SMARANDACHE NEAR-RINGS AND THEIR GENERALIZATIONS

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

SMARANDACHE COSETS

This paper aims to study the Smarandache cosets and derive some interesting results about them. We prove the classical Lagranges theorem for Smarandache semigroup is not true and that there does not exist a one-to-one correspondence between any two right cosets. We also show that the classical theorems cannot be extended to all Smarandache semigroups. This leads to the definition of Smarandache Lagrange semigroup, Smarandache p Sylow subgroup and Smarandache Cauchy elements. Further if we restrict ourselves to the subgroup of the Smarandache semigroup all results would follow trivially hence the Smarandache coset would become a trivial definition

SMARANDACHE GROUPOIDS

In this paper we study the concept of Smarandache Groupoids, subgroupoids, ideal of groupoids, semi-normal subgroupoids, Smarandache-Bol groupoids and Strong Bol groupoids and obtain many interesting results about them

Smarandache Non-Associative (SNA-) rings

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

RESERVATION FOR OTHER BACKWARD CLASSES IN INDIAN CENTRAL GOVERNMENT INSTITUTIONS LIKE IITs, IIMs AND AIIMS – A STUDY OF THE ROLE OF MEDIA USING FUZZY SUPER FRM MODELS

The new notions of super column FRM model, super row FRM model and mixed super FRM model are introduced in this book. These three models are introduced specially to analyze the biased role of the print media on 27 percent reservation for the Other Backward Classes (OBCs) in educational institutions run by the Indian Central Government. This book has four chapters. In chapter one the authors introduce the three types of super FRM models. Chapter two uses these three new super fuzzy models to study the role of media which feverishly argued against 27 percent reservation for OBCs in Central Government-run institutions in India. The experts we consulted were divided into 19 groups depending on their profession. These groups of experts gave their opinion and comments on the news-items that appeared about reservations in dailies and weekly magazines, and the gist of these lengthy discussions form the third chapter of this book. The fourth chapter gives the conclusions based on our study. Our study was conducted from April 2006 to March 2007, at which point of time the Supreme Court of India stayed the 27 percent reservation for OBCs in the IITs, IIMs and AIIMS. After the aforesaid injunction from the Supreme Court, the experts did not wish to give their opinion since the matter was sub-judice. The authors deeply acknowledge the service of each and every expert who contributed their opinion and thus made this book a possibility. We have analyzed the data using the opinion of the experts who formed a heterogeneous group consisting of administrators, lawyers, OBC/SC/ST students, upper caste students and Brahmin students, educationalists, university vice-chancellors, directors, professors, teachers, retired Judges, principals of colleges, parents, journalists, members of the public, politicians, doctors, engineers, NGOs and government staff

Quasi Set Topological Vector Subspaces

In this book the authors introduce four types of topological vector subspaces. All topological vector subspaces are defined depending on a set. We define a quasi set topological vector subspace of a vector space depending on the subset S contained in the field F over which the vector space V is defined. These quasi set topological vector subspaces defined over a subset can be of finite or infinite dimension. An interesting feature about these spaces is that there can be several quasi set topological vector subspaces of a given vector space. This property helps one to construct several spaces with varying basic sets. Further we cannot define quasi set topological vector subspaces of all vector subspaces. We have given the number of quasi set topological vector subspaces in case of a vector space defined over a finite field. It is still an open problem, “Will these quasi set topological vector spaces increase the number of finite topological spaces with n points, n a finite positive integer?”

Subset Polynomial Semirings and Subset Matrix Semirings

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 the semiring or a ring is used we can give the subset collection only the semiring structure. The collection of subsets from the polynomial ring or a polynomial semiring can have only a semiring structure. Several types of subset polynomial semirings are defined described and developed in chapter three of this book

Neutrosophic Super Matrices and Quasi Super Matrices

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution. Here we introduce neutrosophic super matrices and show that the sum or product of two neutrosophic matrices is not in general a neutrosophic super matrix. Another interesting feature of this book is that we introduce a new class of matrices called quasi super matrices; these matrices are the larger class which contains the class of super matrices. These class of matrices lead to more partition of n × m matrices where n \u3e 1 and m \u3e 1, where m and n can also be equal. Thus this concept cannot be defined on usual row matrices or column matrices. These matrices will play a major role when studying a problem which needs multi fuzzy neutrosophic models

Algebraic Structures on Finite Complex Modulo Integer Interval C([0, n))

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 of finite complex modulo integer intervals also we can have only pseudo ring as the distributive law is not true, in general in C([0, n)). Finally the concepts of pseudo vector spaces and pseudo linear algebras are introduced

Special Type of Subset Topological Spaces

In this book we construct special subset topological spaces using subsets from semigroups or groups or rings or semirings. Such study is carried out for the first time and it is both interesting and innovative. Suppose P is a semigroup and S is the collection of all subsets of P together with the empty set, then S can be given three types of topologies and all the three related topological spaces are distinct and results in more types of topological spaces. When the semigroup is finite, S gives more types of finite topological spaces. The same is true in case of groups also. Several interesting properties enjoyed by them are also discussed in this book. In case of subset semigroup using semigroup P we can have subset set ideal topological spaces built using subsemigroups. The advantage of this notion is we can have as many subset set ideal topological spaces as the number of semigroups in P. In case of subset semigroups using groups we can use the subset subsemigroups to build subset set ideal topological spaces over these subset semigroups. This is true in case of subset semigroups which are built using semigroups also. Finally these special subset topological spaces can also be non commutative depending on the semigroup or the group