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

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

7 1.1 Supermatrices and their Properties 7 1.2 Refined Labels and Ordinary Labels and their Properties 30

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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].

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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. Suppose F20000V G W G14000W G23500W G W G12340W FM V P = = G 1 W G54780W HM X G W 2 G80123W G W G11024W HG01203XW

14 is a lower partial triangular matrix. This is a 2 by 1 super matrix where the first element in this super matrix is a square lower triangular matrix and the second element is a rectangular matrix. We have seen column super vectors and row super vectors. Now we can define as in case of simple matrices the concept of transpose of a super matrix. We will only indicate the situation by some examples, however for more refer [47]. Consider F9V G W G3W G1W G W G0W V = G2W G W G4W G W G6W G7W G W H2X be the super column vector. Now transpose of V denoted by Vt = [9 3 1 | 0 2 4 6 | 7 2]. Clearly Vt is a super row vector. Thus if Fv V G 1 W V = Gv2 W G W Hv3 X then FVv t Fvt V 1 G 1 W t GW t F tt tV V = GWv2 = Gv2 W = Hvvv123X . GW G t W HXv3 Hv3 X F0V F9V G W G W t G2W For v1 = G3W and v1 = (9 3 1), v2 = and G4W HG1XW G W H6X F V t 7 t v = (0 2 4 6) and v3 = G W and v = (7 2). 2 H2X 3

15 Hence the claim.

Example 1.1.7: Let P = [3 1 0 2 5 | 7 3 2 | 6] be a super row vector. Now transpose of P denoted by F3V G W G1W G0W G W G2W t G5W P = G W . G7W G W G3W G2W G W HG6XW

Thus if P = [P1 P2 P3] where P1 = (3 1 0 2 5), P2 = (7 3 2) and P3 = (6) then F3V G W G1W F7V G W Pt = G0W , Pt = 3 and Pt = 6. 1 G W 2 G W 3 G2W HG2XW HG5XW Thus F t V P1 t G t W P = GP2 W . G t W HP3 X

Example 1.1.8: Consider the super row vector FV810392 GW X = GW021645. HXGW004101

Now define the transpose of X.

16 FV800 GW t GW120 FV810392 GWGW014 FAt V t = 1 X = GW021645= [A1 A2] = GWG W . GW361 HAt X HXGW004101 2 GW940 GW HXGW251

Example 1.1.9: Let F01234V G W G56078W G90102W G W G31403W FW V W = G16051W = G 1 W G W HW X G27107W 2 G W G38020W G49124W G W H50403X be a super column vector. Now the transpose of W given by

FV059312345 GW GW160167890 t GWF ttV W = 201401014 = HWW12X . GW GW370050220 HXGW482317043

Now having see examples of transpose of a super row vector and super column vector we now proceed onto define the transpose of a super matrix.

17 Example 1.1.10: Let FV9012345 GW GW6789011 FMMMV GW1213141 G 123W M = GW = MMM GW5161718 G 456W HGMMMXW GW1920212 789 GW HXGW2003401 be a super matrix, where FV90 F123V F45V M1 = GW, M2 = G W , M3 = G W , HX67 H890X H11X

FV12 F131V F41V GW G W G W M4 = GW51, M5 = G617W , M6 = G18W , HXGW19 HG202XW HG12XW

M7 = (2, 0), M8 = (0, 3, 4) and M9 = [0 1]. Consider FV961512 GW GW072190 GWF tttV 181620 MMM147 t GWG tttW M = GW293103 = GMMM258W GW301724 GMMMtttW GWH 369X GW414110 GW HX511821 where F18V FV F V t 96 t G W t 41 M1 = GW, M2 = G29W , M3 = G W , HX07 H51X HG30XW F162V FV F V t 151 t G W t 411 M4 = GW , M5 = G310W , M6 = G W HX219 H182X HG172XW

18 F0V FV F V t 2 t G W t 0 M7 = GW, M8 = G3W and M9 = G W . HX0 H1X HG4XW Mt is the transpose of the super matrix M.

Example 1.1.11: Let FV19027 GW GW30405 GW67890 FMMV GWG 12W 23405 M = GW = GMM34W GW 01021 GMMW GWH 56X GW50809 GW HX61080 be the super matrix where F190V F27V M1 = G W , M2 = G W , H304X H05X F678V F90V G W G W G234W G05W M3 = , M4 = , G010W G21W G W G W H508X H09X

t M5 = (6 1 0) and M6 = (8, 0). Consider M , the transpose of M. FV1362056 GW GW9073101 FMMMtttV Mt = GW0484080 = G 135W GWHMMMtttX GW2090208 246 HXGW7505190 where FV13 F6205V F V t GWt 20 t G W M1 = GW90 M2 = G W , M3 = G7310W , H75X HXGW04 HG8408XW

19 F6V FV F V t 9020 t G W t 8 M4 = GW, M5 = G1W , M6 = G W HX0519 H0X HG0XW is the transpose of the super matrix M. We say two 1 × m super row vectors are similar if and only if have identical partition. That is partitioned in the same way. Consider X = (0 1 2 | 3 4 | 5 0 9) and Y = (3 0 5 | 2 1 | 8 1 2) two similar super row vectors. However Z = (0 | 1 2 3 | 4 5 0 | 9) is not similar with X or Y as it has a different partition though X = Z = (0 1 2 3 4 5 0 9) as simple matrices. Likewise we can say two n × 1 super column vectors are similar if they have identical partition in them. For consider F9V F2V G W G W G8W G1W G7W G0W G W G W G0W G3W X = G1W and Y = G7W G W G W G2W G8W G W G W G3W G0W G4W G1W G W G W H5X H2X two super column vectors. Clearly X and Y are similar. However if F 8 V G W G10W G 2 W G W G 3 W Z = G 1 W G W G 4 W G W G 9 W G 0 W G W H 2 X

20 is a 9 × 1 simple column matrix but X and Y are not similar with Z as Z has a different partition.

We now proceed onto give examples of similar super matrices.

Example 1.1.12: Let FV3712 F0123V GWG W GW0101 G4567W GW2010 G8910W GWG W GW0304 G1112W GW4010 G1314W X = GW and Y = G W GW5121 G1516W GWG W GW3111 G1718W GW2222 G1920W GWG W GW4444 G2022W GWG W HX0550 H2324X be two super column vectors. Clearly X and Y are similar super column vectors. Consider F12V G W G34W G56W G W G78W G91W P = G W G12W G W G13W G51W G W G70W G W H92X a super column vector. Though both P and Y have similar partition yet P and Y are not similar super matrices.

21 Thus if P and Y are similar in the first place they must have same natural order and secondly the partitions on them must be identical.

Example 1.1.13: Let FV23702579014 GW X = GW05813610825 HXGW16911181336 be a super row vector. Take FV13018917631 GW Y = GW04001704092 HXGW25702822113 be another super row vector. Clearly X and Y are similar super row matrices. That is X and Y have identical partition on them.

Example 1.1.14: Let F31 2 54V G W G78 9 01W G11038− W X = G W G20 4 09W G32− 101W G W HG10 2− 42XW and F 123 45V G W G 677 90W G 111 23W Y = G W G−−14 5 10W G 700 07W G W HG 01− 231XW be two super matrices. Both X and Y enjoy the same or identical partition hence X and Y are similar super matrices.

22 Example 1.1.15: Let F 12345V G W G 30102W G 780− 12W G W G−−−14 10 2W X = G 05101− W G W G 12− 140W G W G 01201W G 10012W G W H−−24 3 4 1X and F40215V G W G02102− W G71162− W G W G080− 70W Y = G1− 4816W G W G80091W G W G31702W G21640− W G W H04021X be two super matrices. Clearly X and Y are similar super matrices as they enjoy identical partition. We can add only when two super matrices of same natural order and enjoy similar partition otherwise addition is not defined on them.

We will just show how addition is performed on super matrices.

Example 1.1.16: Let X = (0 1 2 | 1 -3 4 5 | -7 8 9 0 -1) and Y = (8 -1 3 | -1 2 3 4 | 8 1 0 1 2) be any two similar super row vectors. Now we can add X with Y denoted by X + Y = (0 1 2 | 1 -3 4 5 | -7 8 9 0 -1) + (8 -1 3 | -1 2 3 4 | 8 1 0 1 2) = (8 0 5 | 0 0 -1 7 9 | 1 9 9 1 1). We say X + Y is also a super row vector which is similar with X and Y.

23 Now we have the following nice result.

THEOREM 1.1.1: Let S = {(a1 a2 a3 | a4 a5 | a6 a7 a8 a9 | … | an-1, an) | ai ∈ R; 1 ≤ i ≤ n} be the collection of all super row vectors with same type of partition, S is a group under addition. Infact S is an abelian group of infinite order under addition.

The proof is direct and hence left as an exercise to the reader. If the field of reals R in Theorem 1.1.1 is replaced by Q the field of rationals or Z the integers or by the modulo integers Zn, n < ∞ still the conclusion of the theorem 1.1.1 is true. Further the same conclusion holds good if the partitions are changed. S contains only same type of partition. However in case of Zn, S becomes a finite commutative group.

Example 1.1.17: Let P = {(a1 | a2 a3 | a4 a5 a6 | a7 a8 a9 a10) | ai ∈ Z; 1 ≤ i ≤ 10} be the group of super row vectors; P is a group of infinite order. Clearly P has subgroups.

For take H = {(a1 | a2 a3 | a4 a5 a6 | a7 a8 a9 a10) | ai ∈ 5Z; 1 ≤ i ≤ 10} ⊆ P is a subgroup of super row vectors of infinite order.

Example 1.1.18: Let G = {(a1 a2 | a3 ) | ai ∈ Q; 1 ≤ i ≤ 3} be a group of super row vectors.

Also we have group of super row vectors of the form given by the following examples.

Example 1.1.19: Let IYCSaaaa a a a LLDT14710131619 ∈≤≤ G = JZDTaaaa2 5 8 11 a 14 a 17 a 20 aQ,1i21 i LLDT K[EUaaaa36912151821 a a a be the group of super row vectors under super row vector addition, G is an infinite commutative group.

Example 1.1.20: Let LLIYCSaaaa a 1357 9 ∈≤≤ V = JZDTaR,1i10i K[LLEUaaaaa246810

24 be a group of super row vectors under addition.

Now having seen examples of group super row vectors now we proceed onto give examples of group of super column vectors. The definition can be made in an analogous way and this task is left as an exercise to the reader.

Example 1.1.21: Let IYCSa LLDT1 LLDTa 2 LLDTa LLDT3 LLDTa LL4 DT ∈≤≤ M = JZaaZ,1i95i LLDT DT LLa6 DT LLa LLDT7 DT LLa8 LLDT K[EUa9 be the collection of super column vector with the same type of partition. Now we can add any two elements in M and the sum is also in M. For take CS3 CS7 DT DT DT2 DT0 DT−1 DT1 DT DT DT4 DT2 DT− DT− x = DT7 and y = DT1 DT8 DT5 DT DT DT0 DT9 DT5 DT2 DT DT EU2 EU−1 in M. Now

25 CS3 CS7 CS10 DTDT DT DT2 DT0 DT2 DT−1 DT1 DT0 DTDT DT DT4 DT2 DT6 DT− DT− DT− x + y = DT7 + DT1 = DT8 DT8 DT5 DT13 DTDT DT DT0 DT9 DT9 DT5 DT2 DT7 DTDT DT EU2 EU−1 EU1 is in M. Thus M is an additive abelian group of super column vectors of infinite order. CS0 DT DT0 DT0 DT DT0 DT Clearly DT0 acts as the additive identity in M. DT0 DT DT0 DT0 DT EU0

Example 1.1.22: Let IYCSa LLDT1 LLDTa 2 LLDTa LLDT3 LLDTa P = JZ4 aQ,1i8∈≤≤ DTa i LLDT5 LLDTa LLDT6 LLDTa 7 LLDT K[EUa8 be a group of super column vectors under addition.

26 Example 1.1.23: Let IYCSaaa LLDT123 LLDTaaa456 LLDTaaa LLDT789 ∈≤≤ M = JZDTaaaaR,1i2110 11 12 i LL DTaaa LLDT13 14 15 LL DTaaa16 17 18 LLDT K[LLEUaaa19 20 21 be a group of super column vectors under addition of infinite order. Clearly M has subgroups. For take IYCSaaa LLDT123 LLDTaaa456 LLDTaaa LLDT789 ∈≤≤ ⊆ V = JZDTaaaaQ,1i2110 11 12 i M LL DTaaa LLDT13 14 15 LL DTaaa16 17 18 LLDT K[EUaaa19 20 21 is a subgroup of super column vectors of M.

Now we proceed onto give examples of groups formed out of super matrices with same type of partition.

Example 1.1.24: Let IYCSaaaaa LLDT12345 LLDTaaaaa678910 LLDTaaaaa DT11 12 13 14 15 ∈≤≤ M = JZaQ,1i30i LLDTaaaaa16 17 18 19 20 LLDT aaaaa21 22 23 24 25 LLDT K[LLEUaaaaa26 27 28 29 30

27 be the collection of super matrices of the same type of partition.

Consider F0 1 234V G W G56012W G10201W x = G W G30120W G1− 4201W G W HG20012XW and FV3010 2 GW GW8136 0 GW7616 2 y = GW GW1251 0 GW3470 2 GW HXGW5601− 2 be two super matrices in M.

F 31336V G W G137372W G 86363W x + y = G W G 4 2630W G 4 0903W G W HG 7 6020XW is in M and the sum is also of the same type.

Thus M is a commutative group of super matrices of infinite order.

28 Example 1.1.25: Let IYCSaaaaa LLDT12345 LLDTaaaaa678910 LLDTaaaaa LLDT11 12 13 14 15 LLDTaaaaa LL16 17 18 19 20 DT∈≤≤ P = JZaaaaaaZ,1i4521 22 23 24 25 i LLDT DT LLaaaaa26 27 28 29 30 DT LLaaaaa LLDT31 32 33 34 35 DT LLaaaaa36 37 38 39 40 LLDT K[EUaaaaa41 42 43 44 45 be a group of super matrices of infinite order under addition. Consider the subgroups H = {A ∈ P | entries of A are from 3Z} ⊆ P; H is a subgroup of super matrices of infinite order. Infact P has infinitely many subgroups of super matrices. Consider IYCS00a 0 0 LLDT1 LLDT00a2 0 0 LLDT00a 0 0 LLDT3 LLDTaa 0a a LL45 8 9 DT∈≤≤ W = JZaa6 7 0a 10 a 11 a3Z,1i15 i LLDT DT LL00a12 0 0 DT LL00a 0 0 LLDT13 DT LL00a14 0 0 LLDT K[EU00a15 0 0

⊆ P is a subgroup of super matrices of infinite order under addition of P. Now we cannot make them as groups under multiplication for product of two super matrices do not in general turn out to be a super matrix of the desired form.

Next we leave for the reader to refer [47], for products of super matrices.

& '!$()#$$!*()#$! %!# In this section we recall the notion of refined labels and ordinary labels and discuss the properties associated with them.

Let L1, L2, …, Lm be labels where m ≥ 1 is an integer. Extend this set of labels where m ≥ 1 with a minimum label L0 and maximum label Lm+1. In case the labels are equidistant i.e., the qualitative labels is the same, we get an exact qualitative result, and the qualitative basic belief assignment (bba) is considered normalized if the sum of all its qualitative masses is equal to Lmax = Lm+1 [48]. We consider a relation of order defined on these labels which can be “smaller” “ less in quality” “lower” etc. L1 < L2 < … < Lm. Connecting them to the classical interval [0, 1] we have 0 ≡ L1 < L2 < … < Li < … < Lm < Lm+1 ≡ 1 and i Li = m1+ for i ∈ {0, 1, 2, …, m, m + 1}. [48] The set of labels L ≅ {L0, L1, …, Lm, Lm+1} whose indices are positive integers between 0 and m+1, call the set 1-tuple labels. The labels may be equidistact or non equidistant [ ] call them as ordinary labels. Now we say a set of labels {L1, …, Lm} may not be totally orderable but only partially orderable. In such cases we have at least some Li < Lj, i ≠j; 1 < i, j < m, with L0 the minimum element and Lm+1 the maximum element. We call {L0, L1, …, Lm, Lm+1} a partially ordered set with L0 = 0 and Lm+1 = 1. Then we can define min {Li, Lj} = Li ∩ Lj and max {Li, Lj} = Li ∪ Lj where Li ∩ Lj = Lk or 0 where Li > Lk and Lj > Lk and Li ∪ Lj = Lt or 1 where Lt > Li and Lt > Lj ; 0 ≤ k, t, i, j ≤ m + 1. Thus {0 = L0, L1, …, Lm, Lm+1 = 1, ∪, ∩ (min or max)} is a lattice.

When the ordinary labels are not totally ordered but partially ordered then the set of labels is a lattice. It can also happen that none of the attributes are “comparable” like in study

30 of fuzzy models only “related” in such cases they cannot be ordered in such case we can use the modulo integers models for the set of modulo integers Zm+1 (m+1 < ∞) (n = m+1) is unorderable.

Now [48] have defined refined labels we proceed onto describe a few of them only to make this book a self contained one. For more please refer [48]. The authors in [48] theoretically extend the set of labels L to the left and right sides of the interval [0, 1] towards - ∞ and respectively to ∞. So define IYj LZ JZjZ∈ K[m1+ where Z is the set of all positive and negative integers zero included.

Thus Lz = {.., L-j, …, L-1, L0, L1, …, Lj, …} = {Lj / j ∈ Z} that is the set of extended labels with positive and negative indexes.

Similarly one can define LQ {Lq / q ∈ Q} as the set of labels whose indexes are fractions. LQ is isomorphic with Q q through the isomorphism fQ (Lq) = , for any q ∈ Q. m1+ Even more general we can define IYr LR JZrR∈ K[m1+ where R is the set of reals. LR is isomorphic with R through the r isomorphism fR (r) = for any r ∈ R. The authors in [ ] m1+ prove that {LR, +, ×} is a field, where + is the vector addition of labels and × is the vector multiplication of labels which is called the DSm field of refined labels. Thus we introduce the decimal or refined labels i.e., labels whose index is decimal.

For instance L3/10 = L0.3 and so on. For [L1, L2], the middle of the label is L1.5 = L3/2.

They have defined negative labels L-i which is equal to –Li.

31 (LR, +, ×, .) where ‘.’ means scalar product is a commutative linear algebra over the field of real numbers R with unit element and each non null element in R is invertible with respect to multiplication of labels.

This is called the DSm field and Linear Algebra of Refined labels (FLARL for short).

Operators on FLARL is described below [48].

Let a, b, c ∈ R and the labels a b c La = , Lb = and Lc = . m1+ m1+ m1+

Let the scalars α, β ∈ R.

a b ab+ La + Lb = + = = La+b. m1+ m1+ m1+

a b ab− La – Lb = – = . m1+ m1+ m1+

La × Lb = L(ab)/m+1 since a b ab / m+ 1 . = . m1+ m1+ m1+ For α ∈ R, we have

α. La = La α = Lαa since aaα α.La = α=. . m1++ m1 For

α = –1; –1La = L– a = –La. Also

32 La 1 = L ÷ β = L = L β. β a β a a/

a 1 a a (β ≠ 0) ( La = and × = ). m1+ β m1+ β(m+ 1)

÷ La Lb = L()ab(m1)+ since a b a(a/b)m1+ ÷ = = m1+ m1+ bm1+

= LCSa . DTm1+ EUb

p (L ) = L − a a(m1)pp1+ since

CSaa(m1)p pp1+ − DT= EUm1++ m1 for all p ∈ R.

k 1/k La = (La)

= L 1 −1 , a(m1)1/k + k

p this is got by replacing p by 1/k in (La) . [ ]

Further (LR, +, ×) is isomorphic with the set of real numbers (R, +, ×); it results that (LR, +, ×) is also a field, called the DSm field of refined labels.

The field isomorphism r fR : LR → R; fR (Lr) = m1+ such that

33 fR (La + Lb) = fR (La) + fR (Lb) since fR (La + Lb) = fR (La+b) ab+ = m1+ so a b ab+ fR (La) + fR (Lb) = + = . m1+ m1+ m1+

fR (La × Lb) = fR (La). fR (Lb) since

fR (La × Lb) = fR (L(ab)/(m+1))

ab = (m+ 1)2 and a b fR (La) × fR (Lb) = . m1+ m1+ a.b = . (m+ 1)2

(LR, +, .) is a vector space of refined labels over R. It is pertinent to mention here that (LR, +, .) is also a vector space over LR as LR is a field. So we make a deviation and since LR ≅ R, we need not in general distinguish the situation for we will treat both (LR, +, .) a vector space over R or (LR, +, .) a vector space over LR as identical or one and the same as LR ≅ R. Now we just recall some more operations [48].

Thus (LR, +, ×, .) is a linear algebra of refined labels over the field R of real numbers called DSm linear algebra of refined labels which is commutative. It is easily verified that multiplication in LR is associative and multiplication is distributive with respect to addition.

34 Lm+1 acts as the unitary element with respect to multiplication.

La. Lm+1 = La.m+1 = Lm+1.a = Lm+1.La = La(m+1)/m+1 = La.

All La ≠ L0 are invertible, 1 (L )-1 = L = . a (m+ 1)2 /a La

L . (L )-1 = L . L a a a (m+ 1)2 /a = L (a(m++ 1)2 /a)/m 1

= Lm+1.

Also we have for α ∈ R and La ∈ LR;

La + α = α + La

= La+α (m+1) since α+(m 1) α + La = + L m1+ a

= Lα(m+1) + La

= Lα(m+1)+a

= La + Lα(m+1)

= La+α (m+1). α+(m 1) La – α = La – m1+

= La – Lα (m+1)

= La – α (m+1)

35 and

α – La = Lα (m+1) – a . Further L a × 1 = La = L. a α α α for a ≠ 0. α ÷ L = L a α+(m 1)2 a as α+(m 1) α ÷ La = ÷ La m1+

= Lα(m+1) ÷ La

= L(α(m+1)/a).m+1

= Lα [48]. (m+ 1)2 a

It is important to make note of the following.

If R is replaced by Q still we see LQ ≅ Q and we can say (LQ, +, .) is a linear algebra of refined labels over the field Q. Further (LR, +, .) is also a linear algebra of refined labels over Q. However (LQ, +, .) is not a linear algebra of refined labels over R. For more please refer [48].

36 +%

In this chapter we for the first time introduce the notion of super matrices whose entries are from LR that is super matrices of refined labels and indicate a few of the properties enjoyed by them.

DEFINITION 2.1: Let X = ( LL,,, L) be a row matrix of aa12 an refined labels with entries L ∈ LR; 1 ≤ i ≤ n. If X is partitioned ai in between the columns we get the super row matrix of refined labels.

Example 2.1: Let X = ()LLLLLLL; ai ∈ LR; aaaaaaa1234567 1 < i < 7, be a super row matrix (vector) of refined labels.

Example 2.2: Let Y = ()LLLLLLLLLL ; bbbbbbbbbb12345678910

L ∈ LR, 1 < i < 10, be a super row matrix of refined labels. bi

Now for matrix of refined labels refer [48]. We can just say if we take a row matrix and replace the entries by the refined

37 labels in LR and partition, the row matrix with refined labels then we get the super row matrix (vector) of refined labels. Now we see that two 1 × n super row matrices (vectors) are said to be of same type or similar partition or similar super row vectors if they are partitioned identically. For instance if x = ()L L L ... L and y = ()LLL...L are two aaa123 a n bbb123 b n super row vectors of refined labels then x and y are similar or same type super row vectors. Suppose z = ()LLLL...LL then z and x cccc1234 c n1n− c are not similar super row vectors as z enjoys a different partition from x and y. Thus we see we can add two super row vectors of refined labels if and only if they hold the same type of partition. For instance if x = ()LLLLLL aaaaaa123456 and y = ()LLLLLL bb123456 bb bb where L,L ∈ LR we can add x and y as both x and y enjoy abii the same type of partition and both of them are of natural order 1 × 6. x + y = ()LLLLLL + aaaaaa123456 ()LLLLLL bb123456 bb bb =++++++()LLLLLLLLLLLL abababababab112233445566

= ()LLLLLL++++++. ab11 ab 22 ab 33 ab 44 ab 55 ab 66 We see x + y is of the same type as that of x and y. In view of this we have the following theorem.

THEOREM 2.1: Let V=∈≤≤{}() LLLLL...LL L L;1in aaaaa12345 a n1n− a a i R be the collection of all super row vectors of refined labels of same type. Then V is an abelian group under addition.

Proof is direct and hence is left as an exercise to the reader. Now if FL V G a1 W GLa W G 2 W L G a3 W X = GL W G a4 W GL W G a5 W G W G W L H an X denotes the super matrix column vector (matrix) of refined labels if (i) Entries of X are elements from LR. (ii) X is a super matrix. Thus if a usual (simple) row matrix is partitioned and the entries are taken from the field of refined labels then X is defined as the super column matrix (vector) of refined labels. We will illustrate this situation by an example.

Example 2.3: Let FL V G a1 W L G a2 W GLa W G 3 W X = L G a4 W L G a5 W GL W G a6 W GL W H a7 X where L ∈ LR; 1 < i < 7 be the super column vector of refined ai labels.

Example 2.4: Consider the super column vector of refined labels given by

39 F L V G a1 W G La W G 2 W L G a3 W G L W G a4 W G W La P = G 5 W G La W G 6 W L G a7 W G L W G a8 W G L W G a9 W GL W H a10 X where L ∈ LR; 1 ≤ i ≤ 10. ai Example 2.5: Let FL V G a1 W GLa W M = G 2 W L G a3 W GL W H a4 X where L ∈ LR; 1 ≤ i ≤ 4 is a super column vector of refined ai labels. Now we can say as in case of super row vectors of refined labels, two super column vectors of refined labels are similar or of same type is defined in an analogous way. We will illustrate this situation by some examples. Consider FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W A = and B = GL W GL W G a4 W G b4 W GL W GL W G a5 W G b5 W GL W GL W H a6 X H b6 X

40 be two super column vectors of same type of similar refined labels; L,L ∈ LR; 1 ≤ i, j ≤ 6 . Take abij FL V G b1 W GLb W G 2 W L G b3 W P = GL W G b4 W GL W G b5 W GL W H b6 X be a super column vector of refined labels, we see P is not equivalent or similar or of same type as A and B. Now having understood the concept of similar type of super column vectors we now proceed onto define addition. In the first place we have to mention that two super column vectors of refined labels are compatible under addition only if; (1) Both the super column vectors of refined labels A and B must be of same natural order say n × 1. (2) Both A and B must be having the same type of partition.

Now we will illustrate this situation by some examples. Let FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W G W G W La Lb A = G 4 W and B = G 4 W GL W GL W G a5 W G b5 W GLa W GLb W G 6 W G 6 W L L G a7 W G b7 W GL W GL W H a8 X H b8 X

41 be two super column vectors of refined labels over LR; that is

L,L ∈ LR; 1 < i, j < 8. abij Clearly A and B are of same type super column vectors of refined labels. Consider FL V G a1 W GLa W G 2 W L G a3 W G W La M = G 4 W GL W G a5 W GLa W G 6 W L G a7 W GL W H a8 X the super column vector of refined labels is not of same type A and B. Clearly all A, B and M are of same natural order 8 × 1 but are not of same type as M enjoys a different partition from A and B.

In view of this we have the following theorem.

THEOREM 2.2: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW K = JZLa LL;1in∈≤≤ GW5 aRi LL LLGW LLGWL LLGWan2− GW LLLa LLGWn1− LLGWLa K[HXn

42 be a collection of super column vectors of refined labels of same type with entries from LR. K is a group under addition of infinite order.

The proof is direct and simple hence left as an exercise to the reader. Let FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W GL W GL W G a4 W G b4 W A = GL W and B = GL W G a5 W G b5 W GLa W GLb W G 6 W G 6 W L L G a7 W G b7 W GL W GL W G a8 W G b8 W GL W GL W H a9 X H b9 X where L,L ∈ LR; 1 ≤ i, j ≤ 9 and abij

FL V FL V F LL+ V F L + V G a1 W G b1 W G ab11W G ab11W + GLa W GLb W GLLabW GLab+ W G 2 W G 2 W G 22W G 22W L L LL+ L + G a3 W G b3 W G ab33W G ab33W G W G W G W G W L L LL+ L + G a4 W G b4 W G ab44W G ab44W A + B = GL W + GL W = GLL+ W = GL + W . G a5 W G b5 W G ab55W G ab55W + GLa W GLb W GLLabW GLab+ W G 6 W G 6 W G 66W G 66W L L LL+ L + G a7 W G b7 W G ab77W G ab77W G W G W G W G W L L LL+ L + G a8 W G b8 W G ab88W G ab88W

GL W GL W GLL+ W GL + W H a9 X H b9 X H ab99X H ab99X

It is easily seen A + B is again of same type as that of A and B. Now the super row vectors of refined labels and super column vectors of refined labels studied are simple super row vectors and simple column vectors of refined labels. We can

43 extend the results in case of super column vectors (matrices) and super row vectors (matrices) of refined labels which is not simple.

We give just examples of them.

Example 2.6: Let FVLLLL L L GWaaaaaa147101116

V = GWLLLLaaaa L a L a GW258121317 GWLLLL L L HXaaaaaa369141518

where L ∈ LR; 1 < i < 18 be a super row vector of refined ai labels.

Example 2.7: Let F LLLLV G aaaa1234W G LLLLaaaaW G 5678W LLLL G aa9101112 aaW GLLLLW G aaaa13 14 15 16 W G W LLLLaaaa M = G 17 18 19 20 W GLLLLaaaaW G 21 22 23 24 W LLLL G aaaa25 26 27 28 W GLLLLW G aaaa29 30 31 32 W GLLLLW G aaaa33 34 35 36 W GLLLLW H aaaa37 38 39 40 X

where L ∈ LR; 1 ≤ i ≤ 40 be a super column vector of refined ai labels. When we have m × n matrix whose entries are refined labels we can partition them in many ways while by partition we get distinct super column vectors (super row vectors); so the study of super matrices gives us more and more matrices.

44 For instance take FL V G a1 W GLa W G 2 W L G a3 W G W La V = G 4 W GL W G a5 W GLa W G 6 W L G a7 W GL W H a8 X where L ∈ LR; 1 < i < 8. Now how many super matrices of ai refined labels can be built using V. FVL FL V FL V FL V GWa1 G a1 W G a1 W G a1 W GWLa GLa W GLa W GLa W GW2 G 2 W G 2 W G 2 W L L L L GWa3 G a3 W G a3 W G a3 W GW G W G W G W La La La La V = GW4 , V = G 4 W , V = G 4 W , V = G 4 W , 1 GWL 2 GL W 3 GL W 4 GL W GWa5 G a5 W G a5 W G a5 W GWLa GLa W GLa W GLa W GW6 G 6 W G 6 W G 6 W L L L L GWa7 G a7 W G a7 W G a7 W GWL GL W GL W GL W HXa8 H a8 X H a8 X H a8 X FV F V F V La La La FL V GW1 G 1 W G 1 W G a1 W L L L GWa2 G a2 W G a2 W GLa W GW G W G W G 2 W La La La L GW3 G 3 W G 3 W G a3 W GW G W G W G W La La La L GW4 G 4 W G 4 W G a4 W V5 = GW, V6 = G W , V7 = G W , V8 = , La La La GL W GW5 G 5 W G 5 W G a5 W GWL GL W GL W a6 a6 a6 GLa W GW G W G W G 6 W La La La L GW7 G 7 W G 7 W G a7 W GW G W G W G W La La La L HX8 H 8 X H 8 X H a8 X

FVL FL V FL V FL V GWa1 G a1 W G a1 W G a1 W GWLa GLa W GLa W GLa W GW2 G 2 W G 2 W G 2 W L L L L GWa3 G a3 W G a3 W G a3 W GW G W G W G W La La La La V = GW4 , V = G 4 W , V = G 4 W , V = G 4 W , 9 GWL 10 GL W 11 GL W 12 GL W GWa5 G a5 W G a5 W G a5 W GWLa GLa W GLa W GLa W GW6 G 6 W G 6 W G 6 W L L L L GWa7 G a7 W G a7 W G a7 W GWL GL W GL W GL W HXa8 H a8 X H a8 X H a8 X

F V FVL FL V La FL V GWa1 G a1 W G 1 W G a1 W L GWLa GLa W G a2 W GLa W GW2 G 2 W G W G 2 W L L La L GWa3 G a3 W G 3 W G a3 W GW G W G W G W L L La L GWa4 G a4 W G 4 W G a4 W V13 = , V14 = , V15 = G W , V16 = , GWL GL W La GL W GWa5 G a5 W G 5 W G a5 W GL W GWLa GLa W a6 GLa W GW6 G 6 W G W G 6 W L L La L GWa7 G a7 W G 7 W G a7 W GW G W G W G W L L La L HXa8 H a8 X H 8 X H a8 X

FVL FL V FL V GWa1 G a1 W G a1 W GWLa GLa W GLa W GW2 G 2 W G 2 W L L L GWa3 G a3 W G a3 W GWL GL W GL W GWa4 G a4 W G a4 W V17 = , V18 = , V19 = GWL GL W GL W GWa5 G a5 W G a5 W GWLa GLa W GLa W GW6 G 6 W G 6 W L L L GWa7 G a7 W G a7 W GWL GL W GL W HXa8 H a8 X H a8 X and so on.

46 Thus using a single ordinary column matrix of refined labels we obtain many super column vectors of refined labels. So we can for each type of partition on V get a group under addition. This will be exhibited from the following: Consider FL V G a1 W GLa W X = G 2 W L G a3 W GL W H a4 X be a column vector of refined labels. F V FVL FL V La FL V GWa1 G a1 W G 1 W G a1 W L L GLa W L GWa2 G a2 W 2 G a2 W X1 = GW, X2 = G W , X3 = G W , X4 = G W , L L La L GWa3 G a3 W G 3 W G a3 W GW G W G W G W L L La L HXa4 H a4 X H 4 X H a4 X

FL V FL V FL V G a1 W G a1 W G a1 W GLa W GLa W GLa W X = G 2 W , X = G 2 W and X = G 2 W . 5 L 6 L 7 L G a3 W G a3 W G a3 W GL W GL W GL W H a4 X H a4 X H a4 X

Thus we have seven super column vectors of refined labels for a given column vector refined label X. Now if IYFVL LLGWa1 LLL LLGWa2 T1 = JZXLL;1i4=∈≤≤GW 1aRL i LLGWa3 LL GWL K[LLHXa4 be the collection of all super column vectors of refined labels. Then T1 is a group under addition of super column vectors of refined labels.

47 Likewise IYFVL LLGWa1 LLL LLGWa2 T2 = JZXLL;1i4=∈≤≤GW 2aRL i LLGWa3 LL GWL K[LLHXa4 is a super column vector group of refined labels.

Thus Ti = {}XL∈≤≤ L;1i4 is a super column vector iai R of refined labels i=1,2, 3, …, 7. Hence we get seven distinct groups of super column vectors of refined labels. Consider Y = ()L L L L L be a super rows aaaaa12345 vector (matrix) of refined labels.

Y1 = ()LLLLL , Y2 = ()LLLLL aaaaa12345 aaaaa12345

Y3 = ()LLLLL , Y4 = ()LLLLL , aaaaa12345 aaaaa12345

Y5 = ()LLLLL , Y6 = ()LLLLL , aaaaa12345 aaaaa12345

Y7 = ()LLLLL , Y8 = ()LLLLL , aaaaa12345 aaaaa12345

Y9 = ()LLLLL , Y10 = ()LLLLL , aaaaa12345 aaaaa12345

Y11 = ()LLLLL , Y12 = ()LLLLL , aaaaa12345 aaaaa12345

Y13 = ()LLLLL , Y14 = ()LLLLL aaaaa12345 aaaaa12345

and Y15 = ()LLLLL . aaaaa12345 Thus we have 15 super row vectors of refined labels built using the row vector Y of refined labels. Hence we have 15 groups associated with the single row vector X = ()LLLLL aaaaa12345 of refined labels. This is the advantage of using super row vectors in the place of row vectors of refined labels. Consider the matrix

48 F LLLV G aaa123W G LLLaaaW P = G 456W LLL G aaa789W GLLLW H aaa10 11 12 X of refined labels. How many super matrices of refined labels can be got from P. FVLLL F LLLV GWaaa123 G aaa123W LLL LLL GWaaa456 G aaa456W P1 = GW, P2 = G W , LLL LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X FVLLL F LLLV GWaaa123 G aaa123W LLL LLL GWaaa456 G aaa456W P3 = GW, P4 = G W , LLL LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X FV LLLaaa F LLLV GW123 G aaa123W GWLLLaaa LLL 456 G aaa456W P5 = GW, P6 = G W , LLLaaa LLL GW789 G aaa789W GWG W LLLaaa LLL HX10 11 12 H aaa10 11 12 X FVLLL F LLLV GWaaa123 G aaa123W LLL LLL GWaaa456 G aaa456W P7 = GW, P8 = G W , LLL LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X FV LLLaaa F LLLV GW123 G aaa123W GWLLLaaa LLL 456 G aaa456W P9 = GW, P10 = G W , LLLaaa LLL GW789 G aaa789W GWG W LLLaaa LLL HX10 11 12 H aaa10 11 12 X

49 F V FVLLL LLLaaa GWaaa123 G 123W LLL G LLLaaaW GWaaa456 456 P11 = GW P12 = G W , LLL LLLaaa GWaaa789 G 789W GWG W LLL LLLaaa HXaaa10 11 12 H 10 11 12 X FVLLL F LLLV GWaaa123 G aaa123W LLL LLL GWaaa456 G aaa456W P13 = GW, P14 = G W , LLL LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X FV LLLaaa F LLLV GW123 G aaa123W GWLLLaaa LLL 456 G aaa456W P15 = GW, P16 = G W , LLLaaa LLL GW789 G aaa789W GWG W LLLaaa LLL HX10 11 12 H aaa10 11 12 X FVLLL F LLLV GWaaa123 G aaa123W GWLLLaaa G LLLaaaW P = GW456, P = G 456W , 17 LLL 18 LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X F V FVLLL LLLaaa GWaaa123 G 123W LLL G LLLaaaW GWaaa456 456 P19 = GW, P20 = G W , LLL LLLaaa GWaaa789 G 789W GWG W LLL LLLaaa HXaaa10 11 12 H 10 11 12 X FV LLLaaa F LLLV GW123 G aaa123W GWLLLaaa LLL 456 G aaa456W P21 = GW, P22 = G W , LLLaaa LLL GW789 G aaa789W GWG W LLLaaa LLL HX10 11 12 H aaa10 11 12 X

50 FVLLL F LLLV GWaaa123 G aaa123W GWLLLaaa G LLLaaaW P = GW456, P = G 456W , 23 LLL 24 LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X FVLLL F LLLV GWaaa123 G aaa123W GWLLLaaa G LLLaaaW P = GW456, P = G 456W 25 LLL 26 LLL GWaaa789 G aaa789W GWLLL GLLLW HXaaa10 11 12 H aaa10 11 12 X and F LLLV G aaa123W G LLLaaaW P = G 456W . 27 LLL G aaa789W GLLLW H aaa10 11 12 X Thus the matrix P of refined labels gives us 27 super matrix of refined labels their by we can have 27 groups of super matrices of refined labels using P1, P2, …, P27. Now suppose we study the super matrix of refined labels using L + = {L | r ∈ R{0}∪ r R+ ∪ {0}} then also we can get the super matrix of refined labels. In this case any super matrix of refined labels collection of a particular type may not be a group but only a semigroup under addition. Let

X = {}()LLLLLL L∈ L+ aa123456 aa aa a i R{0}∪ be a collection of super row vector of refined labels with entries from L + . Clearly X is only a semigroup as no element has R{0}∪ inverse. Thus we have the following theorem.

THEOREM 2.3: Let

X = {}()LLL...LLLL∈≤≤+ ;1in aaa123 a n1ni− aaR{0}∪

51 be a collection of super row vectors of refined labels. X is a commutative semigroup with respect to addition.

The proof is direct and simple and hence left as an exercise to the reader. Now consider IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 M = JZLL∈≤≤+ ;1in GWai R{0}∪ LLLa GW5 LLGWL LLa6 LLGW LLGW LLGWL K[HXan be the super column vector of refined labels. Now we say as in case of other labels that two super column vectors of refined labels x and y from L + are equivalent or R{0}∪ similar or of same type if both x and y have the same natural order and enjoy identical partition on it. We can add only those two super column vector refined labels only when they have similar partition. Consider FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W G W G W X = La and Y = Lb G 4 W G 4 W GL W GL W G a5 W G b5 W GLa W GLb W G 6 W G 6 W L L HG a7 XW HG b7 XW where L ∈ L + ; 1 ≤ i ≤ 7. ai R{0}∪

52 X+Y is defined and X+Y is again a super column vector of refined labels of same type. In view of this we have the following theorem.

THEOREM 2.4: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW ∈≤≤ M = JZLLa + ∪ ;1in LLGWi R{0} GWL LLan2− LLGW GWL LLan1− GW LLL K[HXan be the collection of super column vectors of refined labels of same type with entries from L + . M is a semigroup under R ∪{0} addition.

This proof is also direct and hence left as an exercise to the reader.

Now we show how two super column vectors of refined labels are added.

Let FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W G W G W La Lb X = G 4 W and Y = G 4 W GL W GL W G a5 W G b5 W GLa W GLb W G 6 W G 6 W L L G a7 W G b7 W GL W GL W H a8 X H b8 X

53 be any two super column vectors of refined labels. Both X and Y are of same type and X+Y is also of the same type as X and Y. We see now as in case of usual super column vector of refined labels we have

F LLLV G aaa123W G LLLaaaW G 456W LLL G aaa789W X = GLLLW G aaa10 11 12 W GLLLW G aaa13 14 15 W GLLLW H aaa16 17 18 X

is a super column vector of refined labels from the set L + . R{0}∪ Likewise

IYFVLLL L L L LLGWaa15 a 9131721 a a a LLGWLLL L L L aaa2 6 10 a 14 a 18 a 22 ∈ Y = JZGWLL+ LLL L L L ai R{0}∪ LLGWaaaaaa3 7 11 15 19 23 LL GWLLLLLL K[LLHXaaaaaa4 8 12 16 20 24 is a super row vector of refined labels and it is easily verified Y is not a simple super row vector of refined labels.

