792 research outputs found
SMARANDACHE-GALOIS FIELDS
In this paper we study the notion of Smarandache-Galois fields and homomorphism and the Smarandache quotient ring. Galois fields are nothing but fields having only a finite number of elements. We also propose some interesting problems
SMARANDACHE LOOPS
In this paper we study the notion of Smarandache loops. We obtain some interesting results about them. The notion of Smarandache semigroups homomorphism is studied as well in this paper. Using the definition of homomorphism of Smarandache semigroups we give the classical theorem of Cayley for Smarandache semigroups. We also analyze the Smarandache loop homomorphism. We pose the problem of finding a Cayley theorem for Smarandache loops
Interval Semirings
This book has seven chapters. In chapter one we give the basics needed to
make this book a self contained one. Chapter two introduces the notion of
interval semigroups and interval semifields and are algebraically analysed.
Chapter three introduces special types of interval semirings like matrix
interval semirings and interval polynomial semirings. Chapter four for the
first time introduces the notion of group interval semirings, semigroup
interval semirings, loop interval semirings and groupoid interval semirings and
these structures are studied. Interval neutrosophic semirings are introduced in
chapter five. Applications of these structures are given in chapter six. The
final chapter suggests around 120 problems for the reader.Comment: 155 pages; Published by Kappa & Omega in 201
Interval Algebraic Bistructures
This book has four chapters. In the first chapter interval bistructures
(biinterval structures) such as interval bisemigroup, interval bigroupoid,
interval bigroup and interval biloops are introduced. Throughout this book we
work only with the intervals of the form [0, a] where a \in Zn or Z+ \cup {0}
or R+ \cup {0} or Q+ \cup {0} unless otherwise specified. Also interval
bistructures of the form interval loop-group, interval groupgroupoid so on are
introduced and studied. In chapter two n-interval structures are introduced.
n-interval groupoids, n-interval semigroups, n-interval loops and so on are
introduced and analysed. Using these notions n-interval mixed algebraic
structure are defined and described. Some probable applications are discussed.
Only in due course of time several applications would be evolved by researchers
as per their need. The final chapter suggests around 295 problems of which some
are simple exercises, some are difficult and some of them are research
problems.Comment: 208 page
Neutrosophic Rings
This book has four chapters. Chapter one is introductory in nature, for it
recalls some basic definitions essential to make the book a self-contained one.
Chapter two, introduces for the first time the new notion of neutrosophic rings
and some special neutrosophic rings like neutrosophic ring of matrix and
neutrosophic polynomial rings. Chapter three gives some new classes of
neutrosophic rings like group neutrosophic rings,neutrosophic group
neutrosophic rings, semigroup neutrosophic rings, S-semigroup neutrosophic
rings which can be realized as a type of extension of group rings or
generalization of group rings. Study of these structures will throw light on
the research on the algebraic structure of group rings. Chapter four is
entirely devoted to the problems on this new topic, which is an added
attraction to researchers. A salient feature of this book is that it gives 246
problems in Chapter four. Some of the problems are direct and simple, some
little difficult and some can be taken up as a research problem.Comment: 154 page
n-Linear Algebra of type II
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 n-eigen 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 suggests over a hundred
problems. It is important that the reader is well-versed not only with linear
algebra but also n-linear algebra of type I.Comment: 229 page
Super Fuzzy Matrices and Super Fuzzy Models for Social Scientists
This book introduces the concept of fuzzy super matrices and operations on
them. This book will be highly useful to social scientists who wish to work
with multi-expert models. Super fuzzy models using Fuzzy Cognitive Maps, Fuzzy
Relational Maps, Bidirectional Associative Memories and Fuzzy Associative
Memories are defined here. The authors introduce 13 multi-expert models using
the notion of fuzzy supermatrices. These models are described with illustrative
examples. This book has three chapters. In the first chaper, the basic concepts
about super matrices and fuzzy super matrices are recalled. Chapter two
introduces the notion of fuzzy super matrices adn their properties. The final
chapter introduces many super fuzzy multi expert models.Comment: 280 page
Fuzzy and Neutrosophic Analysis of Periyar's views on Untouchability
This book is organized into four chapters. In Chapter One we just introduce
the basic Fuzzy and Neutrosophic tools used in the analysis of the social evil
of Untouchability. Since the notion of caste is based on themind, it is
appropriate to use Fuzzy and Neutrosophic theory. In Chapter Two we use the
opinion of several experts to analyze the various aspects of untouchability.
Here, we use the tools that have been described in Chapter One. Fuzzy Directed
Graphs and Neutrosophic Directed Graphs of these Fuzzy and Neutrosophic models
happen to be very dense. We have represented 16 such graphs in this book. In
Chapter Three we give a brief introduction about the life and struggle of
Periyar. The fourth and concluding chapter gives observations drawn from our
mathematical results based on the Fuzzy and Neutrosophic analysis.Comment: 385 pages, 16 figure
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