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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 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://www.lib.umi.com/bod/basic Copyright 2011 by Kappa & Omega and the Authors 6744 W. Northview Ave. Glendale, AZ 85303, USA 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 3 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. 6 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 7 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]. 8 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. 9