interval linear algebra, by w. b. vasantha kandasamy, florentin smarandache

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INTERVAL LINEA

ALGEBRA

W. B. Vasantha KandasamyFlorentin Smarandache



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ISBN-13: 978-1-59973-126-1



CONTENTS

Dedication

Preface

Chapter OneINTRODUCTION

Chapter Two

SET INTERVAL LINEAR ALGEBRAS OF

TYPE I AND THEIR GENERALIZATIONS

2.1 Set Interval Linear Algebra of Type I

2.2 Semigroup Interval Vector Spaces



Chapter FourSET INTERVAL BIVECTOR SPACESAND THEIR GENERALIZATION

4.1 Set Interval Bivector spaces and Their Properties4.2 Semigroup Interval Bilinear Algebras

and Their Properties 4.3 Group Interval Bilinear Algebras and their Prope

4.4 Bisemigroup Interval Bilinear Algebras

and their properties

Chapter FiveAPPLICATION OF THE SPECIAL CLASSES OFINTERVAL LINEAR ALGEBRAS

Chapter Six

SUGGESTED PROBLEMS

FURTHER READING

INDEX

ABOUT THE AUTHORS



~ DEDICATED TO ~

Dr C.N Deivanayagam



This book is dedicated to Dr C.N Deivanayagam

Health India Foundation for his unostentatious

all patients, especially those who are econom

impoverished and socially marginalized. He wasin serving people living with HIV/AIDS at the G

Hospital of Thoracic Medicine (Tambaram Ch

When the first author of this book had an oppo

interacting with the patients, she learnt of his

service. His innovative practice of combining tr

Siddha Medicine alongside Allopathic remedie

advocacy of ancient systems co-existing with

health care distinguishes him. This dedication

token of appreciation for his humanitarian s



PREFACE

This Interval arithmetic or interval mathematic

1950’s and 1960’s by mathematicians as an appr

bounds on rounding errors and measurmathematical computations. However no palgebraic structures have been defined or studie

we for the first time introduce several types of

algebras and study them.

This structure has become indispensable forwill find applications in numerical optimization

of structural designs.

In this book we use only special types o



feature of this book is the authors have given

examples.

This book has six chapters. Chapter one is intr

nature. Chapter two introduces the notion of set int

algebras of type one and two. Set fuzzy interval line

and their algebras and their properties are discussed

three.

Chapter four introduces several types of inte

bialgebras and bivector spaces and studies them. T

applications are given in chapter five. Chapter s

nearly 110 problems of all levels.

The authors deeply acknowledge Dr. Kandasa

proof reading and Meena and Kama for the for

designing of the book.

W.B.VASANTHA

FLORENTIN SM



Chapter One

INTRODUCTION

In this chapter we just define some basic proper

used in this book. Throughout this book [a,

interval a d b. If a = b we say the interval degenea. We assume the intervals [a, b] is such that 0 dgive the notations.

N t ti L t



However from the context one can easily follow

set the intervals are taken.While working we further refrain and use main

of the form [0, a] where a Zn or Z+ {0} or Q+ {0}. We add intervals as [[a, b] + [c, d] = [ac, bd]

In case of [0, a] type of intervals [0, a] + [0, b]

and [0, a]. [0, b] =[0, ab] for a, b in Zn or Z+{0} or

use only interval of the form [a, b] where a < bcollection of intervals we do not accept the degenera

except 0. When we say A = (aij) is an interval matrix

aij are intervals.

For example

[0,5] [0,3]

[0,1] [0,4]

[0,2] [0,7]

ª º

« »« »« »¬ ¼

is a 3 u 2 interval matrix.

For more about these concepts please refer [52].



Chapter Two

SET INTERVAL LINEAR ALGEBRA

TYPE I AND THEIR GENERALIZA

In this chapter we for the first time introduce the

set interval linear algebras of type I and their f

This chapter has two sections.



DEFINITION 2.1.1: Let S denote a collection of inte

form {[xi , yi ]; yi , xi Z; 1 d i d n} (This set S need nunder any operation just an arbitrary collection of

Let F be a subset of Z + {0} If for every c F and s

S, we have cs = [cxi , cyi ] S; then we define S interval integer vector space over the subset F. If the

distinct elements in S is finite we call S to be a finite

integer vector space; if |S| = f we say S is an interval vector space of infinite order.

We will illustrate this situation by some example

Example 2.1.1: Let S = {[2n, 2m], n < m | m, n

take F = {2, 4, 8, …, 212} Z. S is a set integer intespace of infinite order over the set F.

Example 2.1.2: Let S = {[1, 2], [0, 0], [4, 7], [–2,

[–45, 37] [3, 7], [147, 2011]} ZI be a subset

intervals. Take F = {0, 1} Z. We see S is a set inte

vector space over the set F. Clearly S is of finite card

o(S) = |S| = eight.

Now having seen the structure of set integer int

space of finite and infinite dimension we now pro

define set rational interval vector space.

DEFINITION 2.1.2: Let S Q I or I Q be a subset of

Q I or I Q . Let F Z + {0} or Q+ {0} be a subset o

( di S i f Q Q )



are defined is assumed to be subsets of Z+ {0}

R +

{0} that is F Z+

{0} (or Q+

{0} or R

We shall illustrate this situation by some exampl

Example 2.1.3: Let

S =1 1

, 3 nn n 2

- ½ª º d d f® ¾« »¬ ¼¯ ¿

I

Q

be a subset of intervals. Take F = {0, 1} Q. C

rational interval vector space over the set

cardinality.

Example 2.1.4: Let S =

7 5 17 22 121,9 , ,4 , ,19 , ,40 ,[0,0], ,14

2 3 5 7 2

- ª º ª º ª º ª º ª ® « » « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¯

I

Q be an interval subset of I

Q .

It is easily verified S is a set rational vector spac

= {0, 1} and the cardinality of S is seven.

DEFINITION 2.1.3: Let S I R (or R I ) be the su

of reals. Let F Z + or Q+ or R+ (Z or Q or R).and c F, sc and cs is in S then we define S interval vector space over F. If the number of e

finite we say S is of finite order otherwise S is of



Example 2.1.6 : Let

S = n n, 1 n7 2

- ½ª º° °d d f® ¾« »¬ ¼° °¯ ¿

IR

be a subset of intervals. Take F = Z+ R +. Clea

infinite set real vector space over F.

Example 2.1.7 : Let S = {[0, 0], [0, 1], [ 2 , 7 ][ 13, 43 ], [5, 8], [ 17 , 41]} R I subset of re

Take F = {0, 1} R. We see S is a set real interval v

over the set F. S is of finite dimension or cardinal

number of elements in S is 7.

Now we will define the concept of set modulo inte

vector spaces.

DEFINITION 2.1.4: Let S = {[x, y] / x, y Z n , x < y}

subset of intervals from the modulo integers. Take F

proper subset of Z n. If for every c F and all s = [x,(mod n), cy (mod n)] and [xc (mod n), yc (mod n)]

say S is a set modulo integer interval vector space ov

{0, 1} Z n (n < f ) any other subset S 1 Z n provided if x < y implies sx < sy s S 1 and [x, y

We will illustrate this situation by some examples.

Example 2.1.8: Let S = {[0, 0], [0, 1], [0, 2], [1, 1I

3Z be the subset of intervals of Z3. Take F = Z3



Thus it is convenient to use these structures wh

just finite.

Now we proceed onto define the notion

interval vector spaces.

DEFINITION 2.1.5: Let S C I subset of interv

numbers. Take F to be a subset of Z

+

{0} or R{0}. If for be the every c in F and for every s = [

S then we call S to be a set complex intervover the set F.


Example 2.1.10: Let S = {[2i, 4i + 2], [7, 3i + 13

1, 27i + 4]} CI be a subset of intervals from {0, 1}; we see S is a set complex interval v

cardinality four over the set F = {0, 1}.

Example 2.1.11: Let S = {[ni, ni + n] | n Z+

} of intervals from CI. Choose F = {1, 2, …, 24}

infinite set complex interval vector space over F.

We now proceed onto describe substrucalgebraic structures.

DEFINITION 2.1.6: Let S I Z (Z I ) be a set

vector space over the set F Z + we say a

interval subset P S to be a set integer



P = {[0, 0], [41, 53], [–5, 17], [3, 9]} S, P is a

interval vector subspace of S over the set F.

Example 2.1.13: Let S = {[0, 0], [(2m)n, (2m)n+1];

f} IZ ; S is a set integer interval vector space ove

{0, 2, 22, …, 240} Z+. Choose P = {[0, 0], [(4m)n,

d n, m d f} S; P is a set integer interval vector su

over F.