Now it is interesting to note L + = {set of all refined R{0}∪ labels with positive value} is a semifield of refined labels or + DSm semifield of refined labels and L + ≅ R ∪ {0} [48]. R{0}∪

Thus we see further F V LL LL G aa12 a 3 a 4W A = GLLLLW G aaa5678 aW GLL L LW H aa9101112 a aX

54 is a super matrix of refined labels with entries from L + . R{0}∪

Suppose F LLLLV G aaaa1234W G LLLLaaaaW A = G 5678W LLLL G aa9101112 aaW GLLLLW H aaaa13 14 15 16 X be a super matrix of refined labels with entries from L + . R{0}∪ Now if we consider super matrices of refined labels with entries from L + of same natural order and similar type of partition R{0}∪ then addition can be defined.

We will just show how addition is defined when super matrices of refined labels are of same type.

F LLLLLV G aaaa12345 aW G LLLLLaaaaaW Let A=G 678910W and LLLLL G aaaaa11 12 13 14 15 W GLLLLLW H aaaaa16 17 18 19 20 X

F LLLLLV G bbbbb12345W G LLLLLbbbbbW B =G 678910W LLLLL G bbbbb11 12 13 14 15 W GLLLLLW H bbbbb16 17 18 19 20 X be any two super matrices of refined labels with entries from

L + ; 1 ≤ i ≤ 20. R{0}∪ F LLLLLV G aaaa12345 aW G LLLLLaaaaaW A + B = G 678910W + LLLLL G aaaaa11 12 13 14 15 W GLLLLLW H aaaaa16 17 18 19 20 X

55 FVLLLLL GWbbbbb12345 GWLLLLLbbbbb GW678910 LLLLL GWbbbbb11 12 13 14 15 GWLLLLL HXbbbbb16 17 18 19 20

FVLL+++++ LL LL LL LL GWabab11 22 ab 33 ab 44 ab 55 +++++ GWLLab LL ab LL ab LLL aba L b = GW66 77 88 991010 LLLLLLLLLL+++++ GWababababab11 11 12 12 13 13 14 14 15 15 GWLLLLLLLLLL+++++ HXababababab16 16 17 17 18 18 19 19 20 20

FVLLLL+++++ L GWab11 ab 22 ab 33 ab 44 ab 55 GWLLLLLab+++++ ab ab ab a b = GW66 77 88 99 1010 . LLLLL+++++ GWab11 11 ab 12 12 ab 13 13 ab 14 14 ab 15 15 GW LLLLL+++++ HXab16 16 ab 17 17 ab 18 18 ab 19 19 ab 20 20

We see A+B, A and B are of same type. In view of this we have the following theorem.

IFaaaV LG 123W THEOREM 2.5: Let A = JGaaa456W ai are submatrices of LG W KHaaa789X refined labels with entries from L + ; 1 ≤ i ≤ 9}. A is a R{0}∪ semigroup under addition.

Here A is a collection of all super matrices of refined labels where a1, a2 and a3 are submatrices of refined labels with same number of rows. Also a4, a5 and a6 are submatrices of refined labels with same number of rows. a7, a8 and a9 are submatrices of refined labels with same number of rows.

Similarly a1, a4 and a7 are submatrices of refined labels with same number of columns. a2, a5 and a6 are submatrices of

56 refined labels with same number of columns. a3, a6 and a9 are submatrices of refined labels with same number of columns. Thus A is a semigroup under addition.

Now L + gives the collection of all refined labels and Q{0}∪ + L + ≅ Q ∪ {0}, infact L + ⊆ L + . Q{0}∪ Q{0}∪ R{0}∪

Suppose we take the labels 0 = L0 < L1 < L2 < . . . < Lm = Lm+1 = 1, that the collection of ordinary labels; then with min or max or equivalently ∩ or ∪, these labels are semilattices or semigroups. That is L = {0 = L0 < L1 < L2 < ... < Lm < Lm+1 = 1} is a semilattice. Infact when both ∪ and ∩ are defined L is a lattice called the chain lattice.

Let P = {}()LLLLLLL∈ = L0 < L1 < aa12345i aa aa

... < Lm < Lm+1= 1} be a super row vector of ordinary labels. We say two super row vector ordinary labels are similar or identical if they are of same natural order and hold same or identical partition on it. We call such super row vectors of ordinary labels as super row vectors of same type. Let X= ()LLLLLLand aaaaaa123456 Y = ()LLLLLL bbbbbb123456 be two super row vectors of ordinary labels; L,L ∈ L = {0 = abij

L0 < L1 < L2 < ... < Lm < Lm+1 = 1}, 1 ≤ i, j ≤ 6. Now ‘+’ cannot be defined. We define ‘∪’which is max ( L,L). abij Thus X ∪ Y = ()LLLLLL ∪ aaaaaa123456 ()LLLLLL bbbbbb123456 =∪∪∪∪∪∪()LLLLLLLLLLLL abababababab112233445566 = (max{ L,L } | max { L,L}, max { L,L }| max ab11 ab22 ab33 { L,L} max { L,L } max { L,L}) ab44 ab55 ab66 = ()LLLLLL. cccccc123456

We see X, Y and X ∪ Y are same type of super row vectors of ordinary labels. In an analogous way we can define X ∩ Y or min {X,Y}. X ∩Y is again of the same type as X and Y. Now we have the following interesting results.

THEOREM 2.6: Let V = {()LLLLLLL...... LLL aaaaaaa1234567 a n1ni− aa

∈ L= {0 = L0 < L1 < . . . < Lm < Lm+1= 1}; 1 ≤ i ≤ an} be the collection of super row vectors of ordinary labels. V is a semigroup under ‘∪’.

THEOREM 2.7: Let V= {()LLLLLLL...... LLL aaaaaaa1234567 a n1ni− aa

∈ L = {0 = L0 < L1 < . . . < Lm = Lm+1 = 1}; 1 ≤ i ≤ an} be the collection of super row vectors of ordinary labels. V is a semigroup under ‘∩’.

The proof is left as an exercise to the reader.

In the theorem 2.7 if ‘∩’ is replaced by ‘∪’, V is a semigroup of super row matrices (vector) of ordinary labels. Now consider FL V G a1 W GLa W G 2 W L G a3 W G W La P = G 4 W GL W G a5 W GLa W G 6 W L G a7 W GL W H a8 X

58 where L ∈ L = {0 = L0 < L1 ai

< . . . < Lm < Lm+1=1}, 1≤ i ≤ 8, P is a super column vector of ordinary labels. Now we cannot in general define ∪ or ∩ on any two super column vectors of ordinary labels even if they are of same natural order. Any two super column vectors of ordinary labels of same natural order can have ∪ or ∩ defined on them only when the both of them enjoy the same type of partition on it. We will now give a few illustrations before we proceed on to suggest some related results.

Let FL V FL V G a1 W G b1 W GLa W GLb W G 2 W G 2 W L L G a3 W G b3 W GL W GL W G a4 W G b4 W G W G W T = La and S = Lb G 5 W G 5 W GLa W GLb W G 6 W G 6 W L L G a7 W G b7 W GL W GL W G a8 W G b8 W GL W GL W H a9 X H b9 X be two super column vector matrices of ordinary labels where

L,L ∈ {L; 0 = L0 < L1 < L2 < ... < Lm < Lm+1}; 1 ≤ i , j ≤ 9. abij Now we define min {T, S}= T ∩ S as follows min {T,S}=

59 FVL FL V F min{L ,L }V GWa1 G b1 W G ab11W GWLa GLb W Gmin{Lab ,L }W GW2 G 2 W G 22W L L min{L ,L } GWa3 G b3 W G ab33W GWL GL W Gmin{L ,L }W GWa4 G b4 W G ab44W GWG W G W min { La , Lb } = min{Lab ,L } GW5 G 5 W G 55W GWLa GLb W Gmin{Lab ,L }W GW6 G 6 W G 66W L L min{L ,L } GWa7 G b7 W G ab77W GWL GL W Gmin{L ,L }W GWa8 G b8 W G ab88W GWL GL W Gmin{L ,L }W HXa9 H b9 X H ab99X

FVL FL V FLL∩ V F L ∩ V GWa1 G b1 W G ab11W G ab11W ∩ GWLa GLb W GLLabW GLab∩ W GW2 G 2 W G 22W G 22W L L LL∩ L ∩ GWa3 G b3 W G ab33W G ab33W GW G W G W G W L L LL∩ L ∩ GWa4 G b4 W G ab44W G ab44W GW G W G ∩ W G W = T ∩ S = La ∩ Lb = LLab = Lab∩ GW5 G 5 W G 55W G 55W ∩ GWLa GLb W GLLabW GLab∩ W GW6 G 6 W G 66W G 66W L L LL∩ L ∩ GWa7 G b7 W G ab77W G ab77W GW G W G W G W L L LL∩ L ∩ GWa8 G b8 W G ab88W G ab88W

GWL GL W GLL∩ W GL ∩ W HXa9 H b9 X H ab99X H ab99X

60 F L V G min{a11 ,b } W GLmin{a ,b } W G 22W L G min{a33 ,b } W GL W G min{a44 ,b } W G W = Lmin{a ,b } . G 55W GLmin{a ,b } W G 66W L G min{a77 ,b } W GL W G min{a88 ,b } W GL W H min{a99 ,b } X

We make a note of the following.

This min = ∩ (or max = ∪) are defined on super column vectors of ordinary labels only when they are of same natural order and enjoy the same type of partition.

THEOREM 2.8: Let IFVL LGWa1 LGWLa LGW2 L LGWa3 L GWL LGWa4 LGW L La K = JGW5 L ∈L GWL ai L a6 LGW L LGWa7 LGW LGW LGWL LGWan−1 LGWL KHXan

= {0 = L0 < L1 < L2 < ... < Lm < Lm+1 = 1}} be the collection of all super column vectors of ordinary labels which enjoy the same form of partition.

61 (i) {K, ∪} is a semigroup of finite order which is commutative ( semilattice of finite order) (ii) {K ,∩} is a semigroup of finite order which is commutative ( semilattice of finite order) (iii) {K ,∪, ∩} is a lattice of finite order.

Now we study super matrices of ordinary labels.

Consider FLLV G aa12W GLLaaW H = G 34W LL G aa56W GLLW H aa78X is a super matrix of ordinary labels. Let FLLV G aa12W GLLaaW M = G 34W LL G aa56W GLLW H aa78X again a super matrix of ordinary labels with entries from L= {0 = L0 < L1 < L2 < . . . < Lm < Lm+1 =1}.

Take FLLV G aa12W GLLaaW P = G 34W , LL G aa56W GLLW H aa78X P is again a super matrix of ordinary labels.

62 FLLV G aa12W GLLaaW R = G 34W LL G aa56W GLLW H aa78X is a supermatrix of ordinary labels. FLLV FLLV G aa12W G aa12W GLLaaW GLLaaW V= G 34W and W= G 34W LL LL G aa56W G aa56W GLLW GLLW H aa78X H aa78X are super matrices ordinary labels.

F V LLaa FLLV G 12W G aa12W LL G aa34W GLLaaW T = G W and B = G 34W LLaa LL G 56W G aa56W G W G W LLaa LL H 78X H aa78X are super matrices of ordinary labels.

F V LLaa FLLV G 12W G aa12W LL G aa34W GLLaaW C = G W and D = G 34W LLaa LL G 56W G aa56W G W G W LLaa LL H 78X H aa78X are super matrices of ordinary labels.

FVLL FLLV GWaa12 G aa12W GWLLaa GLLaaW E = GW34 and F = G 34W LL LL GWaa56 G aa56W GWLL GLLW HXaa78 H aa78X are super matrices of ordinary labels.

F V FVLL LLaa GWaa12 G 12W LL GWLLaa G aa34W G = GW34 and L = G W LL LLaa GWaa56 G 56W GW G W LL LLaa HXaa78 H 78X are super matrices of ordinary labels. Now FLLV G aa12W GLLaaW Q = G 34W LL G aa56W GLLW H aa78X is again a super matrix of ordinary labels. Thus we have H , M , P , R , V , W , T , B , C , D , E , F , G , L and Q; 15 super matrices of ordinary labels of natural order 4 × 2. Now we can also get 15 super matrix semigroups under ‘∪’ and 15 super matrix semigroup under ∩ of finite order.

Thus by converting an ordinary label matrices into super matrices makes one get several matrices out of a single matrix.

THEOREM 2.9: Let A = {all m × n super matrices of ordinary labels of same type of partition}. A is a semigroup under ∩ (or semigroup under ∪) of finite order which is commutative.

Now having seen examples and theorems about super matrices of ordinary labels, we in the chapter four proceed on to define linear algebra and vector spaces of super matrices. It is important to note that in general usual products are not defined among super matrices. In chapter three we define products and transpose of super label matrices. However even the collection of super matrices of same type are not closed in general under product.

In this chapter we proceed on to define transpose of a super matrix of refined labels (ordinary labels) and define product of super matrices of refined labels. Let FL V G a1 W GLa W G 2 W L G a3 W A = GL W G a4 W GL W G a5 W GLa W G 6 W L H a7 X be a super column matrix of refined labels with entries from LR. Clearly the transpose of A denoted by

65 FVL t GWa1 GWLa GW2 L GWa3 t GW FV A = La = LLLLLLL; GW4 HXaaaaaaa1234567 GWL GWa5 GWLa GW6 L HXa7 Thus we see the transpose of A is the super row vector (matrix) of refined labels. If we replace the refined labels by ordinary labels also the results holds good. Consider P = ()LLLLLLLLLL aaaaaaaaaa12345678910 be a super row vector (matrix) of refined labels.

Now L ∈ LR ; 1 ≤ i ≤ 9. Now the transpose of P denoted ai by Pt is F L V G a1 W G La W G 2 W L G a3 W G L W G a4 W G W t La ()LLLLLLLLLL = G 5 W ; aaaaaaaaaa12345678910 G La W G 6 W L G a7 W G L W G a8 W G L W G a9 W GL W H a10 X we see Pt is a super column matrix (vector ) of refined labels. However if we replace the refined labels by ordinary labels still the results hold good.

66 Let F LLLLLV G aaaaa12345W G LLLLLaaaaaW G 678910W A = LLLLL G aaaaa11 12 13 14 15 W GLLLLLW G aaaaa16 17 18 19 20 W GLLLLLW H aaaaa21 22 23 24 25 X be a super matrix of refined labels with entries from LR; ie L ∈ ai

LR ; 1 ≤ i ≤ 25. F LLLLLVt G aaaaa12345W G LLLLLaaaaaW G 678910W At = LLLLL G aaaaa11 12 13 14 15 W GLLLLLW G aaaaa16 17 18 19 20 W GLLLLLW H aaaaa21 22 23 24 25 X FLLLL LV G aaa16111621 a aW GLLLLLaaa a aW G 2 7 12 17 22 W = LLLLL G aaa3 8 13 a 18 a 23 W GLLLLLW G aaa4 9 14 a 19 a 24 W GLLLLLW H aaaaa510152025X is again a supermatrix of refined labels with entries from LR; L ∈ LR ; 1 ≤ i ≤ 25. ai Let F LLLLLLV G aaaaaa123456W G LLLLLLaaaaaaW G 789101112W LLLLLL G aaaaaa13 14 15 16 17 18 W P = GLLLLLLW G aaaaaa19 20 21 22 23 24 W GLLLLLLW G aaaaaa25 26 27 28 29 30 W GLLLLLLW H aaaaaa31 32 33 34 35 36 X

67 be a super matrix of refined labels with entries from LR; L ∈ ai

LR; 1 ≤ i ≤ 36.

FVLLLLLLt GWaaaaaa123456 GWLLLLLLaaaaaa GW789101112 LLLLLL GWaaaaaa13 14 15 16 17 18 Pt = GWLLLLLL GWaaaaaa19 20 21 22 23 24 GWLLLLLL GWaaaaaa25 26 27 28 29 30 GWLLLLLL HXaaaaaa31 32 33 34 35 36

FVLLL L L L GWaaa1 7 13 a 19 a 25 a 31 GWLLLLaaa a LL a a GW2 8 14 20 26 32 LLLL L L GWaaa3 9 15 a 21 a 27 a 33 = GWLL L L L L GWaa41016222834 a a a a GWLLLLLL GWaa51117232935 a a a a GWLL L L L L HXaa61218243036 a a a a is again a supermatrix of refined labels with entries from LR; L ∈ LR; 1 ≤ i ≤ 36. ai Suppose F LLLV G aaa123W G LLLaaaW G 456W LLL G aaa789W GLLLW G aaa10 11 12 W X = GLLLW G aaa13 14 15 W GLLLaaaW G 16 17 18 W LLL G aaa19 20 21 W GLLLW G aaa22 23 24 W GLLLW H aaa25 26 27 X

68 be a super column vector (matrix); with entries from LR (L ∈ ai

LR; 1 ≤ i ≤ 27 ) . The transpose of X denoted by F LLLVt G aaa123W G LLLaaaW G 456W LLL G aaa789W GLLLW G aaa10 11 12 W Xt = GLLLW G aaa13 14 15 W GLLLaaaW G 16 17 18 W LLL G aaa19 20 21 W GLLLW G aaa22 23 24 W GLLLW H aaa25 26 27 X

FVLLLLLLLLL GWaaaaaaaaa147101316192225

= GWLLLLaaaa L a L a L a L a L a GW258111417202326 GWLLLL L L L L L HXaaaa369121518212427 a a a a a is a super row vector (matrix) of refined labels with entries from LR. Let FVLLLLLLL GWaaaaaaa1 7 13 19 25 31 37 GWLLLLLLLaaaaaaa GW2 8 14 20 26 32 38 LLLLLLL GWaaaaaaa3 9 15 21 27 33 39 M = GWLLLLLLL GWaaaaaaa4101622283440 GWLLLLLLL GWaaaaaaa5111723293541 GWLLLLLLL HXaaaaaaa6121824303642 be a super row vector (matrix) of refined labels with entries from LR.

69 FVLLLLLL GWaaaaaa123456 GWLLLLLLaaaaaa GW789101112 LLLLLL GWaaaaaa13 14 15 16 17 18 Mt = GWLLLLLL GWaaaaaa19 20 21 22 23 24 GWLLLLLL GWaaaaaa25 26 27 28 29 30 GWLLLLLLaaaaaa GW31 32 33 34 35 36 LLLLLL HXaaaaaa37 38 39 40 41 42 is the transpose of M which is a super column vector (matrix) of refined labels with entries from LR. Now we find the transpose of super matrix of refined labels which are not super row vectors or super column vectors. Let FVLLLLL GWaaaaa12345 GWLLLLLaaaaa GW678910 P = LLLLL GWaaaaa11 12 13 14 15 GWLLLLL GWaaaaa16 17 18 19 20 GWLLLLL HXaaaaa21 22 23 24 25 where L ∈ LR; 1 ≤ i ≤ 25 be a super matrix of refined labels. ai

FVLLLLL GWaaaaa12345 GWLLLLLaaaaa GW678910 Pt = LLLLL GWaaaaa11 12 13 14 15 GWLLLLL GWaaaaa16 17 18 19 20 GWLLLLL HXaaaaa21 22 23 24 25 is the transpose of P and is again a super matrix of refined labels. Now we will show how product is defined in case of super matrix of refined labels. We just make a mention that if in the super matrix the refined labels are replaced by ordinary labels still the concept of transpose of super matrices remain the same

70 however there will be difference in the notion of product which will be discussed in the following: Suppose A = FLLLLLV be a super H aaaaa12345X matrix (row vector) of refined labels then we find FL V G a1 W GLa W G 2 W At = L , G a3 W GL W G a4 W GL W H a5 X which is a super column vector of refined labels. We find FL V G a1 W GLa W G 2 W AAt = FVLLLLL L HXaaaaa12345G a3 W GL W G a4 W GL W H a5 X FL V F V G a3 W La = FVLLG 1 W + FVLLLLG W HXaa12 HXaaa345 a 4 HGLa XW G W 2 GL W H a5 X = FVL.L+ L.L + FL.L++ L.L L.LV HXaa11 aa 2 2 H aa33 aa 44 aa 55X FV+ F ++V = LL22++ + LLL222+++ HXa/(m1)12 a/(m1) H a345 /(m 1) a /(m 1) a /(m 1) X FVF V = L 22++ + + L 222++ ++ + HXa/(m1)a/(m1)12 H a/(m1)a/(m1)a/(m1)345X FVF V = L 22++ + L 222++ + HXaa/(m1)12 H aaa/(m1)345 X FV = L 22222++++ + . HXaaaaa/(m1)12345

t Clearly AA ∈ LR but it is not a super row vector matrix. Now we see if we want to multiply a super row vector A with a

71 super column vector B of refined labels then we need the following.

(i) If A is of natural order 1 × n then B must be of natural order n × 1. (ii) If A is partitioned between the columns ai and ai+1 then B must be partitioned between the rows bi and bi+1 of B this must be carried out for every 1 ≤ i ≤ n.

We will illustrate this situation by some examples.

Let A = FVLLLLLL be a super row HXaaaaaa123456 FL V G b1 W GLb W G 2 W L G b3 W vector of natural order 1 × 6. Consider B = , a super GL W G b4 W GL W G b5 W GL W H b6 X column vector of natural order 6 × 1. Clearly A is partitioned between the columns 2 and 3 and B is partitioned between the rows 2 and 3 and A is partitioned between the columns 5 and 6 and B is partitioned between the rows 5 and 6.

Now we find the product

FL V G b1 W GLb W G 2 W L FVG b3 W AB = HXLLLLLLaaaaaa G W 123456L G b4 W GL W G b5 W GL W H b6 X

72 FL V F V G b3 W Lb = FVLLG 1 W + FVLLLLG W + FLL× V HXaa12 HXaaa345 b 4H ab66X HGLb XW G W 2 GL W H b5 X

=FVL.L+ L.L + FL.L++ L.L L.LV L.L HXab11 ab 2 2 H ab33 ab 44 ab 55X ab66

FV+ F ++V =HXLLab /(m1)++ a b /(m1) HLLLab /(m1)+++ ab /(m1) ab /(m1)X 11 2 2 33 44 55 + L + ab66 /(m1) FVFV = L +++ L ++ ++ L + HXab11 ab 2 2 /(m1) HX(ababab)/(m1)33 44 55 ab66 /(m1) FV = L +++++ + is in LR. HX(abababababab)/(m1)11 22 33 44 55 66

Suppose FL V G a1 W GLa W G 2 W L G a3 W GL W G a4 W A = GL W G a5 W GLa W G 6 W L G a7 W GL W G a8 W GL W H a9 X and B = FVLLLLLLLLL HXaaaaaaaaa123456789 be super column vector and super row vector respectively of refined labels. BA is not defined though both A and B are of natural order 9 × 1 and 1 × 9 respectively as they have not the same type of partitions for we see A is partitioned between row three and four but B is partitioned between one and two columns and so on.

73 Let us consider FL V G b1 W GL W FVb2 B = HXLLLLaa aa and A = G W 1234 L G b3 W GL W H b4 X be super row vector and super column vector respectively of refined labels. We see FL V G b1 W GL W F V b2 BA = HLLLLaa aaX G W 1234L G b3 W GL W H b4 X

= ()L.L++ L L ()L.L++ L L ab11 a 2 b 2 ab33 a 4 b 4

= ()LL++++ | ()LL++++ ab/(m1)11 a 2 b 2 /(m1) ab/(m1)33 a 4 b/(m1) 4

= L +++ +. (abababab)/m111 2 2 33 4 4

Let us now define product of a super column vector with a super row vector of refined labels.

Let

FVL GWa1 GWLa GW2 L GWa3 GWL GWa4 F V A = and B = HLLLLLLccccccX GWL 123456 GWa5 GWLa GW6 L GWa7 GWL HXa8

74 be a super column vector and super row vector respectively of refined labels. We see the product AB is defined even though they are of very different natural order and distinct in partition.

Consider FVL GWa1 GWLa GW2 L GWa3 GWL GWa4 F V AB = HLLLLLLccccccX GWL 123456 GWa5 GWLa GW6 L GWa7 GWL HXa8

FV GWFVLL FV FV L GWGWaa11 GW GW a 1 GW GWLLLacc() GW LLLL accc() GW LL ac GWGW212 GW 2345 GW 26 GWHXGWLLaa HXGW HXGW L a GW33 3 GW GW GW = GWLL() L LL() L L L.L GWacc412 accc 4345 ac 46 GW GW GW FV FV FV GWLLaa L a GWGW55 GW GW 5 GWGWLLaa GW GW L a 66() 6 GWGW()LLcc GW LLL cc c GWLc LL12 345 L6 GWGWaa77 GW G a 7W GWGW GW GW HXLLaa HX HX L a HX88 8

75 FV GWLL LL LL LL LL LL GWac11 ac 12 ac 13 ac 14 ac 15 ac 16 GWLL LL LL LL LL LL GWac21 ac 22 ac 23 ac 24 ac 25 ac 26 GWLLac LL ac LL ac LL ac LL ac LL ac GW31 32 33 34 35 36 GW GW GW = GWLL LL LL LL LL LL GWac41 ac 42 ac 43 ac 44 ac 45 ac 46 GW GW GWLL LL LL LLLLLL GWac51 ac 52 ac 53 a54cacac 55 56 GWLL LL LL LL LL LL GWac61 ac 62 ac63 ac 64 ac 65 ac 66 GWLLac LL ac LLac LL ac LL ac LL ac GW71 72 73 74 75 76 LL LL LL LL LL LL HXGWac81 ac 82 ac83 ac 84 ac 85 ac 86

FV GW LLLLLL++++++ GWa11 c /m 1 a 12 c /m 1 a 13 c /m 1 a 14 c /m 1 a 15 c /m 1 a 16 c /m 1 GWLLLLLL++++++ GWac/m121 ac/m1 22 ac/m1 23 ac/m1 24 ac/m1 25 ac/m1 26 GWLLLLLLa c /m++++++ 1 a c /m 1 a c /m 1 a c /m 1 a c /m 1 a c /m 1 GW31 32 33 34 35 36 GW GW GW = GWLLLLL++++++L GWac/m141 ac/m1 42 ac/m1 43 ac/m1 44 ac/m145 ac/m1 46 GW GW GW LLLLLL++++++ GWac/m151 ac/m1 52 ac/m1 53 ac/m1 54 ac/m1 55 a 5c/m16 GW LLLLL+++++L + GWa61 c /m 1 a 62 c /m 1 a 63 c /m 1 a 64 c /m 1 a 65 c /m 1 ac/m166 GWLLLLLac/m1+++++ ac/m1 ac/m1 ac/m1 ac/m1 Lac/m1+ GW71 72 73 74 75 76 LLLLL+++++L + HXGWa81 c /m 1 a 82 c /m 1 a 83 c /m 1 a 84 c /m 1 a 85 c /m 1 ac/m186

76 is a super matrix of refined labels. We can multiply any super column vector with any other super row vector and the product is defined and the resultant is a super matrix of refined labels. Let FL V G a1 W GLa W G 2 W L G a3 W G W La C = G 4 W GL W G a5 W GLa W G 6 W L G a7 W GL W H a8 X be a super column vector of refined labels. Ct = ()LLLLLLLL be the aaaaaaaa12345678 transpose of C. Clearly Ct is a super row vector of refined labels. We now find

FVL GWa1 GWLa GW2 L GWa3 GW La CCt = GW4 ()LLLLLLLL GWL aaaaaaaa12345678 GWa5 GWLa GW6 L GWa7 GWL HXa8

77 FV GW GWLLLLLLLLLLLL()( )() GWaa11 a 123 a a a 1456 a a a a 178 a a GW GW GW FVLL FV FV L FV L GWaa22 a 2 a 2 GWGW()La GW()( LL aa GW LLL aaa )() GW LL aa GWLL12345678 GW GW L GW L GWHXaa33 HX HX a 3 HX a 3 GW = GW FVLL FV FV LFV L GWGWaa44 GW GW a 4 GW a4 GW GWLLaa() GW LLL aaa()( GW LLLL aaaa )GWLLLaaa() GWGW51 GW 523 GW 5456GW578 GW HXGWLLaa HXGW HXGW L a HXGWLa GW66 6 6 GW FV FV FV FV GWLLaa L a L a GWGW77()L GW()( LL GW 7 LLL )() GW 7 LL GWLLa12345678 GW aa GW L aaa GW L aa GWHXaa88 HX HX a 8 HX a 8 HXGW

FV LLaa LL aa LL aa LL aa LL aa LL aa LL aa LL aa G 11 12 13 14 15 16 17 18W GLLaa LL aa LL aa LL aa LL aa LL aa LL aa LL aaW G 21 22 23 24 25 26 27 28W LL LL LL LL LL LL LL LL G aa31 aa 32 aa 33 aa 34 aa 35 aa 36 aa 37 aa 38W GLL LL LL LL LL LL LL LLW = G aa41 aa 52 aa 43 aa 44 aa 45 aa 46 aa 47 aa 48W GLL LLLLLLLLLLLLLLW G aa51 aa52 aa 53 aa 54 aa 55 aa 56 aa 57 aa 58W GLLaa LL aa LL aa LL aa LL aa LL aa LL aa LL aaW G 61 62 63 64 65 66 67 68W LL LL LL LL LL LL LL LL G aa71 aa 72 aa 73 aa 74 aa 75 aa 76 aa 77 aa 78W GLL LL LL LL LL LL LL LLW H aa81 aa 82 aa 83 aa 84 aa 85 aa 86 aa 87 aa 88X

78 FV LLLLLLLL2 + aa /m1+++++++ aa/m1 aa /m1 aa /m1 aa /m1 aa /m1 aa /m1 GWa/m11 12 13 14 15 16 17 18

GWLLLLLLLLaa/m1++++++++ aa /m1 aa /m1 aa /m1 aa/m1 aa /m1 aa /m1 aa/m1 GW21 22 23 24 25 26 27 28 GWLLLLLLLaa/m1++++++ aa/m1 aa/m1 aa/m1 aa/m1 aa/m1 aa/m1++L aa/m1 GW31 32 33 34 35 36 37 38 LLLLLLLL++++++++ GWaa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1aa/m141 52 43 44 45 46 47 48 =GW LLLLLLLL++++++++ G aa/m151 aa/m1 52 aa/m1 53 aa/m1 54 aa/m1 55 aa/m1 56 aa/m1 57 aa/m1 58 W G W LLLLL++++++++LLL G aa/m161 aa/m1 62 aa/m1 63 aa/m1 64 aa 65666768/m1 aa /m1 aa /m1 aa/m1W G W LLLLLLLL++++++++ G aa/m171 aa 72 /m1 aa 73 /m1 aa 74 /m1 aa/m1 75 aa 76 /m1 aa 77 /m1 aa/m1 78 W G W LLLLLLLL++++++++ HXaa/m181 aa/m1 82 aa/m1 83 aa/m1 84 aa/m1 85 aa/m1 86 aa/m1 87 aa/m1 88 is a super matrix of refined labels, we see the super matrix is a symmetric matrix of refined labels. Thus we get by multiplying the column super vector with its transpose the symmetric matrix of refined labels. We will illustrate by some more examples. Let FL V G a1 W GLa W G 2 W L G a3 W P = GL W G a4 W GL W G a5 W GL W H a6 X be a super column vector of refined labels. Pt = FVLLLLLL be the transpose of P. HXaaaaaa123456 FL V G a1 W GLa W G 2 W L G a3 W t F V P.P = G W HLLLLLLaaaaaaX L 123456 G a4 W GL W G a5 W GL W H a6 X

F V GFV FV FVW LLaa L a GGW11FVLL GWFV LLL GW 1 LW G HXHaa12 aaa 345 X a 6W HXGWLLaa HXGW HXGW L a G 22 2W G W GWFVLL FV FV L GWGWaa33 GW GW a 3 FVFV = GWGWLLLLLLLLLaaaHXH GW aaaa X GW aa GWGW412434546 GW GW GWLL GW GW L GWHXaa55 HX HX a 5 GW GW GW LLLFV LLLLFV L.L GWaa612HX a aa 6345HX a a aa 66 GW H X

FVLL LL LL LL LL LL GWaa11 aa 12 aa 13 aa 14 aa 15 aa 16 GWLLaa LL aa LL aa LL aa LL aa LL aa GW21 22 23 24 25 26 LL LL LL LL LL LL GWaa31 aa 32 aa 33 aa 34 aa 35 aa 36 = GWLL LL LL LL LL LL GWaa41 aa 42 aa 43 aa 44 aa 45 aa 46 GWLL LL LL LL LL LL GWaa51 aa 52 aa 53 aa 54 aa 55 aa 56 GWLL LL LL LLLLLL HXaa61 aa 62 aa 63 a64aaaaa 65 66

FV LL2 + aa /m1+++++ L aa /m1 L aa /m1 L aa /m1 L aa /m1 GWa/m11 12 13 14 15 16

GWLaa/m1+++++ LLLLL2 + aa/m1 aa/m1 aa/m1 aa/m1 GW21a/m12 23 24 25 26 GWLLa a /m++ 1 a a /m 1 LLLL2 + a a /m +++ 1 a a /m 1 a a /m 1 = GW31 32a/m13 34 35 36 LLL+++ LL2 ++L GWaa/m141 aa/m1 42 aa/m1 43a/m1+ aa 45/ m1 aa/m146 GW4 LLLL++++ LL2 + GWa51 a /m 1 a 52 a /m 1 a 53 a /m 1 a 54 a /m 1a/m1+ a 56 a /m 1 GW5 LLLLLa a /m+++++ 1 a a /m 1 a a /m 1 a a /m 1 a a /m 1 L2 + HXGW61 62 63 64 65 a/m16 is a symmetric super matrix of refined labels.

80 Let FLLLL L LV G aaaa127101112 a aW

A = GLLLLLLaaaa a aW G 348131415W GLLLLLLW H aaaa569161718 a aX be a super matrix of row vectors of refined labels.

F LLLV G aaa135W G LLLaaaW G 246W LLL G aaa789W At = GLLLW G aaa10 13 16 W GLLLW G aaa11 14 17 W GLLLW H aaa12 15 18 X be the transpose the super row vectors. A of refined labels. To find

F LLLV G aaa135W G LLLaaaW FVLLLL L LG 246W GWaaaa127101112 a aLLL G aaa789W A.At = GWLLLLLL aaaa348131415 a aG W GWLLL G aaa10 13 16 W HXGWLLLLLLaaaa a a 569161718GLLLW G aaa11 14 17 W GLLLW H aaa12 15 18 X

FVLL FVL aa12F V a7 GWLLLaaa GW = GWLLG 135W + GWLLLLF V aa34 aaaa8H 789X GWHGLLLaaaXW GW GWLL 246GWL HXaa56 HXa9

FLLLLLLVF V G aaaaaa10 11 12WG 10 13 16 W

+ GLLLLLLaaaWG aaa W G 13 14 15WG 11 14 17 W GLLLLLLWG W H aaaaaa16 17 18XH 12 15 18 X

81 FVLL+++ LL LL LL LL LL GWaa11 a 22 a aa 13 a 24 a aa 15 a 26 a +++ = GWLLaa LL aa LL aa LL aa LL aa LL aa GW31 42 33 44 35 46 GWLL+++ LL LL LL LL LL HXaa51 aa 62 aa 53 aa 64 aa 55 aa 66

FLL LL LLV G aa77 aa 78 aa 79W

+ GLLaa LL aa LL aaW G 87 88 89W GLL LL LL W H aa97 aa 98 aa 99X

FVLL++ LL LL LL ++ LL LL LL ++ LL LL GWaa10 10 aa 11 11 aa 12 12 aa 10 13 aa 11 14 aa 12 15 aa 10 16 aa 11 17 aa 12 18 +++++++GWLL LL LL LL LL LL LL LL LL GWaa13 10 aa 14 11 aa 15 12 aa 13 13 aa 14 14 aa 15 15 aa 13 16 aa 14 17 aa 15 18 GWLL++ LL LL LL + LL +++LL LL LL LL HXaa16 10 aa 17 11 aa 18 12 aa 16 13 aa 17 14 aa18 15 aa 16 16 aa 17 17 aa 18 18

FV+++ LLLL22++aa /m1++++ a a /m1 LL aa /m1 a a /m1 GWa/m112 a/m1 13 2 4 15 26 =+GW + + LLaa/m1++ aa/m1 LLLL22++ aa/m1 ++ aa/m1 GW31 42a/m134 a/m1 35 4 6 GW+++ LLLLaa/m1++++ aa /m1 aa/m1 aa/m1 LL22++ HX51 62 53 6 4 a/m156 a/m1

FV LL2 + aa/m1++ L aa/m1 GWa/m17 78 79 GW + LLLaa/m1++2 + aa/m1 + GW87a/m18 89 GW LLaa/m1++ aa/m1 L2 + HX97 98 a/m19

FV++ ++ ++ LLL222+++ La a /m1++++++ L a a /m1 L a a /m1 L a a /m1 L a a /m1 L a a /m1 GWa/m110 a/m1 11 a/m1 12 10 13 11 14 12 15 10 16 11 17 12 18 ++ ++ ++ LLLLLL+++222 LLL +++ GWa13 a 10 /m1 a 14 a 11 /m1 a 15 a 12 /m1a /m1+++ a /m1 a /m1 a 13 a 16 /m1 a 14 a 17 /m1 a 15 a 18 /m1 GW++13 ++ 14 15 ++ La a /m1++++++LLLLLLLL a a /m1 a a /m1 a a /m1 a a /m1 a a /m1 222+++ HX16 10 17 11 18 12 16 13 17 14 18 15 a16 /m1 a 17 /m1 a 18 /m1

F LL2222+++ + 2 + 2 + aa+++ a a aa a a + a a + a a /m1 + G aaaa127101112 a a/m1 1 3 2 4 7 8 10 13 11 14 12 15 = G LLaa+++ aa aa aa + aa + aa /m1 + 222222+++++ + G 3 1 4 2 8 7 13 10 14 11 15 12 aaaa3 48131415 a a/m1

GLL+++ + + + +++ ++ + H a5 a 1 aa 6 2 aa 9 7 a 16 a 10 a 17 a 11 a 18 a 12 /m1 aa 5 3 a 6 a 4 aa 9 8 a16aaaaa/m1 13 17 14 18 15

82 V L +++ + + + aa1 5 a 2 a 6 aa 7 9 a 10 a 16 a 11 a 17 a 12 a 18 /m1W

L +++ + + +W . aa3 5 aa 4 6 aa 8 9 aa 13 16 aa 14 17 aa 15 18 /m1W L 2222+++++ 2 2 + W aaaa5 6 9 16 a 17 a/m1 18 X

We see AAt is a ordinary matrix of refined labels. Clearly AAt is a symmetric matrix of refined labels. Thus using a product of super row vectors with its transpose we can get a symmetric matrix of refined labels. Consider F LLV G aa12W G LLaaW G 34W LL G aa56W H = GLLW G aa78W GLLW G aa910W GLLW H aa11 12 X a super column vector of refined labels. Clearly F V LLLLLaaaa a L a Ht = G 1357 9 11W GLLLLL LW H aaaaa2 4 6 8 10 a 12 X is the transpose of H which is a super row vector of refined labels. Now we find FVLL GWaa12 GWLLaa GW34 LLaa FLLLLL LV T GW56 aaaa1357 a 9 a 11 H H = GW × G W LLaa GLLLLL LW GW78H aaaaa2 4 6 8 10 a 12 X GWLL GWaa910 GWLL HXaa11 12

83 FVLL FVaa12 LLLaaaGW = GW135GWLL + aa34 HXGWLLLaaaGW 246GWLL HXaa56

FVFVFV LLLLaaaa L GWGW79 78+ a11 F V GWHLLaaX GWGWLL LLGW L 11 12 HXHXaa81091012 aaHX a

FV++ ++ LLaa LL aa LL a a LL aa LL aa LL aa = GW11 33 55 12 34 56 + GWLL++ LL LL LL ++ LL LL HXaa21 aa 43 aa 65 aa 22 aa 44 aa 66

FV++ LLaa LL aa LL aa LL aa GW77 99 78 910 GWLL++ L L LL L L HXaa87 a 109 a aa 88 a 1010 a

F V LLaa LL aa + G 11 11 11 12 W GLL LLW H aa12 11 aa 12 12 X

FV++ + + LLL222+++ La a /m+++ 1 L a a /m 1 L a a /m 1 = GWa/m1135 a/m1 a/m1 12 34 56 ++ ++ GWLLL+++ LLL222 HXaa/m121 aa 43 /m1 aa 65 /m1 a/m1+++ a/m1 a/m1 246

++ FVLL22 L++ L a/m1++ a/m1 aa78 /m1 aa 910 /m1 + GW79 ++ GWLLaa /m1++ a a /m1 LL22++ HX87 109 a/m1810 a /m1

FVLL2 + a/m1+ aa11 12 /m1 + GW11 GWLLaa/m1+ 2 + HX12 11 a/m112

FVLL222222 ++++ + + aaaaaa/m1+++++ + aa12 aa 34 aa 56 aa 78 aa 910 a 1112 a /m1 = GW1357911 GWLLaa++++ aa aa aa a a + a a /m1 + 22222++++ + 2 + HX21 43 65 87 109 1211 aaaaa2 4 6 8 10 a/m1 12 is a symmetric matrix of refined labels which is not a super matrix of refined labels.

