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Dsm Super Vector Space of Refined Labels W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2011 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2011 by Zip Publishing and the Authors Peer reviewers: Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania Prof. Zhang Wenpeng, Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China. Prof. Mihàly Bencze, Department of Mathematics Áprily Lajos College, Bra2ov, Romania Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-167-4 EAN: 9781599731674 Printed in the United States of America 2 5 7 1.1 Supermatrices and their Properties 7 1.2 Refined Labels and Ordinary Labels and their Properties 30 37 65 101 3 165 245 247 289 293 297 4 In this book authors for the first time introduce the notion of supermatrices of refined labels. Authors prove super row matrix of refined labels form a group under addition. However super row matrix of refined labels do not form a group under product; it only forms a semigroup under multiplication. In this book super column matrix of refined labels and m × n matrix of refined labels are introduced and studied. We mainly study this to introduce to super vector space of refined labels using matrices. We in this book introduce the notion of semifield of refined labels using which we define for the first time the notion of supersemivector spaces of refined labels. Several interesting properties in this direction are defined and derived. 5 We suggest over hundred problems some of which are simple some at research level and some difficult. We give some applications but we are sure in due course when these new notions become popular among researchers they will find lots of applications. This book has five chapters. First chapter is introductory in nature, second chapter introduces super matrices of refined labels and algebraic structures on these supermatrices of refined labels. All possible operations are on these supermatrices of refined labels is discussed in chapter three. Forth chapter introduces the notion of supermatrix of refined label vector spaces. Super matrix of refined labels of semivector spaces is introduced and studied and analysed in chapter five. Chapter six suggests the probable applications of these new structures. The final chapter suggests over hundred problems. We also thank Dr. K.Kandasamy for proof reading and being extremely supportive. 6 This chapter has two sections. In section one we introduce the notion of super matrices and illustrate them by some examples. Using these super matrices super matrix labels are constructed in later chapters. Section two recalls the notion of ordinary labels, refined labels and partially ordered labels and illustrate them with examples. These concepts are essential to make this book a self contained one. For more refer [47-8]. !"#$! %!# We call the usual matrix A = (aij) where aij ∈ R or Q or Z or Zn as a simple matrix. A simple matrix can be a square matrix like F 3021V G W G −−47 80W A = G 0219− W G W HG 201/40XW or a rectangular matrix like 7 F 8071 7 3V G W G−258 0 9 5W G W B = G 014− 3 0 7W G W G−720 5 2 0W HG 101− 2 1 8XW or a row matrix like T = (9, 2 , -7, 3, -5, − 7 , 8, 0, 1, 2) or a column matrix like F 2 V G W G 0 W G W G 7 W G −8 W P = G W . G 3 W G 2 W G W G 51W G W H 9 X So one can define a super matrix as a matrix whose elements are submatrices. For instance FaaV G 11 12 W A = Gaa21 22 W G W Haa31 32 X where FV90 F 378V F84− V GW G− W G W a11 = GW12, a12 = G 10 9W , a21 = G45W , HXGW57− HG 27− 6XW HG12XW FV93 F920V GW− G W GW12 F12 3V G012W GWG W G W a31 = 37− , a22 = 45 0 and a32 = 304 GWG W G W GW−90 HG08− 1XW G057W GW G W HX08 H800X 8 are submatrices. So submatrices are associated with a super matrix [47]. The height of a super matrix is the number of rows of submatrices in it and the width of a super matrix is the number of columns of submatrices in it [47]. We obtain super matrix from a simple matrix. This process of constructing