Now we can as in case of set integer interval ve

define for set real (complex, rational, modulo integ

interval vector spaces (complex, rational, modu

interval vector subspaces with appropriate simple cha

We shall however illustrate this situation

examples.

Example 2.1.14 : Let S = {[0, 0], [2, 2], [1, 1], [0, 1

3], [0, 3]} I

4Z be a set modulo integer interval v

built using Z4. Take F = {0, 1, 2, 3} Z4. We semodulo integer interval vector space over F.

Take P = {[0, 0], [1, 1], [2, 2], [3, 3], [0, 2]} S

modulo integer interval vector subspace of S over F.

Example 2.1.15: Let S = {[0, 0], [1, 2 ], [1, 3 ],

[ 17, 23 ]} R I be a set real interval vector space

F = {0, 1}. Choose P = {[0, 0], [1, 3 ], [ 2 , 3 ]}



P = {[0, 0],5 7

,2 2

ª º

« »¬ ¼,

9 11,

2 2

ª º

« »¬ ¼,

23 25,

2 2

ª º

« »¬ ¼,

35 37,

2 2

ª

« ¬ S; it is easily verified P is a set rational

subspace of S over the set F = {0, 1}.

Example 2.1.17 : Let S = {[n 2 , n 23 ], [0, 0

be a set real interval vector space over the sChoose P ={[3n 2 , 3n 23 ], [0, 0]} S;

interval vector subspace of S over the set F = {0,

Example 2.1.18: Let S = {[mi, (m + 3) + (m + 3

Z+} CI be a set complex interval vector space

{0, 1}. Choose P = {[5mi, [5(m + 3) + 5(m + 3)i CI. P is a set complex interval vector subspace

Now we call a set integer (real or complex or rat

integer) interval vector space S to be a simple set

complex or rational or modulo integer) interval v

has no proper set integer (real or complex or rati

integer) interval vector subspace P; where P z [0

We will illustrate by some simple examples t

integer (real or complex or rational or comp

integer) simple vector space.

Example 2.1.19: Let S = {[0, 0], [5, 7]} be a set

vector space over the set F = {0, 1}. We see S

integer interval vector space over F.



Example 2.1.21: Let S = {[0, 0], [ 5 7,2 2

]} QI b

interval vector space over the set F = {0, 1}. S is arational interval vector space over F = {0, 1}.

Example 2.1.22 : Let S = {[0, 0], [1, 3 + i]} be a s

interval vector space over the set F = {0, 1}. S is a

complex interval vector space over F = {0, 1}.

Example 2.1.23 : Let S = {[0, 0], [ 7, 3 40 ]} b

interval vector space over the set F = {0, 1}. Cle

simple set real interval vector space over F.

We now proceed onto define the new notion of sub(real or complex or rational or modulo integer) inte

subspace defined over a subset T F of a set intecomplex or rational or modulo integer) interval v

defined over F.

DEFINITION 2.1.7: Let S Z I be a set integer inte space defined over the set F Z + {0}. Suppose

proper subset of S, P z [0, 0] or P z S) is a set integ

vector space over the subset T F (T z (0) or T z P1) then we define P to be a subset integer inte

subspace of S over the subset T of F. Similar defini

made in case of set real or complex or rational integer interval vector spaces with suitable modificat

However we will illustrate this situation by some exa



[0, 1], [0, 2], [0, 3], [0, 4]} S, P is a subset

interval vector subspace of S over the subset T =

We now proceed onto define the new notion of

set integer (real or rational or complex modulo


DEFINITION 2.1.8: Let S Z I (or Q I or I

n Z or R

integer (rational or modulo integer or real

complex) interval vector space over the subset Suppose S has no proper subset integer (ratiointeger or real) interval vector subspace over a pof F then we define S to be a pseudo simple set i

or modulo integer or real) interval vector space

If S is both a simple set interval vector space as

simple set interval vector space over F then we

doubly simple set interval integer (real or ratio

integer) vector space.

We will give some illustrations before we proc

some properties.

Example 2.1.26 : Let S = {[0, 0], [0, 1], [0, 2],

IZ be a set integer interval vector space over th

Clearly S is a pseudo simple set integer interv

over F. However S is not a simple integer interv

as S has several set integer interval vector subs

{0, 1}. Thus S is not a doubly simple set integer



Proof: The result follows from the fact that the set

has no proper subset T of order greater than or eqThus we cannot have any subset integer (rational

integer or real or complex) vector space over F = {0

the theorem.

THEOREM 2.1.2: Let S = {[0, 0], [x, y]} Z I {or QIor C I ) be a set integer (rational or modulo integercomplex) interval vector space over the set F = {0, 1

a doubly simple set integer (rational or modulo inteor complex) interval vector space over the set F = {0

Proof: Obvious from the very definition and the carS and F. S is a doubly simple set integer (rational

integer or real or complex) interval vector space over

Now we will give an example of a doubly simple

integer vector space.

Example 2.1.27 : Let S = {[0, 0], [ 7,3 19 ]} R I b

interval vector space over the set F = {0, 1}. Cle

doubly simple set real interval vector space over F.

We now proceed onto define the notion of set inte

space interval linear transformations.

DEFINITION 2.1.9: Let S and T be any two set integ

or modulo integer or real or complex) interval ve



Example 2.1.28: Let S = {[0, 0], [0, 2], …, [0, 45

0], [1, 2], [1, 3], …, [1, 45]} be two set integer

spaces defined over the set F = {0, 1}.

Define TI : S o T by

TI {[0, 0]} = {[0, 0]}

TI {[0, n]} = {[1, n]} 2 d

TI is an interval linear transformation of S to T.

Example 2.1.29: Let

S = {[0, 0], [ 2, 7 ], [ 7, 11 ], [ 11, 43 ],

and T = {[0, 0], [7, 9], [3, 11], [24, 45], [10, 29]}

interval vector spaces defined over the set F = {0

Define TI ([0, 0]) = [0, 0].TI ([ 2, 7 ]) = [7, 9]

TI ([ 7, 11 ]) = [3, 11]

TI ([ 11, 43 ]) = [3, 11] and

TI ([ 43,20 45 ]) = [24, 45].

TI is an interval linear transformation of S to

It is important to mention here that S and T can

set interval vector space built using integers or r

or so on but only criteria we need is that both sh

over the same set F. This is evident from the foll

As we do not demand any thing from the se

TI (cs) = cTI (s) for every c F and s S.As in case of usual vector spaces we say a

transformation is an interval linear operator if



spaces defined over the set F = {4, 42, 43, 44, 45}. D

o T by TI [2

n

, 2

n+4

] = [4

n

, 4

n+4

], n = 1, 2, …, f.It is easily verified that TI is a interval linear tran

of S to T.

We see the notion of kernel TI has no meaning as

Now we proceed onto give one example of a lin

operator (interval linear operator) on a set interval ve

Example 2.1.31: Let S = {[3n, 3n+3] | n = 1, 2, …,

integer interval vector space over the set F = {0, 1}.

V o V by TI [3n, 3n+3] o [32n, 32n+3], n = 1, 2, …, f.

It is easily verified that TI is a interval linear ope

Further TI has kernel.

Next we proceed onto define set interval linear al

using integer intervals, real intervals and so on.

DEFINITION 2.1.10: Let S 1 , S 2 , …, S k be a collectio

integer (real, complex, rational or modulo integevector subspaces of S defined over the subsets T 1 , respectively (that is each S i is a subset interval vecto

of S over the subset T i of F; i=1, 2, …, k). If W = S

= T i z I then we call W to be a sectional subset int sectional subspace of S over T.

We will illustrate this situation by an example.

Example 2.1.32: Let S = {[0, 2n], [0, 6n], [0, 5n], [

14n] / n = 0 1 2 f} be a set integer interval v



We have the following interesting theorem the p

left as an exercise for the reader.

THEOREM 2.1.3: Every sectional subset invector subspace W of the set interval vector spac

F is a subset interval vector subspace of a suconversely.

We can as in case of set vector spaces defineinterval set of a set interval vector space.

DEFINITION 2.1.11: Let S be a set interval ve

using interval integers or reals or rationals modulo integers over the set F. We say a subset

S generates S if every interval s of S can be got a

s j z csi and si z cs j for si z s j; si , s j B and c F generating interval set of S over F.

We will illustrate this by some simple examples.