Thus in applications we can multiply two super matrices of refined labels and get a symmetric matrix of refined labels,

84 likewise we can multiply super matrices to get back super matrices. For consider FVLL GWaa12 GWLLaa GW34 LL FVLLL GWaa56 bbb134 X = and Y = GW GWLL GWLLL GWaa78 HXbbb256 GWLL GWaa910 GWLL HXaa11 12 two super matrices of refined labels. Now we find

F LLV G aa12W G LLaaW G 34W LLFLLLV G aa56W bbb134 XY = G W GLLW GLLLW G aa78W H bbb256X GLLW G aa910W GLLW H aa11 12 X

FV GWFVFVLLFVL LLFLLV GWGWGWaa12GWb1 aa 12G bb34W GW HXHXGWGWLLaaHXGWLLLbbb LL aaHG XW GW34256 34 GW GW FV F V GWLb LLbb = GW()()LLGW1 LLG 34W aa56 aa 56 GWHXGWLLLbbbHG XW GW256 GW GWFVFVLL LL GWGWaa78FV aa 78F V GWLb LLbb GWGWLLGW1 LLG 34W GWaa910 aa 910 GWGWGWL GLLW GWHXb2 H bb56X HXHXGWGWLLaa LL aa HXGW11 12 11 12

85 FVLL+++ LL LL LL LL LL GWab11 ab 22 ab 13 a 25 b ab 14 a 26 b +++ GWLLab LL ab LL ab LL ab LL ab LL ab GW31 42 33 45 34 46 LL+++ LL LL LL LL LL GWab51 ab 62 ab 53 ab 65 ab 54 ab 66 = GWLL+++ LL LL LL LL LL GWab71 ab 82 ab 73 ab 85 ab 74 ab 86 GWLL+++ L L LL L L LL L L GWab91 a 102 b ab 93 a 105 b ab 94 a 106 b GWL LLLLLLLLLLL+++ HXa11bababababab 1 12 2 11 3 12 5 11 4 12 6

FVLLL++ ++ ++ GWab11 a 22 b /m1 ab 13 a 25 b /m1 ab 14 a 26 b /m1 GWLLLab++ ab/m1 ab ++ ab/m1 ab ++ ab/m1 GW31 42 33 45 34 46 LLL++ ++ ++ GWab51 ab/m1 62 ab 53 ab/m1 65 ab 54 ab/m1 66 = GW LLL++ ++ ++ GWab71 ab/m1 82 ab 73 ab/m1 85 ab 74 ab/m1 86

GWLLL++ ++ ++ GWab91 a 102 b/m1 ab 93 a 105 b/m1 a94bab/m1 106 GWLLL++ ++ ++ HXab11 1 ab/m1 12 2 ab 11 3 ab/m1 12 5 ab 11 4 ab/m1 12 6 is a super matrix of refined labels. We find FLLV G bb12W t t Y X = GLLbbW G 34W GLLW H bb56X

FV LLLLLLaaaaaa GW1357911 GWLLLLLL HXaaaaaa24681012

FVFV F VF V LLaa LLLLaaaa GW()LLGW13 ()() LLG 57911WG LL W bb12 bb 12 bb 12 GWHXGWLLaa HGLLLLaaaaXHWG XW = GW24 6 8 10 12 FVFV FVFVFVF V GWLLbbLLaa LLL bb a LLL bb a LL a a GWGW35GW13 GWGWGWG 35 5 35 7 911W GWLLGWLL GWGWGW LLL LLLLLG W HXGWHXbb4 6HXaa24 HXHXHXH bb 4 6 a 6 bb 4 6 a 8 a 10 a 12 X

86 FVLL+++ LL LL LL LL LL GWba11 ba 22 ba 13 ba 24 ba 15 ba 26 ++ + GWLLba LL ba LL ba LL ba LL ba LL ba GW31 52 33 54 35 56 LL++ LL LL LL LL + LL GWba41 ba 62 ba 43 ba 64 ba 45 ba 66 = GWLL++ LL LL LL LL + LL GWba17 ba 28 ba 19 b 210111212 a ba ba GWLL++ LL LL LL LL + LL GWba37 ba 58 ba 39 ba 51031151 ba ba2 GWLL++ LL LL LL LL + LL HXba47 ba 68 ba 49 ba 610411612 ba ba

FVLL++ ++ L ++ GWba11 ba/m1 22 ba 13 ba/m1 24 ba 15 ba/m1 26 GWLLba++ ba /m1 ba ++ ba /m1 L ba ++ ba /m1 GW31 52 33 54 35 56 LL++ ++ L ++ GWba41 ba/m1 62 ba 43 ba/m1 64 ba 45 ba/m1 66 = GW. LL++ ++ L ++ GWb17 a b 28 a /m 1 b 19 a b 210 a /m 1 b 111212 a b a /m 1

GWLL++ + +L ++ GWba37 ba 58 /m1 ba 39 ba 510 /m1 ba311 ba 512 /m1 GWLL++ ++ L ++ HXba47 ba/m1 68 ba 49 ba 610 /m1 ba 411612 ba /m1

We see XY = Yt Xt are super matrix of refined labels.

Now we proceed onto define another type of product of super matrix of refined labels.

Let FLLL L L LV G aaa1 8 15222936 a a aW GLLLLLLaaaaaaW G 2 9 16233037W LLLLLL G aaaaaa31017243138W G W M = LLLLLLaaaaaa G 41118253239W GLLLLLLW G aa51219263340 a a a aW GLLLLLLaaaaaaW G 61320273441W LLLLLL HG aa71421283542 a a a aXW be a super matrix of refined labels.

87 Consider FVLLLLLLL GWaaaaaaa1234567 GWLLLLLLLaaaaaaa GW8 9 10 11 12 13 14 LLLLLLL GWaaaaaaa15 16 17 18 19 20 21 MT = GWLLLLLLL GWaaaaaaa22 23 24 25 26 27 28 GWLLLLLLL GWaaaaaaa29 30 31 32 33 34 35 GWLLLLLLL HXaaaaaaa36 37 38 39 40 41 42

FVLLL L L L GWaaa1 8 15222936 a a a GWLLLLLLaaaaaa GW2 9 16233037 LLLLLL GWaaaaaa31017243138 T GW M.M = LLLLLLaaaaaa × GW41118253239 GWLLLLLL GWaa51219263340 a a a a GWLLLLLLaaaaaa GW61320273441 LLLLLL HXGWaa71421283542 a a a a

FVLLLLLLL GWaaaaaaa1234567 GWLLLLLLLaaaaaaa GW8 9 10 11 12 13 14 LLLLLLL GWaaaaaaa15 16 17 18 19 20 21

GWLLLLLLL GWaaaaaaa22 23 24 25 26 27 28 GWLLLLLLL GWaaaaaaa29 30 31 32 33 34 35 GWLLLLLLL HXaaaaaaa36 37 38 39 40 41 42

88 FVL GWa1 GWLa GW2 L GWa3 GW = La FLLLLLLLV + GW4 H aaaaaaa1234567X GWL GWa5 GWLa GW6 L HXGWa7

FVLL GWaa815 GWLLaa GW916 LL GWaa10 17 F LLLLLLLV GWaaaaaaa8 9 10 11 12 13 14 LLaaG W + GW11 18 HGLLLLLLLaaaaaaaXW GWLL 15 16 17 18 19 20 21 GWaa12 19 GWLLaa GW13 20 LL HXGWaa14 21

FVLLL GWaaa22 29 36 GWLLLaaa GW23 30 37 LLLF V GWaaa24 31 38 LLLLLLLaaaaaaa GWG 22 23 24 25 26 27 28 W LLLaaaGLLLLLLLW GW25 32 39 aaaaaaa29 30 31 32 33 34 35 G W GWLLL aaa26 33 40 HGLLLLLLLaaaaaaaW GW36 37 38 39 40 41 42 X GWLLLaaa GW27 34 41 LLL HXGWaaa28 35 42

89 F G FVLL F LLLL V GFVFVaa22 23 aaaa 24 25 26 27 LLLaaaGW LLL aaa G W GGWGW22 29 36GWLL 22 29 36 G LLLL W GGWGWLLLaa29 30 LLL aaaa 31 32 33 34 HXHXaaa23 30 37GW aaa 23 30 37 G W G HXGWLLaa HG LLLL aaaa XW G 36 37 38 39 40 41 G FVFV G LLLaaa LLLaaa GWGW24 31 38 FVLL24 31 38 F LLLL V G aa22 23 aaaa 24 25 26 27 GWGWLLLGWLLL G W G aaa25 32 39 aaa25 32 39 = GWGWGWLL G LLLL W G LLL aa29 30LLL aaaa 31 32 33 34 GWGWaaa26 33 40 GWaaa26 33 40 G W G GWLL G LLLL W GWGWHXaa36 37 H aaaa 38 39 40 41 X GHXHXLa LLaa LLL aaa G 273441 273441 G FV F V G La Laaaaa LLLL G GW2223 G 24252627 W FVLLLGWLLFV LLLLLLL G W GHXaaa28 29 30 aa29 30HX aaaaaaa 28 29 30 31 32 33 34 GW G W G GWLL G LLLL W G HXaa36 37 H aaaa 38 39 40 41 X H

V FL VW FVa28 W LLLaaaG W GW22 29 36 GL WW GWLLL a35 W HXaaa23 30 37 G W HGLa XWW 42 W W FV LLLaaa W GW24 31 38 F V La W GWLLLG 28 W aaa25 32 39 W GWGL W LLL a35 W GWaaa26 33 40 G W GL WW GWH a42 X HXLLLaaa W 27 34 41 W W FL VW G a28 W FVW HXLLLLaaaaG WW 28 29 30G 35 W GL WW H a42 XW X

FFV FV FVV LLaa L a GGW11()LL GW() LLLL GW 1 LW aa12 aaaa 3456 a 7 GHXGWLLaa HXGW HXGW L aW G 22 2W FV FV FV G LLaa L aW GGW33 GW GW 3W GWGWLLaa GW GW L a + GW44()LL GW() LLLL GW 4 L GWLLaa12 aaaa 3456 L a 7 GWGWaa55 GW GW a 5 GWGW GW GW HXLLaa HX HX L a GW66 6 GWLLL() LLLLL() L.L GWaaa712 aaaaa 73456 aa 77 HGWX

FVFVFVFVFLL LL LL L LL L VFV LLF LV GWGWGWGWGaa8 15 a 8 a 9 aa 8 15 a 10 a 11 a 12 a 13 WGW aa 8 15 G a14 W GW HXHXHXHGWGWGWGLLLLaa a a LLLLLL aa a a a a XHXWGW LLL aaHG aXW GW9 16 15 16 9 16 17 18 19 20 9 16 21 GWFVLL FVLL FV LL GWGWaa10 17 GWaa10 17 GW aa 10 17 FV F V F V GWGWLLLLaa aaGWLLLLLLaa aaaa GW LLL aa a +GWGW11 18GW 8 9 GW11 18 G 10 11 12 13 W GW 11 18G 14 W LLLGWL LLLLLL G W LLLG W GWGWaaa12 19HX 1516121917181920121921aGW aaaaaa H X GW aaH aX GWGWLL GWLL GW LL GWHXaa13 20 HXaa13 20 HX aa 13 20 GWFV FVF V LLaa LLLLaaaa L a GWFVLLGW89FV LLGW10 11 12 13 FVLLG 14 W GWHXaa14 21HX a 14 aaa21 HX 14 21 GWLLaa GWLLLLaaaaHG L aXW HXHX15 16 HX17 18 19 20 21

FVLL LL LL LL LL LL LL GWaa11 aa 12 aa 13 aa 14 aa 15 aa 16 aa 17 GWLLaa LL aa LL aa LL aa LL aa LL aa LL aa GW21 22 23 24 25 26 27 LL LL LL LL LL LL LL GWaa31 aa 32 aa 33 aa 34 aa 35 aa 36 aa 37 = GWLL LL LL LL LL LL LL GWaa41 aa 42 aa 43 aa 44 aa 45 aa 46 aa 47 GWLL LL LL LL LL LLLL GWaa51 aa 52 aa 53 aa 54 aa 55 aa56 aa 57 GWLLaa LL aa LL aa LL aa LL aa LL aa LL aa GW61 62 63 64 65 66 67 LL LL LL LL LL LL LL HXGWaa71 aa 72 aa 73 aa 74 aa 75 aa 76 aa 77

FLL++++ LL LL LL LL LL LL LL G aa8 8 aa 15 15 aa 8 9 aa 15 16 aa 8 10 aa 15 17 aa 8 11 aa 15 18 ++++ GLLaa L a L a LL aa L a L a LL aa L a L a LL aa L a L a G 9 8 16 15 9 9 16 16 9 10 16 17 9 11 16 18 LL++ LL LL LL LL + LL LL + LL G aa10 8 aa 17 15 aa 10 9 aa 17 16 aa 10 10 aa 17 17 aa 10 11 aa 17 18 +GLL+ LLLLLLLLLLLLLL+++ G aa11 8 aa18 15 aa 11 9 aa 18 16 aa 11 10 aa 18 17 aa 11 11 aa 18 18 GLL++ LL LL LL LL ++ LL LL LL G aa12 8 aa 19 15 aa 12 9 aa 19 16 aa 12 10 aa 19 17 aa 12 11 aa 19 18 ++ + + GLLaa LL aa LL aa LL aa LL aa LL aa LL aa LL aa G 13 8 20 15 13 9 20 16 13 10 20 17 13 11 20 18 LL+ LLLLLLLLLLLLLL+++ HG aa14 8 a 21aaaaaaaaaaaaa 15 14 9 21 16 14 10 21 17 14 11 21 18

LL+++ LL LL LL LL LL V aa9121619 a a aa 8131520 a a aa 8141521 a aW +++ LLaa L a L a LL aa L a L a LL aa L a L aW 9121619 9131620 9141621W LL+++ LL LL LL LL LL aa10 12 aa 17 19 aa 10 13 aa 17 20 aa 10 14 aa 17 21 W W LL+++ LL LL LL LL LL + aa11 12 aa 18 19 aa 11 13 aa 18 20 aa 11 14 aa 18 21 W L LLLLLLLLLLL+++W a12aaaaaaaaaaa 12 19 19 12 13 19 20 12 14 19 21 W +++ LLaa LL aa LL aa LL aa LL aa LL aaW 13 12 20 19 13 13 20 20 13 14 20 21 W LL+++ LL LL LL LL LL aa14 12 aa 21 19 aa 14 13 aa 21 20 aa 14 14 aa 21 21 XW

FLL++ LL LL LL ++ LL LL LL ++ LL LL G aa22 22 aa 29 29 aa 36 36 aa 22 23 aa 29 30 aa 36 37 aa 22 24 aa 29 31 aa 36 38 ++ ++ ++ GLLaa LL aa LL aa LL aa LL aa LL aa LL aa LL aa LL aa G 23 22 30 29 37 36 23 23 30 30 37 37 23 25 30 31 37 38 LL++ LL LL LL ++ LL LL LL++ LL LL G aa24 22 aa 31 29 aa 38 36 aa 24 23 aa 31 30 aa38 37 aa 24 24 aa 31 31 aa 38 38 GLL++ LL LL LL ++ LL LL LL ++ LL LL G aa25 22 aa 32 29 aa 39 36 aa 25 23 aa 32 30 aa 39 37 aa 25 24 aa 32 31 aa 39 38 GLL++ LL LL LL ++ LL LL LL ++ LL LL G aa26 22 aa 33 29 aa 40 36 aa 26 23 aa 33 30 aa 40 37 aa 26 24 aa 33 31 aa 40 38 + +++++ GLLaa LaaLLLLLLLLLLLLLLL aa aa aa aa aa aa aa G 27 22 34 29 41 36 27 23 34 30 41 37 27 24 34 31 41 38 LL++ LL LL LL ++ LL LL LL ++ LL LL HG aa28 22 aa 35 29 aa 42 36 aa 28 23 aa 35 30 aa 42 37 aa 28 24 aa 35 31 aa 42 38

92 LL++ LL LL LL ++ LL LL LL ++ LL LL aa22 25 aa 29 32 aa 36 39 aa 22 26 aa 29 33 aa 36 40 aa 22 27 aa 29 34 aa 36 41 LL++ LL LL LL ++ LL LL LL ++ LL LL aa23 25 aa 30 32 aa 37 39 aa 23 26 aa 30 33 aa 37 40 aa 23 27 aa 30 34 aa 37 41 LL++ LL LL LL ++ LL LL LL++ LL LL aa24 25 aa 31 32 aa 38 39 aa 24 26 aa 31 33 aa38 40 aa 24 27 aa 31 34 aa 38 41 LL++ LL LL LL ++ LL LL LL ++ LL LL aa25 25 aa 32 32 aa 39 39 aa 25 26 aa 32 33 aa 39 40 aa 25 27 aa 32 34 aa 39 41 LL++ LL LL LL ++ LL LL LL ++ LL LL aa26 25 aa 33 32 aa 40 39 aa 26 26 aa 33 33 aa 40 40 aa 26 27 aa 33 34 aa 40 41 LL+ LLLLLLLLLLLLLLLL+++++ aa27 25 aa34 32 aa 41 39 aa 27 26 aa 34 33 aa 41 40 aa 27 27 aa 34 34 aa 41 41 LL++ LL LL LL ++ LL LL LL ++ LL LL aa28 25 aa 35 32 aa 42 39 aa 28 26 aa 35 33 aa 42 40 aa 28 27 aa 35 34 aa 42 41

LL++ LL LLV aa22 28 aa 29 35 aa 36 42 W ++ LLaa LL aa LL aaW 23 28 30 35 37 42 W LL++ LL LL aa24 28 aa 31 35 aa 38 42 W ++W LLaa LL aa LL aa 25 28 32 35 39 42 W LL++ LL LLW aa26 28 aa 33 35 aa 40 42 W ++ LLaa LL aa LL aaW 27 28 34 35 41 42 W LL++ LL LL aa28 28 aa 35 35 aa 42 42 XW

FV LLLLLLL2 + aa/m1aa/m1aa/m1aa/m1aa/m1aa/m1++++++ GWa/m11 12 13 14 15 16 17

GWLLLLLLLaa/m1+2 + aa/m1 +++++ aa/m1 aa/m1 aa/m1 aa/m1 GW21a/m12 23 24 25 26 27 GWLLaa/m1++ aa/m1 LLLLL2 + aa/m1 ++++ aa/m1 aa/m1 aa/m1 GW31 32a/m13 34 35 36 37 = LL+ ++L LLLL2 +++ GWaa/m141 aa 4243/ m1 aa/m1a/m1+ aa/m1 454647 aa/m1 aa/m1 GW4 LLLL++++ LLL2 ++ GWaa/m1aa/m1aa/m1aa/m151 52 53 54a/m1+ aa/m1aa/m1 56 57 GW5 LLLLLaa/m1+++++ aa/m1 aa/m1 aa/m1 aa/m1 LL2 + aa/m1 + GW61 62 63 64 65a/m16 67 GW LLLaa/m1+++ aa/m1 aa/m1 LLLaa/m1+++ aa/m1 aa/m1 L2 + HXGW71 7 2 73 74 75 76 a/m17

93 F LL22++ aa++ a a /m1 L aa ++ a a /m1 L aa ++ a a /m1 G aa/m1815 89 1516 8101517 8111518

G LLLLaa++ a a /m122++ aa ++ a a /m1 aa ++ a a /m1 G 9 8 16 15aa/m1916 9 10 16 17 9 11 16 18 GLLaa++ aa/m1 aa ++ aa/m1 LL22 + a+ aa ++ aa/m1 G 10 8 17 15 10 9 17 16a/m110 17 10 11 17 18 + L + ++++++LL L22 G aa11 8 aa/m118 15 aa 11 9 aa/m1 18 16 aa 11 10 aa/m1 18 17 a/m1 a + G 11 18 LLL++ ++ ++ L ++ G aa12 8 aa/m1 19 15 aa 12 9 aa/m1 19 16 aa 12 10 aa/m1 19 17 aa 12 11 aa/m1 19 18 G LLL++ ++ ++ L ++ G aa13 8 aa/m1 20 15 aa 13 9 aa/m1 20 16 aa 13 10 aa/m1 20 17 aa 13 11 aa/m1 20 18 G L ++LL ++ ++ L ++ H aa14 8 aa/m1 21 15 aa 14 9 aa/m1 21 16 aa 14 10 aa/m1 21 17 aa 14 11 aa/m1 21 18

LLL++ ++ ++V aa8 12 a 15 a 19 /m1 aa 8 13 a 15 a 20 /m1 aa 8 14 a 15 a 21 /m1W LLLaa++ a a /m1 aa ++ a a /m1 aa ++ a a /m1W 9 12 16 19 9 13 16 20 9 14 16 21 W LLL++ ++ ++ aa10 12 aa/m1 17 19 aa 10 13 aa/m1 17 20 aa 10 14 aa/m1 17 21 W W LLL++ ++ ++ aa11 12 aa/m1 18 19 aa 11 13 aa 18 20 /m1 aa 11 14 aa/m118 21 W + W LL22+ /m1+++++ aa aa /m1 L aa aa/m1 aa12 19 12 13 19 20 12 14 19 21 W

LLLaa++ aa/m122+ /m1 + aa ++ aa/m1W 13 12 20 19aa13 20 13 14 20 21 W LLaa++ aa/m1 aa ++ aa /m1 L22+ /m1 +W 14 12 21 19 14 13 21 20 aa14 21 X

F LL222++ + aa++ aa aa/m1 + L aa ++ aa aa/m1 + G aaa/m122 29 36 22 23 29 30 36 37 22 24 29 31 36 38

GLLLaa++ aa aa/m1 +222++ + aa ++ aa aa/m1 + G 23 22 30 29 37 36aaa/m123 30 37 23 25 30 31 37 38 GLLaa++ aa aa/m1 + aa ++ aa aa/m1 + L22++2 + G 24 22 31 29 38 36 24 23 31 30 38 37 aaa24 31 38 /m 1 GLLLaa++ aa aa/m1 + aa ++ aa aa/m1 + aa ++ aa aa/m1 + G 25 22 32 29 39 36 25 23 32 30 39 37 25 24 32 31 39 38 LLL++ + ++ + ++ + G aa26 22 aa 33 29 aa/m1 40 36 aa 26 23 aa 33 30 aa/m1 40 37 aa 26 24 aa 33 31 aa/m1 40 38 G LL++ + ++ ++++L G aa27 22 aa 34 29 aa/m1 41 36 aa 27 23 aa 34 30 a 41 a/m137 aa 27 24 aa 34 31 aa/m1 41 38 GLLL++ + ++ + ++ + H aa28 22 aa 35 29 aa/m1 42 36 aa 28 23 aa 35 30 aa/m1 42 37 aa 28 24 aa 35 31 aa/m1 42 38

94 LLL++ + ++ + ++ + aa22 25 aa 29 32 aa/m1 36 39 aa 22 26 aa 29 33 aa/m1 36 40 aa 22 27 aa 29 34 aa/m1 36 41

LLL++ + ++ + ++ + aa23 25 aa 30 32 aa/m1 37 39 aa 23 26 aa 30 33 aa/m1 37 40 aa 23 27 aa 30 34 aa/m1 37 41

LL++ + ++ +L ++ + aa24 25 aa 31 32 aa/m1 38 39 aa 24 26 aa 31 33 aa/m1 38 40 aa24 27 aa 31 34 aa/m1 38 41

LL222++ + aa++ aa aa/m1 + L aa ++ aa aa/m1 + aaa/m125 32 41 25 26 32 33 39 40 25 27 32 34 39 41

LLLaa++ aa aa/m1 +222++ + aa ++ aa aa/m1 + 26 25 33 32 40 39aaa/m126 33 40 26 27 33 34 40 41

LLLaa++ aa aa/m1 + a a + aaaa/m1++ L 222++ + 27 25 34 32 41 39 27 26 34 33 41 40 aaa/m127 34 41

LLL++ + ++ + ++ + aa28 25 aa 35 32 aa/m1 42 39 aa 28 26 aa 35 33 aa/m1 42 40 aa 28 27 aa 35 34 aa/m1 42 41

L ++ +V aa22 28 aa 29 35 aa/m1 36 42 W Laa++ aa aa/m1 +W 23 28 30 35 37 42 W L ++ + aa24 28 aa 31 35 aa/m1 38 42 W W L ++ + aa25 28 aa 32 35 aa/m1 39 42 W

L ++ +W aa26 28 aa 33 35 aa/m1 40 42 W Laa++ aa aa/m1 +W 27 28 34 35 41 42 W L 222++ + W aaa/m128 35 42 X

F LL222222++ + + + + aaaaaa++ + aa + aa + aa /m1 + G aaaaaa/m11 8 15 22 29 36 1 2 8 9 15 16 22 23 29 30 36 37

G LLaaaaaa++ + aa + aa + aa /m1 + 222++ + 2 + 2 + 2 + G 2 1 9 8 16 15 23 22 30 29 37 36 aaa2 9 16 a 23 a 30 a/m1 37 GLLaaaa++ aa + aa + aa + aa /m1 + aa + aaaaaaaaaa/m1++++ + G 31 108 1715 2422 3129 3836 32 10 9 17 16 24 23 31 30 38 37 = LL+++++ + +++++ + G aaaa4 1 11 8 aa 18 15 aa 25 22 aa 32 29 aa 39 36 /m1 aaaa 4 2 11 9 aa 18 16 aa 25 23 aa 32 30 aa 39 37 /m1 G LL+++++ + +++++ + G aa5 1 aa 12 8 aa 19 15 aa 26 22 aa 33 29 aa 40 36 /m1 aa 5 2 aa 12 9 aa 19 16 aa 26 23 aa 33 30 a 40a/m137

GLL+++++ + +++++ + G aaaa6 1 13 8 aa 20 15 aa 27 22 aa 34 29 aa 41 36 /m1 aaaa 6 2 13 9 aa 20 16 aa 27 23 aa 34 30 aa 41 37 /m1 GLL++ + + + + ++ + + + + H aaaa7 1 14 8 aa 21 15 aa 28 22 aa 35 29 aa 42 36 /m1 aa 7 2 aa 14 9 aa 21 16 aa 28 23 aa 35 30 aa 42 37 /m1

95 LL+++++ + +++++ + aaaa1 3 8 10 aa 15 17 aa 22 24 aa 29 31 aa 36 38 /m1 aaaa 1 4 8 11 aa 15 18 aa 22 25 aa 29 32 aa 36 39 /m1

LL++++++ ++++++ aaaa2 3 9 10 aa 16 17 aa 23 24 aa 30 31 aa 37 38 /m1 aaaa 2 4 9 11 aa 16 18 aa 23 25 aa 30 32 aa 37 39 /m1

L 22222+++++2 + Laa+++ aa aa aa ++ aa aa /m1 + aaaaaa31017243138/m 1 3 4 10 11 17 18 24 25 31 32 38 39

LLaaaa+++ aa aa ++ aa aa /m1 + 222222+++++ + 4 3 11 10 18 17 25 24 32 31 39 38 aaaaaa/m141118253239

LL+++++ + ++++++ aa5 3 aa 12 10 aa 19 17 aa 26 24 aa 33 31 aa 40 38 /m1 aa 5 4 aa 12 11 aa 19 18 aa 26 25 a 33 aaa/m132 40 39

LL+++++ + +++++ + aaaa6 3 13 10 aa 20 17 aa 27 24 aa 34 31 aa 41 38 /m1 aaaa 6 4 13 11 aa 20 18 aa 27 25 aa 34 32 aa 41 39 /m1

LL++++++++++++ aa7 3 aa 14 10 aa 21 17 aa 28 24 aa 35 31 aa 42 38 /m1 aa 7 4 aa 14 11 aa 21 18 aa 28 25 aa 35 32 aa 42 39 /m1

LL+++++ + +++++ + aaaa1 5 8 12 aa 15 19 aa 22 26 aa 29 33 aa 36 40 /m1 aaaa 1 6 8 13 aa 15 20 aa 22 27 aa 29 34 aa 36 41 /m1

LL+++++ + +++++ + aaaa25 912 aa 1619 aa 2326 aa 3033 aa 3740 /m1 aaaa 26 913 aa 1620 aa 2327 aa 3034 aa 3741 /m1

L +++ +++L ++++++ aa3 5 a 10 a 12 a 17 a 19 a 24 a 26 a31a 33 aa 38 40 /m1 aa 3 6 aa 10 13 aa 17 20 aa 24 27 aa 31 34 aa 38 41 /m1

LL+++ + + + ++ + + + + aaaa4 5 11 12 aa 18 19 aa 25 26 aa 32 33 aa 39 40 /m1 aaaa 6 6 11 13 aa 18 20 aa 25 27 aa 32 34 aa 39 41 /m1

LL222222+++++ + aa++ aa aa +++aa aa aa /m1 + aaaaaa/m151219263340 5 6 12 13 19 20 26 27 33 34 40 41

LLaaaa+++ aa aa ++ aa aa /m1 + 222222+++++ + 6 5 13 12 20 19 27 26 34 33 41 40 aaaaaa/m161320273441

LL+++++ + +++++ + aa7 5 aa 14 12 aa 21 19 aa 28 26 aa 35 33 aa 42 40 /m1 aa 7 6 aa 14 13 aa 21 20 aa 28 27 aa 35 34 aa 42 41 /m1

L ++ + + + +V aaaa17 814 aa 1521 aa 2228 aa 2935 aa 3642 /m1W Laa+++++ aa aa aa aa aa /m1 +W 27 914 1621 2328 3035 3742 W L +++ ++ + aa3 7 aa 10 14 aa 17 21 aa 24 28 aa 31 35 aa 38 42 /m1W W L +++ + + + aaaa4 7 11 14 aa 18 21 aa 25 28 aa 32 35 aa 39 42 /m1W . W L +++ ++ + aa5 7 aa 12 14 aa 19 21 aa 26 28 aa33 35 aa 40 42 /m1W

Laaaa+++ aa aa ++ aa aa /m1 +W 6 7 13 14 20 21 27 28 34 35 41 42 W L 222222+++++ + W aaaaaa/m171421283542 X

We see the product is again super matrix of refined labels but is clearly a symmetric super matrix of refined labels of natural order 7 × 7. Thus by this technique of product we can get symmetric super matrix of refined labels which may have applications in real world problems.

96 We see at times the product of two super matrices of refined labels may turn out to be just matrices or symmetric super matrices of refined labels depending on the matrices producted and the products defined [48].

Now we proceed onto define the notion of M – matrix of labels Z – matrices labels and the corresponding subdirect sums.

Let A = {}()LLL∈ be a m × n refined label matrix aaRiimn× a we say A > 0 if each L = i > 0. (A ≥ 0 if each L = ai m1+ ai a i ≥ 0). Thus if A ≥ B we see A – B ≥ 0. Square matrices m +1 with refined labels which have non positive off – diagonal entries are called Z-matrices of refined labels.

−a We see –La = L-a = . m1+

FVLLL−− GWaaa123 −− A = GWLLaa L a is a Z-matrix of refined labels GW45 6 GW−−LLL HXaaa789 where ai > 0.

We call a Z-matrix of refined labels A to be a non singular M-matrix of refined labels if A-1 ≥ 0.

We have following properties to be true for usual M- matrices which is analogously true in case of M-matrices of refined labels.

(i) The diagonal of a non singular M-matrix of refined labels is positive. (ii) If B is a Z-matrix of refined labels and M is a non singular M-matrix and M ≤ B then B is also a non singular M-matrix.

97 In particular any matrix obtained from a non singular M- matrix of refined labels by setting off-diagonal entries is zero is also a non singular M-matrix. Thus FVLLL GWaaa123 a1 A = GWLLLaaa; La = > 0, GW456 1 m1+ GWLLL HXaaa789

a a L = 5 > 0 and L = 9 > 0 a5 m1+ a9 m1+

a a a L = 2 < 0, L = 3 < 0, L = 4 < 0, a2 m1+ a3 m1+ a4 m1+

a a L = 6 < 0, L = 7 < 0 and a6 m1+ a7 m1+

a L = 8 < 0. a8 m1+

A is a non singular M-matrix and the diagonal elements are positive. A matrix M is non singular M-matrix if and only if each principal submatrix of M is a non singular M-matrix. For instance take FVLLL−− GWaaa123 −− A = GWLLaa L a GW45 6 GW−−LLL HXaaa789 to be a non singular M-matrix then L ≠ 0 ∈ LR and a1 FV− LLaa GW56 are non singular L > 0 GW−LL ai HXaa89

ai that is > 0. 1 ≤ i ≤ 9 and L ∈ LR. m1+ ai

98 Finally a Z-matrix A is non singular M-matrix if and only if there exists a positive vector Lx > 0 such that ALx > 0. Thus all results enjoyed by simple M-matrices can be derived analogously for M-matrices of refined labels.

Now we can realize these structures as super matrices of refined labels. Suppose F LLL−−V G aaa123W −− A = G LLaa L aW G 45 6W G−−LLLW H aaa789X and F V LLL−− G bbb123W B = G−−LL LW G bb45 b 6W G−−LLLW H bbb789X L = L be −L = −L b1 a5 a6 b2 −L = L and −L = L . a9 b5 b4 a8

Two super matrices of refined labels then A ⊕2 B = C is not a M-matrix of refined labels.

F LLL0−− V G aaa123 W −−− G LLa2b2bb L LW C = G 41 23W −−LLL − L G a2b2bb5456W G 0L−− LLW H bbb789X

We can find C-1. We see C is super matrix of refined labels and the sum is not a usual sum of super matrices. This method of addition of over lapping super matrices of refined labels may find itself useful in applications. All results studied can be easily defined for matrices of refined labels with simple appropriate operations.

99 We can also define the P-matrix of refined labels as in case of P-matrix.

Further it is left as an exercise for the reader to prove that in general k-subdirect sum of two P-matrices of refined labels is not a P-matrix of refined labels in general. If FVF V LLLaaa L0L GW123 G bb12W G W A = GWLLaa 0 and B = 0Lbb L GW45 G 34W GWL0Laa GLLLW HX67 H bbb767X be two P-matrices of refined labels then we have the 2-subdirect sum as

FLL L 0V G aa12 a 3 W LL+ 0 L G aab451 b 2W C = A ⊕2 B = G W L0LL+ G aabb6734W G 0L L LW H bbb567X is not a P-matrix of refined labels as det (C) < 0.

Thus the concept of M-matrices, Z-matrices and P-matrices can also be realized as the super matrices of refined labels. Also the k-subdirect sum is also again a super matrix but addition is different. For in usual super matrices addition can be carried out for which the resultant after addition enjoy the same partition. Here we see the k-subdirect sum is defined only in those cases of different partition and however the k-subdirect sum of super matrices do not enjoy even the same natural order as that of the their components. Thus they do not form a group or a semigroup under k-subdirect addition.

100 %

In this chapter we for the first time introduce the notion of supervector space of refined labels using the super matrices of refined labels. By using the notion of supervector space of refined labels we obtain many vector spaces as against usual matrices of refined labels. We proceed onto illustrate the definitions and properties by examples so that the reader gets a complete grasp of the subject.

DEFINITION 4.1 : Let V = {()LLLLL...LL L ...L aaaaa12345 aa rr1r2n++ a a LL;1in∈≤≤} be the collection of all super row vectors of aRi same type of refined labels. V is an additive abelian group. V is a super row vector space of refined labels over R or LR (as LR ≅ R).

We will illustrate this situation by some examples.

101 Example 4.1: Let

V = {}()LLLLLLLL;1i6∈≤≤ aa123456i aa aaaR be a super row vector space of refined labels over the field LR (or R).

Example 4.2: Let V = {}()LLLLLLLL;1i6∈≤≤ be a aa123456i aa aaaR super row vector space of refined labels over the field LR (or R).

Example 4.3: Let W= {}()LLLLLLLL;1i6∈≤≤ be a aa123456i aa aaaR super row vector space of refined labels over the field LR (or R).

Example 4.4: Let V = {}()LLLLLLLL;1i6∈≤≤ be a aa123456i aa aaaR super row vector space of refined labels over the field LR (or R).

We see for a given row vector ()LLLLLL of refined labels we can get aaaaaa123456 several super row vector space of refined labels over the field LR or R. Thus this is one of the main advantage of defining super row vectors of refined labels over LR or R.

Example 4.5: Let

X = {()LLLLLLLLLLaaaaaaaaaa 12345678910 LL;1i10∈≤≤} be a collection of all super row vectors of aRi refined labels. X is a group under addition and X is a super row vector space of refined labels over LR (or R).

Now having seen examples of super row vector spaces of refined labels we now proceed onto define substructures in them.

102 DEFINITION 4.2 : Let V = {()LLLLLLL...LL aaaaaaa1234567 a n1n− a LL∈≤≤;1 in} be a super row vector space of refined labels aRi over the field LR. Suppose W ⊆ V; W a proper subset of V; if W is a super row vector space of refined labels over LR. then we define W to be super vector subspace of V over the field LR.

Example 4.6: Let P = {}()LLLLLLLLL;1i7∈≤≤ aa1234567i aa aaaaR be a super vector space of V over the field LR (or R). Consider S = {}()L0L0L0LLL;1i7∈≤≤ ⊆ P aa1234i aaaR is a super row vector subspace of refined labels of P over the field LR (or R). Take T = {}()0L 0L 0L 0L∈≤≤ L;1i3 ⊆ P, aa123i aaR T is a super row vector subspace of refined labels of P.

Example 4.7: Let X = {}()LLLLLL;1i4∈≤≤ aaaaaR1234i be a super row vector space of refined labels over LR (or R). Consider

X1 = {}()L 000L∈ L ⊆ X, aaR1i

X2 = {}()0L 00L∈ L ⊆ X, aaR2i

X3 = {}()00L 0L∈ L ⊆ X aaR3i and

X4 = {}()000L L∈ L ⊆ X aa4i R be four super row vector subspaces of refined labels of X over LR.

103 4 ∩ ≠ ≤ ≤ We see X = X i and Xi Xj = (0 | 0 0 0) if i j; 1 i, j 4. i=1 However X has more than four super row vector subspaces of refined labels.

For take Y1 = {}()LL00LL;1i2∈≤≤ ⊆ X is aa12 aR i a super row vector subspace of refined labels.

Consider Y2 = {}()L00LLL;1i2∈≤≤ ⊆ X is aaaR12i a super row vector subspace of X of refined labels.

Take Y3 = {}()0L L 0L∈≤≤ L;1i2 ⊆ X is aa12 aR i again a super row vector subspace of X of refined labels over LR.

DEFINITION 4.3: Let V be a super row vector space of refined labels over the field LR (or R). Suppose W1, W2, …, Wm are super row vector subspaces of refined labels of V over the field m ∩ ≠ ≤ ≤ LR (or R). If V = Wi and Wi Wj = (0) if i j, 1 i, j m i=1 then we say V is a direct union or sum of super row vector subspaces of V.

If W1, …, Wm are super row vector subspaces of refined m ∩ ≠ labels of V over LR (or R) and V = Wi but Wi Wj (0) if i=1 i≠j, 1 ≤ i, j ≤ m then we say V is a pseudo direct sum of super vector subspaces of refined labels of V over LR (or R).

Example 4.8: Let V = {}()LLLLLLLLL;1i7∈≤≤ aa1234567i aa aaaaR be a super row vector space of refined labels over LR (or R). Consider

W1 = {}()L L 00000L∈≤≤ L;1i2 ⊆ V, aa12 aR i

104 W2 = {}()L0LL00LLL;1i4∈≤≤ ⊆ V, aaa1234i aaR

W3 = {}()L00LL00LL;1i3∈≤≤ ⊆ V, aaaaR134i

W4 = {}()L0L0LLLLL;1i5∈≤≤ ⊆ aa1 2 aaaaR 435i V and

W5 = {}()0L L 000L L∈≤≤ L;1i3 ⊆ V aa12 aaR 5i be super row vector subspaces of refined labels of V over LR. 5 ∩ ≠ ≠ ≤ Clearly V = Wi but Wi Wj (0 0 | 0 | 0 0 0 | 0) if i j, 1 i, i=1 j ≤ 5. Thus V is only a pseudo direct sum of Wi, i = 1, 2, …, 5 and is not a direct sum. Now we see the supervector space of refined labels as in case of other vector spaces can be written both as a direct sum as well as pseudo direct sum of super vector subspaces of refined labels over LR (or R).

Example 4.9: Let V = {}()LLLLLLLLLL;1i8∈≤≤ aaaaaaaaaR12356784i be a super vector. Consider

W1 = {}()L 0000000L∈ L ⊆ V, aaR1i

W2 = {}()0L 000000L∈ L ⊆ V, aaR1i

W3 = {}()00L 00000L∈ L ⊆ V, aaR1i

W4 = {}()000L 0000L∈ L ⊆ V, aaR1i

W5 = {}()0000L 000L∈ L ⊆ V, aaR1i

W6 = {}()00000LLL 00 ∈ ⊆ V, aaR1 i

W7 = {}()000000L 0L∈ L ⊆ V and aaR1i

105 W8 = {}()0000000L L∈ L ⊆ V; aa1i R are super row vector subspaces of V over the field LR. Clearly 8 ∩ ≠ ≤ ≤ V = Wi but Wi Wj = (0 0 | 0 0 | 0 0 0 0) if i j, 1 i, j 8. i1= V is a direct sum of super row vector subspaces W1, W2, …, W8 of V over LR. Consider

W1 = {}()L L 00000L L∈≤≤ L;1i 3 ⊆ V, aa12 a 3 a i R

W2 = {}()L 0L L 0000 L∈≤≤ L;1i 3 ⊆ V, aaa123 aR i

W3 = {}()0L 00L L 00 L∈≤≤ L;1i 3 ⊆ V aaaaR123i and

W4 = {}()L0L0L0L0LL;1i3∈≤≤ ⊆ V aa1234 aa aR i be super row vector subspaces of V over the refined field LR (or 4

R). Clearly V = Wi is a pseudo direct sum of super row i1= vector subspaces of V over the field LR as Wi ∩ Wj ≠ (0 0 | 0 0 | 0 0 0 0) if i ≠ j, 1 ≤ i, j ≤ 4. Now having seen examples of direct sum and pseudo direct sum of super row vector subspaces of V over LR we now proceed onto define linear operators, basis and linear transformation of super row vector spaces of refined labels over LR. It is pertinent to mention here that LR ≅ R so whether we work on R or LR it is the same.

106 DEFINITION 4.4 : Let V = {}()LL...LLLL;1in∈≤≤ be a super row aa12 a n1ni− aaR vector space of refined labels over the field LR. We say a subset of elements {x1, x2, …, xt} in V are linearly dependent if there exists scalar labels L...L in LR not all of them equal to aa1t zero such that Lx+++= Lx ...Lxt 0. A set which is not a112 a2 a t linearly dependent is called linearly independent.

