Interval Semigroups

167 Pages·2011·0.892 MB·English

by  W. B. Vasantha Kandasamy, Florentin Smarandache

Interval Semigroups - Cover.pdf:Layout 1 1/20/2011 10:04 AM Page 1
INTERVAL SEMIGROUPS

W. B. Vasantha Kandasamy
Florentin Smarandache

KAPPA & OMEGA
Glendale
2011
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Copyright 2011 by Kappa & Omega and the Authors
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Peer reviewers:
Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, Romania
Prof. Mihàly Bencze, Department of Mathematics
Áprily Lajos College, Bra(cid:2)ov, Romania
Dr. Fu Yuhua, 13-603, Liufangbeili
Liufang Street, Chaoyang district, Beijing, 100028 P. R. China

Many books can be downloaded from the following
Digital Library of Science:
http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm

ISBN-10: 1-59973-097-9
ISBN-13: 978-1-59973-097-4
EAN: 9781599730974

Printed in the United States of America
2
CONTENTS
Preface 5
Dedication 6

Chapter One
INTRODUCTION 7

Chapter Two
INTERVAL SEMIGROUPS 9

Chapter Three
INTERVAL POLYNOMIAL SEMIGROUPS 37

Chapter Four
SPECIAL INTERVAL SYMMETRIC SEMIGROUPS 47

Chapter Five
NEUTROSOPHIC INTERVAL SEMIGROUPS 61

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Chapter Six
NEUTROSOPHIC INTERVAL MATRIX
SEMIGROUPS AND FUZZY
INTERVAL SEMIGROUPS 73

6.1 Pure Neutrosophic Interval Matrix Semigroups 73
6.2 Neutrosophic Interval Polynomial Semigroups 94
6.3 Fuzzy Interval Semigroups 118

Chapter Seven
APPLICATION OF INTERVAL SEMIGROUPS 129

Chapter Eight
SUGGESTED PROBLEMS 131

FURTHER READING 159
INDEX 161
ABOUT THE AUTHORS 165
4
PREFACE

In this book we introduce the notion of interval semigroups
using intervals of the form [0, a], a is real. Several types of
interval semigroups like fuzzy interval semigroups, interval
symmetric semigroups, special symmetric interval semigroups,
interval matrix semigroups and interval polynomial semigroups
are defined and discussed. This book has eight chapters.
The main feature of this book is that we suggest 241
problems in the eighth chapter. In this book the authors have
defined 29 new concepts and illustrates them with 231
examples. Certainly this will find several applications.
The authors deeply acknowledge Dr. Kandasamy for the
proof reading and Meena and Kama for the formatting and
designing of the book.

W.B.VASANTHA KANDASAMY
FLORENTIN SMARANDACHE

5
~ DEDICATED TO ~
Ayyankali

Ayyankali(1863–1941) was the first leader of Dalits from
Kerala. He initiated several reforms to emancipate the lives of
the Dalits. Ayyankali organized Dalits and fought against the
discriminations done to Dalits and through his efforts he got the
right to education, right to walk on the public roads and dalit
women were allowed to cover their nakedness in public. He
spearheaded movements against casteism.
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Chapter One
INTRODUCTION
We in this book make use of special type of intervals to build
interval semigroups, interval row matrix semigroups, interval
column matrix semigroups and interval matrix semigroups. We
also introduce and study the Smarandache analogue of them.
The new notion of interval symmetric semigroups and
special interval symmetric semigroups are defined and studied.
For more about symmetric semigroups and their Smarandache
analogue concepts please refer [9].
The classical theorems for finite groups like Lagrange
theorem, Cauchy theorem and Sylow theorem are introduced in
a special way and analyzed. Only under special conditions we
see the notion of these classical theorems for finite groups can
be extended interval semigroups.
The authors also introduce the notion of neutrosophic
intervals and fuzzy intervals and study them in the context of
interval semigroups. I denotes the indeterminate or
inderminancy where I2 = I and I + I = 2I, I + I + I = 3I and so
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on. For more about neutrosophy, neutrosophic intervals please
refer [1, 3, 6-8].
Study of special elements like interval zerodivisors, interval
idempotents, interval units, interval nilpotents are studied and
their Smarandache analogue introduced [9].
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Chapter Two
INTERVAL SEMIGROUPS
In this chapter we for the first time introduce the notion of
interval semigroups and describe a few of their properties
associated with them. We see in general several of the classical
theorems are not true in general case of semigroups. First we
proceed on to give some notations essential to develop these
new structures.
I (Z ) = {[0, a ] | a (cid:2) Z },
n m m n
I(Z+(cid:3) {0}) = {[0, a] | a (cid:2) Z+(cid:3) {0}},
I(Q+(cid:3) {0}) = {[0, a] | a (cid:2) Q+(cid:3) {0}},
I(R+(cid:3) {0}) = {[0, a] | a (cid:2) R+(cid:3) {0}}
and I(C+(cid:3) {0}) = {[0, a] | a (cid:2) C+(cid:3) {0}}.
DEFINITION 2.1: Let S = {[0, a] | a (cid:2) Z ; +} S is a
i i n
semigroup under addition modulo n. S is defined as the interval
semigroup under addition modulo n.
We will first illustrate this by some simple examples.
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