Example 2.1.33: Let S = {[0, 2n], [0, 3n], [0, 5n]

1, 2, …, f} be a set integer interval vector spac

= {0, 1, 2, …, f}.

Take B = {[0, 2], [0, 3], [0, 5], [0, 7]}generating interval subset of S over F.

Example 2.1.34 : Let S = {[2n, 3n], [5n, 7n], [1

29n], [12n, 31n] | n = 0, 1, 2, …, f} be a set

vector space over the set F = Z+ {0}. Take B =

[11 13] [15 29] [12 31]} S B i th i t



D is not a linearly independent interval set as [8, 12

and [10, 14] = 2 [5, 7] for 4, 2 Z+ {0}.

It is left as an exercise for the reader to prove th

theorem.

THEOREM 2.1.4: Let S be a set interval vector spa

set F. Let B S be a generating interval set of S ovis a linearly independent interval set of S over F. Fuand V be any three set interval vector spaces over theand V may be integer interval or real interval

interval or rational interval or modulo integer intthat if T I and M I be interval linear transformations w

T I : S o P and M I : P o V.Then

T I o M I : S o V.

That is (T I o M I ) (s) (for s S)= M I (T I (s))

= M I (p) (p P)= v; v V;

is a interval linear transformation for S to V.

We can define invertible interval linear transform

where T I : S o P then 1 I T : P o S and derive related

It is pertinent to mention that we cannot define finterval vector spaces set interval linear functional;

over which S is defined is not an interval set.



S to be a set integer (real or complex or ratio

integer) interval linear algebra over F.

We will first illustrate this situation by some exam

Example 2.1.35: Let S = {[0, 2n] | n = 0, 1, 2, …

set interval linear algebra over the set F = {0,

closed under interval addition. For if x = [0, 2n] are in S then

x + y = [0, 2n] + [0, 2m]

= [0 + 0, 2n + 2m]

= [0, 2 (n + m)] S.

Example 2.1.36 : Let S = {[0, 5 n] | n = {0, 1

IR be a set interval linear algebra over the set F

f}.

Example 2.1.37 : Let S = {[5n, 9n] | n = 0, 1, 2,

interval linear algebra over the set F = {0, 1}. Fand y = [20, 36] then x + y = [5, 9] + [20, 36] =

9.5] S.

Now having seen examples of set interval

defined using real intervals or integer interv

intervals or modulo integer intervals or complexwe proceed on to define set real (or complex

rational or modulo integer) interval linear subalg

DEFINITION 2 1 13: Let S be a set interval linea



Example 2.1.38: Let S = {[n(1 + i), n (20 + 20i)] | n

set complex interval linear algebra over the set F =

f}. Take P = {[4n(1 + i), 4n(20 + 20i)] | n Z+} complex interval linear subalgebra of S over F.

Example 2.1.39: Let S = {[21n, 43n] | n = 0, 1, 2,

set interval linear algebra over the set F = Z+. Let P =

43 u 5n] | n = 0, 1, 2, …, f } S. P is a set intsubalgebra of S over the set F = Z+.

We illustrate this situation by some examples.

Example 2.1.40: Let S = {[0, 7 n] /n = 0, 1, 2, …,

real interval linear algebra over the set F = {0, 1}. Ta7 u 5n ] / n = 0, 1, 2, …, f} S; P is a set real int

subalgebra of S over F.

Example 2.1.41: Let S = {[n (2 + 3i), n (12 + 17i)] |

…., f} be a set complex interval linear algebra over

{0, 1}. Choose P = {[6n(2 + 3i), 6n(12 + 17i)] | n =

f} S, P is a set complex interval linear subalgebr

F.

Now we proceed onto define subset interval linear

built using integer intervals or complex intervintervals or rational intervals or modulo integer interv

DEFINITION 2.1.14: Let S be a set integer (real or rational or modulo integer) interval linear algebra



Example 2.1.42: Let S = {[0, (3 + 17 )n] be su

2, …, f} be a set real interval linear algebra ove1, 2, …, n = f}. Choose P = {[0, (3 + 17 )n] | n

…., f; that is n is even} S; P is a subset rea

subalgebra of S over the subset T = {4n | n = 0, 1

Example 2.1.43: Let S = {[0, 0], [0, 1], [0, 2], [

5]} be a set modulo integer linear algebra over th

2, 3, 4, 5}. Choose P = {[0, 0], [0, 2], [0, 4]} modulo integer Z6 interval linear subalgebra of S

T = {0, 2, 4} F.

Now if we have a set interval linear algebra S

(real intervals or rational intervals or compl

modulo integer intervals) over the set F and if S

set interval linear subalgebra over F then we

simple set interval linear algebra over F. If S

interval linear subalgebra over any proper subset

define S to be pseudo a simple set interval linear

both a simple set interval linear algebra and a ps

interval linear algebra then we define S to be a

set interval linear algebra.


Example 2.1.44 : Let S = {[0, 0], [0, 1], [0, 2] [

Z5I be a set modulo integer 5 interval linear alge

F = {0, 1, 2, 3, 4} then S is a simple set mo

interval linear algebra. Infact S is also a pse



set modulo integer p interval linear algebra. S

simple set modulo p integer interval linear algebra.

The proof is left as an exercise for the reader.

Now as in case of set interval vector spaces we in

interval linear algebras define interval linear transfor

DEFINITION 2.1.15: Let S and M to two set integcomplex or rational or modulo integer) interval lineover a set F. Suppose T I is a map from S to M, T Iinterval linear transformation if the following condit

T I (cs + s1 ) = cT I (s) + T I (s1 ) for all intervals s, s1 in S and for all c in F.

It is important to mention that interval linear transf

defined if and only if both the set linear algebras

over the same set F. Further set linear interval tran

of set interval vector spaces are different from set int

algebras.

If in the definition 2.1.15, M is replaced by S t

the set interval linear transformation to be a set int

operator on S. As in case of set interval vector space

the notion of generating set linearly independent el

set linearly dependent elements.We see in case of set interval linear algebra S

subset of intervals B S is said to be a linearly i

interval subset if there is no s B such that s can be

s = ¦c s ; si B and ci F;



Example 2.1.45: Let S = {[0, n 2 ] | n = 0, 1, 2,a set real interval linear algebra over the set F =

= {[0, 2 ]} S, B is the generating interv

Consider {[0, 2 ], [0, 5 2 ]} = C S, C

dependent interval of S as [0, 5 2 ] = [0, 2 ]

2 ] + [0, 2 ] + [0, 2 ].We call the set interval linear algebra S ove

dimensional if B is a generating interval subset

the number of elements in B is finite; otherwiseinfinite dimensional set interval linear algebra

dimension of S given in example 2.1.45 is finite

Interested reader can construct and study dimension of set interval linear algebras.

Now having seen only class of set interval lin

now proceed onto define another new class of

algebras.

2.2 Semigroup Interval Vector Spaces

In this section we proceed on to define a new cla

interval vector spaces and discuss a few of t

However every semigroup interval vector space vector space and not vice versa.

DEFINITION 2.2.1: Let S be a subset of intervaIZ Q C F b dditi i



Example 2.2.1: Let S = {[0, 2n] | n = 0, 1, 2, …

semigroup interval vector space over the semigroup{0} under addition.

Example 2.2.2: Let S = {[(1 + i)n, n (2 + 2i)n] | n =

f} be a semigroup interval vector space over the sem

3Z+ {0} under addition.

Example 2.2.3: Let S = {[0, 0], [0, 2], [0, 4], [0, 8]

10], [0, 12], [0, 14], [0, 16], [0, 18]} be a semigro

vector space over the semigroup F = Z20 (semig

addition modulo 20).

Example 2.2.4 : Let S = {[0, 0], [0, 3]} I

9Z be ainterval vector space over the semigroup F = {0, 3,

modulo 9.

Now we proceed on to define semigroup int

subspace of S.

DEFINITION 2.2.2: Let S be a semigroup interval v

over the semigroup F. Suppose I z P S (P z S a prS) is a semigroup interval vector space over the sethen we define P to be a semigroup interval vector sS over the semigroup F.


Example 2.2.5: Let S = {[0, 0], [0, 1], [0, 2] [0, 3], [0

[0 6] [0 ] [0 8] [0 9] [0 10] [0 11]} b



{0} under addition. Take P = {[0, 4n 17 ] such

…, f} S; P is a semigroup interval vector subthe semigroup F.