We say for the super row vector space of refined labels V = {}()LLLLL...LLLL;1in∈≤≤ aaaaa12345 a n1ni− aaR a subset B of V to be a basis of V if (1) B is a linearly independent subset of V. (2) B generates or spans V over LR.

We will give examples of them. It is pertinent to mention in LR; Lm+1 is the unit element of the field LR.

Example 4.10: Let V = {}()LLLLLLLLL;1i7∈≤≤ be a aa1234567i aa aaaaR super row vector space of refined labels over the field LR.

Consider the set B = {(Lm+1, 0 | 0 | 0 0 0 | 0), (0, Lm+1 | 0 | 0 0 0 | 0), (0 0 | Lm+1 | 0 0 0 | 0), (0 0 | 0 | Lm+1 0 0 | 0), (0 0 | 0 | 0 Lm+1 0 | 0), (0 0 | 0 | 0 0 Lm+1 | 0), (0 0 | 0 | 0 0 0 |Lm+1)} ⊆ V is a super basis of over LR. Clearly V is finite dimensional over LR.

Consider P = ()L L 0000L , aa12 a 7 ()00L 0000, ()000L 000, a2 a3 ()0000L L 0 ⊆ V; P is only a super linearly aa45 independent subset of V but is not a basis of V.

107 Example 4.11: Let V = {}()LLLLL;1i3;∈≤≤ aaaaR123i be a super row vector space of refined labels over LR. Consider B = {(Lm+1 | 0 0), (0, | Lm+1 0), (0 | 0 Lm+1)} ⊆ V is a super basis of V over LR.

Clearly dimension of V is three over LR. Consider M ={}()L00,0LL() ⊆ V; M is only a linearly aaa112 independent set of V over LR and is not a basis of V over LR.

Take K = {()L0L,LL0,0LL()(), a1212 a aa aa 12 ()L00,0L0()} ⊆ V, K is a linearly dependent aa12 subset of V. Consider B = {}()0L 0,0L() L ,00L() ⊆ V. B is also a aaaa1122 linearly dependent subset of V over LR.

Now having seen examples of linearly dependent subset, linearly independent subset and not a basis and linear independent subset of V which is a super basis of V, we now proceed onto define the linearly transformation of super row vector spaces of refined labels over LR.

Let V and W be two super row vector space of refined labels over LR. Let T be a map from V to W we say T : V → W is a linear transformation if T (cv + u) = cT (v) + T (u) for all u, v ∈ V and c ∈ LR.

Example 4.12: Let V = {}()LLLLLLLL;1i6∈≤≤ be a super aa123456i aa aaaR row vector space of refined labels over LR. W = {}()LLLLLLLLLL;1i8∈≤≤ aaaaaaaaaR12345678i

be a super row vector space of refined labels over LR. Define a map T : V → W by T ()LLLLLL = aaaaaa123456 ()L 0 L 0 L L L L 0 ; T is easily aaaaaa1 2 3456 verified to be a linear transformation.

108 Let P : V → W be a map such that ()LLLLLL = aaaaaa123456 ()L L L L L L 0 0 ; It is easily verified P aaaaaa123456 is a linear transformation of super row vector space V into W.

Example 4.13: Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be a super row vector space of refined aRi labels over the field LR. Let T be a map from V to V defined by T ()LLLLLLLLLL = aaaaaaaaaa12345678910 ()L0L0L0L0L0; T is a linear aaaaa13579 operator on V. It is clear T is not one to one. Ker T is a non zero subspace of V. Consider a map P : V → V defined by P ()()LLLLLLLLLL = aaaaaaaaaa12345678910 ()LLLLLLL LLL. P is aaaaaaa32154610987 aaa a linear operator with trivial kernel.

Now suppose V = {}()LLLLLLLLLL;1i8∈≤≤ aaaaaaaaaR12345678i be a super row vector space of refined labels over the field LR.

Consider W = {}()L L L 0000L L∈= L;i1,2,3,8⊆V; aaa123 aaR 8i W is a super row vector subspace of V of refined labels over the field LR.

Let T : V → V if T (W) ⊆ W, W is invariant under T, we can illustrate this situation by some examples.

109 T ()()LLLLLLLL = aaaaaaaa12345678 ()L L L 0000L ;is such that T(W)⊆ W. aaa123 a 8

Suppose M = {}()L0L0L0L0LL;1i4∈≤≤ ⊆ V; aa1234i aaaR then M is a super row vector subspace of V of refined labels over LR. Define η : V → V by η ()()LLLLLLLL = aaaaaaaa12345678 ()L0L0L0L0; η is a linear aaaa1234 transformation and M is invariant under η as η (M) ⊆ M.

As in case of usual vector spaces in case of super row vector space of refined labels over the field LR we can have the following theorem.

THEOREM 4.1: Let V and W be two finite dimensional super vector spaces of refined labels over the field LR such that dim V = dim W. If T is a linear transformation from V into W, the following are equivalent. (i) T is invertible (ii) T is non singular (iii) T is onto, that is range of T is W.

The proof is left as an exercise to the reader.

Further we can as in case of usual vector space define in case of super row vector space of refined labels also the following results.

THEOREM 4.2: If {α1, α2, …, αn} is a basis for V then {Tα1, Tα2, …, Tαn} is a basis for W. (dim V = dim W = n is assumed), V a finite dimensional super vector space over LR and Ta is a linear transformation from V to W.

This proof is also direct and hence left as an exercise to the reader.

110 THEOREM 4.3: Let V and W be two super row vector spaces of same dimension over the real field LR. There is some basis {α1, α2, …, αn} for V such that {Tα1, Tα2, …, Tαn} is a basis for W.

The proof of this results is direct.

THEOREM 4.4: Let T be a linear transformation from V into W, V and W super row vector spaces of refined labels over the field LR then T is non singular if and only if T carries each linearly independent subset of V onto a linearly independent subset of W.

We make use of the following theorem to prove the theorem 4.4.

THEOREM 4.5: Let V and W be any two super row vector spaces of refined labels over the field LR and let T be a linear transformation from V into W. If T is invertible then the inverse function T -1 is a linear transformation from W onto V.

Also as in case of usual vector space we see in case of super row vector space V of refined labels over a field F = LR if T1, T2 and P are linear operators on V and for c ∈ LR we have (i) IP = PI = P (ii) P (T1 + T2) = PT1 + PT2 (iii) (T1 + T2) P = T1P + T2P (iv) c (PT1) = (cP)T1 = P (cT1).

All the results can be proved without any difficulty and hence is left as an exercise to the reader.

Also if V, W and Z are super row vector spaces of refined labels over the field LR and T : V → W be a linear transformation of V into W and P a linear transformation of W into Z, then the composed function PT defined by (PT) (α) = P (T(α)) is a linear transformation from V into Z. This proof is also direct and simple and hence is left as an exercise to the reader.

Now having studied properties about super row vector spaces of refined labels we now proceed onto define super column vector spaces of refined labels over LR.

111 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 DEFINITION 4.5: Let V = JZLL1in∈≤≤; be a GW aRi LLGW LL GWL LLGWan1− LLGWL K[HXan super column vector space of refined labels over the field LR (or R); clearly V is an abelian group under addition.

We will illustrate this situation by some examples.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LLLa Example 4.14: Let V = JZGW4 LL;1i8∈≤≤ be group GWL aRi LLa5 LLGW GWL LLa6 GW LLL LLGWa7 LLGWL K[HXa8 under addition. V is a super column vector space of refined labels over the field LR (or R).

IYFVL LLGWa1 LLGWLa LLGW2 Example 4.15: Let V = JZL LL;1i5∈≤≤ be super GWa3 aR LLi GWL LLGWa4 LLGWL K[HXa5 column vector space of refined labels over the field LR.

112 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL ∈≤≤ Example 4.16: Let V = JZa4 LL;1i7 be a super GWaRi LL GWL LLGWa5 LL GWLa LLGW6 LLL K[HXGWa7 column vector space of refined labels over the field LR (or R).

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL ∈≤≤ Example 4.17: Let V = JZa4 LL;1i7 be a super GWaRi LL GWL LLGWa5 LL GWLa LLGW6 LLL K[HXGWa7 column vector space over the field LR.

We can define the super column vector subspace of V, V a super column vector space over LR. We will only illustrate the situation by some examples.

113 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW ∈≤≤ Example 4.18: Let V = JZLa LL;1i7aR be a super LLGW4 i GWL LLa5 LLGW GWL LLa6 GW LLL K[HXa7 column vector space over LR.

IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW ∈≤≤⊆ Let W1 = JZ0 LL;1i4aR V be a super LLGW i GWL LLa3 LLGW LLGW0 GW LLL K[HXa4 column vector subspace of V over LR.

IYFV0 LLGW LLGW0 LLGW L LLGWa1 LL GW ∈≤≤⊆ W2 = JZLa LL;1i3aR V is a super column LLGW2 i GWL LLa3 LLGW LLGW0 LLGW K[HX0 vector subspace of V over LR.

114

IYFV0 LLGW LLGW0 LLGW LLGW0 LL GW ∈≤≤⊆ W3 = JZ0 LL;1i2aR V is a super column LLGW i LLGW0 LLGW GWL LLa1 GW LLL K[HXa2 vector subspace of V of refined labels over LR.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLa4 LLGW GW ∈≤≤ Example 4.19: Let V = JZLa LL;1i9aR be a super LLGW5 i LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 LLGWL K[HXa9 column vector space of refined labels over LR.

115

IYFV0 LLGW LLGW0 LLGW LLGW0 LLGW LL0 LLGW GW ∈≤≤⊆ Take W1 = JZLa LL;1i4aR V; is a super LLGW1 i LLGWLa GW2 LLL LLGWa3 LLGW0 LLGW LLGWL K[HXa4 column vector subspace of refined labels over LR.

IYFV0 LLGW LLGWLa LLGW1 L LLGWa2 LLGW LL0 LLGW GW ∈≤≤⊆ W2 = JZ0 LL;1i3aR V is a super column LLGW i GWL LLa3 LLGW LLGW0 LLGW0 LLGW GW K[LLHX0 vector subspace of refined labels over LR.

116 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa Example 4.20: Let V = JZGW5 LL;1i10∈≤≤ be a aRi LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10 super column vector space of refined labels over LR.

IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW0 LLGW LLGW LLLa GW3 ∈≤≤⊆ Consider W1 = JZLL;1i5aR V, is a super LLGW0 i GW LLL LLGWa4 LLGW0 LLGW LLGW0 LLGW LLGWLa K[HX5 column vector subspace of refined labels over LR.

117 IYFV0 LLGW LLGWLa LLGW1 L LLGWa2 LL GWL LLGWa3 LL LLGW0 Take W2 = JZGWLL;1i7∈≤≤⊆ V is a super aRi LLGWLa GW4 LLL LLGWa5 LLGWL LLGWa6 GW LLLa LLGW7 GW0 K[LLHX column vector subspace of refined labels over LR.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLGWa5 LL LLGWLa Example 4.21: Let V = JZGW6 LL;1i12∈≤≤ be a L aRi LLGWa7 LL GWL LLGWa8 LLGWL LLGWa9 LL GWLa LLGW10 L LLGWa11 LLGW L K[LLHXa12 super column vector space of refined labels over the field LR. Consider the following super column vector subspaces of refined labels over LR of V.

118 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW0 LLGW LLGW0 LLGW LLGW0 W1 = JZGWLL;1i7∈≤≤⊆ V is a super column L aRi LLGWa4 LL GWL LLGWa5 LLGWL LLGWa6 LL GWLa LLGW7 LLGW0 LLGW K[LLHX0 vector subspace of V over LR.

IYFV0 LLGW LLGW0 LLGW LLGW0 LL GWL LLGWa1 LLGWL LLGWa2 LLGWL a3 ∈≤≤⊆ W2 = JZGWLL;1i4aR V is a super column LLGW0 i LL GW0 LLGW LLGW0 LLGW LLGW0 LLGW L LLGWa4 LLGW K[LLHX0 vector subspace of V over LR.

119

IYFV0 LLGW LLGW0 LLGW LLGW0 LL GW0 LLGW LLGW0 LLGW LLGW0 ∈ ⊆ W3 = JZGWLLaR V is a super column vector LLGW0 1 LL GW0 LLGW LLGW0 LLGW LLGW0 LLGW LLGW0 LL GWL K[LLHXa1 3 subspace of refined labels of V over LR. We see V = Wi and i=1 FV0 GW GW0 GW0 GW GW0 GW0 GW GW0 ∩ ≠ ≤ ≤ Wi Wj = GW if i j, 1 i, j 3. GW0 GW0 GW GW0 GW0 GW GW0 HXGW0

120 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 Example 4.22: Let V = JZLL;1i6∈≤≤ be a super GWaRi LLL GWa4 LL GWL LLGWa5 LLGWL K[HXa6 column vector space of refined labels on LR. Take W1 = IYFV0 IYFVL LLGW LLGWa1 LLGWLa LLGW0 LLGW1 LLGW L 0 LLGWa2 LLGW JZLL;1i2∈≤≤ ⊆ V, W2 = JZLL∈ ⊆ V GW0 aRi GW0 aRi LLGW LLGW LLLL GW0 GW0 LLGW LLGW LLLL K[HXGW0 K[HXGW0 IYFV0 LLGW LLGW0 LLGW LLGW0 and W3 = JZLL;1i3∈≤≤⊆ V be three super column GWaRi LLL GWa1 LL GWL LLGWa2 LLGWL K[HXa3 vector subspace of refined labels of V over LR. F0V G W G0W 3 G0W ∩ ≠ ≤ ≤ We see V = Wi and Wi Wj = G W if i j, 1 i, j 3. i1= G0W G0W G W HG0XW Thus V is a direct union of super vector column subspaces of V over LR.

121 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LLLa Example 4.23: Let V = JZGW4 LL;1i8∈≤≤be a super GWL aRi LLa5 LLGW GWL LLa6 GW LLL LLGWa7 LLGWL K[HXa8 column vector space of refined labels over LR (or R). Consider IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LL0 GW ∈≤≤⊆ W1 = JZLL;1i4aR V, be a super column vector LLGW0 i LLGW LLGW0 GW LLL LLGWa4 LLGW K[HX0 subspace of refined labels of V over LR, IYFV0 LLGW LLGWLa LLGW1 LLGW0 LLGW LLL GWa2 W2 = JZLL;1i4∈≤≤ ⊆ V be a super column GWL aRi LLa3 LLGW GWL LLa4 GW LL0 LLGW LLGW K[HX0 vector subspace of refined labels of V over LR.

122 IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW0 LLGW W3 = JZLL;1i4∈≤≤ ⊆ V be a super column GWL aRi LLa3 LLGW LLGW0 GW LL0 LLGW LLGWL K[HXa4 vector subspace of refined labels of V over LR. IYFV0 LLGW LLGWLa LLGW1 L LLGWa2 LL GW0 LLGW W4 = JZLL;1i6∈≤≤⊆ V be a super column GWL aRi LLa3 LLGW GWL LLa4 GW LLL LLGWa5 LLGWL K[HXa6 vector subspace of refined labels over LR of V.

F0V G W G0W G0W G W 4 ∩ ≠ G0W ≠ ≤ ≤ Clearly V = Wi and Wi Wj G W ; i j, 1 i 4. = 0 i1 G W G0W G W G0W HG0XW Thus V is a pseudo direct sum of super column vector subspaces of V over LR.

123 It is pertinent to mention that we do not have super column linear algebras of V of refined labels over LR. However we can also have subfield super column vector subspace of refined labels over LR.

We will illustrate this by some examples. Just we know LQ is a field of refined labels and LQ ⊆ LR. Thus we can have subfield super column vector subspaces as well as subfield super row vector subspaces of refined labels over the subfield LQ of LR (or Q of R).

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 Example 4.24: Let V = JZLL;1i8∈≤≤ be a super GWaRi LLLa GW5 LLGWL LLa6 LLGW GWLa LL7 LLGWL K[HXa8 column vector space of refined labels over the field LR (or R).

IYFVL LLGWa1 LLGW0 LLGW LLGW0 LLGW LLLa GW2 ∈≤≤⊆ Take M = JZLL;1i3aR V be a super LLGW0 i LLGW LLGW0 GW LL0 LLGW LLGWL K[HXa3 column vector space of refined labels over the refined labels field LQ. (LQ ⊆ LR). M is thus the subfield super column vector subspace of refined labels over the subfield LQ of LR.

124

Example 4.25: Let M = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be a super row vector space of refined labels aRi over LR.

Take P = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} ⊆ M; P is a super row vector space of aQi refined labels over LQ; thus P is a subfield super row vector subspace of V of refined labels over the subfield LQ of LR. We have several such subfield vector subspaces of V.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 Example 4.26: Let V = JZLL;1i8∈≤≤ be a super GWaQi LLLa GW5 LLGWL LLa6 LLGW GWLa LL7 LLGWL K[HXa8 column vector space of refined labels over the field LQ. Clearly V has no subfield super column vector subspaces of refined labels of V over the subfield over LQ as LQ has no subfields. In this case we call such spaces as pseudo simple super column vector subspaces of refined labels of V over LQ. However V has several super column vector subspaces of refined labels of

125 IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW0 LLGW V over LQ. For instance take K = JZLL;1i4∈≤≤ ⊆ GWL aRi LLa3 LLGW LLGW0 LLGW GWLa LL4 LLGW0 K[HX

V, K is a super column vector subspace of V over the field LR of refined labels.

Example 4.27: Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be a super row vector space of refined aRi labels over the field LQ.

Clearly LQ has no proper subfields other than itself as LQ ≅ Q. Hence V cannot have subspaces which are subfield super row vector subspace of refined labels over a subfield of LQ so V is a pseudo simple super row vector space of refined labels over LQ.

126 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa Example 4.28: Let V = JZGW5 LL;1i10∈≤≤ be a super aRi LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10 column vector space of refined labels over LQ. V is a pseudo simple super column vector space of refined labels. However V has several super column vector subspaces of refined labels over LQ of V.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW0 LLGW LL LLGW0 GW ∈≤≤⊆ For instance T = JZLL;1i4aR V is a super LLGW0 i LLGW LLGW0 LLGW0 LLGW LLGW0 LLGW LLGWLa K[HX4 column vector subspace of V of refined labels over LQ.

Now we can as in case of other vector spaces define basis linearly independent element linearly dependent element linear

127 operator and linear transformation. This task is left as an exercise to the reader. However all these situations will be described by appropriate examples.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 Example 4.29: Let V = JZLL;1i6∈≤≤ be a super GWaRi LLL GWa4 LL GWL LLGWa5 LLGWL K[HXa6 column vector space of refined labels over the refined field LR.

Consider the set B =

IYFVFVFVFVFVL0000+ LLGWGWGWGWGWm1 LLGWGWGWGWGW0Lm1+ 0 0 0 LLGWGWGWGWGW00L00 JZGWGWGWGWGW,,,,m1+ ⊆ V. It is easily LLGWGWGWGWGW000L0m1+ LLGWGWGWGWGW0000L LLGWGWGWGWGWm1+ K[LLHXHXHXHXHXGWGWGWGWGW00000 verified B is a linearly independent subset of V but is not a basis of V. We see B cannot generate V.

IYFL0VF V F 0 V FVa1 FV LLL0m1+ G WG W G W LLGWG 0LWG W GW G 0 W 0La1 LLGWG WG W GWm1+ G W LLGW0000 GW 0 G WG W G W ⊆ Now consider C = JZGW,,,G WG W GW , G W LLGW0000 GW L G WG W Ga1 W LLGW00G WG W GW G W LLGW00 GW 0 GWG WG W GW G W LLHX00G 0LWG W HX G 0 W K[H XHa2 X H X V; It is easily verifield the subset C of V cannot generate V so is not a basis further C is not a linearly independent subset of V as

128 FL V FV a1 Lm1+ G W GW 0 G 0 W GW G W GW0 G 0 W we see GW and are linearly dependent on each other. 0 G 0 W GW G W GW 0 G 0 W GW G W HXGW0 HG 0 XW These are more than one pair of elements which are linearly dependent.

Now consider H =

IYFVFVFVFVFVFVL+ 00000 LLGWGWGWGWGWGWm1 LLGWGWGWGWGWGW0L0000m1+ LLGWGWGWGWGWGW00L000 JZGWGWGWGWGWGW,,,,,m1+ ⊆ V is a basis LLGWGWGWGWGWGW000L00m1+ LLGWGWGWGWGWGW0000L0 LLGWGWGWGWGWGWm1+ K[LHXHXHXHXHXHXGWGWGWGWGWGW00000Lm1+ L of V and the space is of finite dimension and dimension of V over LR is six.

Now if we shift the field LR from LQ we see the dimension of V over LQ is infinite.

Thus as in case of usual vector spaces of dimension of a super column vector space of refined labels depends on the field over which they are defined.

This above example shows if V is defined over LR it is of dimension six but if V is defined over LQ the dimension of V over LQ is infinite.

Example 4.30: Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be a super row vector space of refined aRi labels over the refined field LQ. Clearly V is of infinite

129 dimension over LQ. Further V has a linearly independent as well as linearly dependent subset over LQ.

Consider B = {()L 000000000, a1 ()00L 0000000, a2 ()00000L 0000, a3 ()000000L 0L 0 aa56 ()0000000L 0L L∈≤≤ L;1i8} aaaR78i

⊆ V, B is a linearly independent subset of V over LQ.

Take P = {()L L 00000000, aa12 ()00L L 000000, aa12 ()0L 00L 00000, aa22 ()L 0L 00L 0000, aa11 a 4 ()000000000L a5 ()00L L L 00000} ⊆ V, P is a aa123 a linearly dependent subset of V over LQ. We have seen though V is of infinite dimension over LQ still V can have both linearly dependent subset as well as linearly independent subset.

Example 4.31: Let V = {}()LLLLLLL;1i5∈≤≤ be a super row aa12345i aa aaQ vector space of refined labels over the field LQ. Clearly V is finite dimensional and dimension of V over LQ is given by B = ()()() { Lm1+++ 0000,0L m1 000,00L m1 00, ()()⊆ 000Lm1++ 0,0000L m1} V, which is the basis of V over LQ.

130 Consider M =

{()L 0000,L()+ 0000,0L() L 00, am1aa1 12

()0L++ L 00,000L() 0,000L() 0 m1 m1 a11 b L,L∈≤≤ L;1i 2} ⊆ V is a subset of V but is clearly not a ba1i Q linearly independent set only a linearly dependent subset of V.

Now we will give a linearly independent subset of V which is not a basis of V.

Consider T =

{}()L 0000,00L()+++ 00,000L() L am1m1m11 in V; clearly T is not a basis but T is a linearly independent subset of V over LQ.

Now having seen the concept of basis, linearly independent subset and linearly dependent subset we now proceed onto define the notion of linear transformation and linear operator in case of the refined space of labels over LR (or LQ).

DEFINITION 4.6: Let V and W be two super row vector spaces of refined labels defined over the refined field LR (or LQ). Let T be a map from V into W. If T (cα + β) = cT (α) + T (β) for all c

∈ LR (or LQ) and α, β ∈ V then we define T to be a linear transformation from V into W.

Note if W = V then we define the linear transformation to be a linear operator on V.

We will illustrate these situations by some examples.

Example 4.32: Let V = {}()LLLLLLLL;1i6∈≤≤ and W aaaaaaaR123456i

={()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be two super row vector spaces of refined aRi labels over LR.

131

Define T : V → W be such that T ()()LLLLLL aaaaaa123456 = ()L L 0 0 0 L L L 0 L ; it is easily aa12 aaa 345 a 6 verified T is a linear transformation of super row vector space of refined labels over LR.

It is still interesting to find the notions of invertible linear transformation and find the algebraic structure enjoyed by the collection of all linear transformations from V to W. IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 Example 4.33: If V = JZLL;1i6∈≤≤ and GWaRi LLL GWa4 LL GWL LLGWa5 LLGWL K[HXa6 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa W = JZGW5 LL;1i10∈≤≤be two super column aRi LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10 vector spaces of refined labels over LR.

132 We can define the map T : V → W by

FVL GWa1 GWLa GW2 L GWa3 T ( ) = ()L L L L L L 0000. GWL aaaaaa123456 GWa4 GWL GWa5 GWL HXa6

It is easily verified T is a linear transformation of V into W. Infact we see kernel T is just the zero super vector subspace of V.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LL GWL ∈≤≤ Example 4.34: Let V = JZa5 LL;1i9aR be a super LLGW i GWL LLa6 GW LLL LLGWa7 LLGWL LLGWa8 LLGWL K[HXa9 column vector space of refined labels over LR.

133 FL V FL V G a1 W G a1 W GLa W G 0 W G 2 W G W L 0 G a3 W G W GL W GL W G a4 W G a2 W Define T : V → V by T = ( GL W ) = GL W ; G a5 W G a3 W GLa W GLa W G 6 W G 4 W L 0 G a7 W G W GL W G 0 W G a8 W G W GL W GL W H a9 X H a5 X it is easily verified that T is a linear operator on V and kerT is a non trivial super vector column subspace of V. Clearly T is not one to one.

Interested reader can define one to one maps and study the algebraic structure enjoyed by Hom (V,V) = {set of all linear LR operators of V to V}. Further it is informed that the reader can derive all the results and theorems proved in the case of super row vector spaces with appropriate modifications in case of super column vector spaces. This task is a matter of routine and hence is left for the reader as an exercise.

Now we proceed onto define the notion of super vector spaces of m × n super matrices of refined labels over the field LR.

DEFINITION 4.7: Let V = {All m × n super matrices of refined labels with the same type of partition with entries from LR}; V is clearly an abelian group under addition. V is a super vector space of m × n matrices of refined labels over the field LR (or R or LQ or Q).

We will illustrate this situation by some simple examples.

134 Example 4.35: Let V =

IYFVLLL LLGWaaa123 LLGWLLLaaa JZGW456LL;1i12∈≤≤ be the super matrix LLL aRi LLGWaaa789 LL GWLLL K[LLHXaaa10 11 12 vector space of refined labels over LR.

IYFVLL LLGWaa12 LLGWLLaa LLGW34 LL LLGWaa56 LL GWLL LLGWaa78 LL GWLL ∈≤≤ Example 4.36: Let M = JZaa910LL;1i18aR be a LLGWi GWLL LLaa11 12 GW LLLL LLGWaa13 14 LLGWLL LLGWaa15 16 LLGWLL K[HXaa17 18 super matrix vector space of refined labels over LR. M is also known as the super column vector vector space of refined labels over LR. These super matrices in M are also known as the super column vectors (refer chapter I of this book).

Example 4.37: Let T =

IYFV LLLLaaaaaa L L LLGW147101316 ∈≤≤ JZGWLLLLaaaa L a L a LL;1i18 aR be a LLGW258111417i GWLLLL L L K[LLHXaaaaaa369121518 super matrix of vector space of refined labels over the refined field LR. We can also call T to be a super row vector – vector space of refined labels over LR.

135 Example 4.38: Let K = IYCSLLLL LLDTaaaa1234 LLDTLLLLaaaa JZDT5678LL;1i16∈≤≤be the super LLLL aRi LLDTaa9101112 aa LL DTLLLL K[LLEUaaaa13 14 15 16 matrix of vector space of refined labels over LR.

Now it is pertinent to mention here that we cannot construct super linear algebra of refined labels even if the natural order of the super matrix is a square matrix. Thus we find it difficult to obtain linear algebras expect those using LR or LQ over LR or LQ respectively.

Thus we see if V = LR = {}LL∈ L then V is a vector aaii R space as well as linear algebra over LR or LQ however it is not a super linear algebra of refined labels over LR or LQ.

Likewise LQ = {}LL∈ L is only a linear algebra over aaii Q

LQ, however it is not super linear algebra over LQ.

Now we can define the notion of super vector subspaces, basis, linear operator and linear transformation as in case of super row vector spaces or super column vector spaces. This task is left as an exercise to the reader. We only give examples of all these concepts so that the reader has no difficulty in following them.

136 IYFVLL LLGWaa12 LLGWLLaa LLGW34 LL LLGWaa56 LL GWLL LLGWaa78 Example 4.39: Let V = JZLL;1i16∈≤≤ be a GWaRi LLLLaa GW910 LLGWLL LLaa11 12 LLGW GWLLaa LL13 14 LLGWLL K[HXaa15 16 super matrix of vector space of refined labels over LR. (super column vector, vector space of refined labels).

IYFVLL LLGWaa12 LLGW00 LLGW LL LLGWaa34 LL GW00 LLGW Consider M = JZLL;1i8∈≤≤⊆ V, is the GWLL aRi LLaa56 LLGW LLGW00 LLGW GWLLaa LL78 LLGW00 K[HX super column vector subspace of V over LR.

137 IYFV00 LLGW LLGW00 LLGW LL LLGWaa12 LL GWLL LLGWaa34 Take P = JZLL;1i6∈≤≤⊆ V is a super GWLL aRi LLaa56 LLGW LLGW00 GW LL00 LLGW LLGW00 K[HX column vector, vector subspace of V over LR of V.

Example 4.40: Let K =

IYCSLLLLLL LLDTaaaaaa123456 LLDTLLLLLLaaaaaa JZDT7 8 9 10 11 12 LL;1i24∈≤≤ be LLLLLL aRi LLDTaaaaaa13 14 15 16 17 18 LL DTLLLLLL K[LLEUaaaaaa19 20 21 22 23 24 a super row vector, vector space of refined labels over LR.

Consider M = IYCS00L 0L L LLDTaaa159 LLDT00Laaa 0L L JZDT2610LL;1i12∈≤≤ ⊆ K, is a 00L 0L L aRi LLDTaaa3711 LL DT00L0LL K[LLEUaaa4812 super row vector, vector subspace of K over LR.

We can have many such super row vector, vector subspaces of K over R.

138 IYCS LLaa LLDT14 ∈≤≤ Example 4.41: Let V = JZDTLLLL;1i6aa aR be a LLDT25 i DTLL K[LLEUaa36 super matrix vector space of refined labels over the field LR (or R).

Example 4.42: Let P = IYCSLLL LLDTaaa123 LLDTLLLaaa LLDT456 LLDTLLLaaa JZ789LL;1i18∈≤≤ be a super matrix DTaRi LLLLL DTaaa10 11 12 LL DTLLL LLDTaaa13 14 15 LLDTLLL K[EUaaa16 17 18 vector space of refined labels over the field LR. Take M = IYCS00L LLDTa1 LLDT00La LLDT2 LLDTLLaa 0 JZ34 LL;1i8∈≤≤ ⊆ P is a super matrix DTaRi LL00L DTa5 LL DTLL 0 LLDTaa67 LLDT00L K[EUa8 vector subspace of P over LR.

IYCS LLL LLDTaaa123 LL DT∈≤≤ Example 4.43: Let V = JZLLLLL;1i9aaa aR LLDT456 i DT LLLLLaaa K[EU789 be a super matrix vector space of refined labels over LR.

139 IYCS 00L LLDTa1 LL DT∈≤≤⊆ Take M = JZ00LLL;1i4aaR V, be a LLDT2i DT LLLLaa 0 K[EU34 super matrix vector subspace of refined labels over LR of V.

Now as in case of super row vector space of refined labels we can in the case of general super m × n matrix of refined label of vector space define the notion of direct sum of vector subspaces and pseudo direct sum of super vector subspaces. This task is simple and hence is left as an exercise to the reader.

However we illustrate this situation by some examples.

Example 4.44: Let V =

IYFVLLL LLGWaaa123 LLGWLLLaaa JZGW456LL;1i12∈≤≤ be a super matrix LLL aRi LLGWaaa789 LL GWLLL K[LLHXaaa10 11 12 vector space of refined labels over the field LR. Consider W1 = IYCSL00 LLDTa1 LLDTL00a JZDT2 LL;1i3∈≤≤⊆ V, be a super matrix row 0L 0 aRi LLDTa3 LLDT K[LLEU000 vector subspace of V over LR. IYCS0L 0 LLDTa3 LLDT000 W2 = JZDTLL;1i3∈≤≤⊆ V, L00aRi LLDTa1 LL DTL00 K[LLEUa2

140 IYCS00L LLDTa1 LLDT00L a2 ∈≤≤⊆ W3 = JZDTLL;1i2aR V, LLDT00 0 i LLDT K[LLEU00 0 IYCS00 0 LLDT LLDT0L 0 LLa1 W4 = JZDTLL;1i2∈≤≤ ⊆ V and 00L aRi LLDTa2 LLDT K[LLEU00 0 IYCS00 0 LLDT LLDT00 0 ∈≤≤⊆ W5 = JZDTLL;1i2aR V, be super matrix LLDT00 0 i LL DT0L L K[LLEUaa12 5 vector subspaces of V over LR. We see clearly V = Wi and i1= CS000 DT DT000 Wi ∩ Wj = if i ≠ j, 1 ≤ i, j ≤ 5. Thus V is the direct DT000 DT EU000 sum of super matrix vector subspaces of V over LR.

Example 4.45: Let V = IYCS LLLLLLLaaaaaaa LLDT1234567 ∈≤≤ JZDTLLLLLLLLL;1i21aaaaaaa aR LLDT8 9 10 11 12 13 14 i DTLLLLLLL K[LLEUaaaaaaa15 16 17 18 19 20 21

be a super row vector, vector space of refined labels over LR.

141 Consider IYCS La 000000 LLDT1 ∈≤≤⊆ W1 = JZDTLaaR 000000L L;1i3 V, LLDT2i DTL 000000 K[LLEUa3 IYCS 0La 00000 LLDT1 ∈≤≤⊆ W2 = JZDT0LaaR 00000L L;1i3 V, LLDT2i DT0L 00000 K[LLEUa3 IYCS 00Laa L 000 LLDT12 ∈≤≤⊆ W3 = JZDT00 0 0 000LaR L;1i4 V, LLDTi DT00L L 000 K[LLEUaa34 IYCS 00 0 0 La 00 LLDT3 ∈≤≤⊆ W4 = JZDT00LLL00LL;1i5aa a aR V, LLDT124 i DT00 0 0 L 00 K[LLEUa5 IYCS 00000La 0 LLDT1 ∈≤≤⊆ W5 = JZDT00000LaaR 0L L;1i3 V and LLDT2i DT00000L 0 K[LLEUa3 IYCS 000000La LLDT1 ∈≤≤⊆ W6 = JZDT000000LaaR L L;1i3 V be LLDT2i DT000000L K[LLEUa3 super row vector, vector subspaces of V over the field LR.

142 6 ∩ Clearly V = Wi and Wi Wj = i1= CS0000000 DT DT0000000; if i ≠ j. Thus we see V is a direct DT EU0000000 sum of super row vector subspace of V over LR.

Example 4.46: Let V = IYCSLLLL LLDTaaaa1234 LL LLDTLLLLaaaa JZDT5678LL;1i16∈≤≤ be a super LLLL aRi LLDTaa9101112 aa LLDTLLLL K[LLEUaaaa13 14 15 16 matrix vector space of refined labels over LR. IYCSL 000 LLDTa1 LLDTL 000 LLa2 Consider W1 = JZDTLL;1i3∈≤≤ ⊆ V, L 000 aRi LLDTa3 LLDT0 000 K[LLEU IYCS0 000 LLDT LLDT0 000 W2 = JZDTLL;1i4∈≤≤⊆ V, 0L 00 aRi LLDTa2 LLDTLL0L K[LLEUaa13 a 4 IYCS00 0 L LLDTa2 LLDT00 0 L LLa3 W3 = JZDTLL;1i4∈≤≤ ⊆ V; 00 0 L aRi LLDTa4 LLDT00L 0 K[LLEUa1

143 IYCS0L 0 0 LLDTa1 LLDT0L 0 0 LLa2 W4 = JZDTLL;1i3∈≤≤ ⊆ V and 00L 0 aRi LLDTa3 LLDT00 00 K[LLEU IYCS00L 0 LLDTa1 LL LLDT00La 0 2 ∈≤≤⊆ W5 = JZDTLL;1i3aR V LLDT00 0 0 i LLDT00 0 0 K[LLEU be super matrix vector subspaces of V over the refined field LR.

CS0000 DT 5 DT0000 Clearly V = W and W ∩ W = if i ≠ j, i i j DT0000 i1= DT DT EU0000 1 ≤ i, j ≤ 5. Thus V is the direct sum of super vector subspaces of V over LR.

Now having seen the concept of direct sum we now proceed onto give examples of pseudo direct sum of subspaces.

Example 4.47: Let V = IYCS LLLLLaaaaa LLDT12345 ∈≤≤ JZDTLLLLLLL;1i15aaaaaaR be a super LLDT678910i DTLLLLL K[LLEUaaaaa11 12 13 14 15 row vector, vector space of refined labels over LR.

144 Consider IYCS La 0000 LLDT1 ∈≤≤⊆ W1 = JZDTLaaR 0000L L;1i5 V, LLDT2i DTL0L0L K[LLEUaaa345 IYCS0 0 000 LLDT ∈≤≤⊆ W2 = JZDT00L00LL;1i4aaR V, LLDT3i DTLL 00L K[LLEUaa12 a 4 IYCS 0La 000 LLDT2 ∈≤≤⊆ W3 = JZDT0LaaR 000LL;1i4 V, LLDT3i DTLL000 K[LLEUaa14 IYCS 0Laaaa L L L LLDT2345 ∈≤≤⊆ W4 = JZDTLaaR 0000LL;1i6 V and LLDT1i DT0000L K[LLEUa6 IYCS 0Laa L 0 0 LLDT23 ∈≤≤⊆ W5 = JZDT00LLLLL;1i9aaa aR V are LLDT456 i DTL0LLL K[LLEUaaaa1789 super row vector subspaces of V over LR. 5 ∩ ≠ We see clearly V = Wi and Wi Wj i1=

CS00000 DT DT00000if i ≠ j, 1 ≤ i, j ≤ 5. DT EU00000

Thus V is not the direct sum of super row vector subspaces of V but only a pseudo direct sum of super row vector subspaces of V over LR.

145

Example 4.49: Let V =

IYCSLLL LLDTaaa123 LLDTLLLaaa LLDT456 LLL LLDTaaa789 LL DTLLL LLaaa10 11 12 LLDT DT∈≤≤ JZLLLaaaLL;1i27aR LLDT13 14 15 i LLDTLLLaaa DT16 17 18 LLLLL LLDTaaa19 20 21 DT LLLLL LLDTaaa22 23 24 LLDTLLL K[EUaaa25 26 27 be a super column vector, vector space of refined labels over LR.

IYCSLLL LLDTaaa123 LLDT000 LLDT LLL LLDTaaa456 LLDT LL000 LLDT DT∈≤≤⊆ Consider W1 = JZ000LL;1i9aR V, LLDTi DTLLL LLaaa789 LLDT LLDT000 DT LL000 LLDT DT K[LLEU000

146 IYCSL00 LLDTa1 LLDT000 LLDT 0L 0 LLDTa2 LLDT LL000 LLDT DT∈≤≤⊆ W2 = JZ000LL;1i9aR V, LLDTi LLDT00La DT3 LLLLL LLDTaaa456 DT LL000 LLDT LLDTLLL K[EUaaa789

IYCSL00 LLDTa1 LLDTLLLaaa LLDT234 0L 0 LLDTa5 LL DTLLL LLaaa678 LLDT DT∈≤≤⊆ W3 = JZ000LL;1i8aR V, LLDTi LLDT000 LLDT LLDT000 DT LL000 LLDT DT K[LLEU000

147 IYCSL00 LLDTa1 LLDT00La LLDT2 LLDT000 LL DT00L LLa3 LLDT DT∈≤≤⊆ W4 = JZLLLaaaLL;1i7aR V and LLDT456 i LLDT000 LLDT LLDT000 DT LL000 LLDT LLDT00L K[EUa7

IYCSL00 LLDTa1 LLDT00La LLDT2 L00 LLDTa3 LL DT0L 0 LLa4 LLDT DT∈≤≤⊆ W5 = JZ00La LL;1i11aR V are super LLDT5 i LLDT0La 0 DT6 LLL00 LLDTa7 DT LLLLL LLDTaaa8910 LLDTL00 K[EUa11 column vector, vector subspaces of V over LR.

148 F000V G W G000W G000W G W 000 5 G W ∩ ≠ G W ≠ ≤ Clearly V = Wi and Wi Wj 000if i j, 1 i, = G W i1 G000W G W G000W G000W G W H000X j ≤ 5.

Thus V is only a pseudo direct sum of super column vector, vector subspaces of V over LR.

Example 4.49: Let V = IYCSLLLL LLDTaaaa1234 LLDTLLLLaaaa LLDT5678 LLDTLLLLaa a a 9101112 ∈≤≤ JZDTLL;1i24aR be a super LLLL i LLDTaaaa13 14 15 16 LLDTLLLL LLDTaaaa17 18 19 20 LLDTLLLL K[EUaaaa21 22 23 24 matrix vector space of refined labels over LR. Consider W1 = IYCSL00L LLDTaa13 LLDTL00Laa LLDT24 LLDT0000 JZLL;1i4∈≤≤⊆ V, DTaRi LLDT0000 LLDT0000 LLDT LLDT0000 K[EU

149 IYCSLLL 0 LLDTaa123 a LLDT0Laaa L L LLDT456 LLDT0000 W2 = JZLL;1i6∈≤≤⊆ V, DTaRi LLDT0000 LLDT0000 LLDT LLDT0000 K[EU IYCS0L 0L LLDTaa24 LLDTL0L0aa LLDT13 LLDT0000 W3 = JZLL;1i7∈≤≤⊆ V, DTaRi LLDT0000 LLDT0000 LLDT LLDTL0LL K[EUaaa567 IYCSLL0L LLDTaa12 a 3 LLDT0 000 LLDT LLDTL00Laa 47∈≤≤⊆ W4 = JZDTLL;1i9aR V, L00L i LLDTaa58 LLDTL00L LLDTaa69 LLDT0 000 K[EU IYCSL000 LLDTa1 LLDT00LLaa LLDT910 LLDTLLaa 0 0 24 ∈≤≤⊆ W5 = JZDTLL;1i10aR V, and 0L 0 0 i LLDTa5 LLDT0L 0 0 LLDTa6 LLDTL0LL K[EUaaa378

150 IYCSL000 LLDTa1 LLDT00L0a LLDT4 LLDT00L0a 6 ∈≤≤⊆ W6 = JZDTLL;1i9aR V, be super L0L0 i LLDTaa27 LLDT00LL LLDTaa89 LLDTLL 00 K[EUaa53 6 ∩ ≠ vector subspace of V over LR. V = Wi and Wi Wj i=1 FV0000 GW GW0000 GW0000 GWif i ≠ j, 1 ≤ i, j ≤ 6. GW0000 GW0000 GW HXGW0000

Thus V is a pseudo direct sum of super matrix vector subspaces of V over LR.

