseudo special ring as the distributive law is …

Set Theoretic Approach To Algebraic Structures In Mathematics - A Revelation, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out how sets in algebraic structure can be used to construct most generalized algebraic structures, like set linear algebra/vector space, set ideals in rings and semigroups. This sort of study is not only innovative but infact very helpful in cases instead of working with a large data we can work with a considerably small data. Thus instead of working with a vector space or a linear algebra V over a field F we can work with a subset in V and a needed subset in F, this can save both time and economy. The concept …

Variance On Topics Of Plane Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

This book contains 21 papers of plane geometry. It deals with various topics, such as: quasi-isogonal cevians, nedians, polar of a point with respect to a circle, anti-bisector, aalsonti-symmedian, anti-height and their isogonal. A nedian is a line segment that has its origin in a triangle’s vertex and divides the opposite side in Q equal segments. The papers also study distances between remarkable points in the 2D-geometry, the circumscribed octagon and the inscribable octagon, the circles adjointly ex-inscribed associated to a triangle, and several classical results such as: Carnot circles, Euler’s line, Desargues theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s theorem, …

Dual Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Dual numbers was first introduced by W.K. Clifford in 1873. This nice concept has lots of applications; to screw systems, modeling plane joint, iterative methods for displacement analysis of spatial mechanisms, inertial force analysis of spatial mechanisms etc. In this book the authors study dual numbers in a special way. The main aim of this book is to find rich sources of new elements g such that g2 = 0. The main sources of such new elements are from Zn, n a composite number. We give algebraic structures on them. This book is organized into six chapters. The final chapter …

Exploring The Extension Of Natural Operations On Intervals, Matrices And Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book extends the natural operations defined on intervals, finite complex numbers and matrices. The intervals [a, b] are such that a ≤ b. But the natural class of intervals [a, b] introduced by the authors are such that a ≥ b or a need not be comparable with b. This way of defining natural class of intervals enables the authors to extend all the natural operations defined on reals to these natural class of intervals without any difficulty. Thus with these natural class of intervals working with interval matrices like stiffness matrices finding eigenvalues takes the same time as …

Fuzzy Linguistic Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Amal

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Supermodular Lattices, Florentin Smarandache, Iqbal Unnisa, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In lattice theory the two well known equational class of lattices are the distributive lattices and the modular lattices. All distributive lattices are modular however a modular lattice in general is not distributive.

In this book, new classes of lattices called supermodular lattices and semi-supermodular lattices are introduced and characterized as follows: A subdirectly irreducible supermodular lattice is isomorphic to the two element chain lattice C2 or the five element modular lattice M3. A lattice L is supermodular if and only if L is a subdirect union of a two element chain C2 and the five element modular lattice M3.

Special Quasi Dual Numbers And Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new notion called special quasi dual number, x = a + bg.

Among the reals – 1 behaves in this way, for (– 1)2 = 1 = – (– 1). Likewise –I behaves in such a way (– I)2 = – (– I). These special quasi dual numbers can be generated from matrices with entries from 1 or I using only the natural product ×n. Another rich source of these special quasi dual numbers or quasi special dual numbers is Zn, n a composite number. For instance 8 in Z12 is such that …

Special Dual Like Numbers And Lattices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b …

Non Associative Linear Algebras, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce the notion of non associative vector spaces and non associative linear algebras over a field. We construct non associative space using loops and groupoids over fields. In general in all situations, which we come across to find solutions may not be associative; in such cases we can without any difficulty adopt these non associative vector spaces/linear algebras. Thus this research is a significant one.

This book has six chapters. First chapter is introductory in nature. The new concept of non associative semilinear algebras is introduced in chapter two. This structure is …

Non Associative Algebraic Structures Using Finite Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book for the first time have constructed nonassociative structures like groupoids, quasi loops, non associative semirings and rings using finite complex modulo integers. The Smarandache analogue is also carried out. We see the nonassociative complex modulo integers groupoids satisfy several special identities like Moufang identity, Bol identity, right alternative and left alternative identities. P-complex modulo integer groupoids and idempotent complex modulo integer groupoids are introduced and characterized. This book has six chapters. The first one is introductory in nature. Second chapter introduces complex modulo integer groupoids and complex modulo integer loops.