If a semigroup interval vector space S over the sno proper semigroup interval vector subspace ov

P = {[0, 0]}, then we call S to be a simple sem

vector space over F.We will illustrate this situation by some example

Example 2.2.7 : Let S = {[0, 0], [0, 1], [0, 2], [0

the semigroup interval vector space over the sem

under addition modulo 5. S is a simple sem

vector space over F.

Example 2.2.8: Let S = {[0, 0], [0, n] / n = 1, 2,

be a semigroup interval vector space over the sem

under addition modulo 23. S is a simple semvector space over Z23.

In view of these examples we have the follwhich guarantees the existence of a class of sim


THEOREM 2.2.1: Let S = {[0, n] / n = 0, 1, 2, …

a prime. F = Z p a semigroup under addition m simple semigroup interval vector space over F.

Proof: Follows from the fact that no proper inte

S (P z [0 0] or P z S) can be a semigroup interv



In view of this we have the following theorem.

THEOREM 2.2.2: Let Z m = {0, 1, 2, …, m – 1}; m

where p1 , …, pt are t distinct primes and D i t 1, 1 d

set of integers modulo m. S = {[0, n] | n Z m }

semigroup interval vector space over the semigro

Infact S is not a simple semigroup interval vector sp= {[0, npi ] / pi a prime such that i

i pD

/ m and 1

/i

i pD

t, n, pi Z m } S are semigroup interval vector subover F = Z m.

The proof is straight forward and left as an exerc

reader.

We will illustrate the above theorem by some examp

Example 2.2.10: Let Z30 = {0, 1, 2, …, 29} be

integer 30 and 30 = 2.3.5.

S = {[0, n] | n Z30} be a semigroup interval vover the semigroup F = Z30. Take P1 = {[0, 0] [0, 2

6], …, [0, 28]} = {[0, 2n] | 2, n Z30} S.

It is easily verified P1 is a semigroup inte

subspace of S over F.

Take P2 = {[0, 3n] | 3, n Z30} S, P2 is ainterval vector subspace of S over F.

P3 = {[0, 5n] | 5, n Z30} S; P3 is a semigrovector subspace of S.



= Z36. P3 = {[0, 3n] / n Z36} S is a semigroup

subspace of S over F = Z36.

P4 = {[0, 0], [0, 9], [0, 18], [0, 27]} S

interval vector subspace of S over F = Z36.

Now we will proceed onto define the notiolinearly independent linearly dependent interv

semigroup interval vector space.

DEFINITION 2.2.3: Let S be a semigroup intervover the semigroup F. A set of interval elements

sn } of S is a said to be a semigroup linearly indep

subset if si z cs j; for all c F and si , s j B; i z j;

If for some si = cs j , c F; iz j; si , s j B t semigroup interval subset is linearly dependentindependent.

If B is a semigroup linearly independent inteand B generates S, the semigroup interval vecto

that is if every element s B can be got as s = c

S; 1 d i d n.

We will illustrate this by some examples.

Example 2.2.12: Let S = {[0, n] | n = 0, 1,

semigroup interval vector space over the semig

{0}. Take B = {[0, 1]} S, B generates S interval vector space over F.

Example 2.2.13: Let S = {[0, n] | n Z12} b



Take B = {[0, 2], [0, 0], [0, 4], [0, 8]} S; B i

dependent interval subset of S over F.

Example 2.2.15: Let S = {[0, n] | n = 1, 2, …

semigroup interval vector space over the semigroup F

…, 10} Z12, semigroup under addition modulo 12

{[0, 1], [0, 3], [0, 5], [0, 7]} S, B is a linearly i

interval subset of S over F but is not a generating int

of S over F.

We will now proceed onto define the notion of interval linear algebra.

DEFINITION 2.2.4: Let S be a semigroup interval vover the semigroup F. If S is also an interval semigaddition then we define S to be semigroup inte

algebra over the semigroup F if c (s1 + s2 ) = cs1 + c

and c1 , c2 F.


Example 2.2.16 : Let S = {[0, n] | n Z+ {0}} be a

interval linear algebra over the semigroup Z+ {0}

= {[0, 5n] | n Z+ {0}} S; P is a semigroup int

subalgebra of S over the semigroup F = Z+ {0}.

Example 2.2.17 : Let S = {[na, (n + 5) a] | a Q+

{0}} be a semigroup interval linear algebra over F =

Take P {[na, (n + 5)a | a, n Z+ {0}} S; P is a



We will give some examples of simple se

algebras.

Example 2.2.19: Let S = {[0, n] | n Z7} I

7Z

interval linear algebra over the semigroup F = Z

simple semigroup interval linear algebra over F.

Example 2.2.20: Let S = {[0, n]/ n Z p, p any pa semigroup interval linear algebra over the set F

It is easily verified S is a simple semigroupalgebra over the set F.

Now we define new concepts of substructuralgebraic structures.

DEFINITION 2.2.5: Let S be a semigroup interva

over the semigroup F. If P S (P = {0} or P z

subsemigroup of S. If T be a proper subsemigrou

a semigroup interval linear algebra over the sewe call P to be a subsemigroup interval linear sover the subsemigroup T of F.

If S has no subsemigroup interval linear subaldefine S to be a pseudo simple semigroup

algebra over F.

We will first illustrate this situation by

examples.



Now we define a semigroup interval linear algeb

both simple and pseudo simple as a doubly simpleinterval linear algebra over F.

We will illustrate this situation by some simple e

Example 2.2.23: Let S = {[0, n] | n Z5} I

5

Z be a

interval linear algebra over the semigroup F = Z5. S

simple semigroup interval linear algebra of over F.

Example 2.2.24 : Let S = {[0, n] | n Z11} I

11Z be a

interval linear algebra over the semigroup F = Z11. S

simple semigroup interval linear algebra over the sem

In view of this we give a class of semigroup int

algebras which are doubly simple semigroup inte

algebras.

THEOREM 2.2.3: Let S = {[0, n] | n Z p , p a prime}

semigroup interval linear algebra over the semigroudoubly simple semigroup interval linear algebra over

The proof is left as an exercise to the reader.

THEOREM 2.2.4: Let S = {[0, n]| n Z + {0}}

semigroup interval linear algebra over the semigrou{0}. S has both subsemigroup interval linear subalsemigroup interval linear subalgebras



DEFINITION 2.2.6: Let R and S be two semigroupalgebras defined over the same semigroup F

mapping from R to S such that T(cD + E ) = cT( D

c F and D , E R, then we define T to be a semlinear transformation from R to S.

If R = S we define T to be a semigroup operator on R.

We will illustrate this by some simple examples

Example 2.2.25: Let R = {[0, n] | n Z+ {0}}

/ n Q+ {0}} be two semigroup interval line

the semigroup F = Z+ {0}. The map T: R o

T([0, n]) = [0, n], n Z

+

{0} is a semigrouptransformation.

Example 2.2.26 : Let R = {[n, 5n] | n Z+ {0

5n] | n R + {0}} be two semigroup interval

defined over the semigroup F = Z+ {0}. Defin

{[n, 5n]} = [n, 5n], for all [n, 5n] R.

It is easily verified T is a semigroup

transformation of R to S and infact T is an embed

We will give an example of a semigroup

operator.

Example 2.2.27 : Let S = {[n, 2n] | n Z+

semigroup interval linear algebra on the semig



linear subalgebra of S over the semigroup F. Let T fbe a semigroup interval linear operator over F. T is

semigroup interval linear projection on P if T(v) =

and T( D u + v) = D T(u) + T( Q ), T(u) and T(v) P fo

and u, Q S.


Example 2.2.28: Let S = {[n, 5n]| n Q+ {0}} be a

interval linear algebra over the semigroup F = Z+ = {[n, 5n] | n Z+ {0}} S; P is a semigroup int

algebra over F.

Define T: S o S by

T ([n, 5n]) = [n,5n] if n Z

[0,0] if n Z

- °®

°̄We see T is a semigroup interval linear projection.

Example 2.2.29: Let S = {[0, n] | n Z30} be a

interval linear algebra over the semigroup F = Z30. Tan] | n {0, 5, 10, 15, 20, 25} Z30} S. P is a

interval linear subalgebra of S over F.

Define K: S o S by K{[0, n]} = [0, 5n]; Ksemigroup interval projection of S on P.

Now we proceed on to define the notion of pseudointerval linear operator on V.

DEFINITION 2.2.8: Let S be a semigroup interval lin

h F L P S b b



DEFINITION 2.2.9: Let S be a semigroup interv

over the semigroup F. Let W 1 , W 2 , …, W n be semvector subspaces of S over F.