Now we can as in case of vector spaces define linear transformation, linear operator and projection. This task is left as an exercise to the reader. Also all super matrix vector spaces of refined labels over LR.

Example 4.50: Let V and W be two super matrix vector spaces of refined labels over LR, where V =

IYCSLLLL LLDTaaaa1234 LLDTLLLLaaaa JZDT5678LL;1i16∈≤≤be a super matrix LLLL aRi LLDTaa9101112 aa LL DTLLLL K[LLEUaaaa13 14 15 16 vector space of refined labels over LR.

151

Consider W = IYCS LLLLL LLDTaaaaa12345 LL DT∈≤≤ JZLLLLLLL;1i15aaaaaaR LLDT678910i DT LLLLLLLaaaaa K[EU11 12 13 14 15 be a super matrix vector space of refined labels of LR.

Define T : V → W by CSFVLLLL aaaa1234F V DTGWLLLLL G aaaaa12345W DTGWLLLLaaaa T DTGW5678= GLLLLLW LLLL G aaaaa678910W DTGWaa9101112 aa G W DTLLLLLaaaaa GWLLLL H 11 12 13 14 15 X EUHXaaaa13 14 15 16 Clearly T is a linear transformation of super matrix vector space of refined labels over LR.

Example 4.51: Let V =

IYCSLLL LLDTaaa123 LLDTLLL aaa456 LLDT LLL ∈≤≤ JZDTaaa789LL;1i15aR be a super matrix LLi DTLLL LLDTaaa10 11 12 LLDTLLL K[LLEUaaa13 14 15 vector space of refined labels over LR.

Consider T : V → V a map defined by

CSLLL CSLL 0 DTaaa123DTaa12 DTLLLaaa DT00La DT456DT3 LLL 0L L T DTaaa789 = DTaa46 DTLLL DTL00 DTaaa10 11 12 DTa5 DTLLL DT0L L EUaaa13 14 15 EUaa78

152 Clearly T is a linear operator on V.

IYCSLLL LLDTaaa123 LLDTLLL aaa456 LLDT LLL ∈≤≤ Now consider W = JZDTaaa789LL;1i9aR LLDTi LLDT000 LLDT000 K[LLEU ⊆ V, W is a super matrix vector subspace of V of refined labels over LR.

CSFVLLL CSLLL DTGWaaa123 DTaaa123 DTGWLLLaaa DTLLLaaa DTGW456 DT456 LLL LLL P DTGWaaa789 = DTaaa789. DT GWLLL DT000 DTGWaaa10 11 12 DT DTGWLLL DT000 EUHXaaa13 14 15 EU

It is easily verified that P is a projection of V to W and P (W) ⊆ W; with P2 = P on V.

We can as in case of super row vector space derive all the related theorems given in this chapter with appropriate modifications and this work can be carried out by the reader.

However we repeatedly make a mention that super matrix linear algebra of refined labels cannot be constructed as it is not possible to get any collection of multiplicative wise compatible super matrices of labels.

Now we proceed onto define the notion of linear functional of refined labels LR.

Here once again we call that the refined labels LR is isomorphic with R. So we can take the refined labels itself as a field or R as a field.

153 Let V be a super matrix vector space of refined labels over LR. Define a linear transformation from V into the field of refined labels called the linear functional of refined labels on V.

If f : V → LR then f (Lc α + β) = Lc f (α) + f (β) where α, β ∈ V and Lc ∈ LR.

We will first illustrate this concept by some examples.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LLLa Example 4.52: Let V = JZGW4 LL;1i8∈≤≤ be a super GWL aRi LLa5 LLGW GWL LLa6 GW LLL LLGWa7 LLGWL K[HXa8 column vector space of refined labels over the field of refined labels LR.

FVL GWa1 GWLa GW2 L GWa3 GW La T = ( GW4 ) = L...L++ GWL aa18 GWa5 GWLa GW6 L GWa7 GWL HXa8

= L +++ ∈ LR. aa...a12 8

T is a linear functional on V.

154 Infact the concept of linear functional can lead to several interesting applications.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW ∈≤≤ Example 4.53: Let V = JZLa LL;1i7aR be a super LLGW4 i GWL LLa5 LLGW GWL LLa6 GW LLL K[HXa7 column vector space of refined labels over LR.

Define f : V → LR by

FVL GWa1 GWLa GW2 L GWa3 GW ++ f ( La ) = LLL GW4 aaa123 GWL GWa5 GWLa GW6 L HXa7

= L ++ ∈ LR. f is a linear functional on V. aa123 a

Consider f1 : V → LR by

155 FVL GWa1 GWLa GW2 L GWa3 GW f1 ( La ) = L ; f1 is also a linear functional on V. We GW4 a4 GWL GWa5 GWLa GW6 L HXa7 can define several linear functional on V.

FVL GWa1 GWLa GW2 L GWa3 GW ++ Consider f2 ( La ) = LLL GW4 aaa147 GWL GWa5 GWLa GW6 L HXa7

= L ++ ∈ LR; aa147 a

f2 is a linear functional on V. We see all the three linear functionals f1, f2 and f are distinct.

Example 4.54: Let V = {}()LLLLLLLLLL;1i8∈≤≤ aaaaaaaaaR12345678i

be a super row vector space of refined labels over LR. Define f : V → LR by f ( ()LLLLLLLL) aaaaaaaa12345678 = LLLL+++ aaaa1247

= L +++ ; f is a linear functional on V. aa1247 aa We can define several linear functional on V.

156

Example 4.55: Let V = IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LL GWLLL LLGWaaa10 11 12 LLGW LLLLLaaa JZGW13 14 15 LL;1i30∈≤≤ be a super column GWLLL aRi LLaaa16 17 18 LLGW LLL LLGWaaa19 20 21 LLGWLLL LLGWaaa22 23 24 LLGWLLL LLGWaaa25 26 27 LLGWLLL K[HXaaa28 29 30 vector, vector space of refined labels over LR, the field of refined labels.

Define f : V → LR by

FVLLL GWaaa123 GWLLLaaa GW456 LLL GWaaa789 GWLLL GWaaa10 11 12 GW LLLaaa f( GW13 14 15 ) GWLLLaaa GW16 17 18 LLL GWaaa19 20 21 GWLLL GWaaa22 23 24 GWLLL GWaaa25 26 27 GWLLL HXaaa28 29 30 = LLLLLLLLL++++++++ aaaaaaaaa15 7111317192329

= L +++ + + + + + , f is a linear functional on V. aaaaaaaaa15 7111317192329

157 Example 4.56: Let V = IYFVLLL L L L LLGWaa15 a 9131721 a a a LLGWLLLaaa L a L a L a JZGW2 6 10 14 18 22 LL;1i24∈≤≤ be a LLL L L L aRi LLGWaaaaaa3 7 11 15 19 23 LL GWLLLLLL K[LLHXaaaaaa4 8 12 16 20 24 super row vector space of refined labels over LR.

Consider f : V → LR by FVLLLLLL GWaa159131721 a a a a GWLLLLLLaaa a a a f ( GW2 6 10 14 18 22 ) = LLL++ + L LLLLLL aaa161124 a GWaaaaaa3 7 11 15 19 23 GWLLL L L L HXaaaaaa4 8 12 16 20 24

= L ++ + is a linear functional on V. aa1 6 a 11 a 24

Example 4.57: Let V = IYFVLLL LLGWaaa123 LLGWLLL aaa456 LLGW LLL ∈≤≤ JZGWaaa789LL;1i15aR be a super matrix LLi GWLLL LLGWaaa10 11 12 LLGWLLL K[LLHXaaa13 14 15 super vector space of refined labels over LR, the refined field of labels.

Define f : V → LR by

FVLLL GWaaa123 GWLLLaaa GW456 LLL f ( GWaaa789) GWLLL GWaaa10 11 12 GWLLL HXaaa13 14 15

158 = LLLLLL+++ ++ + L aaaa1571161213 aa a

=L +++ ++ + ; aaa1571161213 a aa a f is a linear functional on V.

We as in case of vector space define the collection of all linear functionals on V, V a super matrix vector space over LR as the dual space and it is denoted by L (V, LR).

Here we define Lδ = 0 if i ≠ j if i = j then Lδ = Lm+1. ij ij

Using this convention we can prove all results analogous to L (V, F), F any field.

For instance if V = {}()LLLLLL;1i4∈≤≤ be a super row aa1234i aa aR vector space of refined labels over LR then B = {(Lm+1 | 0 | 0 0) = v1, v2 = (0 | Lm+1 | 0 0), v3 = (0 | 0 | Lm+1 | 0), v4 = (0 | 0 | 0 Lm+1)} ⊆ V is a basis of V over LR.

Define fi (vj) = Lδ ; 1 ≤ i, j ≤ 4; (f1, f2, f3, f4) is a basis of ij

L (V, LR) over LR.

We as in case of usual vector spaces define if f : V → LR is a linear functional on super vector space of refined labels with appropriate modifications then null space of f is of dimension n – 1.

It is however pertinent to mention here that we may have several super vector spaces of dimension n which need not be isomorphic as in case of usual vector spaces with this in mind we can say or derive results with suitable modifications.

However for any super matrix vector space V of refined labels over LR the refined hyper super space of V as a super subspace of dimension n – 1 where n is the dimension of V.

159 IYCS LLLLaa Consider V = JZDT12LL;1i4∈≤≤ be a super DTaRi LLLL K[EUaa34 matrix vector space of dimension four over the refined field LR.

IYCS LLLLaa Consider W = JZDT12LL;1i3∈≤≤ ⊆ V, W is DTaRi LL0L K[EUa3 the hyper super space of V and dimension of W is three over LR.

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa Example 4.58: Let V = JZGW5 LL;1i10∈≤≤be a super aRi LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10 column vector space of dimension 10 over the field LR of refined labels.

160 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LL LLGW0 Consider M = JZGWLL;1i9∈≤≤ ⊆ V is a hyper aRi LLGWLa GW5 LLL LLGWa6 LLGWL LLGWa7 GW LLLa LLGW8 LLGWLa K[HX9 super space of refined labels of V over LR. Infact V has several hyper super spaces.

Example 4.59: Let V =

IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 JZLL;1i18∈≤≤ be a super matrix GWaRi LLLLL GWaaa10 11 12 LL GWLLL LLGWaaa13 14 15 LLGWLLL K[HXaaa16 17 18 vector space of refined labels over LR.

IYFV0L L LLGWaa12 LLGWLLLaaa LLGW345 LLL LLGWaaa678 Consider W = JZLL;1i17∈≤≤⊆ V, GWaRi LLLLL GWaa91011 a LL GWLLL LLGWaaa12 13 14 LLGWLLL K[HXaaa15 16 17 be a super matrix vector subspace of V over LR.

161 Clearly W is a hyper super space of refined labels over LR of V. Dimension of V is 18 where as dimension of W over LR is 17.

Let V be a super vector space of refined labels over LR. S be a subset of V, the annihilator of S is the set So of all super linear functionals f on V such that f (α) = 0 for every α ∈ S.

Let V = {}FVLLL LL;1i3∈≤≤ be a super HXaaa123 aR i row vector space of refined labels over LR. We will only make a trial. Suppose W = {}FVL00LL∈ ⊆ V be a subset of HXaaR1i V. Define f : V → V by f ( ()L00) = ()000 every ()L00 ∈ W. a1 a1 o W = {f ∈ L (V, LR) | f ( ()L00) a1

= ()000 for every L ∈ LR}. a1

o We see W is a subspace of L (V, LR) as it contains only f and fo the zero function. However one has to study in this direction to get analogous results.

We will give some more examples.

Consider K =

IYFVLLLLL LLGWaaaa12345 a LLGWLLLLLaaaaa JZGW678910LL;1i20∈≤≤ LLLLL aRi LLGWaaaaa11 12 13 14 15 LL GWLLLLL K[LLHXaaaaa16 17 18 19 20 be a super matrix vector space of refined labels over the field LR.

162 IF 00000V FL L 000V LG W G aa12 W LG 00000W GLaa L 000W Let S = JG W , G 34 W , 0L 000 0 0 000 LG a1 W G W L GL0000W G 0 0 000W KH a2 X H X FV00L L L Y GWaaa123L GW00 0 0 0L Z where L ∈ LR; 1 ≤ i ≤ 4} ⊆ V be a GWai GW00 0 0 0L GWL HX00 0 0 0[ proper subset of V. Clearly S is not a super vector subspace of V it is only a proper subset of V.

Define fi : V → V as follows:

FVLLLLL GWaaaa12345 a LLLLL GWaaaaa678910 f1 ( GW ) LLLLL GWaaaaa11 12 13 14 15 GWLLLLL HXaaaaa16 17 18 19 20

FV000 0 0 GW GW00LLLaaa = GW12 13 14 , L0LLL GWaaaa5151617 GW0LLLL HXaaaa8181920

FVLLLLL F00 0 00V GWaaaa12345 a G W LLLLL 00L 00 GWaaaaa678910 G a8 W f2 ( GW) = G W , LLLLL 00 0 00 GWaaaaa11 12 13 14 15 G W GWLLLLL G00 0 00W HXaaaaa16 17 18 19 20 H X

163 FVLLLLL GWaaaa12345 a LLLLL GWaaaaa678910 f3 ( GW) LLLLL GWaaaaa11 12 13 14 15 GWLLLLL HXaaaaa16 17 18 19 20 FV00 0 0 0 GW GW00Laa L 0 = GW18 19 , GW00 0 0 0 GW HX00 0 0 0 FVLLLLL GWaaaa12345 a LLLLL GWaaaaa678910 f4 ( GW) = LLLLL GWaaaaa11 12 13 14 15 GWLLLLL HXaaaaa16 17 18 19 20 FV00 0 0 0 GW GW00 0 0 0 GW and so on. 00L L L GWaaa123 GW00L L L HXaaa456

We see f1, f2, f3 and f4 are defined such that fi (x) = CS00000 DT DT00000 for every x ∈ S, 1 ≤ i ≤ 4. Now having seen DT00000 DT EU00000 o how S looks like we can give a subspace structure of L (V, LR).

We can as in case of usual vector space derive properties related with the dual super space and properties enjoyed by the super linear functionals or linear functionals on super matrix vector spaces of refined labels. The only criteria being LR is isomorphic to R. Also we have the properties used in the super linear algebra regarding super vector spaces [47].

164 !,

In this chapter we for the first time introduce the notion of super semivector space of refined labels.

We have defined the notion of semigroup of refined labels.

Also we have seen the set of labels L ∈ L + forms a a R{0}∪ semifield of refined labels. Also L + is again a semifield of Q{0}∪ refined labels. We would be using the semifield of refined labels to build super semivector spaces of different types. Let M = {()LLLLLL...LL aaaaaa123456 a n1n− a

LL∈≤≤+ ;1in} be the collection of all super row ai R{0}∪ vectors of refined labels.

Clearly M is a semigroup under addition. We see M is infact only an infinite commutative semigroup under the operation of addition. Clearly M is not a group under addition. Further we cannot define product on M as it is not compatible

165 under any form of product being a super row vector. We know

L + and L + are semifields. Now we can define super R{0}∪ Q{0}∪ semivector space of refined labels.

DEFINITION 5.1: Let

V=∈{() LLLLLL...LLL L+ ; aaaaaa123456 a n1n− a a i R{0}∪ 1 ≤ i ≤ n} be a semigroup under addition with () LLLLLLLLL000000000 as the zero row vector of refined labels. V is a super semivector space of refined labels over L + ; the semifield of refined labels isomorphic R{0}∪ with R+ ∪ {0}.

We will first illustrate this situation by some simple examples.

Example 5.1: Let

V = {}()LLLLLL∈≤≤+ ;1i4 aaaaa1234iR{0}∪ be a super semivector space of refined labels over the semifield

L + . R{0}∪

Example 5.2: Let V = {()LLLLLLLLLL aaaaaaaaaa 12345678910

LL∈≤≤+ ;1i10} be a super semivector space of refined ai R{0}∪ labels over the semifield L + . We also call V to be a super R{0}∪ row semivector space of refined labels. Since from the very context one can easily understand we do not usually qualify these spaces by row or column.

Example 5.3: Let

S=∈≤≤{}() LLLLLLLLL+ ;1i7 aaaaaaaa1234567iR{0}∪ be a super semivector space of refined labels over the semifield

L + . R{0}∪

166 Example 5.4: Let

M = {}()LLLLLLLL∈≤≤+ ;1i6 aaaaaaa123456iR{0}∪ be a super semivector space of refined labels over the semifield

L + . Q{0}∪

Example 5.5: Let V = {()LLLLLLLLL aaaaaaaaa 123456789

LL∈≤≤+ ;1i9} be a super semivector space of refined ai R{0}∪ labels over L + the semifield of refined labels isomorphic Q{0}∪ with Q+ ∪ {0}.

Example 5.6: Let

K=∈≤≤{}() LLLLLLLLL L+ ;1i8 aaaaaaaa12345678 a i Q{0}∪ be a super semivector space of refined labels over L + . Q{0}∪ Clearly K is not a super semivector space of refined labels over

L + . R{0}∪

Example 5.7: Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910

LL∈≤≤+ ;1i10} be a super semivector space of refined ai Q{0}∪ labels over L + . It is pertinent to mention here that it does Q{0}∪ not make any difference whether we define the super semivector + + space over R ∪ {0} or L + (or Q ∪ {0} or L + ) as R{0}∪ Q{0}∪ + + R ∪ {0} is isomorphic to L + (Q ∪ {0} is isomorphic with R{0}∪

L + ). Q{0}∪

We can as in case of semivector spaces define substructures on them. This is simple and so left as an exercise to the reader. We give some simple examples of substructures and illustrate their properties also with examples.

167 Example 5.8: Let

V = {()LLLLLLLLLaaaaaaaaa 123456789

LL∈≤≤+ ;1i9} be a super semivector space of refined ai R{0}∪ labels over L + . Consider R{0}∪

M = {()Laa L L a 0 0L a 0L a 0 123 4 5

LL∈≤≤+ ;1i5} ⊆ V, M is a super semivector subspace ai R{0}∪ of V over the semifield L + . R{0}∪

Example 5.9: Let

V = {()LLLLLLLLLLLaaaaaaaaaaa 1234567891011

LL∈≤≤+ ;1i11} be a super semivector space of refined ai R{0}∪ labels over the semifield L + . Consider R{0}∪

W = {()LLLLLLaaaaaa 0 0 0 0 0 123456

LL∈≤≤+ ;1i6} ⊆ V, is a super semivector subspace of ai R{0}∪

V over the refined label semifield L + . R{0}∪

Example 5.10: Let

V = {()LLLLLLLLLLLLaaaaaaaaaaaa 123456789101112

LL∈≤≤+ ;1i12} be a super semivector space of refined ai R{0}∪ labels over the semifield L + . Q{0}∪ Consider W={() LLLLLLLLLLLL aaaaaaaaaaaa123456789101112

LL∈≤≤+ ;1i12} ⊆ V; W is a super semivector subspace ai Q{0}∪ of V of refined labels over the semifield L + . Q{0}∪

168

Example 5.11: Let

V = {}()LLLLLLLL∈≤≤+ ;1i6 aaaaaaa123456iQ{0}∪ be a super semivector space of refined labels over the semifield

L + . Q{0}∪ Consider

M=∈≤≤{}() L 0L 0L 0 L L+ ;1i3 aaa123 a iQ{0}∪ ⊆ V, is a super semivector subspace of V of refined labels over

L + . Q{0}∪

Now having seen super semivector subspaces of a super semivector space we can now proceed onto give examples of the notion of direct sum and pseudo direct sum of super semivector subspaces of a super semivector space.

Example 5.12: Let

V = {}()LLLLLLL∈≤≤ L+ ;1i6 aaaaaa123456 a i R{0}∪ be a super semivector space of refined labels over the semifield of refined labels L + . R{0}∪ Consider ∈≤≤⊆ M1 = {}()L 0L 000L L+ ;1i2 V, aa12 a iR{0}∪

M2 = {}()0L 000L L∈≤≤ L+ ;1i2⊆ V, aaa12iR{0}∪ and ∈≤≤⊆ M3 = {}()000L L 0L L+ ;1i2 V aa12 a iR{0}∪ be three super semivector subspace of V of refined labels over 3

L + . Clearly V = M with M ∩ M = {(0 0 0 | 0 0 | 0)} if R{0}∪ i i j i1= i ≠ j, 1 ≤ i, j ≤ 3. Thus V is a direct sum super semivector subspaces M1, M2 and M3 of V.

169 Example 5.13: Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910

LL∈≤≤+ ;1i10} be a super semivector space of refined ai Q{0}∪ labels over L + . Q{0}∪ Take ⊆ W1 = {}()L 000L 00000 L∈≤≤ L+ ;1i 2 V, aa12 a iQ{0}∪ ⊆ W2 = {()00000L L L 00 L∈≤≤ L+ ;1i 3} V, aaa123 a i Q{0}∪ ⊆ W3 = {}()0L L L 000000 L∈≤≤ L+ ;1i 3 V aaa123 a i Q{0}∪ and ⊆ W4 = {}()00000000L L L∈≤≤ L+ ;1i 2 V; aa12 a i Q{0}∪ be super semivector subspaces of V over the refined label 4 semifield L + . We have V = W with W ∩ W = {(0 0 0 Q{0}∪ i i j i1= 0 | 0 | 0 0 | 0 0 0)} if i ≠ j, 1 ≤ i, j ≤ 4. Thus V is a direct sum super semivector subspaces W1, W2, W3 and W4.

Example 5.14: Let V = {(LLLLLL aaaaaa123456 LLLL L L L L) aaaa7 891011121314 a a a a

LL∈≤≤+ ;1i14} be a super semivector space of refined ai R{0}∪ labels over the semifield of refined labels L + . R{0}∪ Consider W= {() L 00L L L 0000000L 1 a1 aaa 234 a 5

LL∈≤≤+ ;1i5} ⊆ V, ai R{0}∪

W2 = {()0L L L 0L 00L L 0000 aa123 a a 4 aa 56

170 LL∈≤≤+ ;1i6} ⊆ V, ai R{0}∪

W3 = {()L 000L L L 00L 00L L aaaaaaa1 234 5 76

LL∈≤≤+ ;1i7} ⊆ V, ai R{0}∪

W4 = {()LL0000LLLLL0L0 aa12 aaaaa 34567 a 7

LL∈≤≤+ ;1i8} ⊆ V ai R{0}∪ and

W5 = {()0L 0L 0L L 0L 0L L L 0 a12345678 a aa a aaa

LL∈≤≤+ ;1i8} ⊆ V ai R{0}∪ be super semivector subspaces of V of refined labels over 5

L + . We see V = W but W ∩ W ≠ {(0 | 0 0 | 0 0 0 | 0 0 R{0}∪ i i j i1= 0 0 | 0 0 | 0 0)} if i ≠ j, 1 ≤ i, j ≤ 5. Thus we see V is only a pseudo direct sum of super semivector subspaces of V over LR and is not a direct sum of super semivector subspaces of V over LR.

Example 5.15: Let

V=∈≤≤{}() LLLLLLLLL L+ ;1i8 aaaaaaaa12345678 a i Q{0}∪ be a super semivector space of refined labels over the refined labels semifield L + . Q{0}∪ Take

P1 = {}()000L L 00L L∈≤≤ L+ ;1i 3 ⊆ V, aa12 a 3 a i Q{0}∪ ∈≤≤⊆ P2 = {}()L0LL0L00LL+ ;1i4 V, aaaa1234 a iQ{0}∪ ⊆ P3 = {}()L0LL0LL0LL∈≤≤+ ;1i5 V, aaaaaa12534iQ{0}∪ ∈≤≤ P4 ={}()0L L L L 0L L L L+ ;1i 6 aa1253 aa a 46 a a i Q{0}∪

171 and ∈≤≤ P5 = {}()0L 0L 0L 0L L L+ ;1i 4 ⊆ V aaaaa1234iQ{0}∪ be super semivector subspaces of V of refined labels over 5

L + . We see V = P and P ∩ P ≠ {(0 | 0 0 | 0 | 0 | 0 | 0 Q{0}∪ i i j i1= 0 | 0)}, if i ≠ j, 1 ≤ i, j ≤ 8. Thus V is only a pseudo direct sum of super semivector subspaces P , P , …, P of V over L + . 1 2 5 Q{0}∪ Now it may so happen sometimes for any given super semivector space V of refined labels over the semifield L + Q{0}∪ or L + the given set of super semi vector subspaces of V R{0}∪ n n ≠ ⊄ may not be say as V Pi only Pi V. This situation can i1= i1= occur when we are supplied with the semivector subspaces of refined labels. But if we consider the super semivector subspace of refined labels we see that we can have either the direct union or the pseudo direct union. Now we discuss about the basis. Since we do not have negative elements in a semifield we need to apply some modifications in this regard.

Let V = {}()LLLL...LL∈≤≤ L+ ;1in aaaa1234 a n a i R{0}∪ be a super vector semivector space of refined labels over

L + . We see B = {[L 0 | 0 0 | … | 0], [0 L | 0 0 | … | 0], R{0}∪ m+1 m+1

…, [0 0 | 0 0 | … | L ]} acts as a basis of V over L + . We m+1 R{0}∪ define the notion of linearly dependent and independent in a different way [42-45] For if (v1, …, vm) are super semivectors from the super semivector space V over L + . We declare v ’s are linearly R{0}∪ i dependent if vi = B Lv ; LL∈ + , 1 ≤ j ≤ m, j ≠ i, other ajj a j R{0}∪ wise linearly independent.

For take V = {}()LLLLL L∈≤≤ L+ ;1i5 to aa12345 aa a a i R{0}∪ be a super semivector space of refined labels over L + . R{0}∪

Consider (Lm+1 0 | 0 0 | Lm+1) and ( L 0 | 0 0 | L ) in V. an an

172 Clearly ( L 0 | 0 0 | L ) = L (Lm+1 0 | 0 0 | Lm+1) so, the an an an given two super semivectors in V are linearly dependent over

L + . Now consider (Lm+1 0 | 0 0 | 0), (0 0 | L 0 | 0) and (0 R{0}∪ a1 0 | 0 0 | L ) in V we see this set of super semivectors in V are b1 linearly independent in V over L + . R{0}∪ Thus with this simple concept of linearly independent super semivectors in V we can define a basis of V as a set of linearly independent super semivectors in V which can generate the super semivector space V over L + . R{0}∪ Consider

V = {}()LLLLLL L∈≤≤ L+ ;1i6 aaaaaa123456 a i R{0}∪ to be a super semivector space of refined labels over L + . R{0}∪

Consider the set B = {(Lm+1 | 0 | 0 0 | 0 0), (0 | Lm+1 | 0 0 | 0 0), (0 | 0 | Lm+1 0 | 0 0), (0 | 0 | 0 Lm+1 | 0 0), (0 | 0 | 0 0 | Lm+1 0), (0 | 0 | 0 0 | 0 Lm+1)} ⊆ V is a set of linearly independent elements of V which generate V. Infact B is a basis of the super semi vector space V over L + . We can give more examples. R{0}∪ We would give the definition of super column vector space of refined labels over L + . R{0}∪

DEFINITION 5.2: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 V = JZLL∈≤≤+ ;1in GW... ai R{0}∪ LLGW LL GWL LLGWan1− LLGWL K[HXan be an additive semigroup of super column vectors of refined labels. Clearly V is a super column semivector space of refined labels over LR.

173 We will illustrate this situation by some examples.

Example 5.16: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 V = JZLL∈≤≤+ ;1i6 GWai R{0}∪ LLL GWa4 LL GWL LLGWa5 LLGWL K[HXa6 be a super column semivector space of refined labels over

L + . R{0}∪

Example 5.17: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLGWa5 LLGWL a6 LLGW L ∈≤≤ V = JZGWa7 LLa + ∪ ;1i13 LLi R{0} GWL LLGWa8 LLGWL LLGWa9 LLGWLa LLGW10 L LLGWa11 LLGW L LLGWa12 LLGWL K[LLHXa13 be a super column semivector space of refined labels over

L + . R{0}∪

174 Example 5.18: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa 4 LLGW LLLa P = GW5 JZLL∈≤≤+ ;1i10 a i Q{0}∪ LLGWLa GW6 LLL LLGWa 7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10 be the class of super column semivector space of refined labels over L + . It is to be noted P is not a super column Q{0}∪ semivector space of refined labels over L + . As in case of R{0}∪ general semivector spaces the definition depends on the semifield over which the semivector space is defined. We will illustrate this situation also by some examples.

Example 5.19: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW La LLGW4 LLGWL V = LLa 5 JZGWLL∈≤≤+ ;1i10 GWL a i R{0}∪ LLa 6 LLGW L LLGWa 7 LLGWL LLGWa8 LLGW La LLGW9 LLGWL K[HXa10

175 be a super column semivector space of refined labels over the semifield L + . Q{0}∪

Example 5.20: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLa4 LLGW GW ∈≤≤ P = JZLa LLa + ∪ ;1i9 LLGW5 i Q{0} LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 LLGWL K[HXa9 is a super column semivector space of refined labels over

L + . Q{0}∪

Now we will give examples of basis and dimension of these spaces.

Example 5.21: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LLL GWa4 V = JZLL∈≤≤+ ;1i8 GWL ai Q{0}∪ LLa5 LLGW GWL LLa6 GW LLL LLGWa7 LLGWL K[HXa8 be a super column vector space of refined labels over L + . Q{0}∪

176 Consider

IYFVFVFVFVFVL+ 0 0000F VF00 VF V LLGWGWGWGWGWm1 G WG WG W LLGWGWGWGWGW0Lm1+ 0000G WG00 WG W LLGWGWGWGWGW00L000G WG00 WG W LLGWGWGWGWGWm1+ G WG WG W LLGWGWGWGWGW000L00+ G WG00 WG W M,,= JZm1 GWGWGWGWGW0000L0G WG00 WG W LLGWGWGWGWGWm1+ G WG WG W LL GWGWGWGWGW00000LG m1+ WG00 WG W LLGWGWGWGWGWG WG WG W LLGWGWGWGWGW0 0 0000G WGL0m1+ WG W LLGWGWGWGWGWG WG WG W K[HXHXHXHXHX0 0 0000H XH0L XHm1+ X

⊆ V is a basis of V over L + . Q{0}∪

Example 5.22: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLa4 LLGW GW ∈≤≤ V = JZLa LLa + ∪ ;1i9 LLGW5 i R{0} LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 LLGWL K[HXa9 be a super column semivector space of refined labels over

L + . Clearly V is of dimension nine over L + . R{0}∪ R{0}∪ Consider V as a super column semivector space of refined labels over L + , then we see V is of infinite dimension over Q{0}∪

L + . This clearly shows that as we change the semifield Q{0}∪ over which the super column semivector space is defined is

177 changed then we see the dimension of the space varies as the semifield over which V is defined varies.

Example 5.23: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 ∈≤≤ V = JZGWLLa + ∪ ;1i6 L i Q{0} LLGWa4 LLGWL LLGWa5 LLGWL K[HXa6

be a super semivector space of refined labels over L + of Q{0}∪ dimension six over L + . Clearly V is not defined over the Q{0}∪ refined label field L + . R{0}∪

Example 5.24: Let IYFVL LLGWa1 LLGWLa LLGW2 M = JZL LL∈≤≤+ ;1i5 GWa3 a ∪ LLi R{0} GWL LLGWa4 LLGWL K[HXa5 be a super column semivector space of refined labels over the field L + . R{0}∪

Clearly M is of dimension five over L + ; but M is of R{0}∪ dimension infinite over L + . Q{0}∪ Now we proceed onto give examples of linear transformations on super column semivector space of refined labels.

178 Example 5.25: Let

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GW ∈≤≤ V = JZLa LLa + ∪ ;1i7 LLGW4 i R{0} GWL LLa5 LLGW GWL LLa6 GW LLL K[HXa7 and

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLGWa5 LLGWL a6 LLGW L ∈≤≤ W = JZGWa7 LLa + ∪ ;1i13 LLi R{0} GWL LLGWa8 LLGWL LLGWa9 LLGWLa LLGW10 L LLGWa11 LLGW L LLGWa12 LLGWL K[LLHXa13 be two super column semivector spaces defined over the same semifield L + . Q{0}∪ Define a map η : V → W given by

179 FL V G a1 W GLa W G 2 W L G a2 W CSFV G W La L DTGW1 G a3 W DTGWLa GL W DTGW2 G a3 W DTLa GL W GW3 a3 η DTGW G W La = L . DTGW4 G a4 W DTGW G W La L DTGW5 G a5 W G W DTGWLa La DTGW6 G 6 W DTL GL W EUHXa7 a7 G W L G a7 W G W L G a7 W GL W H a1 X It is easily verified η is a linear transformation of super semivector spaces or super semivector space linear transformation of V to W.

Example 5.26: Let FV V=∈≤≤{} LLLLLLLL L L+ ;1i7 HXaaaaaaaa12345678 a i Q{0}∪ IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLa4 LLGW GW ∈≤≤ and W = JZLa LLa + ∪ ;1i9 LLGW5 i Q{0} LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 LLGWL K[HXa9

180 be two super semivector spaces of refined labels over L + . Q{0}∪ Let η: V → W be defined by FL V G a1 W GLa W G 2 W L G a3 W GL W G a4 W η G W (()LLLLLLLL) = La . aaaaaaaa12345678 G 5 W GLa W G 6 W L G a7 W GL W G a8 W HG 0 XW

It is easily verified η is a super semivector space linear transformation of V into W.

Example 5.27: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLa5 P = JZGWLL∈≤≤+ ;1i10 GWL ai Q{0}∪ LLa6 LLGW L LLGWa7 LLGWL LLGWa8 LLGWL LLGWa9 LLGWL K[HXa10

be a super semivector space of refined labels over L + . Q{0}∪ Consider T : V → V defined by

181 F V F L V La G a1 W G 1 W L L G a W G a 2 W G 2 W G W L 0 G a 3 W G W G W G W L La G a 4 W G 4 W G L W G 0 W T ( G a 5 W ) = G W ; G La W GLa W G 6 W G 5 W L L G a 7 W G a3 W G L W G 0 W G a8 W G W G L W GL W G a 9 W G a6 W GL W G 0 W H a10 X H X T is a linear operator of super semivector spaces. Now having seen examples of linear transformations and linear operators on super column semi vector spaces of refined labels. We proceed onto define super matrix semivector spaces.

DEFINITION 5.4: Let IFVLL...L LGWaa11 12 a 1 m LGWLL...L L aa21 22 a 2m V = GW∈ J LLa ++∪∪ (or L); LGW i R{0} Q{0} LGW GWLL...Laa a KLHXnn12 n m 1 ≤ i ≤ n and 1 ≤ j ≤ m} be the additive semigroup of super matrices of refined labels. V is a super matrix semivector space of refined labels over L + . R ∪{0}

Example 5.28: Let IYFV LLLaaa LLGW123 LLL LLGWaaa456 LLGWLLL M = aaa789 JZGWLL∈≤≤+ ;1i18 LLL ai R{0}∪ LLGWaaa10 11 12 LLGWLLL LLGWaaa13 14 15 GWLLL K[LLHXaaa16 17 18

182 be a super matrix semivector space of refined labels over

L + . M is also known as the super column vector R{0}∪ semivector space of refined labels over the field of refined labels L + . R{0}∪

Example 5.29: Let IYFV LLLLLaaa LLGW123 GW∈≤≤ V = JZLLLLLaaaa + ∪ ;1i9 LLGW456iR{0} LLGWLLLaaa K[HX789 be the super square matrix semivector space of refined labels over L + . R{0}∪

Example 5.30: Let IFVLLLLLLLLL LGWaaaaaaaaa123456789 LGWLLLLLLLLLaaaaaaaaa W = JGW10 11 12 13 14 15 16 17 18 LLLLLLLLL LGWaaaaaaaaa19 20 21 22 23 24 25 26 27 L GWLLLLLLLLL KLHXaaaaaaaaa28 29 30 31 32 33 34 35 36

LL∈≤≤+ ;1i36} be a super row vector semivector space ai R{0}∪ of refined labels over L + or super matrix semivector space R{0}∪ of refined labels over L + . R{0}∪

Example 5.31: Let IFVLLLLLLLL LGWaaaaaaaa2345678 LGWLLLLLLLLaaaaaaaa LGW9 10111213141516 LLLLLLLL LG aaaaaaaa17 18 19 20 21 22 23 24 W V = J GLLLLLLLLW LG aaaaaaaa25 26 27 28 29 30 31 32 W LGLLLLLLLLW LG aaaaaaaa33 34 35 36 37 38 39 40 W LGLLLLLLLLW KHXaaaaaaaa41 42 43 44 45 46 47 48

183 LL∈≤≤+ ;1i48} be a super matrix vector space of refined ai R{0}∪ labels over the semifield of refined labels L + . R{0}∪ We as in case of other super semivector spaces define the notion of super semivector subspaces, basis, dimension and linear transformations. However this task is direct and hence is left as an exercise to the reader we only give examples of them.

Example 5.32: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 V = JZLL∈≤≤+ ;1i18 GWai R{0}∪ LLLLL GWaaa10 11 12 LL GWLLL LLGWaaa13 14 15 LLGWLLL K[HXaaa16 17 18 be a super matrix semivector space or super column vector semivector space over L + . Clearly R{0}∪ IYFV000 LLGW LLGWLLLaaa LLGW123 LLGW000 P = JZLL∈≤≤+ ;1i9 ⊆ V GWai R{0}∪ LLLLL GWaaa456 LL GW000 LLGW LLGWLLL K[HXaaa789 is a super column vector, semivector subspace of refined labels of V over L + . We have several but only finite number of R{0}∪ super matrix semivector subspaces of V over L + . If V be R{0}∪ defined over the refined semifield of labels L + then V has Q{0}∪ infinite number of super matrix semivector subspaces. Thus even the number of super matrix semivector subspaces depends on the semifield over which it is defined. This is evident from example 5.32.

184 Example 5.33: Let IFVLLLLLLL Y LLGWaaaaaaa1234567 LLGWLLLLLLLaaaaaaa LLGW8 9 10 11 12 13 14 LLLLLLL ∈ LLGWaaaaaaa15 16 17 18 19 20 21 LL+ ; V = JZai Q{0}∪ GWLLLLLLL ≤≤ LLGWaaaaaaa22 23 24 25 26 27 28 1i42 LLGWLLLLLLL LLGWaaaaaaa29 30 31 32 33 34 35 LLGWLLLLLLL K[HXaaaaaaa36 37 38 39 40 41 42 be a super matrix of semi vector space of refined labels over

L + . V is a finite dimensional super matrix semivector Q{0}∪ subspace over L + having infinite number of semivector Q{0}∪ subspaces of refined labels over L + . Q{0}∪ Now we proceed onto define new class of super matrix semivector spaces over the integer semifield Z+ ∪ {0}; known as the integer super matrix semivector space of refined labels. We will illustrate them before we proceed onto derive properties related with them.

Example 5.34: Let

V = {}()LLLLLLLL∈≤≤ L+ ;1i7 aaaaaaa1234567 a i Q{0}∪ be a integer super row semivector space of refined labels over Z+ ∪ {0}, the semifield of integers.

Example 5.35: Let

IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 V = JZLLLLL∈≤≤+ ;1i15 GWaaa789a ∪ LLi R{0} GWLLL LLGWaaa10 11 12 LLGWLLL K[HXaaa13 14 15

185 be an integer super matrix semivector space of refined labels over Z+ ∪ {0}, the semifield of integers.

Example 5.36: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 V = JZLL∈ + GWai R{0}∪ LLLa GW5 LLGWL LLa6 LLGW GWLa LL7 LLGWL K[HXa8 be an integer super row semivector space over the semifield Z+ ∪ {0}.

Example 5.37: Let IFVLLLLLLLL LGWaaaaaaaa12345678 LGWLLLLLLLLaaaaaaaa LGW9 10111213141516 V = J LLLLLLLL GWaaaaaaaa17 18 19 20 21 22 23 24 L GWLLLLLLLL LGWaaaaaaaa25 26 27 28 29 30 31 32 LGWLLLLLLLL KHXaaaaaaaa33 34 35 36 37 38 39 40

LL∈≤≤+ ;1i40} be a integer super matrix semivector ai Q{0}∪ space of refined labels over the semifield Z+ ∪ {0}. We can define substructures, linear transformation a basis; this task can be done as a matter of routine by the interested reader.

But we illustrate all these situations by some examples.

186 Example 5.38: Let IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLa5 M = JZGWLL∈≤≤+ ;1i10 GWL ai Q{0}∪ LLa6 LLGW L LLGWa7 LLGWL LLGWa8 LLGWL LLGWa9 LLGWL K[HXa10 be an integer super column semivector space of refined labels over the semifield Z+ ∪ {0}. Take IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW0 LLGW LL LLGWL GWa3 ∈≤≤⊆ K = JZLLa + ∪ ;1i5 M, LLGW0 i Q{0} LLGW L LLGWa4 LLGW0 LLGW LLGWL LLGWa5 K[LLHXGW0 K is a integer super column semivector subspaces of refined labels over the semifield Z+ ∪ {0}. Infact we can have infinite number of integer super column semivector subspaces of V of refined labels over Z+ ∪ {0}. For take

187 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa P = JZGW5 L∈∪≤≤ L where 6Z+ {0};1 i 10 ⊆ M GWL aTi LLa6 LLGW L LLGWa7 LLGWL LLGWa8 LLGWL LLGWa9 LLGWL K[HXa10 is again an integer super column semivector subspace of M of refined labels over Z+ ∪ {0}. Infact 6 can be replaced by any positive integer in Z+ and K will continue to be a super column semivector subspace of M of refined labels over Z+ ∪ {0}.