super matrix from a simple matrix will be known as the partitioning. A simple matrix is partitioned by dividing or separating the simple matrix of a specified row and specified column. When division is carried out only between the columns then those matrices are called as super row vectors; when only rows are partitioned in simple matrices we call those simple matrices as super column vectors. We will first illustrate this situation by some simple examples. Example 1.1.1: Let FV07578423012 GW A = GW23034567890 HXGW51311121314 be a 3 × 11 simple matrix. The super matrix or the super row vector is obtained by partitioning between the columns. FV07578423012 GW A1 = GW23034567890 HXGW51311121314 is a super row vector. FV07578423012 GW A2 = GW23034567890 HXGW51311121314 is a super row vector. We can get several super row vectors from one matrix A. Example 1.1.2: Let V = (1 2 3 4 5 6 7 8 9 1 0 9 4 7 2 4 3) be a simple row vector we get a super row vector V1 = (1 2 3 | 4 5 6 | 7 8 9 1 0 | 9 4 7 | 2 4 3) and many more super row vectors can be found by partitioning V differently. 9 Example 1.1.3: Let F301V G W G057W G123W G W V = G457W G890W G W G107W G W H014X be a simple matrix. We get the super column vector by partitioning in between the columns as follows: FV301 F301V GW G W GW057 G057W GW123 G123W GW G W V1 = GW457 and V2 = G457W GW 890 G890W GW G W GW107 G107W GW G W HX014 H014X are super column vectors. Example 1.1.4: Let F1V G W G2W G3W G W G4W G5W X = G W G6W G W G7W G8W G W G9W HG0XW be a simple matrix. 10 F1V G W G2W G3W G W G4W G5W X′ = G W G6W G W G7W G8W G W G9W HG0XW is a super column vector. We can get several such super column vectors using X. Example 1.1.5: Let us consider the simple matrix; F01 2 3 4 5V G W 6 7 8 9 10 11 P = G W , G14−− 8 4 7 3W G W H25 1− 5 8 2X we get the super matrix by partitioning P as follows: F01 2 3 4 5V G W G6 7 8 9 10 11W P1 = G14−− 8 4 7 3W G W H25 1− 5 8 2X is a super matrix with 6 submatrices. FV012 F3V F 45V a1 = GW, a2 = G W , a3 = G W , HX678 H9X H10 11X FV14− 8 F−4V F73V a4 = GW, a5 = G W and a6 = G W . HX25 1 H−5X H82X FV aaa123 Thus P1 = GW. Clearly P1 is a 4 × 6 matrix. HXaaa456 11 Example 1.1.6: Let F3631V G W 1382− V = G W G2493W G W H7144− X be a super matrix. FaaV V = G 12W Haa34X where a1, a2, a3 and a4 are submatrices with FV36 F31V F24V F93V a1 = GW, a2 = G W , a3 = G W and a4 = G W . HX13− H82X H71− X H44X We can also have the notion of super identity matrix as follows. Let FI V G t1 W G I0 W S = t2 G W G W G 0IW H tr X where I are identity matrices of order ti × ti where 1 < i < r. ti We will just illustrate this situation by some examples. Consider FVFVFV1000 000 GWGWGW GWGWGW0100 000 GWGWGW0010 000 GWGWGWFI0V P = GWHXHX0001 000 = G 4 W GWH 0I3 X GWFVFV0000 100 GWGW GW0000 010 GWGWGW HXGWHXHXGWGW0000 001 is a identity super diagonal matrix. 12 Consider FVFVF10 000 VFV 00 GWGWG WGW GWHXH01 000 XHX 00 GW FVF00 100 VFV 00 FI00V GWGWG WGWG 2 W GW S = GWG00 010 WGW 00 = G 0I3 0W GWGWG WGWG W GWHXH00 001 XHX 00 H 00I2 X GWFVF00 000 VFV 10 GWGWG WGW HXGWHXH00 000 XHX 01 is the super diagonal identity matrix [47]. We now proceed onto recall the notion of general diagonal super matrices. Let FV30 0 00 0 GW GW170000 GW890000 GWFm0V S = GW12 0 00 0 = G 1 W H 0mX GW34 0 00 0 2 GW GW00 4 53 2 GW HX00−− 107 4 where FV30 GW 17 GW F V GW 4532 m1 = 89 amd m2 = G W , GW H−−107 4X GW12 HXGW34 S is a general diagonal super matrix. Take FV3457840000 GW GW1240310000 GW0102050000 Fk0V K = GW = G 1 W GW0000009620 H 0k2 X GW0000000412 GW HXGW0000005781 13 is a general diagonal super matrix. Also take F1000V G W G1000W G0000W G W G0100W G0100W G W G0010W V = G W G0010W G0010W G W G0010W G0001W G W G0001W HG0001XW is again a general diagonal super matrix. Now [47] defines also the concept of partial triangular matrix as a super matrix. Consider FV6214621 GW GW0789034 S = = [M1 M2] GW0014405 GW HX0003789 is a upper partial triangular super matrix.
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