If S = * i

i

W and W i W j = I or {0}, if i z j

the direct union of the semigroup interval vectothe semigroup interval vector space S over F.


Example 2.2.30: Let S ={[0, n]| n Z4} be a inte

linear algebra over the semigroup F = Z4. S cann

a union of semigroup interval sublinear algebras

Example 2.2.31: Let S ={[0, n]| n Z6} be a inte

vector space over F = {0, 3}. Take W1 = {[0, n

5} Z6} and W2 = {[0, n] | n {0, 2, 4} Z6};

interval semigroup vector subspace of V ove

Clearly V = W1

W2

and W1

W2

= {0}. Thuunion of semigroup interval vector subspaces of

Example 2.2.32: Let G = {[0, n]| n Z10}

semigroup vector space over the semigroup S =

= {[0, n] | n {0, 2, 4, 6, 8}} G and W2 = {[0

3, 5, 7, 9}} G be interval semigroup vector over the semigroup S = {0, 5}. Clearly V = W1 W2 = {0}. Thus V is a direct sum of the inte

vector subspaces W1 and W2.



Example 2.2.33: Let

V =1 2

i

3 4

5 6

[0, a ] [0, a ]a Z {0}

[0, a ] [0, a ]1 i 6

[0, a ] [0, a ]

- ½ª º ° °« »® ¾« » d d° °« »¬ ¼¯ ¿

be an interval semigroup linear algebra over F = 3Z+

Let

W1 =

1 2

1 2

[0 a ] [0 a ]

0 0 a ,a Z {0}

0 0

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

W2 = 1 1 2

2

0 0

[0 a ] 0 a ,a Z {0}

[0 a ] 0

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

and

W3 = 1 1 2

2

0 0

0 [0 a ] a ,a Z {0}

0 [0 a ]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

be interval semigroup linear subalgebras of V over

{0}. Clearly V = W1 + W2 + W3 and Wi W j = (0);

d 3. Thus V is the direct sum of interval linearsubalgebras.

Example 2 2 34: Let



W1 =1

1 8

a 0a Z

0 0

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

,

W2 =2

2 8

0 aa Z

0 0

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿,

W3 = 3 8

3

0 0a Z

a 0

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿and

W4 = 4 8

4

0 0a Z

0 a

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

be semigroup interval linear subalgebras of V o

see V = W1 + W2 + W3 + W4 and Wi W j = (

Thus V is a direct sum of semigroup interval line

Now we proceed on to define Group interval line

DEFINITION 2.2.11: Let V be a set of intervals wis non empty. Let G be a group under addition. a group interval vector space over G if the follo

are true;

(a) For every Q V and g V gv and vg are

(b) 0Q =0 for every Q V and 0 is the additiv




Example 2.2.37 : Let

V =

1

2

1 2

1 2 3

3 4

4

5

[0, a ]

[0, a ][0, a ] [0, a ]

, [0, a ],[0, a ] , [0, a ][0, a ] [0, a ]

[0, a ]

[0, a ]

- ª º° « »° « »ª º° « »® « »

« »¬ ¼°

« »° « »° ¬ ¼¯

be a group interval vector space over the group Z90 =

addition modulo 90.

Example 2.2.38: Let

V = 1 i

1 2 3

2

[0, a ] a Z, [0, a ],[0, a ][0, a ]

[0, a ] 1 i

- ª º° ® « » d d¬ ¼° ¯

be the group interval vector space over the group Z1

addition modulo 14.

Now we proceed on to define substructures of gro

vector spaces.

DEFINITION 2.2.12: Let V be a group interval vectorthe group G. Let P V be a proper subset of V andinterval vector space over G. We define P to be a grovector subspace over G.



be a group interval vector space over the group G

P = i 15

1

0 0a Z

[0, a ] 0

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿ V

P is a group interval vector subspace of V over

Z15.

Example 2.2.40: Let V = {[0, ai]| ai Z40} be a

vector space over the group G = Z40. Take P =

2, 4, 6, 8, 10, ..., 38} Z40} V; P is a group

subspace of V over G.

Example 2.2.41: Let

V =10

i 7i

i

i 0

a Z[0,a ]x

0 i 10

- ½® ¾

d d¯ ¿¦

be a group interval vector space over the additiv

Let

W =5

i

i i 7

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦ V

be a group interval vector subspace of V over G =

Example 2.2.42: Let

V =

> @ > @> @ > @

1 20,a 0,a

0 a 0 a a Z ;1 i

- ª º° « » d® « »



is a group interval vector subspace of V over the grou

DEFINITION 2.2.13: Let V be a group interval vector

a group G. We say a proper subset P of V to be

dependent subset of V if for any p1 , p2 P (p1 z p2 ) p

p2 = ac p1 for some a, ac G.

If for no distinct pair of elements p1 , p2 P we hG such that p1 = ap2 or p2 = a1 p1 then we say thelinearly independent set.

Example 2.2.43: Let

V => @

> @ > @

> @

> @

1 1

i 12

2 3 2

0,a 0 0,a 0, a Z ;1

0,a 0,a 0,a 0

- ª º ª º° d® « » « »

¬ ¼ ¬ ¼° ¯

be a group interval vector space over the group G = ZConsider

x =[0,1] 0

[0, 2] [0, 4]

ª º

« »¬ ¼, y =

[0, 3] 0

[0, 6] 0

ª º

« »¬ ¼

in V. Clearly x and y are linearly dependent as 3x =

= Z12.


V => @ > @ > @> @ > @ > @

1 2 3

i 15

4 5 6

0,a 0,a 0,aa Z ;1 i

0,a 0,a 0,a

- ª º° d ® « »

¬ ¼°̄



We see {x, y} forms a linearly dependent s

we see x = 4y where 4 Z15 = G.

Example 2.2.45: Let V be a group interval vect

group G. Let H be a proper subgroup of G. If

that W is a group interval vector space over theG then we define W to be a subgroup interval v

of V over the subgroup H of G.

If W happens to be both a group interval vec

well a subgroup interval vector subspace then we

duo subgroup interval vector subspace. If V h

interval vector subspace then we define V to be

interval vector space.

We will first illustrate this situation by

examples.


V = > @> @> @

1

2 i 24

3

0,a0,a a Z ;1 i 3

0,a

- ½ª º° °« » d d® ¾« »

° °« »¬ ¼¯ ¿

be a group interval vector space over the g

Consider

W =

> @> @

1

2 i 24

0,a

0,a a Z

0

- ½ª º° °« » ® ¾« »° °« »¬ ¼

V.



Example 2.2.47 : Let V = {([0, a1], [0, a2], [0, a3], [0,

[0, a6], [0, a7])| ai Z19, 1 d i d 7} be a group inte

space over the group G = Z19. It is easy to verify thsubgroup interval subspaces as G = Z19 has no

However V has several group interval vector sub

take W1 = {([0, a1], [0, a2], [0, a3], 0, 0, 0, 0) | ai Z1

V is a group interval vector subspace of V over th

W2 = {([0, a], [0, a], …, [0, a]) where a Z19} V

interval vector subspace of V over the group G. W3

…, 0 [0, a7])| a1, a7 Z19} V is a group inte

subspace of V.

Example 2.2.48: Let

V = 13

[0,a][0,a]

a Z[0,a]

[0,a]

[0,a]

- ½ª º° °« »° °« »° °« » ® ¾

« »° °« »° °« »° °¬ ¼¯ ¿

be a group interval vector space over the group G

easily verified V has no proper group interval vect

as well as subgroup interval vector subspace.

We cannot define the notion of pseudo semigro

vector subspace. However we can define the notion

set interval vector subspace of a group interval vecto

DEFINITION 2 2 14: Let V be a group interval vector



[0, 7], [0, 14], [0, 21], [0, 28], [0, 35], [0, 42]

pseudo set interval vector subspace of V over th

7} Z49.

Example 2.2.50: Let V = {[0, n] / n Z40} be avector space over the group G = Z40. W = {[0,

20], [0, 30]} V is pseudo set interval vector

over the set S = {0, 1, 2, 3} Z40.

Now we proceed onto define group interval l

DEFINITION 2.2.15: Let V be a group interval vethe group G. If V is a group under addition then

a group interval linear algebra.


Example 2.2.51: Let V = {[0, n] | n Z25} be a

linear algebra over the group G = Z25.