Example 5.39: Let

P = {}()LLLLLL L∈≤≤ L+ ;1i6 aa123456 aa a a a i R{0}∪ be an integer super row semivector space of refined label over the semifield Z+ ∪ {0}.

K = {}()L L L 00L L∈≤≤ L+ ;1i 4 ⊆ P is aa123 a a 4 a i R{0}∪ an integer super row semivector subspace of P of refined labels over the integer semifield Z+ ∪ {0}.

Example 5.40: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 S = JZLLLLL∈≤≤+ ;1i15 GWaaa789a ∪ LLi Q{0} GWLLL LLGWaaa10 11 12 LLGWLLL K[HXaaa13 14 15

188 be an integer super matrix semivector space of refined labels over the semifield Z+ ∪ {0}. Consider IYFV00L LLGWa1 LLGW00La LLGW2 H = JZLL 0LL∈≤≤+ ;1i8⊆ S; GWaa34 a ∪ LLi R{0} GWLL 0 LLGWaa56 LLGWLL 0 K[HXaa78 H is an integer super matrix semivector subspace of S over the integer semifield Z+ ∪ {0}. It is pertinent to mention here that S has infinitely many integer super matrix semivector subspace of refined labels over the integer semifield Z+ ∪ {0}.

Example 5.41: Let

V = {}()LLLLLL L∈≤≤ L+ ;1i6 aaaaaa123456 a i R{0}∪ be an integer super row vector semivector space of refined labels over the semifield of integers Z+ ∪ {0}. Take ∈≤≤⊆ W1 = {}()L L 0000L L+ ;1i2 V, aa12 a iR{0}∪ ∈ ⊆ W2 = {}()00L 000L L+ V, aa1iR{0}∪ ∈≤≤⊆ W3 = {}()000L L 0L L+ ;1i2 V aa12 a iR{0}∪ and ∈ ⊆ W4 = {}()00000L L L+ V aa1iR{0}∪ be integer super row vector semi subspaces of V. 4 ∩ ≠ Clearly V = Wi ; with Wi Wj = (0 | 0 0 0 | 0 0); i j, 1 i1= ≤ i, j ≤ 4. V is the direct union of integer super row semivector subspaces of V over the semifield Z+ ∪ {0}.

189 Example 5.42: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LLGW LLLLL GWaaa10 11 12 V = JZLL∈≤≤+ ;1i24 GWLLL ai R{0}∪ LLaaa13 14 15 LLGW GWLLL LLaaa16 17 18 GW LLLLL LLGWaaa19 20 21 LLGWLLL K[HXaaa22 23 24 be an integer super column vector semivector space of refined labels over the semifield Z+ ∪ {0} = S. Consider IYFVLLL LLGWaa123 a LLGW000 LLGW LLGW000 LLGW LL000 GW∈≤≤⊆ V1 = JZLLa + ∪ ;1i3 V, LLGW000 i R{0} LLGW LLGW000 GW LL000 LLGW LLGW K[HX000 IYFV000 LLGW LLGWLLLaaa LLGW123 LLL LLGWaaa456 LLGW LL000 GW∈≤≤⊆ V2 = JZLLa + ∪ ;1i6 V, LLGW000 i R{0} LLGW LLGW000 GW LL000 LLGW LLGW K[HX000

190 IYFV000 LLGW LLGW000 LLGW LLGW000 LLGW LLLLL GWaaa123 ∈≤≤ V3 = JZLL+ ;1i9 ⊆ V GWLLL ai R{0}∪ LLaaa456 LLGW GWLLL LLaaa789 GW LL000 LLGW LLGW K[HX000 and IYFV000 LLGW LLGW000 LLGW LLGW000 LLGW LL000 GW∈≤≤⊆ V4 = JZLLa + ∪ ;1i6 V LLGW000 i R{0} LLGW LLGW000 GW LLLLL LLGWaaa123 LLGWLLL K[HXaaa456 be integer super column vector semivector subspaces of V over 4 + ∪ the semifield S = Z {0}. Clearly V = Vi with i1= F000V G W G000W G000W G W G000W Vi ∩ Vj = if i ≠ j, 1 ≤ i, j ≤ 4. G000W G W G000W G W G000W HG000XW

191 Thus V is a direct sum of integer super column vector semivector subspaces of V over S.

Example 5.43: Let

IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 V = JZLL∈≤≤+ ;1i18 GWai R{0}∪ LLLLL GWaaa10 11 12 LL GWLLL LLGWaaa13 14 15 LLGWLLL K[HXaaa16 17 18 be an integer super matrix semivector space of refined labels over the semifield S = Z+ ∪ {0}.

Consider IYFVLL 0 LLGWaa12 LLGW0La 0 LLGW3 L00 LLGWa4 P1 = JZLL∈≤≤+ ;1i6 ⊆ V, GWai R{0}∪ LL00L GWa6 LL GW000 LLGW LLGW0L 0 K[HXa5

IYFVL0L LLGWaa12 LLGW000 LLGW LL 0 LLGWaa34 P2 = JZLL∈≤≤+ ;1i6 ⊆ V, GWai R{0}∪ LL00L GWa6 LL GW000 LLGW LLGWL00 K[HXa5

192 IYFVL00 LLGWa1 LLGWLLLaaa LLGW423 LLGW000 P3 = JZLL∈≤≤+ ;1i6 ⊆ V, GW000 ai R{0}∪ LLGW LL GW000 LLGW LLGWLL 0 K[HXaa56 IYFVL00 LLGWa1 LLGW000 LLGW L0L LLGWaa24 P4 = JZLL∈≤≤+ ;1i7 ⊆ V, GWai R{0}∪ LLL00 GWa5 LL GW0L 0 LLGWa7 LLGWL00 K[HXa6 IYFVL00 LLGWa1 LLGW000 LLGW LLGW000 P5 = JZLL∈≤≤+ ;1i3 ⊆ V GW000 ai R{0}∪ LLGW LL GWL00 LLGWa2 LLGWL00 K[HXa3 IYFVLL 0 LLGWaa12 LLGW000 LLGW LLGW000 and P6 = JZLL∈≤≤+ ;1i5 ⊆ V GWai R{0}∪ LL0L 0 GWa3 LL GW00L LLGWa4 LLGW00L K[HXa5 are integer super matrix semivector subspaces of refined labels of V over the semifield Z+ ∪ {0}.

193 F000V G W G000W 6 G000W ∩ ≠ ≠ ≤ ≤ Clearly V = Pi and Vi Vj G W if i j, 1 i, j 6. i1= G000W G000W G W HG000XW Thus V is a pseudo direct sum of integer super matrix semivector subspaces of V over Z+ ∪ {0} of refined labels.

Example 5.44: Let

IYFVLLLLL LLGWaaa1 6 11 a 12 a 13 LLGWLLLLLaaa a a LLGW27141516 B= JZLLLLLLL∈≤≤+ ;1i25 GWaaaaa38171819a ∪ LLi R{0} GWLLL LL LLGWaaa4 9 20 a 21 a 22 LLGWLLLLL K[HXaa510232425 a a a be a integer super row vector semivector space of refined labels over the semi field Z+ ∪ {0}. Consider IYFVL 000 0 LLGWa1 LLGWLa 000 0 LLGW2 H = JZL 000L LL∈≤≤+ ;1i6 ⊆ B, 1 GWaa36a ∪ LLi R{0} GWL 000 0 LLGWa4 LLGWL 000 0 K[HXa5 IYFVLLLL 0 LLGWaaaa1234 LLGW0000La LLGW5 H = JZ0000LLL∈≤≤+ ;1i7⊆B, 2 GWa6 a ∪ LLi R{0} GW0000L LLGWa7 LLGW K[HX00000

194 IYFV0L 0 0 0 LLGWa2 LLGWLLLLLaaaaa LLGW14356 ∈≤≤⊆ H3 = JZGW00000LLa + ∪ ;1i6 B, LLi R{0} GW00000 LLGW LLGW K[HX00000 IYFVL000L LLGWaa19 LLGWL000Laa LLGW210 H = JZ0L L L 0 LL∈≤≤+ ;1i10⊆ B, 4 GWaaa345 a ∪ LLi R{0} GW00000 LLGW LLGW0L L L 0 K[HXaaa678 and IYFV0L 0 0 0 LLGWa2 LLGWL0L00aa LLGW13 H = JZ000L0LL∈≤≤+ ;1i9 ⊆ B, 5 GWa4 a ∪ LLi R{0} GW0LLL 0 LLGWaaa567 LLGW000LL K[HXaa89 be an integer super row vector semivector subspaces of B of refined labels over the integer semifield S = Z+ ∪ {0}. 5

B = Hi and i1= F00000V G W G00000W H ∩ H ≠ G00000W if i ≠ j, 1 ≤ i, j ≤ 5. i j G W G00000W HG00000XW

Thus H is only a pseudo direct sum of integer super row vector semivector subspaces of B of refined labels over Z+ ∪ {0}.

195 Now having seen examples of pseudo direct sum of integer super matrix semivector subspaces and direct sum of integer super matrix semivector subspaces we now proceed onto give examples of integer linear transformation and integer linear operator of integer super matrix semivector spaces defined over the integer semifield Z+ ∪ {0}.

Example 5.45: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LL GW∈≤≤ V = JZLLLaaaLLa + ∪ ;1i21 LLGW10 11 12 i Q{0} GWLLL LLaaa13 14 15 LLGW GWLLL LLaaa16 17 18 GW LLLLL K[HXaaa19 20 21 be an integer super matrix semivector space of refined labels over the integer semifield Z+ ∪ {0}. IYFVLLLLLLLL LLaaaaaaaa1 4 7 10 13 16 19 22 GWLL∈ + ; LLai R{0}∪ W = JZGWLLLLLaaaaaaaa L L L LLGW25811141720231i≤≤ 24 GWLLLL L L L L K[LLHXaaaa3 6 9 12 a 15 a 18 a 21 a 24 be an integer super matrix semivector space of refined labels over the integer semifield Z+ ∪ {0}. Define T a integer linear transformation from V into W as follows. CSF LLLV DTG aaa123W DTGLLLaaaW DTG 456W LLL DTG aaa789W DT T GLLLW DTG aaa10 11 12 W DTGLLLW DTG aaa13 14 15 W DTGLLLaaaW DTG 16 17 18 W DTLLL EUH aaa19 20 21 X

196

FVLLLLLLL0 GWaaaa14710131619 a a a

= GWLLLLaaaaaaa L L L 0. GW2 5811141720 GWLLLL L L L 0 HXaaaa36912151821 a a a

Example 5.46: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LL GWLLL LLaaa10 11 12 LLGW GW∈≤≤ V = JZLLLaaaLLa + ∪ ;1i27 LLGW13 14 15 i Q{0} LLGWLLLaaa GW16 17 18 LLLLL LLGWaaa19 20 21 LLGWLLL LLGWaaa22 23 24 LLGWLLL K[HXaaa25 26 27 be an integer super semivector space of refined labels over the semifield S = Z+ ∪ {0}. Define T : V → V by

CSFVLLL F 000V DTGWaaa123 G W DTGWLLLaaa G LLLaaaW DTGW456 G 123W LLL 000 DTGWaaa789 G W DT GWLLL G LLLW DTGWaaa10 11 12 G aaa456W T DTGWLLL = G 000W . DTGWaaa13 14 15 G W DTGWLLLaaa G LLLaaaW DTGW16 17 18 G 789W LLL 000 DTGWaaa19 20 21 G W DT GWLLL GLLLW DTGWaaa22 23 24 G aaa10 11 12 W DTGWLLL G 000W EUHXaaa25 26 27 H X

197 Clearly T is a integer linear operator on V where kernel T is a nontrivial semivector subspace of V. Now we give an example of a projection the concept is direct and can define it as in case of semivector spaces.

Example 5.47: Let IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 V = JZLL∈≤≤+ ;1i18 GWai Q{0}∪ LLLLL GWaaa10 11 12 LL GWLLL LLGWaaa13 14 15 LLGWLLL K[HXaaa16 17 18 be an integer super matrix semivector space of refined labels over the integer semifield S = Z+ ∪ {0}. Consider IYFV0L 0 LLGWa1 LLGWLLLaaa LLGW234 00L LLGWa5 W = JZLL∈≤≤+ ;1i9 ⊆ V GWai Q{0}∪ LLLL 0 GWaa67 LL GW000 LLGW LLGWLL 0 K[HXaa89 be an integer super matrix semivector space of refined labels over the semifield Z+ ∪ {0} of V. Define T an integer linear operator from V into V given by CSFVLLL F 0L 0V DTGWaaa123 G a1 W DTGWLLLaaa GLLLaaaW DTGW456 G 234W LLL 00L DTGWaaa789 G a5 W T DT = . GWLLL GLL 0W DTGWaaa10 11 12 G aa67 W DTGWLLL G 000W DTGWaaa13 14 15 G W DTGWLLL GLL 0W EUHXaaa16 17 18 H aa89 X

198 Clearly T is a linear operator with nontrival kernel. Further T (W) ⊆ W that is W is invariant under the integer linear operator T on V. Consider FVLLL FVLLL GWaaa123 GWaaa123 GWLLLaaa GW000 GW456 GW LLL LLL GWaaa789 GWaaa456 T1 ( ) = . GWLLL GW000 GWaaa10 11 12 GW GWLLL GWLLL GWaaa13 14 15 GWaaa789 GWLLL GW000 HXaaa16 17 18 HX

We see T1 is a integer linear operator with nontrivial kernel but clearly T1 is not an integer linear operator which keeps W invariant; that is T1 (W) ⊄ W. Thus all integer linear operators in general need not keep W invariant.

Now we proceed onto define various types of super matrix semivector spaces of refined labels.

DEFINITION 5.4: Let V = {set of super matrices whose entries are labels from L + or L + } be a set of super matrices R ∪{0} Q ∪{0} refined labels. If V is a such that for a set S subset of positive reals (R+ ∪ {0} or Z+ ∪ {0} or Q+ ∪ {0}) sv and vs are in V for every v ∈ V and s ∈ S; then we define V to be a set super matrix semivector space of refined labels over the set S or super matrix semivector space of refined labels over the set S, of subset of reals.

We will first illustrate this situation by some examples.

Example 5.48: Let IFVL LGWa1 LGWLLLFV V = aaa212 JGW,GW ,LLLLLLLL() LLLGWaaaaaaaa12345678 LGWaaa334HX L GWL KLHXa 4

199 LL∈≤≤+ ;1i8} be a set of super matrices of refined ai Q{0}∪ labels. Let S = {a | a ∈ 3Z+ ∪ 2Z+ ∪ {0}. V is a set super matrix semivector space of refined labels over the set S.

Example 5.49: Let

IFV LLLF LLLLLLV LGWaaa123G aaaaaa 147101316W LGW V = J LLL,LLLLLaaaG aaaa a L, aW LGW456G 258111417W GWLLLG LLLL L LW LHXaaa789H aaaaaa 369121518X K FVL Y GWa1 L GWLa L GW2 L L L GWa3 ∈ GW LLa + ∪ ;L L ()i Q{0} a4 LLLLLLLL Z GWaaaaaaaa12345678 1i18≤≤ L GWL GWa5 L L GWLa GW6 L L L HXGWa7 [ be a super matrix set semivector space over the set S = {a | a ∈ + + 3Z ∪ 25Z ∪ {0}} ⊆ L + of refined labels. R{0}∪ Clearly V is of infinite order we can define two substructure for V, a super matrix set semivector space of refined labels over the set S, S ⊆ R+ ∪ {0}.

DEFINITION 5.5: Let V be a super matrix semivector space of + refined labels with entries from L + over the set S (S ⊆ R ∪ R ∪{0} {0}). Suppose W ⊆ V, W a proper subset of V and if W itself is a set super matrix semivector space over the set S then we define W to be a set super matrix semivector subspace of V of refined labels over the set S.

We will first illustrate this situation by some examples.

200 Example 5.50: Let IFVLL F LL L V LGWaa12 G a123 a a W LGWLL G LLL W aa34CS a456 a a LGWLLLLaaa a L a L a G W V,= J LLDT123456 , LLL GWaa56DT G a789 a a W L EULLLLaaaa L a L a GWLL789101112 G LLL W LGWaa78 G a10 aa 11 12 W LGWLL G LLL W KHXaa910 H aaa13 14 15 X

LL∈≤≤+ ;1i15} be a set super matrix semivector space ai Q{0}∪ of refined labels over the set S = {a ∈ 3Z+ ∪ 5Z+ ∪ {0}} ⊆ Q+ ∪ {0}. Consider IYFVFVLL L0L LLGWGWaa12 a 1 a 6 LLGWGWLLaa L0L a a LLGWGW34 2 7 W = JZ00L0L,LL;1i10∈≤≤+ GWGWaa38a ∪ LLi Q{0} GWGW00L0L LLGWGWaa49 LLGWGWLL L0L K[HXHXaa56 a 5 a 10 ⊆ V, W is a set super matrix semivector subspace of refined labels over the set S ⊆ Q+ ∪ {0}.

Example 5.51: Let IFVL LGWa1 LGWLFV LLLLLLL a2 aaaaaaa 12345613FL LLLV LGWGWa1237aaa L LGW LLLLLLL G W GWa3 aaaaaaa 7 8 9 10 11 12 14 GLLLLW L GWaaaa4568 V,=JGWL LLLLLLL ,G W GWa4GW aaaaaaa 15161718192021 L LLLLaa aa GWGWG 9101112W L La LLLLLLL aaaaaaa GW5GW 22232425262728GLLLLW H aaaa13 14 15 16 X LGWLGW LLLLLLL L a6HX aaaaaaa 29303132333435 GW L L KHXa7

LL∈≤≤+ ;1i35} be a set super matrix semivector space ai Q{0}∪ of refined labels over the set S = 3Z+ ∪ 5Z+ ∪ 8Z+ ∪ {0}} ⊆ Q+ ∪ {0}.

201 Consider IFV La LGW1 F V LGW000L000Laa FVL00L G 511W LGW aa12 LL00LLLLGWG W LGWaaaaaa2 1 8 9 10 12 L GW0Laa 0L = GW 86G W P,J 00LL0000GW ,aa GW G 26 W L 00LLaa GWGW97G W Laaa 0L000L0 L 3313GWG W GWLLLLaaaa L HX34516G W GWLL0LL00Laaaaa L 4471514H X LGW KHX0

LL∈≤≤+ ;1i15} ⊆ V; P is a set super matrix semivector ai Q{0}∪ subspace of V over the set S of refined labels.

Now we proceed onto define the notion of subset super semivector subspace of refined labels.

DEFINITION 5.6: Let V be a set super matrix semivector space of refined labels over the set S, (S ⊆ R+ ∪ {0}). Let W ⊆ V; and P ⊆ S (W and P are proper subsets of V and S respectively). If W is a super matrix semivector space of refined labels over the set P then we define W to be a subset super matrix semivector subspace of refined labels over the subset P of the set S.

We will illustrate this situation by some examples.

Example 5.52: Let

IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLLFLLLLLLV LGWaaa789 aaaa a a G 1 4 7 10 13 16 W LGW P = J LLLaaa,LLLLG aaaa L a L, aW LGW10 11 12 G 25 8111417W GWLLLGLLLLLLW L aaa13 14 15 H aaaa369121518 a aX LGW GWLLL L aaa16 17 18 GW L LLL KHXaaa19 20 21

202

FVLLL Y GWaaa123 L GWLLL L aaa456 ∈≤≤ GWLL+ ;1i21Z LLL ai Q{0}∪ GWaaa789 L L GWLLL HXaaa10 11 12 [L be a set super matrix semivector space of refined labels over the set S = {a ∈ 3Z+ ∪ 8Z+ ∪ 7Z+ ∪ 13Z+ ∪ {0}} ⊆ Q+ ∪ {0}.

Consider

IFVLLL LGWaaa123 LGW000 LGW LLLFL0L0L0V LGWaaa456 aaa G 137W LGW K = J 000,0G Laaa 0 L 0 LW , LGWG 458W GWLLLGL0L0L0W L aaa789H aaa269X LGW LGW000 GW L LLL KHXaaa11 12 13

FVLLL Y GWaaa123 L GWLLL L aaa456 ∈≤≤⊆ GWLL+ ;1i12Z P LLL ai Q{0}∪ GWaaa789 L L GWLLL HXaaa10 11 12 [L and T = a ∈ 3Z+ ∪ 7Z+ ∪ {0}} ⊆ S ⊆ Q+ ∪ {0}.

K is a subset super matrix semivector subspace of P over the subset T of S.

203 Example 5.53: Let IFVL LGWa1 LGWLa LGW2 L LGWa3 LGWL LGWa4 L = GWL () VJ a5 ,LLLLLLLLL,aaaaaaaaa LGW 123456789 GWL L a6 GW L L LGWa7 LGWL LGWa8 LGWL KHXa9 FVLL Y GWaa12 L FVLLLL L aaaa1234GWLLaa GWGW34 L GWLLLLaaaaLL 5678GWaa56 L GW,LL;1i16∈≤≤+ Z LLLLGWai Q{0}∪ GWaa9101112 aaLLaa L GW78 GW L LLLLaaaaGWLL HX13 14 15 16 GWaa910 L GWLL L HXaa11 12 [ be a set super matrix semivector space of refined labels over the set S = a ∈ 5Z+ ∪ 3Z+ ∪ 7Z+ ∪ 8Z+ ∪ 13Z+ ∪ 11Z+ ∪ {0}}. Consider I F 0 V L G W L L G a1 W L G 0 W L G W L GLa W L 2 W = J()L00LLL0L0,G 0 W , aaaaa12345G W L 0 L G W G W L La L G 3 W GL W L a4 G W L L K H a5 X

204

FVL0 Y GWa1 L FVL0L0 L GW0La aa12 GW2 GW L LLGW0Laa 0L GWaa34 34 L ,LL;1i7GW∈≤≤+ Z ⊆ V GW0000 ai Q{0}∪ L0a GW L GW5 GW L GW0L L0L0aa GWa6 HX56 L GWL0 L HXa7 [ and K = a ∈ 3Z+ ∪ 8Z+ ∪ 13Z+ ∪ {0}} ⊆ S ⊆ Q+ ∪ {0}. Clearly W is a subset super matrix semivector subspace of V of refined labels over the subset K of S. As in case of usual vector spaces we can define direct union and pseudo direct union of set super matrix semivector space. The definition is easy and direct and hence is left for the reader as an exercise.

We now illustrate these two situations by some examples.

Example 5.54: Let IFVL LGWa1 LGWLa L 2 GWLLLLLFV LGWaaaaa3GW 1234 LGW LLLLLaaaaaGW LGW45678FLLLLLLV GWaaaaaa1 4 7 10 13 16 LGWLLLLLaaaaaG W V,= J 5GW 9 10 11 12 ,LG LLLLLW GWLGW LLLL a2 aaa5 8 11 a 14 a 17 L aaaaa6 13141516G W GWGWLLLLLL L H aaaaaa3 6 9 12 15 18 X GWLLLLLaGW aaaa LGW7 17181920 L LGW LLLL GWaaaaa8HX 21222324 L GWL LGWa9 LGWL KHXa10

LL∈≤≤+ ;1i24} be a set super matrix of semivector ai Q{0}∪ space of refined labels over the set S = {a ∈ 3Z+ ∪ 25Z+ ∪ 11Z+ ∪ 32Z+ ∪ {0}} ⊆ Q+ ∪ {0}.

205 Consider IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGW LLLa GW5 ∈≤≤⊆ W1 = JZLL+ ;1i10 V, ai Q{0}∪ LLGWLa GW6 LLL LLGWa7 LLGWL LLGWa8 GW LLLa LLGW9 LLGWLa K[HX10

IYFVLLLL L L LLGWaaaaaa147101316 W = ∈≤≤ 2 JZGWLLLLaaaa L a L a LL a + ∪ ;1i18 LLGW258111417iQ{0} GWLLLL L L K[LLHXaaaa369121518 a a ⊆ V and IYFVLLLL LLGWaaaa1234 LLGWLLLLaaaa LLGW5678 LLLL LLGWaa9101112 a a ∈≤≤⊆ W3 = JZGWLLa + ∪ ;1i24 V LLLL i Q{0} LLGWaaaa13 14 15 16 LLGWLLLL LLGWaaaa17 18 19 20 LLGWLLLL K[HXaaaa21 22 23 24 be set super matrix semivector subspaces of refined labels over the set S. 3 ∩ φ ≠ ≤ ≤ Now V = Wi and Wi Wj = if i j, 1 i, j 3. Thus i1= V is the direct union of set super matrix semivector subspaces.

206 Example 5.55: Let

IFVL LGWa1 LGWLa LGW2 L LGWa3 L GWL LGWa4 LGWL LGWa5 LGWLa LGW6 LGWLa 7 F LLLLLLLV LGW aaaaaaa135791113 V,= J L G W GWa8 L HGLLLLLLLaaaaaaaXW GW 2 4 6 8 101214 L La GW9 L GWLa LGW10 L L GWa11 LGW L L GWa12 L GWL L a13 LGW GWLa LGW14 LGWL KHXa15 FVLLL Y GWaaa123 L FVGWLLL L LLLLaaaa aaa456 GW1234GWL GWLLLL LLLaaaLL∈ + ;L aaaa5678GW789ai Q{0}∪ GW,GWZ LLLL LLLaaa ≤≤ GWaa9101112 aaGW10 11 12 1i18L GWGWLLL L HXLLLLaaaa aaa13 14 15 13 14 15 16 GWL GWLLL L HXaaa16 17 18 [ be a set super matrix semivector space of refined labels over the set S = {3Z+ ∪ 7Z+ ∪ 13Z+ ∪ 11Z+ ∪ 19Z+ ∪ 8Z+ ∪ {0}} ⊆ Q+ ∪ {0}.

Consider

207 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLGWa5 LL GWLa LLGW6 LLL GWa7 ∈ LLFVL 00000L LL+ ; GWaa13ai Q{0}∪ M,= JZL GW 1 GWa8 LLHXGWLaa 00000L 1i15≤≤ GW 24 L La L GW9 L L GWLa LGW10 L L L L GWa11 LGW L L L L GWa12 L L GWL L a13 L LGW L GWLa LGW14 L LGWL L K[HXa15 ⊆ V,

IY∈ LLFVLLLLLLL LL+ ; aaaaaaa14578910ai Q{0}∪ M2 = JZGW ⊆ V, GWLLLLLLL ≤≤ K[LLHXaaaaaaa2 3 6 11121314 1i14

IFVLLLL LGWaaaa1234 L LLLLF LL 0000LV LGWaaaa5678 aa 13 a 5 M3= JGW,G W LLLLG LL 0000LW LGWaa910111224 aaH aa a 6X L GWLLLL KLHXaaaa13 14 15 16

LL∈≤≤+ ;1i16} ⊆ V ai Q{0}∪ and

208 IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 ∈≤≤ M4 = JGWLLa + ∪ ;1i18 LLL i Q{0} LGWaaa10 11 12 LGWLLL LGWaaa13 14 15 LGWLLL KHXaaa16 17 18 and FVL 00000L Y aa15L GWLL∈≤≤+ ;1j16Z ⊆ V GWLL 00L 0L a j Q{0}∪ HXaa23 a 4 a 6 [L be set super matrix semivector subspaces of refined labels of V over the set S ⊆ Q+ ∪ {0}. 4 ∩ ≠ φ ≠ ≤ ≤ Clearly V = Mi but Mi Mj if i j, 1 i, j 4. i1= Thus V is only a pseudo direct union of set super matrices semivector subspace of refined labels over S. We can define set linear transformation of set super matrix semivector spaces only if both the set super matrix semivector space of refined labels are defined over the same set S ⊆ R+ ∪ {0}.

We will just illustrate this situation by some examples.

Example 5.56: Let

IFVLLL LGWaaa1815 LGWLLLaaa LGW2916 LLL LGWaa31017 a L = GWLLL () VJ aaa41118 ,LLLLLLLL, GWaaaaaaaa12345678 L GWLLL LGWaa51219 a L GWLLLaaa LGW61320 L LLL KHXGWaa71421 a

209 FVLLLLL Y GWaaaaa12345 L GWLLLLLaaaaa L GW678910 L LLLLLLL∈≤≤+ ;1i25Z GWaaaaa11 12 13 14 15 a ∪ i Q{0} L GWLLLLL GWaaaaa16 17 18 19 20 L GWLLLLL L HXaaaaa21 22 23 24 25 [ be a set super matrix semivector space of refined labels over S = a ∈ 5Z+ ∪ 3Z+ ∪ {0}} ⊆ Q+ ∪ {0}. Let I F LLLLLLLV LCSaaaaaaa1 4 7 10131619 L LLLLaaaaG W W= JDT1234 ,LLLLLLL,G W DTaaaaaaa25811141720 LEULLLLaaaaG W 5678G LLLLLLLW KL H aaaaaaa3 6 9 12151821X FVLLL Y GWaaa123 L GWLLLaaa L GW456 L LLL GWaaa789 L GW L LLL L GWaaa10 11 12 LL∈≤≤+ ;1i24Z GWLLL a i Q{0}∪ aaa13 14 15 L GW L GWLLL aaa16 17 18 L GW LLL L GWaaa19 20 21 L GWLLL L HXaaa22 23 24 [ be a set super matrix semivector space of refined labels over S = {a ∈ 5Z+ ∪ 3Z+ ∪ {0}} ⊆ Q+ ∪ {0}. Define T : V → W a set linear transformation of refined label set super semivector spaces as follows. F LLLV G aaa123W G LLLaaaW G 456W LLL G aaa789W T ( GLLLW ) G aaa10 11 12 W GLLLW G aaa13 14 15 W GLLLaaaW G 16 17 18 W LLL HG aaa19 20 21 XW

210 FVLLLLLLL GWaaaaaaa1 4 7 10 13 16 19

= GWLLLLLLLaaaaaaa, GW2 5 8 11 14 17 20 GWLLLLLLL HXaaaaaaa3 6 9 12 15 18 21 T ()()LLLLLLLL = aaaaaaaa12345678 CS LLLLaaaa DT1234 DTLLLL EUaaaa5678 and F LLLV G aaa123W G LLLW CSFVaaa456 LLLLLaaaaa G W DTGW12345 LLL G aaa789W DTGWLLLLLaaaaa DT678910 GLLLW GWG aaa10 11 12 W T(DTLLLLLaaaaa = . GW11 12 13 14 15 G W DTLLLaaa GWLLLLL G 13 14 15 W DTGWaaaaa16 17 18 19 20 GLLLW DTGWLLLLL aaa16 17 18 EUHXaaaaa21 22 23 24 25 G W LLL G aaa19 20 21 W GLLLW H aaa22 23 24 X T is a set super linear transformation of set super linear semivector spaces of refined labels defined over the set S ⊆ Q+ ∪ {0}.

Example 5.57: Let

IFVLLL LGWaaa1815 F LLLLLV LGWLLLaaa aaaaa12345 LGW2916G W LLLG LLLLLaaaaaW LGWaa31017 a 678910 L G W GWLLLLL V = J LLLaaa,G aaaaa11 12 13 14 15 W, LGW41118 GWGLLLLLW L LLLaa a aaaaa16 17 18 19 20 GW51219G W L G W GWLLL LLLLaaaaLa L aaa61320H 21 22 23 24 25 X GW L LLL KHXaa71421 a

211 FVL Y GWa1 L GWLa L GW2 L L GWa3 L L GWL GWa4 L GWFVL La LLLLLLL 5 aaaaaaa1 4 7 10131619 ∈ L GWGWLL+ ;L ai Q{0}∪ GWLa ,LLLLLLLGWaaaaaaa Z GW6 GW25811141720 1i≤≤ 25L L GWLLLLLLL GWa7 HXaaaaaaa36912151821 L GW L GW L GWL L GWa23 L GWLa L GW24 L L HXa 25 L[ be a set super matrix semivector space of refined labels over the set S = {a ∈ 3Z+ ∪ {0} ∪ 2Z+}.

Define a map T : V → V given by

CSF LLLV DTG aaa1815W DTG LLLaaaW DTG 2916W LLL DTG aa31017 aW DT T G LLLW DTG aaa41118W DTG LLLW DTG aa51219 aW DTG LLLaaaW DTG 61320W DTLLL EUH aa71421 aX

FVLLLLLLL GWaaaaaaa1 4 7 10 13 16 19

= GWLLLLLLLaaaaaaa GW2 5 8 11 14 17 20 GWLLLLLLL HXaaaaaaa3 6 9 12 15 18 21

212 F L V G a1 W G La W G 2 W L G a3 W CSFVG W LLLLLaaaaa L DTGW12345 G a4 W DTLLLLL G W GWaaaaa678910 La DTGWG 5 W T DTLLLLLaaaaa = G L W , GW11 12 13 14 15 a6 DTGWG W LLLLLaaaaa L DTGW16 17 18 19 20 G a7 W DTGWLLLLL G W EUHXaaaaa21 22 23 24 25 G W GL W G a23 W GLa W G 24 W L H a25 X CSFVL DTGWa1 DTGWLa DTGW2 L DTGWa3 DTGW F V L LLLLLaaaaa DTGWa4 G 12345W DTGW LLLLL La G aaaaa678910W DTGW5 G W T DTGWL = LLLLLaaaaa a6 G 11 12 13 14 15 W DTGW G W L LLLLLaaaaa DTGWa7 G 16 17 18 19 20 W DTGW GLLLLLW DTGW H aaaaa21 22 23 24 25 X DTGWL DTGWa23 DT GWLa DTGW24 DTL EUHXa25 and

CSFVLLLLLLL DTGWaaaaaaa1 4 7 10 13 16 19 DT T GWLLLLLLLaaaaaaa DTGW2 5 8 11 14 17 20 DTGWLLLLLLL EUHXaaaaaaa3 6 9 12 15 18 21

213 F LLLV G aaa1815W G LLLaaaW G 2916W LLL G aa31017 aW = G LLLW . G aaa41118W G LLLW G aa51219 aW G LLLaaaW G 61320W LLL H aa71421 aX Clearly T is a set super matrix semivector space set linear operator of refined labels. As in case of usual vector spaces we can talk of a set super matrix semivector subspace of refined labels which is invariant under a set super linear operator T of V.

Example 5.58: Let IFVL LGWa1 LGWLa LGW2 L LGWa3 FVLLLL L aaaa1234 GWL GW LGWa4 CS L GWLLLLaaaa LL aa GW 5678 13 V = J La ,,,GWDT GW5 LLLLDT LL L GWaa910111224 a a aa GWL EU L a6 GW GWLLLL L L HXaaaa13 14 15 16 LGWa7 LGWL LGWa8 LGWL KHXa9

FVY LLLLLLLaaaaaaa GW1 4 7 10131619 L ∈≤≤ GWLLLLLLLaaaaaaa LL a + ∪ ;1i21Z GW25811141720iQ{0} L GWLLLLLLL HXaaaaaaa3 6 9 12151821 [L be a set super matrix semivector space of refined labels over the set S = a ∈ 3Z+ ∪ 2Z+ ∪ 7Z+ ∪ {0}} ⊆ Q+ ∪ {0}.

214 Consider IYFVL LLGWa1 LLGWLa LLGW2 LLL GWa3 FV LLLLLLaaaa GWL GW1234 LLGWa4 CS ∈ LLLLaaGW LLLL a a a aLL+ ; GW 13 5678 ai Q{0}∪ W = JZLa ,,DTGW GW5 DTLL LL L L ≤≤ LLaa24GW a 9 a 101112 a a 1i16 GWL EU LLa6 GW GW LLLL LLL HXaaaa13 14 15 16 LLGWa7 LLGWL LGWa8 L LGWL L K[HXa9

⊆ V is a set super matrix semivector subspace of V of refined labels over the set S. Define a set super linear transformation T : V → V by

CSF V F V La L DTG 1 W G a1 W DTGLa W GLa W DTG 2 W G 2 W L L DTG a3 W G a3 W DT GL W GL W DTG a4 W G a4 W DTG W G W T La = La DTG 5 W G 5 W DTGLa W GLa W DTG 6 W G 6 W L L DTG a7 W G a7 W DT GL W GL W DTG a8 W G a8 W DTGL W GL W EUH a9 X H a9 X

CSCSCS LLaa LLaa T DTDT13 = DT13, DTDTLL DTLL EUEUaa24 EUaa24

215 CSFVLLLL F LLLLV DTGWaaaa1234 G aaaa1234W DTGWLLLLaaaa G LLLLaaaaW T DTGW5678 = G 5678W LLLL LLLL DTGWaa9101112 aa G aa9101112 aaW DT DTGWLLLL GLLLLW EUHXaaaa13 14 15 16 H aaaa13 14 15 16 X and

CSFVLLLLLLL DTGWaaaaaaa1 4 7 10 13 16 19 DT T GWLLLLLLLaaaaaaa = DTGW2 5 8 11 14 17 20 DTGWLLLLLLL EUHXaaaaaaa3 6 9 12 15 18 21

CS0000000 DT DT0000000. DT EU0000000

T is a set linear super operator on V. Further T (W) ⊆ W. Thus W is invariant under the set linear super operator of V.

Now we proceed onto define the notion of semigroup super matrix semivector space of refined labels over a semigroup S ⊆ R+ ∪ {0}.

DEFINITION 5.7: Let V be a set super matrix semivector space of refined labels over the set S ⊆ R+ ∪ {0}. If S is a semigroup of refined labels contained in R+ ∪ {0} then we call the set super matrix semivector space of refined labels as semigroup super matrix semivector space of refined labels over the semigroup S ⊆ R+ ∪ {0}.

We will illustrate this situation by some examples.

216 Example 5.59: Let

IFVLL LGWaa12 LGWLLaa LGW34 LL LGWaa56 F LLLLLV L aaaaa12345 GWLLG W LGWaa78 L G LLLLLaaaaaW GW678910 V = J LLaa,,G W GW910LLLLL L G aaaaa11 12 13 14 15 W GWLL L aa11 12 G W GWLLLLL L LLH aaaaa16 17 18 19 20 X LGWaa13 14 LGWLL LGWaa15 16 LGWLL KHXaa17 18

()LLLLLLL L∈≤≤ L+ ;1i20} aaaaaaa1234567 a i R{0}∪ be a semigroup super matrix semivector space of refined labels over the semigroup S = 3Z+ ∪ {0}.

Example 5.60: Let

IFVLL LGWaa12 LGWLLaa M = JGW34,LLLLLLLL,() LL aaaaaaaa12345678 LGWaa56 L GWLL KHXaa78

FVLLLLLL Y GWaaaaaa123456 L GWLLLLLLaaaaaa L GW789101112 L LLLLLLLL∈≤≤+ ;1i30Z GWaaaaaa13 14 15 16 17 18 a ∪ i Q{0} L GWLLLLLL GWaaaaaa19 20 21 22 23 24 L GWLLLLLL L HXaaaaaa25 26 27 28 29 30 [

217 is a semigroup super matrix semivector space of refined labels over the semigroup Z+ ∪ {0} under addition.

It is pertinent to mention here that the semigrioup must be only a subset of R+ ∪ {0} but it can be a semigroup under addition or under multiplication.

Example 5.61: Let

IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 L GWLLLF LLLLLLV LGWaaa10 11 12G aaaaaa 1 5 9 13 17 21 W LGW L LLLaaaG LLLLLL aaaaaaW K = JGW13 14 15,G 2 6 10 14 18 22 W, GWLLL LLLLLL L aaa16 17 18G aaa 3 7 11151923aaaW LGW LLLG LLLLLLW LGWaaa19 20 21 H aaaaaa4 8 12 16 20 24 X LGWLLL LGWaaa22 23 24 LGWLLL LGWaaa25 26 27 GWLLL KLHXaaa28 29 30

FVLLL Y GWaaa123 L GWLLL L aaa456 L GWLL∈≤≤+ ;1i30Z LLL ai Q{0}∪ GWaaa789 L GWLLL L HXaaa10 11 12 [L be a semigroup super matrix semivector space over the semigroup L + under addition. Q{0}∪

218 Example 5.62: Let IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL () T = JGWaaa789,LLLLLLL,aaaaaaa L 1234567 GWLLL LGWaaa10 11 12 LGWLLL KHXaaa13 14 15 FVLLLLLL Y GWaaaaaa123456 L GWLLLLLLaaaaaa L GW789101112 L LLLLLLLL∈≤≤+ ;1i30Z GWaaaaaa13 14 15 16 17 18 a ∪ i Q{0} L GWLLLLLL GWaaaaaa19 20 21 22 23 24 L GWLLLLLL L HXaaaaaa25 26 27 28 29 30 [ be a semigroup super matrix semivector space of refined labels over S = L + . S a semigroup under multiplication. R{0}∪ We can define substructure which is left as an exercise to the reader.

However we give some examples of them.

