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 ...

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 ...

In this book for the first time authors introduce the concept of merged lattice, which gives a lattice or a graph. The resultant lattice or graph is defined as the pseudo lattice graph of type I. Here we also merge a graph with a lattice or two or more graphs which call as the pseudo lattice graph of type II. We merge either edges or vertices or both of a lattice and a graph or a lattice and a lattice or graph with itself. Such study is innovative and these mergings are ...

For the first time authors have ventured to study, analyse and investigate the properties of the fuzzy models, the experts opinion and so on. Here the concept of merged Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps are carried out, which are based on merged graphs and merged matrices. This concept is better than the usual combined Fuzzy Cognitive Maps. Further by this new technique we are able to give equal importance to all the experts who work with the problem. ...

In this book, the authors 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. The authors 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...

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

In this book authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers was introduced by the authors in 2011. Now using this finite complex modulo integer intervals several algebraic structures are built. Further the concept of finite complex modulo integers itself happens to be new and innovative for in case of finite complex modulo integers the square value of the finite complex number varies with varying n of Zn. In case o...

In this book authors for the first time introduce the notion of distance between any two m x n matrices. If the distance is 0 or m x n there is nothing interesting. When the distance happens to be a value t; 0 < t < m x n the study is both innovating and interesting. The three cases of study which is carried out in this book are: 1. If the difference between two square matrices is large, will it imply the eigen values and eigen vectors of those matrices are distinct? Se...

This volum includes 37 papers of mathematics or applied mathematics written by the author alone or in collaboration with the following co-authors: Cătălin Barbu, Mihály Bencze, Octavian Cira, Marian Niţu, Ion Pătraşcu, Mircea E. Şelariu, Rajan Alex, Xingsen Li, Tudor Păroiu, Luige Vlădăreanu, Victor Vlădăreanu, Ştefan Vlăduţescu, Yingjie Tian, Mohd Anasri, Lucian Căpitanu, Valeri Kroumov, Kimihiro Okuyama, Gabriela Tonţ, A. A. Adewara, Manoj K. Chaudhary, Mukesh Kumar, S...

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. The semigroup under × has no finite subsemigroups but under min or max we ...

In this book authors for the first time introduce the notion of special type of topological spaces using the interval [0, n). They are very different from the usual topological spaces. Algebraic structure using the interval [0, n) have been systemically dealt by the authors. Now using those algebraic structures in this book authors introduce the notion of special type of topological spaces. Using the super subset interval semigroup special type of super interval topologi...

An AG-groupoid is an algebraic structure that lies in between a groupoid and a commutative semigroup. It has many characteristics similar to that of a commutative semigroup. If we consider x2y2= y2x2, which holds for all x, y in a commutative semigroup, on the other hand one can easily see that it holds in an AG-groupoid with left identity e and in AG**-groupoids. This simply gives that how an AG-groupoid has closed connections with commutative agebras. We extend now for...

This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when Florentin Smarandache was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), ...

Acest volum este o versiune nouă, revizuită și adăugită, a "Problemelor Compilate şi Rezolvate de Geometrie şi Trigonometrie" (Universitatea din Moldova, Chișinău, 169 p., 1998), și include probleme de geometrie și trigonometrie, compilate și soluționate în perioada 1981-1988. Gradul de dificultate al problemelor este de la mediu spre greu. Cartea se dorește material didactic pentru elevi, studenți și profesori. This book is a translation from Romanian of "Probleme Co...

The authors define MOD transformations and discuss several interesting features about them. So chapter one of this book serves the purpose of only recalling these properties. Various properties of MOD interval transformation of varying types are discussed. Next the mod planes are introduced mainly to make this book a self contained one. Finally the new notion of higher and mixed multidimensional MOD planes are introduced and their properties analysed. Several problems ar...

In this book authors for the first time elaborately study the notion of MOD vector spaces and MOD pseudo linear algebras. This study is new, innovative and leaves several open conjectures. In the first place as distributive law is not true one can define only MOD pseudo linear algebras. Secondly most of the classical theorems true in case of linear algebras are not true in case of MOD pseudo linear algebras. Finding even eigen values and eigen vectors happens to be a cha...

In this book for the first time authors study MOD planes using modulo intervals. These planes unlike the real plane have only one quadrant so the study is carried aut in a compact space but infinite in dimension. The authors have given seven MOD planes viz real MOD planes (MOD real plane) finite complex MOD plane, neutrosophic MOD plane, fuzzy MOD plane, (or MOD fuzzy plane), MOD dual number plane, MOD special dual like number plane and MOD special quasi dual number plan...

In this book authors answer the question proposed by Florentin Smarandache “Does there exist neutrosophic numbers which are such that they take values differently and behave differently from I; the indeterminate?”. The authors have constructed a class of natural neutrosophic numbers. Further there are more natural neutrosophic numbers in the MOD interval [0, m). This concept is thoroughly analysed in the second chapter. Using all types of MOD planes and MOD intervals, th...

In this book the notion of MOD functions are defined on MOD planes. This new concept of MOD functions behaves in a very different way. Even very simple functions like y = nx has several zeros in MOD planes where as they are nice single line graphs with only (0, 0) as the only zero. Further polynomials in MOD planes do not in general follows the usual or classical laws of differentiation or integration. These drawbacks becomes a challenging issue. So study in this directi...

In this book authors study the properties of finite real quaternion ring which was introduced in [2000]. Here a complete study of these finite quaternion rings are made. Also polynomial quaternion rings are defined, they happen to behave in a very different way. In the first place the fundamental theorem of algebra, “a nth degree polynomial has n and only n roots”, n is untrue in case of polynomial in polynomial quaternion rings in general. Further the very concept of de...