Smarandache Fuzzy Algebra

W. B. Vasantha Kandasamy, "Smarandache Fuzzy Algebra"
merican Research Press | 2003 | ISBN: 1931233748 | 453 pages | PDF | 2,7 MB

PREFACE
In 1965, Lofti A. Zadeh introduced the notion of a fuzzy subset of a set as
a method for representing uncertainty. It provoked, at first (and as
expected), a strong negative reaction from some influential scientists and
mathematicians—many of whom turned openly hostile. However, despite
the controversy, the subject also attracted the attention of other
mathematicians and in the following years, the field grew enormously,
finding applications in areas as diverse as washing machines to
handwriting recognition. In its trajectory of stupendous growth, it has also
come to include the theory of fuzzy algebra and for the past five decades,
several researchers have been working on concepts like fuzzy semigroup,
fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings
and so on.
In this book, we study the subject of Smarandache Fuzzy Algebra.
Originally, the revolutionary theory of Smarandache notions was born as a
paradoxist movement that challenged the status quo of existing
mathematics. The genesis of Smarandache Notions, a field founded by
Florentine Smarandache, is alike to that of Fuzzy Theory: both the fields
imperatively questioned the dogmas of classical mathematics.
Despite the fact that Fuzzy Algebra has been studied for over fifty years,
there are only two books on fuzzy algebra. But both the books do not
cover topics related to fuzzy semirings, fuzzy near-rings etc. so we have
in this book, two parts: In Part 1 we have recalled all the definitions and
properties of fuzzy algebra. In Part II we give Smarandache fuzzy
algebraic notions. This is the first book in fuzzy algebra which covers the
notions of fuzzy semirings and fuzzy near-rings though there are several
papers on these two concepts.
This book has seven chapters, which are divided into two parts. Part I
contains the first chapter, and Part II encloses the remaining six chapters.
In the first chapter, which is subdivided into twelve sections, we deal with
eleven distinct fuzzy algebraic concepts and in the concluding section list
the miscellaneous properties of fuzzy algebra. The eleven fuzzy algebraic
concepts which we analyze are fuzzy sets, fuzzy subgroups, fuzzy subbigroups,
fuzzy rings, fuzzy birings, fuzzy fields, fuzzy semirings, fuzzy
near-rings, fuzzy vector spaces, fuzzy semigroups and fuzzy halfgroupoids.
The results used in these sections are extensive and we have
succeeded in presenting new concepts defined by several researchers. In
the second chapter we introduce the notion of Smarandache fuzzy
semigroups and its properties and also study Smarandache fuzzy
bisemigroups. In the third chapter, we define the notion of Smarandache
fuzzy half-groupoids and their generalizations (Smarandache fuzzy
groupoids and bigroupoids, Smarandache fuzzy loops and biloops)...

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