Example 5.63: Let IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 L GWLLL LGWaaa10 11 12 LGWF V L LLLaaa LLLLLL aaaaaa V,= JGW13 14 15G 1 2 3 4 5 6 W , GWLLLG LLLLLLW L aaa16 17 18H aaaaaa 7 8 9 10 11 12 X LGW LLL LGWaaa19 20 21 LGWLLL LGWaaa22 23 24 LGWLLL LGWaaa25 26 27 GWLLL KLHXaaa28 29 30

219 FVLLLLLL Y GWaaaaaa123456 L GWLLLLLL L CSLL aaaaaa789101112 ∈ aa13GWLL+ ;L DT, LLLLLL ai Q{0}∪ Z DTLL GWaaaaaa13 14 15 16 17 18 ≤≤ EUaa24 1i30L GWLLLLLL GWaaaaaa19 20 21 22 23 24 L GWLLLLLL L HXaaaaaa25 26 27 28 29 30 [ be a semigroup super matrix semivector space of refined labels over the semigroup S = L + . Take Q{0}∪ IYFVCSLL ∈ LLLLLLaaaaaa L L aa13LL+ ; JZGW123 4 5 6DTai R{0}∪ ⊆ W = ,DT V, LLGWLLLL LL LLaa 1i12≤≤ K[HXaaaa7 8 9 10 a 11 a 12 EU24

W is a semigroup super matrix semivector subspace of refined labels over the semigroup S = L + . Take Q{0}∪ IFVLLL Y LGWaaa123 L LGWLLLaaa L LGW456 L LLL LGWaaa789 L L L GWLLL LGWaaa10 11 12 L LGWCS∈ L L LLLaaaLLaaLL+ ;L JGW13 14 15 DT13ai R{0}∪ Z ⊆ B = ,DT V GWLLL LLaa ≤≤ L aaa16 17 18 EU241i30L LGW L LLL LGWaaa19 20 21 L LLGWLLL LLGWaaa22 23 24 LLGWLLL LLGWaaa25 26 27 LLGWLLL KHXaaa28 29 30 [ is a semigroup super matrix semivector subspace of refined labels over the semigroup S. Clearly IYCS LLLLaa DT13 B ∩ W = JZLL∈≤≤+ ;1i4 ⊆ V DTLL ai R{0}∪ K[LLEUaa24

220 is again a semigroup super matrix semivector subspace of refined labels over the semigroup S.

Example 5.64: Let IFVLLL LGWaaa123 F V LGWLLLaaa LLLLLLaaaaaa LGW456G 123456W M,LLLLLL,= J LLL G W GWaaa789 aaaaaa7 8 9 10 11 12 LGWG W LLLaaaGLLLLLLW LGW10 11 12 H aaaaaa13 14 15 16 17 18 X LGWLLL KHXaaa13 14 15 FVLLLL Y GWaaaa1234 L LLLL L GWaaaa5678 L GWLL∈≤≤+ ;1i18Z LLLL ai Q{0}∪ GWaa9101112 a a L GWLLLL L HXaaaa13 14 15 16 [L be a semigroup super matrix semivector space of refined labels over the semigroup S = L + under multiplication. Consider Q{0}∪ IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 W = JZLLLLL∈≤≤+ ;1i15 ⊆ V, 1 GWaaa789a ∪ LLi Q{0} GWLLL LLGWaaa10 11 12 LLGWLLL K[HXaaa13 14 15 IYFVLLLLLL LLaaaaaa123456 ∈ LLGWLL+ ; ai Q{0}∪ ⊆ W2 = JZGWLLLLLLaaaaaa V LLGW7 8 9 10 11 12 1i18≤≤ GWLLLLLL K[LLHXaaaaaa13 14 15 16 17 18 IYFVLLLL LLGWaaaa1234 LLLLLL LLGWaaaa5678 and W3 = JZGWLL∈≤≤+ ;1i16 LLLL ai Q{0}∪ LLGWaaaa9101112 LLGWLLLL K[LLHXaaaa13 14 15 16

221 ⊆ V be semigroup super matrix semivector subspaces of V of refined labels over S. Clearly Wi ∩ Wj = φ if i ≠ j, 1 ≤ i, j ≤ 3. 3

Also V = Wi . Thus is a direct union of semigroup super i1= matrix semivector subspaces of refined labels over L + = S. R{0}∪

Example 5.65: Let

IFVLL LGWaa12 F V LGWLa L a LLLLLL aaaaaa LGW3 4G 111111W L LG LLLLLLW LGWa5 a 6 aaaaaa 222222 LGWG W M = J La L a LLLLLL aaaaaa GW7 8,,G 333333W L GWLLG LLLLLLW LGWa910444444 aG aaaaaaW LGWL LG LLLLLLW a1112555555 aH aaaaaaX LGW L LL KHXGWaa13 14

()LLLLLLLL L∈≤≤ L+ ;1i14} aaaaaaaa12121212 a i R{0}∪ be a semigroup super matrix semivector space of refined labels over the semigroup S = L + . Q{0}∪ Consider

IFVLL LGWaa12 F V LGWLa L a LLLLLL aaaaaa LGW3 4G 111111W LLG 000000W LGWaa56 LGWG W J LL 000000 P1 = GWaa78,G W L GWLLG 000000W LGWaa910G W LGWL LG LLLLLLW a1112555555 aH aaaaaaX LGW L LL KHXGWaa13 14

LL∈≤≤+ ;1i14} ⊆ M, ai R{0}∪

222 IYFVLLLLLL LLGWaaaaaa111111 LLGWLLLLLL aaaaaa222222 LLGW LLLLLL ∈≤≤ P2 = JZGWaaaaaa333333LLa + ∪ ;1i5 LLi R{0} GWLLLLLL LLGWaaaaaa444444 LLGWLLLLLL K[LLHXaaaaaa555555 ⊆ M and

P3 = {()LLLLLLLL, aa12121212 aaaaaa

FVLLLLLL Y GWaaaaaa111111 L GW000000 L GWL 000000 ∈≤≤⊆ GWLLa + ∪ ;1i2Z M i R{0} L GW000000 GWL GW000000 L HX[L be semigroup super matrix semivector subspaces of M.

We see

P1 ∩ P2 = IYFVLLLLLL LLGWaaaaaa111111 LLGW000000 LLGW 000000 ∈=⊆ JZGWLLa + ∪ ;i1,5 M LLi R{0} GW000000 LLGW LLGWLLLLLL K[LLHXaaaaaa555555

is again a semigroup super matrix semivector subspace of M.

223 IYFVLLLLLL LLGWaaaaaa111111 LLGW000000 LLGW P ∩ P = 000000 ∈ ⊆ M 2 3 JZGWLLa + ∪ LL1 R{0} GW000000 LLGW LLGW000000 K[LLHX and

P3 ∩ P1 = IYFVLLLLLL LLGWaaaaaa111111 LLGW000000 LLGW 000000 ∈ ⊆ M JZGWLLa + ∪ LL1 R{0} GW000000 LLGW LLGW000000 K[LLHX are again semigroup super matrix semivector subspaces of M over the semigroup S = L + . Q{0}∪ 3 ∩ ≠ φ ≠ ≤ ≤ We see thus Pi Pj ; if i j; 1 i, j 3. But M = Pi i1= so M is a pseudo direct union of semigroup super matrix semivector subspaces of M over S.

Example 5.66: Let

IFVLLL LGWaa123 a LGWLLLaaa LGW456 LLL LGWaaa789 LGW P = J LLLaaa,LLLLLLLL,()aaaaaaaa LGW10 11 12 12345678 GWLLL L aaa13 14 15 LGW GWLLL L aaa16 17 18 GW L LLL KHXaaa19 20 21

224 FVLLLLLLL Y GWaaaaaaa1234567 L GWLLLLLLL L aaaaaaa1234567 GWL LLLLLLL ∈≤≤ GWaaaaaaa1234567LLa + ∪ ;1i21Z i R{0} L GWLLLLLLL GWaaaaaaa1234576 L GWLLLLLLL L HXaaaaaaa7324561 [L be a semigroup of super matrix semivector space of refined labels over L + = S. R{0}∪ Consider IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 LGW M = J LLLaaa,LLLLLLLL()aaaaaaaa LGW10 11 12 11111111 GWLLL L aaa13 14 15 LGW GWLLL L aaa16 17 18 GW L LLL KHXaaa19 20 21

LL∈ + } ⊆ P and T = { L + } ⊆ S be a subsemigroup of ai R{0}∪ Q{0}∪ S. We see M is a semigroup super matrix semivector space of refined labels over T of P called the subsemigroup super matrix semivector subspace of refined labels over the subsemigroup T of S.

Now we can give the related definition.

DEFINITION 5.8: Let V be a semigroup super matrix semivector space of refined labels over the semigroup S. Let W (⊆ V) and P (⊆ S) be subsets of V and S respectively, if P be a subsemigroup of S and W is a semigroup super matrix semivector space over the semigroup P of refined labels then we

225 define W to be a subsemigroup of super matrix semivector subspace of V refined labels over the subsemigroup P of the semigroup S. If V has no subsemigroup super matrix semivector subspace then we define V to be a pseudo simple semigroup super matrix semivector space of refined labels over the semigroup S.

Interested reader can supply examples of them.

We can define linear transformation of semigroup super matrix semivector spaces of refined labels provided they are defined over the same semigroup. This task is let to the reader, however we give some examples of them.

Example 5.67: Let

IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 V = J ,LLLLLLLLLL,() GWaaaaaaaaaa12345678910 L LLL GWaaa10 11 12 L GWLLL LGWaaa13 14 15 LGWLLL KHXaaa16 17 18

FVY L LLLLLaaaaa GWa1 23123 L GWLLLLLLaaaaaa L GW312231 L LLLLLL GWaaaaaa123321 L LL∈≤≤+ ;1i18Z GWai R{0}∪ LLLLLL L GWaaaaaa231132 L GWLLLLLL GWaa121233 aaaa L GWLLLLLL L HXaaaaaa223231 [ be a semigroup super matrix semivector space of refined labels over the semigroup S = L + . Q{0}∪

226 IF LLLLLLV LG aaaaaa123456W LG LLLLLLaaaaaaW LG 7 8 9 10 11 12 W LLLLLL LG aaaaaa13 14 15 16 17 18 W W = JG W, L LLLLLL G aaaaaa19 20 21 22 23 24 W L GLLLLLLW LG aaaaaa25 26 27 28 29 30 W LGLLLLLLW KH aaaaaa31 32 33 34 35 36 X

CS LLLLLLLLLaaaaaaaaa DT123456789, DTLLLLLLLLL EUaaaaaaaaa10 11 12 13 14 15 16 17 18

FVLL Y GWaa12 L GWLL L aa34 GW L LL ∈≤≤ GWaa56LLa + ∪ ;1i36Z i R{0} L GWLL GWaa78 L GWLL L HXaa910 [L be a semigroup super matrix semivector space of refined labels over the same semigroup S = L + . Q{0}∪

Define T : V → W as follows:

CSF LLLV DTG aaa123W DTGLLLaaaW DTG 456W LLL DTG aaa789W T DT = GLLLW DTG aaa10 11 12 W DTGLLLW DTG aaa13 14 15 W DTGLLLW EUH aaa16 17 18 X

227 CS LLLLLLLLLaaaaaaaaa DT123456789, DTLLLLLLLLL EUaaaaaaaaa10 11 12 13 14 15 16 17 18

T ()()LLLLLLLLLL = aaaaaaaaaa12345678910

F LLV G aa12W G LLaaW G 34W LL G aa56W G LLW G aa78W G LLW H aa910X and

CSFV L LLLLLaaaaa DTGWa1 23123 DTGWLLLLLLaaaaaa DTGW312231 LLLLLL DTGWaaaaaa123321 T DT) GWLLLLLL DTGWaaaaaa231132 DTGWLLLLLL DTGWaaaaaa121233 DTGWLLLLLL EUHXaaaaaa223231

FV L LLLLLaaaaa GWa1 23123 GWLLLLLLaaaaaa GW331312 LLLLLL GWaaaaaa122331 = . GWLLLLLL GWaaaaaa231122 GWLLLLLL GWaaaaaa113311 GWLLLLLL HXaaaaaa332132

It is easily verified T is a semigroup super matrix semivector space linear transformation of refined labels over the semigroup S.

228 Example 5.68: Let IFVLLL LGWaaa123 LGWLLLaaa LGW456 LLL LGWaaa789 L GWLLL LGWaaa10 11 12 LGWCS L LLLaaa LLLLLLLLL aaaaaaaaa V = JGW13 14 15,DT 1 2 3 4 5 6 7 8 9 , GWLLLDT LLLLLLLLL L aaa16 17 18EU aaaaaa 10 11 12 13 14 15 aaa16 17 18 LGW LLL LGWaaa19 20 21 LGWLLL LGWaaa22 23 24 LGWLLL LGWaaa25 26 27 GWLLL KLHXaaa28 29 30 FVLLLL Y GWaaaa1234 L GWLLLL L aaaa5678 L GWLL∈≤≤+ ;1i30Z LLLL ai R{0}∪ GWaaaa9101112 L GWLLLL L HXaaaa13 14 15 16 [L be a semigroup super matrix semivector space of refined labels over the semigroup S = L + . Define a map T : V → V Q{0}∪ CSFVLLL DTGWaaa123 DTGWLLLaaa DTGW456 LLL DTGWaaa789 DTGWF V LLL LLLLaaaa DTGWaaa10 11 12 G 1357W DTGWLLLL LLLaaa G aaaa9111315W T DTGW13 14 15 = G W , DTGWLLL LLLLaaaa aaa16 17 18 G 17 19 21 23 W DTGWG W LLL LLLLaaaa DTGWaaa19 20 21 H 25 27 29 30 X DT GWLLL DTGWaaa22 23 24 DTGWLLL DTGWaaa25 26 27 DTGWLLL EUHXaaa28 29 30

229 CSCS LLLLLLLLLaaaaaaaaa T DTDT123456789 DTDTLLLLLLLLL EUEUaaaaaaaaa10 11 12 13 14 15 16 17 18

F L0LV G aa13W G LLaa L aW G 2106W LL L G aa3119 aW G LLLW G aaa41212W G W LLLaaa = G 51315W G LLLaaaW G 61418W LL 0 G aa715 W G LL LW G aa8161 aW G LL LW G aa9172 aW G 0L LW H aa18 3 X and CSF LLLLV DTG aaaa1234W DTG LLLLaaaaW T DTG 5678W = LLLL DTG aa9101112 aaW DT DTGLLLLW EUH aaaa13 14 15 16 X

CS LLLLLLLLaaaaaaaa 0 DT12345678 . DT0LLLLLLLL EUaaaaaaaa910111213141516

It is easily verified T is a semigroup super matrix semivector space linear operator on V.

Now we proceed onto give examples of the notion of integer semigroup super matrix semivector space of refined labels.

230 Example 5.69: Let

IFVLLL LGWaaa123 LGWLLL aaa456CS LGWLLLLLLLaaaaaaa V = J LLL,,DT1234567 GWaaa789DT L EULLLLLLLaaa a a a a GWLLL 891011121314 LGWaaa10 11 12 LGWLLL KHXaaa13 14 15

FVL Y GWa1 L GWLa L GW2 L L GWa3 L L GWL GWa4 L GWL L a5 L GW L GWL ∈≤≤ a6 LLa + ∪ ;1i14Z GWi R{0} L L GWa7 L GWL L GWa8 L GWL L GWa9 L GWLa L GW10 L GWLa HX11 [L be an integer group super matrix semivector space of refined labels over the semigroup S = Z+ ∪ {0}.

231 Example 5.70: Let

IFVLL LGWaa112 LGWLLaa LGW213 LL LGWaa314 L GWLLF LLLLLV LGWaa41512345G aa a a aW LGWLL LLLLL aa5166G aaaaa 7 8 910W LGWG W V = GWLL LLLLL J aa6 171112131415,G aaaaaW, LGW LLG LLLLLW LGWaa7 181617181920G aaaaaW LGWLLG LL L LLW LGWaa8192122H a a aaa23 24 25 X LGWLL LGWaa920 GWLL L aa10 21 LGW LL KLHXGWaa11 22

FV Y LLaa L GW12 L GW∈≤≤ LLLLaa a + ∪ ;1i25Z GW34iR{0} L GWLLaa L HX56 [ be an integer semigroup super matrix semivector space of refined labels over the semigroup 3Z+ ∪ {0}.

Substructures, linear transformation can be defined as in case of usual semigroup super matrix semivector spaces of refined labels.

Now we proceed onto define the notion of group super matrix semivector spaces of refined labels over a group.

DEFINITION 5.9: Let V be a semigroup super matrix semivector space of refined over the semigroup S = L + ; ( L + ) under Q R multiplication. If S is a group under multiplication then we

232 define V to be a group super matrix semivector space of refined labels over the multiplication group L + or L + . Q R

We will first illustrate this situation by some examples.

Example 5.71: Let I F LLLLLLLV L G aaaaaaa1234567W L LLLLLLL G aaaaaaa2 8 9 10111213W LFVG W LL LLLLLLLaaaaaaa LGWaa12G 3 9 14 15 16 17 18 W LGWG W V = J LL,aaLLLLLLLaaaaaaa, LGW34G 4 101519202122W G W GWLL LLLLLLLaaaaaaa LHXaa56 5 111620232425 L G W G LLLLLLLW L aaaaaaa6 121721242627 G W L LLLLLLL K H aaaa7131822aaa25 27 28 X FV LLLLLLLL L∈≤≤ L+ ;1i28} HXaaaaaaaa12345678 a i R{0}∪ be a group super matrix semivector space of refined labels over the multiplicative group L + . Q

Example 5.70: Let

IFVLL LGWaa112 LGWLLaa LGW213 LL LGWaa314 L GWLL LGWaa415 LGWF V L LLaa LLLLLLL aaaaaaa M = JGW5161234567,,G W GWLLG LLLLLLLW L aa6 17891011121314H aaaaaaaX LGW LL LGWaa718 LGWLL LGWaa819 LGWLL LGWaa920 GWLL KLHXaa10 21

233 FVLLLLLLLLL GWaaaaaaaaa123456789 GWLLLLLLLLLaaaaaaaaa GW10 11 12 13 14 15 16 17 18 LLLLLLLLL GWaaaaaaaaa19 20 21 22 23 24 25 26 27

GWLLLLLLLLL GWaaaaaaaaa28 29 30 31 32 33 34 35 36 GWLLLLLLLLL GWaaaaaaaaa37 38 39 40 41 42 43 44 45 GWLLLLLLLLL HXaaaaaaa46 47 48 49 50 51 52 aa53 54

LL∈≤≤+ ;1i54} ai R{0}∪ be a group super matrix semivector space of refined labels over

L + the group of positive refined labels. R

We can define as a matter of routine the substructures in them. Here we present them by some examples.

Example 5.73: Let

IFVLLL LGWaaa123 LGWLLLaaa V= JGW456 ,LLLLLLLLLL() , LLL aaaaaaaaaa12345678910 LGWaaa789 L GWLLL KHXaaa10 11 12

FVFLL LLLL L V LY aa12 aaaa 1234 a 5 ∈≤≤ GWG,LL;1i12 Wa + ∪ Z GWGLL LLLLL Wi R{0} L HXHaa34 aaaaa 678910 X [ be a group super matrix semivector space of refined labels over the group G = L + . Q Now consider

234 IYFVLLL LLGWaaa123 LLGWLLLFV LL aaa456 aa 12 ∈≤≤ P = JZGW,LL;1i12GW + LLLGW LL ai R{0}∪ LLGWaaa789HX aa 34 LL GWLLL K[LLHXaaa10 11 12

⊆ V is a group super matrix semivector subspace of refined labels over the group G = L + under multiplication. Q

Example 5.72: Let IFLLV LG aa12W LGLLaaW LG 34W LL V = JG aa56W, L GLLW LG aa78W LGLLW KH aa910X

FV LLLLLLLLaaaaaaaa GW12345678, GWLLLLLLLL HXaaaaaaaa910111213141516

FVLL L L L L Y GWaaaaaa123456 L GWLLLLLaaaaaa L L GW2 7 8 9 10 11 L LLLLL L LL∈ + ; GWaaaaaa3 8 12 13 14 15 ai R{0}∪ L GWZ LLLLL L 1i≤≤ 21L GWaaaaaa4 9 13 16 17 18 L GWLLLLLL GWaaaaaa51014171920 L GWLLLLLL L HXaa61115182021 a a a a [ be a group super matrix semivector space of refined labels over the group G = L + , L + group under multiplication. Q Q

235 Consider

IYFVLL LLGWaa12 LLGWLL aa34 LLGW LL ∈≤≤⊆ P1 = JZGWaa56LLa + ∪ ;1i10 V, LLi R{0} GWLL LLGWaa78 LLGWLL K[LLHXaa910

IY FVLLLLLLLL LL∈ + ; LLaa12345678 aa a a a a ai R{0}∪ P2 = JZGW ⊆ V LLGWLLLLLLLL ≤≤ K[HXaaaaaaaa910111213141516 1i16 and

IYFVLLLLLL LLGWaaaaaa12 3 4 5 6 LLGWLLLLLaa aaa L a LLGW2 7 8 9 10 11 LLLLLL LL∈ + ; LLGWaaaaaa3 8 12 13 14 15 a ∪ i R{0} ⊆ P3 = JZGW V LLLLLLL L 1i≤≤ 21 GWaaaaaa4 9 13 16 17 18 LL GWLLLLLL LLGWaaaaaa51014171920 LLGWLLLLLL K[HXaa61115182021 a a a a be group super matrix semivector subspaces of refined labels of

V over the multiplicative group L + . Q We see 3 ∩ φ ≠ ≤ ≤ V = Pi ; Pi Pj = if i j, 1 i, j 3, i1= thus V is a direct sum of group super matrix semivector subspaces of refined labels of V.

236 Example 5.75: Let

I F LLLLV L G aaaa1234W FV L LLaaG LLLL aaaaW V = JGW12,,G 5678W GWLL LL L L LHXaa34G a 9101112 a a aW L GLLLLW KL H aaaa13 14 15 16 X

F LLLLLV G aaaa12345 aW G LLLLLaaaaaW G 678910W LLLLL G aaaaa11 12 13 14 15 W GLLLLLW H aaaaa16 17 18 19 20 X

FVLL GWaa12 GWLLaa GW34 LL GWaa56 GWLL GWaa78 GW LLaa,LLLLLLL() GW910aaaaaaa1234567 GWLLaa GW11 12 LL GWaa13 14 GWLL GWaa15 16 GWLL HXaa17 18

LL∈≤≤+ ;1i20} be a group super matrix semivector ai R{0}∪ space of refined labels over the group G = L + , L + = G is a Q Q multiplicative group.

Consider

237 IYFV LLaa LL∈ + ; LL12 ai R{0}∪ H1 = JZGW,LLLLLLL() ⊆ V, GWLL aaaaaaa1234567 ≤≤ K[LLHXaa34 1i 7

IF LLLLV LG aaaa1234W LG LLLLaaaaW H = JG 5678W, 2 LLLL LG aa9101112 a aW LGLLLLW KLH aaaa13 14 15 16 X ()LLLLLLL | aaaaaaa1234567

LL∈≤≤+ ;1i16} ⊆ V, ai R{0}∪

I F LLLLLV L G aaaaa12345W L LLLLL L G aaaaa678910W H3 = J()LL 0000L,G W aa12 a 3LLLLL L G aaaaa11 12 13 14 15 W L GLLLLLW KL H aaaaa16 17 18 19 20 X

LL∈≤≤+ ;1i20} ⊆ V, ai R{0}∪

IYFVLL LLGWaa12 LLGWLLaa LLGW34 LL LLGWaa56 LL GWLL H4 = LLGWaa78 ∈ LLLL+ ; GWLL ()a i R{0}∪ JZaa910,LLLLLaaaaa 0L a LLGW12345 7 1i18≤≤ GWLL LLaa11 12 GW LLLL LLGWaa13 14 LLGWLL LLGWaa15 16 LLGWLL K[HXaa17 18 and

238

∈≤≤⊆ H5 = {}()LLLLLLL L L+ ;1i7 aaaaaaa1234567 a i R{0}∪ V be group super matrix semivector subspaces of refined labels of V over the group L + . Q

We see = Hi ∩ Hj = φ if i ≠ j, 1 ≤ i, j ≤ 5. Further V = 5

Hi ; thus V is the pseudo direct union of group super matrix i1= semivector subspaces of refined labels over the group L + = G. Q

We can define group super matrix semivector spaces V and W of refined labels linear transformations provided V and W are defined over the same group G. We can also define subgroup super matrix semivector subspace of refined labels of V over the subgroup of G.

We will illustrate this situation by some examples.

Example 5.76: Let F V La G 1 W GLa W G 2 W L G a3 W G W La G 4 W GL W IFV a5 L LLaa G W V = GW12()GL W J ,LLLLLLLLL,aaaaaaaaa a6 , LL 123456789 LGWaa34 G W KHX L G a7 W G W La G 8 W G W La G 9 W GLa W G 10 W La HG 11 XW

239 FVLLLLL Y GWaaaaa12345 L GWLLLLLaaaaa L GW12345 L LLLLL GWaaaaa12345 L L GW∈≤≤ LLLLLaaaaaLLa + ∪ ;1i11Z GW12345 i R{0} L GWLLLLL aaaaa12345 L GWL GWLLLLL aaaaa12345 L GW LLLLL L HXaaaaa12345 [ be a group super row semivector space of refined labels over the group G = L + under multiplication. R

Consider IFV L LLaa W = JGW12,LLLLLLLLL() GWLL aaaaaaaaa123456789 LHXaa34 K

LL∈≤≤+ ;1i9} ⊆ V; consider H = L + ⊆ L + = G, L + ai R{0}∪ Q R Q = H is a subgroup of G under multiplication.

We see W is a group super matrix semivector space of refined labels over the group H, thus W is a subgroup super matrix semivector subspace of refined labels over the subgroup H of the group G. If G has no proper subgroups then we define V to be a pseudo simple group vector space of refined labels over G.

Likewise if V the group super matrix semivector space of refined labels over the group G has a subspace S which is over a semigroup H ⊆ G we call S a pseudo semigroup super matrix semivector subspace of V over the semigroup H of the group G.

We will just illustrate this situation by some examples.

240 Example 5.77: Let

IFV La LGW1 LGWLa LGW2 L LGWa3 LGW La LGW4 LGW L La GW5 V = J ,LLLLLLLL,()aaaaaaaa GWL 12345678 L a6 LGW La LGW7 LGW La LGW8 GW L La LGW9 GWLa KLHX10

FVLLLLLLL Y GWaaaaaaa12345613 L GWLLLLLLLaaaaaaa L GW7 8 9 10111214 L LLLLLLL LL∈≤≤+ ;1i35Z GWaaaaaaa15 16 17 18 19 20 21 a ∪ i R{0} L GWLLLLLLL GWaaaaaaa22 23 24 25 26 27 28 L GWLLLLLLL L HXaaaaaaa29 30 31 32 33 34 35 [

be a group super matrix semivector space of refined labels over the group G = L + under multiplication. Q Take P = {()LLLLLLLL aaaaaaaa12345678

241 FVL GWa1 GWL a GW2 L GWa 3 GWL GWa 4 GWL GWa 5 LL∈≤≤+ ;1i10} ⊆ V ai R{0}∪ GWL a GW6 L GWa 7 GWL GWa 8 GWL GWa 9 GWL HXa10 be a pseudo semigroup super matrix semivector subspace of refined labels of V over the semigroup IY S = JZL n=∞ 0,1,2,..., ⊆ L + 1 Q K[2n under multiplication.

It is interesting to note that this V has infinitely many pseudo semigroup super matrix semivector subspaces of refined labels.

The following theorem guarantees the existence of pseudo semigroup super matrix semivector subspaces of refined labels. Recall a group G is said to be a Smarandache definite group if it has a proper subset H ⊂ G such that H is the semigroup under the operations of G. We see L + and L + are Q R Smarandache definite groups as they have infinite number of semigroups under multiplications.

THEOREM 5.1: Let V be a group super matrix semivector space of refined labels over the group G = L + (or L + ). V has Q R infinitely many pseudo semigroup super matrix semivector

242 subspaces of refined labels of V over subsemigroups H under multiplication of the group G = L + (or L + ). Q R

Interested reader can derive several related results. However we can define pseudo set super matrix semivector subspace of refined labels of a group super matrix semivector space of refined labels over the multiplicative group L + and L + . Q R

We will only illustrate this situation by an example or two.

Example 5.78: Let IF V LLLaaa LG 123W V = JGLLL,aaaW LG 456W GLLLW KLH aaa789X

CSLLLLLLLL DTaaaaaaaa12345678 DTLLLLLLLL DTaaaaaaaa910111213141516 DTLLLLLLLL EUaaaaaaaa12345678

FVLLLL L L Y GWaaa123 a 4 a 5 a 6 L GWLLLLaaaa L a L a L GW789101112 L LLLL L L GWaaa123 a 4 a 5 a 6 L LL∈≤≤+ ;1i16Z GWai R{0}∪ LLLL L L L GWaaaa789101112 a a L GWLLLL L L GWaaa123 a 4 a 5 a 6 L GWLLLL L L L HXaaaa789101112 a a [

be a group super matrix semivector space of refined labels over the multiplicative group G = L + . Q Consider IY P = JZL∪= L n 0,1,2,... ⊆ L + ; 11 Q K[25nn

243

P is only a subset of L + = G. Q Take IFVC S LLLaaa L00L0LL0 a a a a LGW123DT 1 4 6 7 M = JGWLLL,L00L0LLDT 0 aaa4 5 6 a 9 a 12 a 14 a 15 LGWDT LHXEGWLaaa L L L a 00L00 a 00 U K 789 1 4

L∈= L+ ;i 1,2,...,9,12,14,15} ⊆ V. ai R{0}∪ M is a set super matrix semivector space of refined labels over the set P ⊆ G. Thus M is a pseudo set super matrix semivector space of refined labels of V over the set P of the multiplicative group G.

This pseudo set super matrix semivector space of refined labels can also be defined in case of semigroup super matrix semivector space of refined labels over the semigroup L + Q{0}∪ or L + . R{0}∪

The task of studying this notion and giving examples to this effect is left as an exercise to the reader.

244 !-

Applications of super matrix vector spaces of refined labels and super matrix semivector space of refined labels is at a very dormant state, as only in this book such concepts have been defined and described. The study of refined labels is very recent. The introduction of super matrix vector spaces of refined labels (ordinary labels) are very new. Certainly these new notions will find applications in the super fuzzy models, qualitative belief function models and other places were super matrix model can be adopted.

245 Since DSm are applied in several fields the interested researcher can find application in appropriate models where DSm finds its applications. When one needs the bulk work to save time super matrix of refined labels can be used for information retrieval, fusion and management. Further as the study is very new the applications of these structures will be developed by researches in due course of time.

246 ,

In this chapter we have suggested over hundred problems some of which are open research problems, some of the problems are simple, mainly given to the reader for the better understanding of the definitions and results about the algebraic structure of the refined labels. We suggest problems which are both innovative and interesting.

1. Obtain some interesting properties about DSm super row vector space of refined labels over LQ.

2. Obtain some interesting properties about DSm super column vector space of refined labels over LR.

3. Find differences between the DSm super vector spaces of refined labels defined over LR and LQ.

247 4. Let V be a super row vector space given by; IYFV LLLLaaaa LLGW14710 ∈≤≤ V = JZGWLLLLaaaa LL;1i12 aR ; LLGW25811i GWLLLL K[LLHXaaaa36912

a) Find a basis of V over LR. b) What is the dimension of V over LR? c) Give a subset of V which is linearly dependent.

5. Let V = {}()LLLLLLL;1i5∈≤≤ be aa12345i aa aaR

a super row vector space of refined labels over the field LR. a) Find a non invertible linear operator on V. b) Find an invertible linear operator on V. c) Find the algebraic structure enjoyed by L(V,V). LR d) Does T : V → V be such that ker T is a hyper subspace of V?

IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LLGW LLL GWa4 6. Let V = JZLL∈≤≤+ ;1i8be a super column GWL ai R{0}∪ LLa5 LLGW GWL LLa6 GW LLL LLGWa7 LLGWL K[HXa8

vector space of refined labels over LR.

Study questions (a) to (d) described in problem (5).

248 7. Obtain some interesting properties about super row matrix vector spaces of refined labels. IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLa5 LLGW GWL ∈≤≤ Let V = JZa6 LL;1i11aR be a super column LLGWi L LLGWa7 LLGWL LLGWa8 LLGWL LLGWa9 GWL LLa10 LLGW GWLa K[LLHX11 vector space of refined labels over LR. a) Write V as a direct sum of super column vector subspaces of V over LR. b) Write V as a pseudo direct sum of super column vector subspaces of V over LR. c) Define a linear operator on V which is a projection. d) Find dimension of V over LR. e) Find a basis of V over LR.

249 8. Let IYCSLLLLL LLDTaaaaa12345 LLDTLLLLLaaaaa LLDT678910 LLLLL LLDTaaaaa11 12 13 14 15 LLDT JZLLLLL ∈≤≤ W = DTaaaaa16 17 18 19 20 LL;1i35aR LLi DTLLLLL LLDTaaaaa21 22 23 24 25 LL DTLLLLLaaaaa LLDT26 27 28 29 30 LLDTLLLLL K[EUaaaaa31 32 33 34 35 be a super matrix vector space of refined labels over the field LR.

a) Find subspaces Wi in V so that V = Wi is a direct i1= sum. b) Find a basis for V over LR. c) Find dimension of V over LR.

d) Suppose M = IYCS00L00 LLDTa1 LLDT00L00a LLDT2 LL 0L 0 LLDTaa34 a 12 LLDT JZLL 0 0L ∈≤≤ ⊆ DTaa56 a 13LL;1i15aR LLi DTLL 0L 0 LLDTaa78 a 14 LL DTLLaa 0 0L a LLDT910 15 LLDT00L00 K[EUa11

W be a super matrix vector subspace of V over LR find T : W → W such that T (W) ⊆ W. e) Is T invertible? f) Find a non invertible T on W. g) Write V as a pseudo direct sum.

250 IYCSLLLL LLDTaaaa1234 LL LLDTLLLLaaaa 9. Let P = JZDT5678LL;1i16∈≤≤be a LLLL aRi LLDTaa9101112 aa LLDTLLLL K[LLEUaaaa13 14 15 16

super matrix vector space of refined labels over LR. a) Find a basis for P over LR. b) If LR is replaced by LQ what will be the dimension of P? c) Write P as a pseudo direct sum of super vector subspaces of refined labels over LR.

IYCSLLLL LLDTaaaa1234 LLDTLLLL aaaa5678 LLDT LLLL ∈≤≤ 10. Let M = JZDTaa9 101112 a aLL;1i20aR be LLi DTLLLL LLDTaaaa13 14 15 16 LLDTLLLL K[LLEUaaaa17 18 19 20 a super matrix vector space over the refined labeled field LR. a) Prove M is infinite dimensional over LQ. b) Write M as a direct union of super matrix vector spaces. c) Find S = L(M,M); what is the algebraic structure LR enjoyed by S.

IYCS LLL LLDTaaa123 LL DT∈≤≤ 11. Let V = JZLLLLL;1i9aaa aR be a super LLDT456 i DT LLLLLaaa K[EU789

matrix vector space over LR of refined labels.

251 IYFVLL LLGWaa12 LLGWLLaa LLGW34 LL LLGWaa56 LL GW∈≤≤ M = JZLLaaLL;1i14aR be a super matrix LLGW78i GWLL LLaa910 LLGW GWLL LLaa11 12 GW LLLL K[HXaa13 14

vector space of refined labels over LR. a) Define a T : V → M so that ker T is a nontrivial subspace of V. F000V G W b) Define p : V → M such that ker p = G000W . G W H000X c) Find the algebraic structure enjoyed by S = Hom (V,M) = L(V,M). LR LR

d) What is the dimension of M over LR? F00V G W G00W G00W G W e) Find η : M → V so that ker η ≠ G00W . G00W G W G00W G W H00X f) Find B = Hom (M,V) = L(M,V). LR LR g) Compare B and S.

252 12. Let IYCSLLLLL LLDTaaaaa12345 LLDTLLLLLaaaaa LLDT678910 LLDTLLLLLaaaaa M = JZ11 12 13 14 15 LL;1i30∈≤≤ DTaRi LLLLLLL DTaaaaa16 17 18 19 20 LL DTLLLLL LLDTaaaaa21 22 23 24 25 LLDTLLLLL K[EUaaaaa26 27 28 29 30

be a super matrix vector space over LR.

a) Let IYCS000LL LLDTaa12 LLDT0000 0 LLDT LLDTLLLaaa 0 0 P = JZ345 LL;1i11∈≤≤ ⊆ DTaRi LL000LL DTaa67 LL DT000LL LLDTaa89 LLDT000LL K[EUaa10 11 M. Find T : M → M so that T (P) ⊆ P. find η : M → M such that η (P) ⊄ P.

LLIYFVLLLL aaaa1357 ∈≤≤ b) Let V = JZGWLL;1i8aR be LLGWLLLL i K[HXaaaa2468 a super row vector, vector space of refined labels over LR. i) Define a linear function f : V → LR. ii) Is ker f a super hyper space?

13. Obtain some interesting results on super matrix vector space of refined labels over LR.

14. Let V = LR be a vector space (linear algebra) over LQ. Prove dimension of V is infinite over LQ.

253 15. Prove V = LR is finite dimensional over LR.

IYFV LLLLLaaa LLGW123 GW∈≤≤ 16. Let V = JZLLLLL;1i9aaaaR be a super LLGW456i LLGWLLLaaa K[HX789

matrix vector space of refined labels over LR. Prove V is not a super matrix linear algebra of refined labels over LR.

17. Prove a super matrix vector space of refined labels over LR can never be a super matrix linear algebra of refined labels over LR.

18. Let M = {}()LLLLLLL;1i5∈≤≤be aa12345i aa aaR

a super row vector space of refined labels over LR. How many such super row vector spaces of refined labels can be constructed using 1 × 5 super row vectors by varying the partition on ()LLLLL? aaaaa12345

19. Prove those classical theorems which can be adopted on super matrix vector space of refined labels.

20. Let IYFVLLLLL LLGWaaaaa123417 LLGWLLLLL aaaaa567818 ∈≤≤ V = JZGWLL+ ;1i20 LLLL L ai R{0}∪ LLGWaa9 10111219 aa a LL GWLLLLL K[HXaaaaa13 14 15 16 20 21. be a super row vector semivector space of refined labels

over L + . R{0}∪

a) Find a basis for V over L + . R{0}∪

254 b) Find a set of linearly independent elements of V which

is not a basis of V over L + . R{0}∪ c) Find a linearly dependent subset of V over the semifield

L + . R{0}∪ d) Prove V has linearly independent subsets whose cardinality is greater than that of the basis.

IYFVLLLL LLGWaaaa1234 LLGWLLLLaaaa LLGW5678 LLLL LLGWaa9101112 aa LLGW LLLLLL GWaaaa13 14 15 16 21. Let V = JZLL∈≤≤+ ;1i32 GWLLLL ai R{0}∪ LLaaaa17 18 19 20 LLGW GWLLLL LLaaaa21 22 23 24 GW LLLLLL LLGWaaaa25 26 27 28 LLGWLLLL K[HXaaaa29 30 31 32 be super column vector, semivector space of refined labels

over the semifield L + . R{0}∪ a) Find two super column vector semivector subspaces of

V which has zero intersection over L + . R{0}∪ b) Write V as a direct sum of super column vector

semivector subspaces over L + . R{0}∪ c) Write V as a pseudo direct sum of super column vector

semivector subspaces over L + . R{0}∪ d) Find a T : V → W which has a non trivial kernel. e) Find a η : V → V so that the η is an invertible linear transformation on V. f) Can η : V → V be one to one yet η-1 cannot exist? Justify.

255 IYFVL LLGWa1 LLGWLa LLGW2 L LLGWa3 LL GWL LLGWa4 LLGWL LLGWa5 LLGWL a6 LLGW L ∈≤≤ 22. Let V = JZGWa7 LLa + ∪ ;1i13 be a super column LLi Q{0} GWL LLGWa8 LLGWL LLGWa9 LLGWLa LLGW10 L LLGWa11 LLGW L LLGWa12 LLGWL K[LLHXa13

semivector space of refined labels over L + . Q{0}∪

a) Find a basis of V over L + . Q{0}∪ b) Can V have more than one basis?

c) Find dimension of V over L + . Q{0}∪ d) Write V as a pseudo direct sum of super column

semivector subspaces over L + . Q{0}∪

256 IYFVL LLGWa1 LLGW0 LLGW L LLGWa2 LL GW0 LLGW LLGWL LLGWa3 LLGW0 LLGW L ∈≤≤⊆ e) Suppose W = JZGWa4 LLa + ∪ ;1i7 V be a LLGW i Q{0} LLGW0 LLGWL LLGWa5 LLGW0 LLGW L LLGWa6 LLGW LLGW0 LLGWL K[LLHXa7 super column semivector subspace of V of refined

labels over L + . Q{0}∪ i) Find a T : V → V such that T (W) ⊆ W.

ii) Find T1 : V → V such that T1 (W) ⊄ W. f) Find a subset of V with more than 14 elements which is

a linearly independent subset of V over L + . Q{0}∪

IYFVLLLL LLGWaaaa1234 LLGWLLLLaaaa LLGW5678 LLLL LLGWaa9101112 a a ∈≤≤ 23. Let V = JZGWLLa + ∪ ;1i24 LLLL i R{0} LLGWaaaa13 14 15 16 LLGWLLLL LLGWaaaa17 18 19 20 LLGWLLLL K[HXaaaa21 22 23 24 be a super matrix semivector space of refined labels over

L + . Q{0}∪

257 a) Is V finite dimensional over L + ? Q{0}∪ b) Can V have finite linearly independent subset? c) Can V have finite linearly dependent subsets?

d) Find HomL + (V,V) = S. Q{0}∪ e) What is the algebraic structure enjoyed by S? f) Is S a finite set? g) Give two maps T : V → V and P : V → V so that TP = PT.

h) Find T : V → L + . f Q{0}∪

i) If K = {f | f : V → L + }, what is the cardinality of Q{0}∪ K? j) Does K have any nice algebraic structure on it?