Example 2.2.52: Let V = {([0, a1], [0, a2], [0, a

Z18} be a group interval linear algebra over the g

Example 2.2.53: Let

V =

1

2

1 2 3 4 143

3

4

[0,a ]

[0,a ]a ,a , a ,a Z

[0,a ]

[0,a ]

- ½ª º

° « »° « » ® « »° « »° ¬ ¼¯ ¿



Now having seen examples of group interval line

we now proceed onto define group interval linear sub

DEFINITION 2.2.16: Let V be a group interval lin

over the group G. Let W V (W a proper subset itself is a group interval linear algebra over the gr

we define W to be a group interval linear subalgebrthe group G.


Example 2.2.55: Let V = {[0, a] | a Z144} be a gro

linear algebra over the group G = Z144. Consider W

{2Z144}} V; W is a group interval linear subal

over the group G = Z144.

Example 2.2.56 : Let V = {([0, a1], [0, a2], [0, a3], [0,

a1, a2, a3, a4, a5 Z48 be a group interval linear algeb

group G = Z48. Consider W = {([0, a1], 0, 0, 0, [0, a5

Z48} V; W is a group interval linear subalgebra of

group G = Z48.

Now we proceed onto define the notion of dir

group interval linear algebras.

DEFINITION 2.2.17: Let V be a group interval linover the group G. Let W 1 , W 2 , …, W n be a group int

subalgebras of V over the group G. We say V is a dthe group interval linear subalgebras W 1 , W 2 , …, W n

(a) V = W + + W



W1 =

1 2

3 1 2 3

[0,a ] [0,a ] 0

0 0 [0,a ] a ,a ,a0 0 0

-§ ·°¨ ¸

® ̈ ¸° ̈ ¸© ¹¯

W2 =

1

2 1 2 3

3

0 0 [0,a ]

[0,a ] 0 0 a ,a ,a

0 [0,a ] 0

- § ·° ̈ ¸® ̈ ¸° ̈ ¸© ¹¯

W3 = 1 1 48

0 0 0

0 [0,a ] 0 a Z

0 0 0

- ½§ ·° °¨ ¸ ® ¾¨ ¸° °¨ ¸

© ¹¯ ¿and

W4 = 1 2

1 2

0 0 0

0 0 0 a ,a

[0,a ] 0 [0,a ]

- § ·° ̈ ¸ ® ̈ ¸° ̈ ¸

© ¹¯

be group interval linear subalgebras of V overZ48. Clearly V = W1 + W2 + W3 + W4 and

Wi W j =

0 0 0

0 0 0

0 0 0

§ ·¨ ¸¨ ¸

¨ ¸© ¹if i z j; 1 d i, j d n.

Thus V is the direct sum of group

subalgebras W1 W2 W3 and W4



W2 = 1 2

1 2

3

0 [0,a ] 0 [0,a ]a ,a Z

[0,a ] 0 0 0

- ª º° ® « »

¬ ¼° ¯

W3 =1

1 2 7

2

0 0 [0,a ] 0a ,a Z

0 0 0 [0,a ]

- ª º° ® « »

¬ ¼° ¯ and

W4 = 1 7

1

0 0 0 0a Z

0 0 [0,a ] 0

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

be group interval linear subalgebras of V over the gsee V = W1 + W2 + W3 + W4 and

Wi W j =0 0 0 0

0 0 0 0

§ ·¨ ¸© ¹

; 1 d i, j d 4.

Thus V is a direct sum of group interval linear subalg

Let

P1 =1 2 i

3 4

[0,a ] 0 [0,a ] 0 a

0 [0,a ] 0 [0,a ] 1 i

- ª º° ® « » d d¬ ¼° ¯

P2 =

1 2 i 7

3 4

0 0 [0,a ] [0,a ] a Z

0 [0,a ] 0 [0,a ] 1 i 4

- ª º°

® « » d d¬ ¼° ¯

P1 2 i

[0,a ] [0,a ] 0 0 a Z- ª º°®« »



Pi P j z0 0 0 0

0 0 0 0

§ ·¨ ¸

© ¹

if i z j ; 1 d i; j d 4. Thus any collection of grou

subalgebras may not in general give a direct sum

In view of this we have the following interes

DEFINITION 2.2.18: Let V be a group intervalover the group G. Let W 1 , W 2 , …, W n be ninterval linear subalgebras of V over the group G

We say V is a pseudo direct sum if

(a) V = W 1 + … + W n(b) W i W j z {0} even if i z j

(c) We need W i’s to be distinct that is W i

W j = W j even if iz j i.e., W i W j = W p…, n} that is W p does not belong to th

group interval linear subalgebras of V.


Example 2.2.59: Let V = {Collection of all

matrices with entries from Z11} be the group

algebra over the group G = Z11.

Consider

[0 a ] 0- ½ª º



W2 =

1 6

2

i 11

3

4

5

[0,a ] [0,a ]

[0,a ] 0a Z

[0,a ] 01 i 6

[0,a ] 0

[0,a ] 0

- ½ª º° °« »° °« » ° °« »® ¾

d d« »° °« »° °« »° °¬ ¼¯ ¿

,

W3 =

2

3

i 11

4

5

1 6

0 [0,a ]

0 [0,a ]a Z

0 [0,a ]1 i 6

0 [0,a ]

[0,a ] [0,a ]

- ½ª º° °« »° °« » ° °« »® ¾

d d« »° °« »

° °« »° °¬ ¼¯ ¿and

W4 =

1

2i 11

3

4

5

0 [0,a ]

[0,a ] 0a Z

0 [0,a ]1 i 5

[0,a ] 0

0 [0,a ]

- ½ª º° °« »

° °« » ° °« »® ¾d d« »° °

« »° °« »° °¬ ¼¯ ¿

be group interval linear subalgebras of V over G = ZV = W1 + W2 + W3 + W4 and Wi W j z 0. If i z j.

W2, W3 and W4 are all distinct. Thus V is a pseudo d

W1, W2, W3 and W4.

-



W1 =

1 2

1 2

3 4 5

6 7

[0,a ] [0,a ] 0 0

0 0 0 0 a ,a[0,a ] [0,a ] [0,a ] 0

0 [0,a ] 0 [0,a ]

- ª º° « »

° « »® « »° « »° ¬ ¼¯

W2 =

2 3

1 6

i 3

4 5

7

0 [0,a ] 0 [0,a ]

0 [0,a ] [0,a ] 0a Z

0 0 [0,a ] [0,a ]

0 0 0 [0,a ]

- ª º

° « »° « » ® « »° « »° ¬ ¼¯

W3 =

1 2

i 3

3 5

6 4

[0,a ] [0,a ] 0 00 0 0 0

a Z[0,a ] 0 0 [0,a ]

[0,a ] 0 0 [0,a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

and

W4 =

1 2

3 6 4

i

8 5 7

[0,a ] [0,a ] 0 0

[0,a ] 0 [0,a ] [0,a ]a

0 0 0 0

[0,a ] 0 [0,a ] [0,a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

be group interval linear subalgebras of V over th

We see Wi W j z (0) if i z j 1 d i, j d 4

W d fi li i d d i i t



We now define linear independence in group int

algebras.


over the group G. Let X V be a proper subset of V,a linearly independent subset of V if X = {x1 , …, xn }

[0, ai ], 1 d i d n) and for some ni G; 1 d i d n; D 1

… + D n xn = 0 if and only if each D i = 0.

A linearly independent subset X of V is said to gevery element of v V can be represented as

v = ; ;

d d¦n

i i i

i 1

x G 1 i nD D .


Example 2.2.61: Let V = {([0, a1], [0, a2], [0, a3], [0

Z5, 1 d i d 4} be a group interval linear algebra over

= Z5. Consider X = {x1 = ([0, 1], 0, 0, 0), x2 = (0, [0,

= (0, 0, [0, 1], 0) and x4 = (0, 0, 0, [0, 1]) V. X i

independent set and generates V over G so X is aover G.

Example 2.2.62: Let V = {set all 4 u 2 interval m

entries from Z12} be a group interval linear algeb

group G.

Consider

[0,1] 0 0 [0,1] 0 0-ª º ª º ª º

0 0 0 0ª º ª º



0 0 0 0

0 [0,1] 0 0,0 0 [0,1] 0

0 0 0 0

ª º ª º« » « »

« » « »« » « »« » « »¬ ¼ ¬ ¼

,

0 0 0 0 0 0

0 0 0 0 0 0, ,0 [0,1] 0 0 0 0

0 0 [0,1] 0 0 [0,1]

½ª º ª º ª º°« » « » « »°« » « » « »¾« » « » « »°« » « » « »°¬ ¼ ¬ ¼ ¬ ¼¿

X is a linearly independent set and generates V

basis of V.