Set Ideal Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors for the first time introduce a new type of topological spaces called the set ideal topological spaces using rings or semigroups, or used in the mutually exclusive sense. This type of topological spaces use the class of set ideals of a ring (semigroups). The rings or semigroups can be finite or infinite order. By this method we get complex modulo finite integer set ideal topological spaces using finite complex modulo integer rings or finite complex modulo integer semigroups. Also authors construct neutrosophic set ideal toplogical spaces of both finite and infinite order as well as …

Dsm Vector Spaces Of Refined Labels, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

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 [35]. This book has six chapters. The first one is introductory in nature just giving only the needed concepts to make this book a self contained one. Chapter two introduces the notion of refined plane of labels, the three dimensional space of refined labels DSm vector spaces. Clearly …

Uniform And Partially Uniform Redistribution Rules, Florentin Smarandache, Jean Dezert

Branch Mathematics and Statistics Faculty and Staff Publications

This paper introduces two new fusion rules for combining quantitative basic belief assignments. These rules although very simple have not been proposed in literature so far and could serve as useful alternatives because of their low computation cost with respect to the recent advanced Proportional Conflict Redistribution rules developed in the DSmT framework.

G-Neutrosophic Space, Mumtaz Ali, Florentin Smarandache, Munazza Naz, Muhammad Shabir

Branch Mathematics and Statistics Faculty and Staff Publications

The Concept of a G-space came into being as a consequence of Group action on an ordinary set. Over the history of Mathematics and Algebra, theory of group action has emerged and proven to be an applicable and effective framework for the study of different kinds of structures to make connection among them.

Generalized Interval Neutrosophic Soft Set And Its Decision Making Problem, Said Broumi

Branch Mathematics and Statistics Faculty and Staff Publications

In this work, we introduce the concept of generalized interval neutrosophic soft set and study their operations. Finally, we present an application of generalized interval neutrosophic soft set in decision making problem.

Algebraic Structures Using Super Interval Matrices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix.

This book has seven chapters. Chapter one is introductory in nature, just introducing the super interval matrices or interval super matrices. In chapter two essential operations on super interval matrices are defined. Further in this chapter algebraic structures are defined on these super interval matrices using these operation. Using these super interval matrices semirings and semivector spaces …

Finite Neutrosophic Complex Numbers, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time the authors introduce the notion of real neutrosophic complex numbers. Further the new notion of finite complex modulo integers is defined. For every C(Zn) the complex modulo integer iF is such that 2 Fi = n – 1. Several algebraic structures on C(Zn) are introduced and studied. Further the notion of complex neutrosophic modulo integers is introduced. Vector spaces and linear algebras are constructed using these neutrosophic complex modulo integers. This book is organized into 5 chapters. The first chapter introduces real neutrosophic complex numbers. Chapter two introduces the notion of finite complex …

Interval Algebraic Bistructures, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Authors in this book construct interval bistructures using only interval groups, interval loops, interval semigroups and interval groupoids. Several results enjoyed by these interval bistructures are described. By this method, we obtain interval bistructures which are associative or non associative or quasi associative. The term quasi is used mainly in the interval bistructure B = B1 ∪ B2 (or in n-interval structure) if one of B1 (or B2) enjoys an algebraic property and the other does not enjoy that property (one section of interval structure satisfies an algebraic property and the remaining section does not satisfy that particular property). The …

Study Of Natural Class Of Intervals Using (–∞,∞) And (∞, –∞), Florentin Smarandache, W.B. Vasantha Kandasamy, D. Datta, H.S. Kushwaha, P.A. Jadhav

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors study the properties of natural class of intervals built using (–∞, ∞) and (∞, –∞). These natural class of intervals behave like the reals R, as far as the operations of addition, multiplication, subtraction and division are concerned. Using these natural class of intervals we build interval row matrices, interval column matrices and m × n interval matrices. Several properties about them are defined and studied. Also all arithmetic operations are performed on them in the usual way. The authors by defining so have made it easier for operations like multiplication, addition, finding determinant and …