24. Let IYFVLLLLL LLGWaaaa12345 a LLLLLLL LLGWaaaaa678910 V = JZGWLL∈≤≤+ ;1i20 LLLLL ai Q{0}∪ LLGWaaaaa11 12 13 14 15 LLGWLLLLL K[LLHXaaaaa16 17 18 19 20 be a super matrix semivector space of refined labels over

L + } and Q{0}∪ IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LL GWLLL LLaaa10 11 12 LLGW GW∈≤≤ W = JZLLLaaaLLa + ∪ ;1i27 LLGW13 14 15 i R{0} LLGWLLLaaa GW16 17 18 LLLLL LLGWaaa19 20 21 LLGWLLL LLGWaaa22 23 24 LLGWLLL K[HXaaa25 26 27

258 be a super column vector semivector space of refined labels

over L + . Q{0}∪ a) Define T : V → W such that T is an invertible linear transformation on V. b) Prove W is an infinite dimensional space. c) If η : V → W does η enjoy any special property related with dimensions of V and W? d) Can P : V → W such that ker P is different from the zero space? e) What is the algebraic structure enjoyed by

HomL + (V,W) ? Q{0}∪

25. Let V be a super row vector space of refined labels over LR; where V ={}()LLL...LLL;1in∈≤≤; aa123 a a ni aR a) Find Hom (V,V) . LR b) Is Hom (V,V) a super row vector space? LR c) Is Hom (V,V) just a vector space of refined labels? LR Justify your claim. 26. Let IYFVLLLLL LLGWaaaaa12345 LLGWLLLLL aaaaa678910 LLGW LLLLL ∈≤≤ V = JZGWaaaaa11 12 13 14 15 LL;1i25aQ LLi GWLLLLL LLGWaaaaa16 17 18 19 20 LLGWLLLLL K[LLHXaaaaa21 22 23 24 25 be a super matrix vector space of refined labels over the refined field LQ. a) What is the dimension of V over LQ? b) Does V have super matrix row vector subspace over LQ? c) Find a set linearly dependent elements in V over LQ. d) Find the algebraic structure enjoyed by L (V,V) = LQ Hom (V,V) . LQ

259 IYFVLLLL LLGWaaaa1234 LGWLLLLaaaa L LGW5678 L LLLL LGWaaaa9101112 L L L GWLLLL L aaaa13 14 15 16 L LGWL GW∈≤≤ 27. Let V = JZLLLLaaaaLL;1i36aR be LGW17 18 19 20 i L LGWLLLLaaaa L GW21 22 23 24 L LLLL L LGWaaaa25 26 27 28 L LGWLLLL L LGWaaaa29 30 31 32 L LGWLLLL L KHXaaaa33 34 35 36 [

a super matrix vector space of refined labels over LR. a) Find a super matrix vector subspace W of V dimension 19 of refined labels over LR. b) Find T : V → V so that T (W) ⊆ W. c) Find P : V → V so that P (W) ⊄ W. d) Compare the super linear operators P and T of V. e) What is ker T? f) Find ker P and compare it with ker T. g) Find a super matrix vector subspace of dim 5 of V of refined labels over LR. h) Write V as a pseudo direct sum of super matrix vector subspaces.

28. Let IYFVLL LL L L LLGWaa159131721 a a a a LL∈ GWLLLaaa L a L a L aLL;aR V = JZGW2 6 10 14 18 22 i LLL L L L 1i≤≤ 24 LLGWaaaaaa3 7 11 15 19 23 LL GWLLLLLL K[HXaaa4 8 12 a 16 a 20 a 24

be a super matrix vector space of refined labels over LQ. a) Find dimension of V over LQ. b) Find Hom (V,V) . LQ

260 6

c) Write V = Wi so that V is a direct union of super i1= matrix vector subspaces of V over LQ.

d) Write V = Wi so that V is the pseudo direct union of i1= super matrix vector subspaces of V. IYFVLLLLL LLGWaaaaa12345 LLGWLLLLLaaaaa LLGW26789 29. Let M = JZLLLLLLL;1i9∈≤≤ GWaaaaa37123aR LLi GWLLLLL LLGWaaaaa48945 LLGWLLLLL K[HXaaaaa56356

be a super matrix vector space of refined labels over LR. a) Find dimension of V over LR. b) Is dim V over R less than 25? c) Can V have a super matrix vector subspaces of refined labels of dimension 9 over LR? d) Find the dimension of the super matrix vector subspace IYFVL 0000 LLGWa1 LLGW0Laa L 0 0 LLGW67 P= JZ0L L 0 0L,L,L,L∈ L ⊆ V GWaa71 aaaa R LL1745 GW000LL LLGWaa45 LLGW000LL K[HXaa56 e) Find a linear operator T on V such that T (P) ⊆ P. f) Let M = IYFV0LLLL LLGWaaaa2345 LLGWL00LLaaa LLGW289 JZL00LLL,L,L,L,L,L∈ L GWaaa323aaaaaa R LL234589 GWL 0000 LLGWa4 LLGWL 0000 K[HXa5

261 ⊆ V, be a super matrix vector space of refined labels over LR of V. Find dimension of M over LR. g) Find T : V → V so T (M) ⊄ M.

IYFVLLLLL LLGWaaaaa12345 LLGWLLLLLaaaaa LGW12345 L LLLLL LGWaaaaa12345 L L L GWLLLLL L aaaaa12345 L LGWL GW∈≤≤ 30. Let V = JZLLLLLaaaaaLL;1i5aR LGW12345 i L LGWLLLLLaaaaa L GW12345 L LLLLL L LGWaaaaa12345 L LGWLLLLL L LGWaaaaa12345 L LGWLLLLL L KHXaaaaa12345 [ be a super matrix vector space of refined labels over LR. a) What is dimension of V over LR? b) Is V of dimension 45 over LR? Justify your answer. F00000V G W G00000W G00000W G W G00000W c) Define T : V → V so that ker T = G00000W . G W G00000W G W G00000W G00000W G W H00000X

d) Find P1 and P2 two distinct linear operators of V so that P1P2 = P2 P1 = Identity operator on V.

31. Give some interesting applications of super matrix vector spaces of refined labels over LR or LQ.

262

32. Prove we cannot in general construct super matrix linear algebras of refined labels defined over LR or LQ.

33. Obtain some interesting properties enjoyed by integer set super matrix semivector space of refined labels defined over the set S = 3Z+ ∪ 5Z+ ∪ {0}.

34. Obtain some interesting properties by group super matrix

semivector spaces of refined labels defined over L + or Q

L + . R

35. Let V = IFV La LGW1 LGWLa LGW2 L LGWa3 L FVLLL GWL GWaaa123 LGWa4 GWLLLaaa LGWL 456 a5 GW LGWLLLF LLLLLLLLV L GWaaa789 aaaaaaaa 12345678 JGWL G W a6 GW LGWLLLG LLLLLLLLW GWaaa10 11 12H aaaa 9 10 11 12 aaaa13 14 15 16 X GWLa L 7 GW GWLLLaaa L L GW13 14 15 LGWa8 HXGWLLLaaa LGWL 16 17 18 LGWa9 LGWLa LGW10 GWLa KLHX11

LL∈≤≤+ ;1i18} be a integer super matrix semivector ai R{0}∪ space of refined labels over 3Z+ ∪ 5Z+ ∪ {0} = S. a) Find a basis of V over S. b) Find integer super matrix semivector subspace of V of refined labels. 3

c) Write V = Wi as a direct sum. i1=

263 36. Let V = IYFVFVFV LLLLLaaa LLL aaa LLL aaa LLGWGW123 123GW 123 GW∈≤≤ JZGWGWLLL,LLL,LLLLL;1i9aaa aaa aaaaQ LLGWGW456 456GW 456i LLHXHXGWGWLLLaaa LLL aaaGW LLL aaa K[789 789HX 789 be a set super matrix vector space of refined labels over

the set S = L + ∪ L + . Q{0}∪ R{0}∪ a) Find subspaces of V. b) Write V as a pseudo direct sum of subspaces.

37. Let V =

I FVL L GWa1 L GWLa L GW2 L L GWa3 L GWL LCSGWa4 L LLaa FLLLLLLLV 12GW aa1234567 aa aa a JDT,,La G W DTLLGW5 L aa34 HGLLLLLLLaa aa aa aXW EUGWL 1234567 L a6 GW L L L GWa7 L GWL L GWa8 L GWL K HXa9

LL∈≤≤+ ;1i9} be a group super matrix semivector ai R{0}∪

L + space of refined labels over R the multiplicative group.

a) Find V= Wi as a pseudo direct sum. i1= b) Find subgroup super matrix semivector subspace of V

of refined labels over L + . R c) Find pseudo semigroup super matrix semivector

subspaces of V of refined labels over L + . R

264 d) Find pseudo set super matrix semivector subspace of V

of refined labels over L + . R

38. Let V =

IFV LLaa LGW12 LGWLLaa LGW12 LL LGWaa34 L GWLLFV L aa34LLaa GWGW12 LGWLL L aa34GWLLaa JGW,34 ,LLLLLLLLLF V GWH aaaaaaaaa123456789X GWLLaaLL L 56GWaa56 LGWGW LLaaLL LGW56HXaa78 LGWLL LGWaa56 GW L LLaa LGW56 LGWLLaa KHX78

LL∈≤≤+ ;1i9} be a semigroup super matrix semivector ai R{0}∪ space of refined labels over the semigroup S = L + under Q{0}∪ addition.

a) Write V = Wi as a pseudo direct sum of subspaces. b) Find two pseudo set super matrix semivector subspaces of refined labels of V over S. c) Find T : V → V with non trivial kernel. d) Find P : V → V so that P-1 exists. IYFVLL LLGWaa12 LLGWLL aa34 ∈≤≤⊆ e) If K = JZGWLL+ ;1i2 V; find LL ai Q{0}∪ LLGWaa56 LL GWLL K[LLHXaa78

dimension of K over L + . Q{0}∪

265 39. Obtain some nice applications of semigroup super matrix semivector space of refined labels over the semigroup

L + under addition. R{0}∪

IYFVLLLL LLGWaaaa1234 LLGWLLLLaaaa LLGW5678 LLLL LLGWaa9 101112 a a LL GWLLLL ∈≤≤ 40. Let V = JZaaaa13 14 15 16 LL;1i28 be GWaRi LL GWLLLL LLGWaaaa17 18 19 20 LL GWLLLLaaaa LLGW21 22 23 24 LL GWLLLLaaaa K[HX25 26 27 28

a super matrix vector space of refined labels over LR. a) Find dimension of V over LR. b) Find subspaces of dimensions 5, 7 and 20.

c) Write V=W1⊕ … ⊕ Wt where Wi ’s are subspaces of V. d) Find T : V → V so that T is an idempotent operator on V. e) Find T : V → V so that T-1 exists.

41. Let I FVL Y L GWa1 L L GWLa L L GW2 L LLLLLFV L GWaaaaa31234 L L GWGWL LGW LLLL LFVLL GWa45678 aaaa ∈ L L aa12 GWLL;L GWGWLLLLL aRi P = J ,,aaaaa5GW 9 10 11 12 Z LGWLLaaGW 1i19≤≤ L HX34GWLGW LLLL L aaaaa6 13141516 L GWGW L LLLLLGWL L GWaaaaa717182019HXL L GWL L L GWa8 L L GWL L K HXa9 [ be a set super matrix vector space of refined labels over the set

266 IY ∪∈∞⊆ S = JZLLn0,1,2,...,11 LR. K[2{0}nn∪∪ 5{0} 3

a) Write V = Wi as a direct sum of subspaces. i1= 5

b) Write V = Wi as a pseudo direct sum of subspaces? i1= c) Find two disjoint subspaces W1 and W2 and find two linear operators T1 and T2 so that T1 (W1) ⊆ W1 and T2 (W2) ⊆ W2. d) Find a subset super matrix vector subspace of refined labels of V.

42. Write down any special feature enjoyed by the super matrix

semivector space of refined labels over L + or L + . R{0}∪ Q{0}∪

43. Can we have finite super matrix semivector space of refined

labels over the semifield L + or L + ? Justify your R{0}∪ Q{0}∪ claim.

44. Is it possible to construct set super matrix semivector space

of refined labels over L + of finite order? Q{0}∪

IYFVLLLLL LLGWaaaaa12345 LLGWLLLLL aaaaa21534 LLGW LLLLL ∈≤≤ 45. Let P = JZGWaaaaa35412LL;1i5aR be LLi GWLLLLL LLGWaaaaa43125 LLGWLLLLL K[LLHXaaaaa54253 a super matrix vector space of refined labels over the field LR. a) Find dimension of P over LR. b) Write P = ∪ Wi, Wi, subspaces as a direct sum. c) Can P have ten dimensional super matrix vector subspace of refined labels over LR? Justify your answer.

267 d) Define T : P → P so that T2 = T.

e) Find S : P → P so that S2 = (0). f) Find hyper subspaces of refined labels in P over LR.

46. Obtain some interesting properties about L(V,L) where LRR

V is a super matrix vector space of refined labels over LR.

IYFVLLLL LLGWaa1234 aa LLGWLLLLaaaa LLGW5678 LLLL LLGWaa1234 aa 47. Let V = JZLL;1i8∈≤≤be a GWaRi LLLLLL GWaaaa5678 LL GWLLLL LLGWaa1234 aa LLGWLLLL K[HXaaaa5678

super matrix vector space of refined labels over LR. Define f : V → LR so that f is a linear functional on V. Suppose v1, …, v6 are some six linearly independent elements of V; find f (vi); i = 1, …, 6. Study the set {f (v1) ,…, f (v6)} ⊆ LR. What is the algebraic structure enjoyed by L(V,L)? How many hyper subspaces exist in case of LRR V?

IYFVLLLL LLGWaaaa1234 LLGWLLLL LLaaaa2341 48. Let V = JZGWLL∈≤≤+ ;1i4 be LLLL ai R{0}∪ LLGWaaaa3421 LLGWLLLL K[LLHXaaaa4133 a integer super matrix semivector space of refined labels over Z+ ∪ {0}. a) Find a linear operator T on V which is non invertible. b) Write V as a direct sum of semivector subspaces of refined labels over Z+ ∪ {0}.

268 IYFVLLL LLGWaaa123 LLGWLLLaaa LLGW456 LLL LLGWaaa789 LL GWLLL LLGWaaa10 11 12 LLGWLLL LLaaa13 14 15 49. Let M = JZGWLL∈≤≤+ ;1i30 be a GWLLL ai Q{0}∪ LLaaa16 17 18 LLGW LLL LLGWaaa19 20 21 LLGWLLL LLGWaaa22 23 24 LGWLLL L LGWaaa25 26 27 L LGWLLL L K[HXaaa28 29 30 super matrix semivector space of refined labels over the

semifield L + . Q{0}∪

a) Find the dimension of M over L + . Q{0}∪ b) Write M as a pseudo direct sum of semivector

subspaces of refined labels over L + . Q{0}∪

c) Write M = Wi as a pseudo direct sum of super i matrix semivector subspaces of refined labels over

L + . Q{0}∪ d) Show in M we can have more number of linearly independent elements than the cardinality of the basis of

M over L + . Q{0}∪

IYFVLLLL LLGWaaaa1234 LLGWL0LLaaa LLGW245 50. Let V = JZ00L0LL;1i11∈≤≤ be a GWa3 aR LLi GWLL 0L LLGWaa75 a 9 LLGW0L L 0 K[HXaa10 1

super matrix vector space over LR.

269 a) Find dimension of V over LR. b) Write ∪ Wi = V as a pseudo direct sum of subspaces. c) Is the number of subspaces of V over LR finite or infinite? d) Give a linearly dependent subset of order 10.

IYFVLLLL LLGWaaaa1234 LLGWL00L aa25∈≤≤ 51. Let V = JZGWLL+ ;1i7 be L0L0ai R{0}∪ LLGWaa36 LL GWLL 00 K[LLHXaa47 a group super matrix semivector space of refined labels over

multiplicative group G = L + . R a) What is the dimension of V over G? b) Prove there exist infinite number of pseudo set super matrix semivector subspaces of V over LR. c) Prove there exists infinite number of pseudo semigroup super matrix semivector subspaces of V over LR. d) Prove V is not a simple group super matrix semivector space over LR. 7

e) Write V = Wi as direct sum of subspaces of V. i1= 6

f) Write V= Wi as pseudo direct sum of subspaces of V i1=

52. Let IYFV LLLLaaaa L a L a L a L a LLGW1 4 7 10 13 16 19 22 LL;∈ aRi V = JZGWLLLLaaaa L a L a L a L a LLGW2 5 8 11 14 17 20 23 1i24≤≤ GWLLLL L L L L K[LLHXaaaa3 6 9 12 a 15 a 18 a 21 a 24

be a super row vector space over LR of refined labels. a) What is the dimension of V? b) Find Hom (V,V) . LR c) Find T : V → V so that T2 = T. → 2 d) Find T1 : V V so that T1 = (0).

270 IYFVFVFV FVL LLLaaa LLGWa1 GWGWGW111 LLGWGWGWLLL GWLa aaa222 LLGW2 GWGWGW LLL LLLaaa GWa3 GWGWGW333 53. Let P = JZ,,, LL;1i6∈≤≤+ GWGWGWGWai R{0}∪ LLL LLLaaa GWa4 GWGWGW444 LLGWGWGW GWL LLLaaa LGWa5 GWGWGW555 L L L GWL GWGWGWLLLaaa K[HXa6 HXHXHX666 be a set super matrix semivector space of refined labels over the set S = LLL∪∪. 2nnn∪∪ {0} 3 {0} 5 a) Find subspaces of P. b) What is dimension of P over S? 4

c) Express P = Pi as a direct sum of subspaces of V. i1= d) Define T : P → P with T.T = T. → −1 e) Define T1 : P P so that T1 exists.

54. Let I FVLL LFVFV0LLLGWaa12 0LLL LGWGWaaa234LL aaa 123 GWaa34 LGWGWLaaa 0LLGW L aaa 0LL V = J 234,,LL 145 GWGWGWaa56 LL 0L LL 0L LGWGWaa33 a 2 aa 24 a 6 GWLL LGWGWGWaa78 HXHXLLLaaa 0 LLL aaa 0 L 442GWLL 356 K HXaa910

LL∈≤≤+ ;1i6} be a integer set super matrix semivector ai R{0}∪ space of refined labels over the set of integers S = 3Z+ ∪ {0} ∪ 2Z+. Suppose W = {FVLLLLLLLLLL, HXaaaaaaaaaa12345678910

271 FVLL Y FVaa12 L LLLaaaGW GW123GWLL L GWaa34 ∈≤≤ LLL,GW LL+ ;1i6Z GWaaa456LL a iR{0}∪ GWaa12 L GWLLLaaa L HX789GWLL HXaa34 [L be an integer set super matrix semivector space of refined labels over the set S = 3Z+ ∪ 2Z+ ∪ {0}. a) Find a linear transformation T from V into W so that T-1 exists. b) Find the algebraic structure enjoyed by HomS (V, W). c) Find T1 and T2 two linear transformation from V into V such that T1 . T2 is an idempotent linear transformation. d) Find two linear transformation L1 and L2 from W into W so that L1 . L2 = L1 . L2 but L1 . L2 = L1 . L2 ≠ I.

55. Let IYF LLLLLV LLG aaaaa12345W LLG LLLLLaaaaaW LLG 678910W LLLLL LLG aaaaa11 12 13 14 15 W LL GLLLLLW LLG aaaaa16 17 18 19 20 W LLG W ∈ LLLLLLLaaaaaLL+ ; M = JZGW21 22 23 24 25 ai Q{0}∪ be a LLGWLLLLLaaaaa 1i50≤≤ GW26 27 28 29 30 LLLLLLL LLGWaaaaa31 32 33 34 35 LLGWLLLLL LLGWaaaaa36 37 38 39 40 GW LLLLLLLaaaaa LLGW41 42 43 44 45 LLGWLLLLLaaaaa K[H 46 47 48 49 50 X

super matrix vector space of refined labels over L + . Q{0}∪

a) Write M = Wi ; Wi super matrix vector subspaces of i

refined labels over L + of M as a direct sum. Q{0}∪

272 IYFV00000 LLGW LLGWLLLLLaaaa a LLGW12345 LLGW00000 LL GWLLLLL LLGWaaaaa678910 LLGW∈ LL00000LL+ ; b) Let W = JZGWai Q{0}∪ ⊆ LLGW00000 1i20≤≤ LLGW LLGW00000 LLGWLLLLL LLGWaaaaa11 12 13 14 15 GW LLLLLLLaaaaa LLGW16 17 18 19 20 GW00000 K[LLHX M be a super matrix vector subspace of M of refined

labels over L + . Q{0}∪ Find T : M → M so that T (W) ⊆ W. c) What is dimension of W?

d) Find T1, T2 : M → M so that (T1 . T2) (W) ⊆ W.

56. Obtain some special properties enjoyed by set super matrix vector space of ordinary labels over a suitable set.

57. Let V = {}FVLLLLL LL;1i5∈≤≤ be HXaaaaaaR12345 i

a super row matrix vector space of refined labels over LR. Will V ≅ LR × LR × LR × LR × LR? Justify your answer.

58. How many super row matrices of dimension seven can be built using the row matrix ()LLLLLLL of refined labels? aaaaaaa1234567

59. Prove or disprove that if V is a super matrix vector space of dimension n of refined labels defined over LR then V is isomorphic with all super matrix vector space of dimension n of refined labels defined over LR.

273

IYFVLLL LLGWaaa123 LLGWLLL aaa456 LLGW LLL ∈≤≤ 60. Let V = JZGWaaa789LL;1i15aR be a super LLi GWLLL LLGWaaa10 11 12 LLGWLLL K[LLHXaaa13 14 15

matrix vector space of refined labels defined over LR. W = IYFV LLaaaaa LL L LLGW14567 ∈≤≤ JZGWLLLLLaaaa a LL;1i15 aR be a LLGW28 91011i GWLLLLL K[LLHXaaaaa312131514

super matrix vector space of refined labels defined over LR. Is W ≅ V?

61. Let IFVL LGWa1 LGWL a2 F V LGW LLLLLaaaaaa L L FVLL G 159131721W LGWa3 aa L GW12GLLLLLLW = GW aaa2 6 10 a 14 a 18 a 22 V,LL,J La GWaaG W LGW4 GW34LLLLLL GW G aaaaaa3711151923W L La HXGWLLaa GW5 56GLLL L L LW L H aaaaaa4 8 12 16 20 24 X GWL L a6 GW L L KHXa7

LL∈≤≤+ ;1i24} ai R{0}∪ and IFV L LLLaaa W = JGW123,LLLLLLL,() GWLLL aaaaaaa1234567 KLHXaaa456

274 FVLLLLLL Y GWaaaaaa123456 L GWLLLLLL L aaaaaa7 8 9 10 11 12 ∈≤≤ GWLL+ ;1i24Z LLLLLL ai R{0}∪ GWaaaaaa13 14 15 16 17 18 L L GWLLLLLL HXaaaaaa19 20 21 22 23 24 [L be a group super matrix semivector space of refined labels

over the multiplicative group G = L + . R a) Is V ≅ W? b) Find dimension V. c) Find T : V → W so that T-1 exists.

d) Find T1 : V → W so that T1 . T1 = T1. e) Find pseudo set super matrix semivector subspaces M and P of V and W respectively over the set B =

LLL∪⊆+ such that M ≅ P. 11R 25nn f) Define f : V → W so that f (M) = P.

62. Let I F V FVLLLLL LLL L aa12789 a a a aaa 123 LGWG W = G W VLLLLL,LLL,JGWaaa a a aaa LGW3 4 10 11 12G 4 5 6 W LGWLLLLLaaaaaG LLL aaaW KHX5 6 13 14 15H 7 8 9 X FVLLL Y GWaaa123 L GWLLLaaa L GW456 L LLLFV LLLL GWaaa789 aaaa 1234 L GW L GWLLLGW LLLL GWaaa1011125678 aaaa L ,LL;1iGW∈≤+ ≤ 24Z GWLLL LLLL ai R{0}∪ aaa13 14 15GW aaaa 9 10 11 12 L GW L GWLLLGW LLLL aaa16 17 18HX aaaa 13 14 15 16 L GW L GWLLLaaa 19 20 21 L GWLLL L HXaaa22 23 24 [

be a semigroup super matrix semivector space of refined

labels over the semigroup L + under addition. R{0}∪

275 a) Find dimension V. b) Find T : V → V so that T . T = T. c) Find P : V → V so that P2 = 0. d) Find M : V → V so that ker M is a semigroup super matrix semivector subspace of V of refined labels over

the additive semigroup L + . R{0}∪ e) Find subsemigroup super matrix semivector subspace of refined labels of V. f) Find pseudo set super matrix semivector subspaces of V of refined labels over subsets of the additive

semigroup L + . R{0}∪ 63. Give an example of a group super matrix semivector space of refined labels which is simple.

64. Obtain some interesting properties about group super square matrix semivector space of refined labels over the

multiplicative group L + . Q

65. Let V = {()LLLLLLLLLL aaaaaaaaaa12345678910 LL;1i10∈≤≤} be a super row matrix vector space of aRi

refined labels over LR. a) Find for f : V → LR, a linear functional defined by f()LLLLLLLLLL aaaaaaaaaa12345678910

= L;+++ the ker f. aa...a12 10 b) Find L(V,L). LRR c) What is dimension of L(V,L)? LRR d) What is the algebraic structure enjoyed by L(V,L)? LRR

e) What is the dimension of V over LR? f) Find Hom (V,V) . LR g) What is the algebraic structure enjoyed by Hom (V,V) ? LR

276 66. Find some interesting properties enjoyed by L(V,L) = LRR

{f : V → LR} where V is a super matrix vector space of refined labels defined over LR.

67. Find the algebraic structure enjoyed by HS (V,V) where V is a set super matrix semivector space of refined labels defined

over the set S = L + . R{0}∪

IYFVLLLL LLGWaaaa1234 LLGWLLLLaaaa 68. Let V = JZGW5678LL;1i16∈≤≤ be a LLLL aRi LLGWaa9101112 aa LL GWLLLL K[LLHXaaaa13 14 15 16

super matrix vector space of refined labels over LR. IYFV 0LLLaaa LLGW234LL;∈ aRi LLGWL0LL−aaa W = JZGW2561i7;≤≤ LL−− 0L − LLGWaa35 a 7 LL− =− LLGWaaii LLL−− 0 K[LLHXaaa467

be a super matrix vector space of refined labels over LR. a) Is V ≅ W? b) Find dim W over LR. c) Find dim V over LR. d) Study Hom (V,W) and Hom (W,V) . LR LR e) Find T : V → W so that T . T = T. → −1 f) Find T1 : V W so that T1 exists.

277 IFVLL LGWaa12 LGWLLaa LGW34 LL LGWaa56 F LLLLV LGWaaaa1234 LLaaG W LGW78G LLLLW LGWaaaa5678 L LLaaG W JGW910,,LLLL 69. Let V = G aaa9 101112 aW LGWLL aa11 12 G W LGWLLLLaaaa LLG 13 14 15 16 W LGWaa13 14 G W HLLLLaaaaX LGWLL 17 18 19 20 LGWaa15 16 LGWLL LGWaa17 18 GWLL KLHXaa19 20 FVLLLLL 0L GWaaaaa12345 a 6

GWLLLLLLLaaaaaaa, GW78910111213 GWLLLLLLL HXaaaaaaa14 15 16 17 18 19 20 FV0L L L L Y GWaa1234 aa L GWL0LLLaaaa L GW5 678 L LL 0LL LL;1i20∈≤≤Z GWaa910 a 1112 a aR i L GWLLL 0L GWaaa13 14 15 a 16 L GWLLLL 0 L HXaaaa17 18 19 20 [ be a semigroup super matrix semivector space of refined

labels over L + under addition. R{0}∪ a) Find dim V.

b) Find HomL + (V,V) . R{0}∪

c) Find W1, …, Wt in V so that V = ∪ Wi where Wi’s are semigroup super matrix semivector subspaces of V over

L + where W ∩ W = φ if i ≠ j 1 ≤ i, j ≤ t. R{0}∪ i j

70. Find some nice applications of set super matrix semivector

spaces defined over the set S = LL⊆ + . 2n ∪∪ {0} R {0}

278

71. Does there exists a set super matrix semivector space which has no subset super matrix semivector subspace of refined labels? Justify your claim.

72. Does there exist a semigroup super matrix semivector space of refined labels which has no subsemigroup super matrix semivector subspace of refined labels.

73. Prove every integer set super matrix semivector space of refined labels has integer subset super matrix semivector subspace of refined labels!

IYFVLLLL LLGWaaaa1111 LLGWLLLLaaaa 74. Let V = JZGW1111LL∈ be a super LLLL aQ1 LLGWaaaa1111 LL GWLLLL K[LLHXaaaa1111

matrix vector space of refined labels over the field LQ. a) Can V have non trivial super matrix vector subspace of refined labels? b) Find Hom (V,V) . LQ c) Does T : V → V exist so that T2 = T? (T ≠ I). d) Find L(V,L). LQQ IYFVLLLLL LLGWaaaaa11111 LLGWLLLLLaaaaa LLGW11111 LLLLL LLGWaaaaa11111 LLGW 75. Let V = JZLLLLLaaaaaLL∈ + be a GW11111a1 R{0}∪ LL GWLLLLL LLGWaaaaa11111 LL GWLLLLLaaaaa LLGW11111 LLLLLLL K[HXGWaaaaa11111 semigroup super matrix semivector space of refined labels

over the additive semigroup L + . R{0}∪

279 a) Find dim V. b) Can V have non trivial semigroup submatrix semivector

subspaces of refined labels over L + ? R{0}∪ c) Can V have non trivial subsemigroup submatrix semivector subspaces of refined labels over

subsemigroups in L + ? R{0}∪ d) Find pseudo set super matrix semivector subspaces of

refined labels over subsets in L + . R{0}∪

e) Find HomL + (V,V) . R{0}∪

IYFVLLLLLLL LLGWaaaaaaa1111111 LLGWLLLLLLLaaaaaaa 76. Let V = JZGW1111111LL∈ LLLLLLL aR1 LLGWaaaaaaa1111111 LL GWLLLLLLL K[LLHXaaaaaaa1111111

be a super matrix vector space of refined labels over LR. a) Can V have proper super matrix vector subspaces of refined labels over LR? b) Is V simple?

IYFVLLLL LLGWaaaa1111 LLGWLLLLaaaa 77. Let V = JZGW1111LL∈ be the super LLLL aQ1 LLGWaaaa1111 LL GWLLLL K[LLHXaaaa1111

matrix vector space of refined labels over LQ. a) Is V simple? b) Find Hom (V,L ) . LQQ

IYFVLLL LLGWaaa111 ∈ 78. Let V = JZGWLLLLLaaaa + ∪ LLGW111 1Q{0} GWLLL K[LLHXaaa111

280 be a super matrix semivector space over the semifield L + of Q{0}∪ refined labels.

a) Find dim of V over L + . Q{0}∪ b) Is V simple?

c) Find HomL + (V,V) . Q{0}∪

IYFVLLLLL LLGWaaaaa11111 LLLLLLL LLGWaaaaa11111 79. Let V = JZGWLL∈ + be a LLLLL a1 R{0}∪ LLGWaaaaa11111 LLGWLLLLL K[LLHXaaaaa11111 super matrix semivector space of refined labels over

L + . R{0}∪ a) Is V simple?

b) Can V have substructures if instead of L + ; V is R{0}∪

defined over L + . R{0}∪

c) Find HomL + (V,V) . R{0}∪ d) Find T : V → V so that T-1 exist. (Is this possible). e) Can T : V → V be such that T . T = T (T ≠ I)?

IYFVLLLL LLGWaaaa1111 LLGWLLLLaaaa LLGW1111 LLLL LLGWaaaa1111 LLGW 80. Let V = JZLLLLaaaaLL∈ + be a group GW1111a1 R{0}∪ LL GWLLLL LLGWaaaa1111 LL GWLLLLaaaa LLGW1111 LLLLLL K[HXGWaaaa1111 super matrix semivector spaces refined labels over the

multiplicative group L + . R a) Is V simple?

281 b) Does V have subgroup super matrix semivector subspaces? c) Does V contain pseudo semigroup super matrix semivector subspaces? d) Can V have pseudo set super matrix semivector subspaces of refined labels?

e) Find HomL + (V,V) . R

81. Characterize simple super matrix vector spaces of refined labels over LR or LQ.

82. Let V = IFVL LGWa1 LGWLa LGW2 L LGWa3 L GWL LGWa4 LGWFV La LL L 5 aa12 LGWGWFLLLL L LV GWL aaa123 a 4 a 5 a 6 J a6 ,LGW L ,G W GWaa34GLLLL L LW L GWH aaaa7 8 9101112 a aX La GWLL LGW7 HXaa56 LGWL LGWa8 LGWL LGWa9 LGWLa LGW10 GWLa KLHX11

LL∈≤≤+ ;1i12} be a integer set super matrix ai R{0}∪ semivector space of refined labels over Z+ ∪ {0}.

a) Find Wi’s in V so that V = ∪ Wi; Wi ∩ Wj = φ if i ≠ j.

b) Find Hom+ (V,V) . Z{0}∪ c) Is V simple? d) Is V finite dimensional? e) Can V have subset integer super matrix semivector subspaces of refined labels over S ⊆ Z+ ∪ {0}?

282

83. Find the algebraic structure enjoyed by Hom (V,V) LR where V = ()LL...LLL;LL∈ . aa11 aaaaR 1111

84. Find the algebraic structure of L(V,L) where V is LRR given in problem 83.

85. Find the algebraic structure of where V = IYFV LLLLLaaa JZGW111LL∈ . GWLLL aQ1 K[LLHXaaa111

86. Find HomL + (V,V) where V = Q I FV LF LLLLLV LFV GWaaaaaa1 11111 L LLaa G W JGW11,LLL,L,LLLLL()GWG W GWLL aaa111GW a 1 aaaaa 11111 LHXaa11 G W L GWLaaaaaaHG LLLLLXW K HX1 11111

LL∈ + } group super matrix semivector space of a1 Q{0}∪

refined labels over the multiplicative group L + . Q

87. If L + is replaced by the semigroup L + under addition, Q Q{0}∪ V becomes the semigroup super matrix semivector space of

refined labels over L + ; semigroup under addition. Q{0}∪

Find HomL + (V,V) . Q{0}∪

a) Compare HomL + (V,V) in problem 86 with Q{0}∪

HomL + (V,V) . Q{0}∪

283 IYFVL LLGWa1 LLGWLa LLGW1 L LLGWa1 LL GWL LLGWa1 LLGW LLLa 88. Let V = JZGW1 LL∈ be a super matrix (column GWL aR1 LLa1 LLGW L LLGWa1 LLGWL LLGWa1 LLGWL LLGWa1 LLGWL K[HXa1

vector) vector space of refined labels over the field LR. a) Find Hom (V,V) . LR b) L(V,L). LRR

89. Obtain some interesting properties enjoyed by Hom (V,V) , V is a super row vector, vector space of LQ

refined labels over LQ.

90. What are the applications of super square symmetric matrix vector space of refined labels over LR?

91. What are the applications of super skew symmetric matrix vector space of refined labels over LQ ?

92. If V is a diagonal super matrix collection of refined labels vector space over LR; can we speak of eigen values and eigen vectors of refined labels over LR?

93. Can the theorem of diagonalization be adopted for super square matrix of vector space of refined labels over LR?

94. Study super model of refined labels using the refined labels as attributes.

284

IYFVLLL LLGWaaa123 LLGWLLL aaa456 LLGW LLL ∈≤≤ 95. Let V = JZGWaaa789LL;1i15aR be a super LLi GWLLL LLGWaaa10 11 12 LLGWLLL K[LLHXaaa13 14 15

matrix vector space of refined labels over LQ. a) Is V finite dimensional? b) Does V have a subspace of finite dimension? Justify.

96. Obtain some interesting results related with hyperspace of super matrix vector space of refined labels.

97. Obtain some applications of super matrix semivector spaces

of refined labels over L + or L + . R{0}∪ Q{0}∪

IYFVLL LL LLGWaaaa12 3 4 LL LLGWLLLLaa aa 98. Prove L = JZGW58 910LL;1i16∈≤≤ is LLLL aQi LLGWaa6111213 a a LLGWLLLL K[LLHXaaaa7141516 a group of super matrix of refined labels under addition.

99. Prove a super matrix vector space of refined labels over LQ or LR is never a super matrix linear algebra of refined labels over LQ or LR.

100. Let V = {}FVLL...LLL;1i8∈≤≤ be a HXaa12 a 8 aQ i

super row matrix vector space of refined labels over LQ. V is of dimension 8. How many super row matrix vector spaces of refined labels of dimension 8 can be constructed over LQ?

285 101. If V = {m × n super matrix with entries from LQ} be a super matrix vector space of dimension mn over LQ. How many such spaces be constructed with m natural rows and n natural columns?

FVLLLL GWaaaa1234 GWLLLLaaaa GW58910 LLLL × 102. Let T = GWaaa6111213 a be 5 4 matrix of refined GWLLLL GWaaa7141516 a GWLLLL HXaaaa17 18 19 20 labels. In how many ways can T be partitioned to form super matrix of refined labels.

103. Does the partition of a matrix of refined labels affect the basis algebraic structure?

IYFV LLLLaaaa LLGW12310 ∈≤≤ 104. If V = JZGWLLLLaaaa LL;1i12 aR and LLGW45611i GWLLLL K[LLHXaaaa78912 IYFV LLaa LL aa LLGW12 3 4 ∈≤≤ W = JZGWLLLLLL;1i12aaa aaR be two LLGW5678i GWLL LL K[LLHXaa7101112 a a

super matrix vector space of refined labels over LR. Is V ≅ W?

105. Let IYFVLLLLL LLGWaaaa12345 a LLGWLLLLLaaaaa V = JZGW678910LL;1i20∈≤≤ LLLLL aRi LLGWaaaaa11 12 13 14 15 LL GWLLLLL K[LLHXaaaaa16 17 18 19 20 and

286

IYFV LLLLLL...Laaaa a WLL;1i20=∈≤≤JZGW1234 10 GWLLLL...L aRi K[LLHXaaaa11 12 13 14 a 20

be two super matrix vector spaces of refined labels over LR. a) Is V ≅ W? b) Is dim V = dim W? c) Is Hom (V,V) ≅ Hom (W,W)? LR LR

106. Let V = {}()LLLLLLL;1i5∈≤≤ aa12345i aa aaR IYCS LLLLLaaaaa LLDT12345 ∈≤≤ and W = JZDTLLLLLaaaaa LL;1i5 aR LLDT12345 i DTLLLLL K[LLEUaaaaa12345

be two super matrix vector space of refined labels over LR? a) Is V ≅ W? b) Is L(V,L) ≅ L(W,L)? LRR LRR c) Is Hom (V,V) ≅ Hom (W,W)? LR LR

IYFVLL 0L LLGWaa12 a 1 LLGW0Laa L 0 107. Let V = JZGW34 LL;1i4∈≤≤ and W L00L aRi LLGWaa21 LL GW0L L L K[LLHXaaa432 IYFVL0L0L0 LLGWaaa123 LLGW0Laaa 0L 0L LLGW412 L0L0L0 LLGWaaa341 ∈≤≤ = JZGWLL;1i4aR LLLL 0 0 i LLGWaaaa1324 LLGW0L 0 0L 0 LLGWaa41 LLGWL0LL0L K[HXaaaa2123

be two super matrix vector spaces of refined labels over LR?

287 a) Is dim V = dim W? b) Is V ≅ W? c) Find Hom (V,W) . LR d) Is Hom (V,V) ≅ Hom (W,W)? LR LR e) Is L(V,L) ≅ L(W,L)? LRR LRR f) Find a basis for V and a basis for W.

288

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292

Abelian group of super column vectors of refined labels, 112-4

Diagonal super matrix, 12-3 Direct sum of integer super matrix semivector subspaces of refined labels, 190-7 Direct sum of super semivector subspaces of refined labels, 168-173 Direct sum of super vector subspaces of refined labels, 143-9 Direct union or sum of super row vector subspaces, 104-5 DSm field of refined labels, 34-5 DSm field of refined labels, 32-3 DSm linear algebra of refined labels, 34-6 DSm semifield, 53-5

FLARL(DSm field and Linear Algebra of Refined Labels), 32-4

Group of super column vectors, 27-8 Group of super matrices, 28-30 Group of super row vectors, 25-6 Group super matrix semivector space of refined labels, 236-9

293 Group super matrix semivector subspace of refined labels, 237- 242 H

Height of a super matrix, 8-9

Integer super column semivector space of refined labels, 181-9

Linear functional of refined labels, 153-7 Linear transformation of set super matrix of semivector spaces of refined labels, 211-8 Linear transformation of super vector spaces, 110-3 Linearly independent super semivectors of refined labels, 173-8 Linearly dependent set, 106-7 Linearly independent set, 106-7

M-matrix of refined labels, 96-8

Non equidistant ordinary labels, 31-3

Ordinary labels, 31-3

P-matrix of refined labels, 98-100 Pseudo direct sum of integer super matrix semivector subspaces of refined labels, 190-8 Pseudo direct sum of super vector subspaces of refined labels, 144-8

294 Pseudo semigroup super matrix semivector subspace of refined labels, 243-8. Pseudo simple semigroup super matrix semivector space of refined labels, 230-5 Pseudo simple super column vector subspaces of refined labels, 125-7

Qualitative labels, 31-3

Refined labels, 31-3

Semifield of refined labels, 166-9 Semigroup of super column vectors, 52-4 Semigroup super matrix semivector space of refined labels, 220- 229 Semigroup super matrix semivector subspace of refined labels, 222-9 Semivector space of column super vectors of refined labels, 174-9 Semivector spaces of super matrices of refined labels, 180-5 Set super matrices semivector space of refined labels, 200-8 Set super matrix semivector subspace of refined labels, 201-8 Simple matrix, 9-10 Subgroup super matrix semivector subspace of refined labels, 240-6 Submatrices, 8-9 Subsemigroup super matrix semivector subspace of refined labels, 230-5 Subset super matrix semivector subspace of refined labels, 204-8 Super column vectors of refined labels, 39-42 Super column vectors, 9-11 Super hyper subspace of refined labels, 159-163

295 Super matrix, 7-10 Super row matrix of refined labels, 37-8 Super row vector space of refined labels, 101 Super row vector subspace of refined labels, 103-5 Super row vectors of refined labels of same type, 37-9 Super row vectors, 9-10 Super semivector space of refined labels, 165-8 Super semivector subspaces of refined labels, 169-173 Super vector space of m × n matrices of refined labels, 134-6 Super vector subspaces of m × n matrices of refined labels, 140-3

Transpose of a super matrix 17-18

Upper partial triangular super matrix, 14-15

Z-matrix of refined labels, 96-98

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297