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W. B. Vasantha Kandasamy, "Smarandache Fuzzy Algebra"

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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