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 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

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

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

Special Type of Topological Spaces Using [0, n)

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 topological spaces are built. Several interesting results in this direction are obtained. Next six types of topological spaces using subset interval pseudo ring semiring of type I is built and they are illustrated by examples. Strong Super Special Subset interval subset topological spaces (SSSS-interval subset topological spaces are constructed using the algebraic structures semigroups, pseudo groups or semirings or pseudo rings

Subset Non Associative Topological Spaces

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 to satisfy special identities is also discussed and determined. Chapter three develops subset non associative topological spaces by using non associative ring or semirings.Over 90 problems are suggested for this chapter. These non associative subset topological spaces can be got by using matrices

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