ALGEBRAIC STRUCTURES USING SUPER INTERVAL MATRICES W. B. Vasantha Kandasamy Florentin Smarandache THE EDUCATIONAL PUBLISHER INC Ohio 2011 This book can be ordered from: The Educational Publisher, Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 E-mail: [email protected] Website: www.EduPublisher.com Copyright 2011 by The Educational Publisher, Inc. and the Authors Peer reviewers: Prof. Ion Goian, Department of Algebra, Number Theory and Logic, State University of Kishinev, R. Moldova. Prof. Mihàly Bencze, Department of Mathematics Áprily Lajos College, Braşov, Romania Professor Paul P. Wang, Department of Electrical & Computer Engineering Pratt School of Engineering, Duke University, Durham, NC 27708, USA Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-153-7 EAN: 9781599731537 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two INTERVAL SUPERMATRICES 9 Chapter Three SEMIRINGS AND SEMIVECTOR SPACES USING SUPER INTERVAL MATRICES 83 Chapter Four SUPER INTERVAL SEMILINEAR ALGEBRAS 137 Chapter Five SUPER FUZZY INTERVAL MATRICES 203 5.1 Super Fuzzy Interval Matrices 203 5.2 Special Fuzzy Linear Algebras Using Super Fuzzy Interval Matrices 246 3 Chapter Six APPLICATION OF SUPER INTERVAL MATRICES AND SET LINEAR ALGEBRAS BUILT USING SUPER INTERVAL MATRICES 255 Chapter Seven SUGGESTED PROBLEMS 257 FURTHER READING 281 INDEX 285 ABOUT THE AUTHORS 287 4 PREFACE In this book authors for the first time introduce the notion of super interval matrices using the special intervals of the form [0, a], a belongs to Z+ ∪ {0} or Z or Q+ ∪ {0} or R+ ∪ {0}. n The advantage of using super interval matrices is that one can build only one vector space using m × n interval matrices, but in case of super interval matrices we can have several such spaces depending on the partition on the interval matrix. This book has seven chapters. Chapter one is introductory in nature, just introducing the super interval matrices or interval super matrices. In chapter two essential operations on super interval matrices are defined. Further in this chapter algebraic structures are defined on these super interval matrices using these operation. Using these super interval matrices semirings and semivector spaces are defined in chapter three. This chapter gives around 90 examples. In chapter four two types of super interval semilinear algebras are introduced. Super fuzzy interval matrices are introduced in chapter five. This chapter has two sections, in section one super fuzzy interval matrices are introduced using the fuzzy interval [0, 1]. In section two special fuzzy linear algebras using super fuzzy interval matrices are defined and described. Chapter six suggests the probable applications to interval 5 analysis. The final chapter suggests around 110 problems some of which are at research level. We thank Dr. K.Kandasamy for proof reading and being extremely supportive. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 6 Chapter One INTRODUCTION In this chapter we just indicate how we build super interval matrices using intervals of the form [0, a] where a ∈ R+ ∪ {0} or Q+ ∪ {0} or Z+ ∪ {0} or Z or intervals of the form [a, b] with a ≤ b, a, b ∈ R or Z or Q. n Consider a m × n matrix ⎡a a ... a ⎤ 11 12 1n ⎢ ⎥ a a ... a A = ⎢ 21 22 2n⎥ ⎢ (cid:35) (cid:35) (cid:35) ⎥ ⎢ ⎥ ⎢⎣am am ... am ⎥⎦ 1 2 n where a = [0, a], a ∈ R+ ∪ {0} or Z+ ∪ {0} or Z or Q+ ∪ {0} or [a, b] ij n = a with a ≤ b, a, b ∈ R or Q or Z or Z . ij n We partition the m × n matrix A and get the super interval matrix. We in this book only use intervals of the form [0, a]; a ∈ Z+ ∪ {0} or Z n or R+ ∪ {0} or Q+ ∪ {0}. 7 Just we wish to indicate given a 3 × 2 matrix ⎡a a ⎤ 1 2 ⎢ ⎥ a a = A ⎢ 3 4⎥ ⎢⎣a a ⎥⎦ 5 6 a’s intervals, we can partition A as i ⎡a a ⎤ ⎡a a ⎤ 1 2 1 2 ⎢ ⎥ ⎢ ⎥ A = a a , A = a a , 1 ⎢ 3 4⎥ 2 ⎢ 3 4⎥ ⎢⎣a a ⎥⎦ ⎢⎣a a ⎥⎦ 5 6 5 6 ⎡a a ⎤ ⎡a a ⎤ 1 2 1 2 ⎢ ⎥ ⎢ ⎥ A = a a , A = a a , 3 ⎢ 3 4⎥ 4 ⎢ 3 4⎥ ⎢⎣a5 a6⎥⎦ ⎢⎣a5 a6⎥⎦ ⎡a a ⎤ ⎡a a ⎤ ⎡a a ⎤ 1 2 1 2 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = a a , A = a a and A = a a . 5 ⎢ 3 4⎥ 6 ⎢ 3 4⎥ 7 ⎢ 3 4⎥ ⎢a a ⎥ ⎢a a ⎥ ⎢a a ⎥ ⎣ 5 6⎦ ⎣ 5 6⎦ ⎣ 5 6⎦ Thus using one 3 × 2 matrix A we can get 7 super interval matrices. Hence such study is not only innovative very useful in applications. For operations on super matrices refer [17, 47]. 8 Chapter Two INTERVAL SUPERMATRICES In this chapter we for the first time introduce the notion of interval super matrices and derive several of their related properties. We will illustrate this by some examples. DEFINITION 2.1: Let X = ([a , b ] [a , b ] [a , b ] | [a , b ], [a , b ]| … | 1 1 2 2 3 3 4 4 5 5 [a , b ] [a , b ]) where a, b ∈ R or Z or Q or Z ;1 ≤ i ≤ n; a < b. n-1 n-1 n n i i m i i We define X to be a super interval row matrix. Thus we can say in a usual super row matrix X = (x x x | x x | x 1 2 3 4 5 6 x | … | x x x ) if we replace each x by an interval [a, b], 1 ≤ i ≤ n, 7 n-2 n-1 n i i i we get the super interval row matrix. We will illustrate this situation by some examples. Example 2.1: Let X = ([0, 2] | [1, 5] [2, 7] [-9, 13] | [-8, 2] [3, 12] [-1 +1] [3, 20]) be the row interval super matrix. Example 2.2: Let Y = ([3, 5] [0, 12] | [3, 7], [2, 4] [7, 9]| [-2, 3]) be the row interval super matrix. 9