Here also we cannot define the notion of pse

interval linear subalgebras of a group interval lin

However we can define the notion of pseudo

vector subspace of a group interval linear algebra

DEFINITION 2.2.20: Let V be a group intervaover the group G. If P is just a subset of V and

structure but is a group interval vector space ovthen we call P to be a pseudo group interval vec

V.


Consider



Consider

W =

1 1 1 2

2 2

3 4 3

[0,a ] 0 0 [0,a ] [0,a ] [0,a

[0,a ] 0 , 0 [0,a ] , 0 0

[0,a ] [0,a ] 0 0 [0,a ] 0

- ª º ª º ª ° « » « » « ® « » « » « ° « » « » « ¬ ¼ ¬ ¼ ¬ ¯

W is a pseudo group interval vector subspace of

group G. Now we will define in the next chapter the noti

interval linear algebras.



Chapter Three

SET FUZZY INTERVAL LINEAR

ALGEBRAS AND THEIR PROPERT

In this chapter we introduce the notion of set

linear algebras, semigroup fuzzy interval lineagroup fuzzy interval linear algebras and study t

This chapter has two sections. First section intro

t t d di th i ti

DEFINITION 3 1 1: A fuzzy vector space (V K) or K



DEFINITION 3.1.1: A fuzzy vector space (V, K ) or K

an ordinary vector space V defined over the field F

K : V o [0, 1] satisfying the following conditions.

(a) K (a + b) t min { K (a), K (b)}

(b) K (–a) = K (a)

(c) K (0) = 1

(d) K (ra) t K (a)

for all a, b V and r F, where F is the field. V

K V will denote the fuzzy vector space.

For more about these notions refer [53].

DEFINITION 3.1.2: Let V be a set vector space ove

We say V with the map K is a fuzzy set vector space

vector space if K : V o [0, 1] and K (ra) t K (a) for al

r S. We call V K or K V or V K to be the fuzzy set vover the set S.

For more about these notions please refer [52].

Likewise we define a set fuzzy linear algebra (or fuzz

algebra) (V, K) or VK or KV to be an ordinary set lin

V with a map K : V o [0, 1] such that K(a + b) >

K(b)) for a, b V.

Notation: We say an interval [0, a] to be a fuzzy inte

d 1. [0, 0] = (0) and [0, 1] is the fuzzy set. We incl



DEFINITION 3.1.3: Let V be a set interval vector

set S. We say V with the map K is a fuzzy set int space or set fuzzy interval vector space if I K : V

I K (r[0, a]) > I K ([0, a]) for all [0, a] V and r

or I K V to be the fuzzy set interval vector space ov


Example 3.1.1: Let V = {[0, a1], [0, a2], [0, a3],

ai Z5, 1 d i d 5} be a set interval vector space

{0, 1, 2, 3}. IK: V o I [0, 1] is defined as follow

IK ([0, ai]) =i

i

i

1[0, ] if a 0a

[0,1] if a 0.

-z°®

° ¯

VIK is a set fuzzy interval vector space.

Example 3.1.2: Let V = {[0, ai] | ai Z+ {0}}

vector space over the set S = {0, 1, 2, 3, 4, 5, 8

o I [0, 1] as follows:

IK [0, ai] =i

i

i

1[0, ] if a 0

a

[0,1] if a 0.

- z°®

° ¯

IKV is a fuzzy set interval vector space.

1-



IK [0, ai] =i

i

i

1[0, ] if a 0

a

[0,1] if a 0.

-z°

®° ¯

;

IKV is a set fuzzy interval vector space.

DEFINITION 3.1.4: Let V be a set interval linear athe set S. A set fuzzy interval linear algebra (o

interval linear algebra) (V, K I) or V K I is a map K I: V such that K I(a + b) t min( K I(a), K I(b)) for every a, b


Example 3.1.4 : Let

V = i

i i

i 0

[0,a ]x a Z {0}f

- ½ ® ¾¯ ¿¦

be a set interval linear algebra over the set S = {0, 1,

16}.

Define KI : V o I [0, 1] as

KI (p(x) =n

i

i

i 0

[0,a ]x¦ )

=

1[0, ] if p(x) is not a constan

degp(x)

[0,1] if p(x) is a constant

-°®

°̄

VKI is a set fuzzy interval linear algebra.



Example 3.1.6 : Let

V =1 3

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]-§ ·°®¨ ¸°© ¹¯

where ai Z+ {0}} be a set interval linear alge

S = {3Z+, 2Z+, 0} Z+ {0}.

Define KI : V o I [0, 1] by

KI1 3

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]

§ ·¨ ¸© ¹

=

1

1

2 1

2

3 1

3

4 1

4

1 2 3

1[0, ] if a 0

a

1[0, ] if a 0 and a

a1

[0, ] if a 0 and aa

1[0, ] if a 0 and a

a

[0,1] if a a a

- z°°°

z °°°

z ®°°

z °°

° °¯

VKI is a fuzzy set interval linear algebra.

Now we proceed onto define set fuzzy interval su

DEFINITION 3.1.5: Let V be a set interval vector set S. Let W be a set interval vector subspace of V

{0, 2, 4, 6, 8, 10, 12, 14, 16} Z18; 1 d i d 12} be a



{ , , , , , , , , } ; }

vector subspace of V over S.

Define KI : W o I[0, 1] by

KI ([0, a1], [0, a2], …, [0, a12]) =

i

i

i

1[0, ] if no a

12

1[0, ] if some a10

[0,1] if all a

-°°°®°°°̄

(W, KI) is the set fuzzy interval vector subsubspace o

Note: It is important and interesting to note that W

be extendable to VKI in general.

Example 3.1.8: Let V = {([0, a1], [0, a2], …, [0, a8)]

{0}; 1 d i d 8} be a set interval vector space over the

5, 12, 13, 90, 184, 249, 1000} Z+ {0}. Choose

a1], [0, a2], …, [0, a8] | ai 5Z+ {0}} V be a

vector subspace of V over the set S.

Define KI : W o I [0, 1] by

KI (x) = ii

i 1

i

1[0, ] if a 0

a

[0,1] if a 0

- ¦ z°°®°

¦ °̄¦

interval vector subspace of V over Z24. Define K



as follows.

KI ([0, ai]) =i

i

i

1[0, ] if a 0a

[0,1] if a 0

- z°®° ¯

(W, KI) is a set fuzzy interval subspace of V. Cl

be extended to whole of V.Suppose

T =

1

i 24

2

3

[0,a ]a Z

[0,a ]1 i 3

[0,a ]

- ½ª º° °« »

® ¾« » d d° °« »¬ ¼¯ ¿

V

be a set interval vector subspace of V.

Define KI : T o I [0, 1] by

KI

1

2

3

[0,a ][0,a ]

[0,a ]

ª º« »« »« »¬ ¼

=

i

i

i

1[0, ] if a 0, i 1,2,3

31[0, ] if atleast one of a

2

[0,1] if a 0, 1 i 3

- z °

°°z®

° d d°

°¯

(T, KI) is a fuzzy set interval vector subspace o

cannot be extended to whole of V.

b l l

Example 3.1.10: Let

-



V =

1 3

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]

-§ ·°

®¨ ¸°© ¹¯

where ai Z+ {0}; 1 d i d 4} be a set interval lin

over the set S = {0, 2, 5, 8, 11, 16} Z+ {0} .

Let

W =1 3

3

[0,a ] [0,a ]

0 [0,a ]

-§ ·°®¨ ¸°© ¹¯

where ai Z+ {0}; 1 d i d 3} V be a set int

subalgebra of V over the set S.

Define KI : W o I [0, 1] as follows.

KI1 3

3

[0,a ] [0,a ]

0 [0,a ]

§ ·

¨ ¸© ¹=

1 2

1 2 3

1

1[0, ] if a a

a a a

[0,1] if 0 a

- ° ®° ¯

(W, KI) or KI W is a set fuzzy interval linear subalge

Example 3.1.11: Let

1

2

[0,a ]

[0,a ]

- ½ª º° °« »° °« »° °

1[0, a ]- ½ª º° °« »



W =

1

2

i 12

3

0

[0,a ]a Z ;1 i 3

0

[0, a ]

0

° °« »

° °« »° °« »° ° d d« »® ¾

« »° °« »° °« »° °« »° °¬ ¼¯ ¿

be a set interval linear subalgebra of V.

Define KI : W o I [0, 1] by

KI =

1

2

3

[0,a ]

0

[0,a ]

0

[0,a ]

0

ª º« »« »« »« »« »« »

« »« »¬ ¼

=

1 2 3

1

2 1 3

2

3 1 2

3

i

1

1[0, ] if a 0; a a 0

a1

[0, ] if a 0; a 0 aa

1[0, ] if a 0; a 0 a

a

1[0, ] if a 0; 1 i 3 or an3

[0,1] if a 0;i 1,2,3

- z °

°°z °

°°®

z °°

° z d d°°

°̄

(W, KI) is a fuzzy set interval linear subalgebra o

Now we proceed onto define the notion of s




V =

1 2

1

3 4

1 2 2

5 6

3

7 8

[0,a ] [0,a ] [0,a ][0,a ] [0,a ]

, ([0,a ],[0,a ]), [0,a ][0,a ] [0,a ]

[0, a ][0,a ] [0,a ]

- ª º ª ° « »° « « »® « « »° « « » ¬ ° ¬ ¼¯

be a semigroup interval vector space over the se

Define KI : V o I [0, 1] as follows:

KI

1 2

3 4

5 6

7 8

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ]

§ ·ª º¨ ¸« »¨ ¸« »¨ ¸« »¨ ¸« »¨ ¸¬ ¼© ¹

=i

10, if atleast o

5[0,1] if all a

-ª º°« »

¬ ¼®° ¯

KI (([0, a1], [[0, a2]) =

1

1

2

2

1

1 2

10, if a 0 an

a1

0, if a 0 ana

10, if a 0 an

10

[0,1] if a a

-ª ºz °« »

¬ ¼°°ª º° z °« »®¬ ¼°

ª º° z « »°¬ ¼

° °̄and

1[0,a ] 1§ ·ª º -ª º¨ ¸ °

Now we proceed onto illustrate only by exam

semigroup interval linear algebras and leave the sim



semigroup interval linear algebras and leave the sim

defining semigroup fuzzy interval linear algebras to t

Example 3.1.15: Let

1 2

3 4 i

5 6

[0,a ] [0,a ]

V [0,a ] [0,a ] a Z {0}; 1 i

[0,a ] [0,a ]

-ª º°« » d d® « »° « »¬ ¼¯

be a semigroup interval linear algebra over the sem

Z+ {0}.

Define KI : V o I[0, 1] as follows:

1 2

3 4

5 6

[0,a ] [0,a ]

I [0,a ] [0,a ]

[0,a ] [0,a ]

§ ·ª º¨ ¸« »K ¨ ¸« »¨ ¸« »¬ ¼© ¹

1 2

1 2 6

1 2 6

10, if atleast one of a aa a a

[0,1] if a a a 0

-ª º °« » ®¬ ¼° ¯

!!

!

(V, KI) or KI V is a fuzzy semigroup interval linear

semigroup fuzzy interval linear algebra.


1

10, if a 0

-ª ºz°« »



KI1 2

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]§ ·ª º¨ ¸« »¨ ¸¬ ¼© ¹

=

1

1

1

2

1

0, if a 0a

10, if a

a

[0,1] if a

z°« »¬ ¼

°°ª º®« »°

¬ ¼°° ¯

VKI is a semigroup fuzzy interval linear algebra.


V = i

i i

i 0

[0,a ]x a Q {0}f

- ½ ® ¾

¯ ¿

¦

be a semigroup interval linear algebra over the

Z+ {0}.

Define KI : V o I [0, 1] as follows:

KI i

i

i 0

[0, a ] xf

§ ·¨ ¸© ¹¦ =

if the degree of the interval polyno10,

greater than or equal to three8

[0,1] if the degree of the polynomial is l

three this includes zero polynomia

-ª º°« »¬ ¼°°®°°°̄

Example 3.1.18: Let



V =

1 2 3

7 4 5 i 8

8 9 6

[0,a ] [0,a ] [0,a ][0,a ] [0,a ] [0,a ] a Z ;1 i

[0,a ] [0,a ] [0,a ]

- ª º° « » d d® « »° « »¬ ¼¯

be a semigroup interval linear algebra define

semigroup S = {0, 2, 4, 6}, under addition modulo 8.Consider

W =

1 2 3

4 5 i 8

6

[0,a ] [0,a ] [0,a ]

0 [0,a ] [0,a ] a Z ;1 i 6

0 0 [0,a ]

- ½ª º° « » d d® « »

° « »¬ ¼¯ ¿

W is a semigroup interval linear subalgebra of V.

Define KI : W o I [0, 1]

KI

1 2 3

4 5

6

[0,a ] [0,a ] [0,a ]

0 [0,a ] [0,a ]

0 0 [0,a ]

§ ·ª º¨ ¸« »¨ ¸« »¨ ¸« »¬ ¼© ¹

=

1

1

2

2

33

3

10, if a 0

a

10, if a 0 if

a

1

0, if a 0 ifa

10, if a 0 if

a

-ª º z°« »¬ ¼°

°ª º° z « »°¬ ¼°

ª º®

z « »°¬ ¼°°ª º° z « »°¬ ¼

be a semigroup interval linear algebra de



be a semigroup interval linear algebra de

semigroup S = {0, 10, 20, 30} Z40.

W =10

i

i i 40

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦ V

be a semigroup interval linear subalgebra of V ovDefine KI : W o I [0, 1] as follows:

KI10

i

i

i 0

p(x) [0,a ]x

- ½® ¾

¯ ¿¦ =

> @

i

i

[0,a ] corresponds to the coefficie1

0, ; of the highest degree of x in p(x)a

if p (x) is a constant polynomial0,1

-ª º°« »°°¬ ¼

®°°

°̄

(W, KI) is a fuzzy semigroup interval linear suba

Example 3.1.20: Let V = {[0, ai] | ai Z+

semigroup interval linear algebra over the semi

{0}}. Consider W = {[0, ai] | ai 5Z+ {0semigroup interval linear subalgebra over the

{3Z+ {0}}.

D fi I W I [0 1] f ll

Now we can define for group interval vector

notion of group fuzzy interval vector spaces or f



g p y p



defined over the group G.

Let K I : V o I [0, 1] such that

K (a + b) t min { K (a), K (b)}

K (–a) = K (a)K (0) = 1

K (ra) t K (a)

for all a, b V and r G.

We call V K I or (V, K I) to be the group fuzzy intalgebra.


Example 3.1.21: Let V = {([0, a1], [0, a2], [0, a3], [0

Z40; 1 d i d 4} be a group interval linear algebra ove

G = {0, 10, 20, 30} Z40.

Define K I : V o I [0, 1] as follows:

KI ([0, a1], [0, a2], [0, a3], [0, a4]) =

> @

1

10, if a

a

0,1 if a

-ª º°« »®¬ ¼°

¯

(V, KI) is a group fuzzy interval vector space.

I ( ( ))

10,

deg(p(x))

-ª º°« »

¬ ¼®



KI (p(x)) =

> @

deg(p(x))

0,1 if deg p(x) 0

« »¬ ¼®

° ¯

(V, KI) is a fuzzy group interval vector space;


Example 3.1.23: Let

V =

1

2

i 25

3

4

[0, a ]

[0,a ]a Z ;1 i 4

[0,a ]

[0,a ]

- ½ª º° °« »° °« » d d® ¾« »° °

« »° °¬ ¼¯ ¿

be a group interval linear algebra over the group

KI : V o I [0, 1] as follows:

KI

1

2

3

4

[0,a ]

[0,a ]

[0,a ]

[0,a ]

§ ·ª º¨ ¸« »¨ ¸« »¨ ¸« »¨ ¸« »¨ ¸¬ ¼© ¹

=

1

1

2 1

2

3 1

3

10, if a 0

a

10, if a 0 if a 0

a

10, if a 0 if a aa

10 if 0 if

-ª º z°« »¬ ¼°

°ª º° z « »°¬ ¼°

ª º® z « »°¬ ¼°

°ª º°« »




V =

1 2

3 4 i 21

5 6

[0,a ] [0,a ]

[0,a ] [0,a ] a Z ,1 i 6

[0,a ] [0,a ]

- ½ª º° °« » d d® ¾« »° °« »¬ ¼¯ ¿

be group interval vector space.Define KI : V o I [0, 1] as follows:

KI

1 2

3 4

5 6

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ]

§ ·ª º¨ ¸« »¨ ¸« »

¨ ¸« »¬ ¼© ¹

=i

1

10, ;1 i

max{a }

[0,1] if a 0,i 1,2

-ª ºd d°« »

®¬ ¼

° ¯That is if

x =

[0,8] [0,17]

[0,4] [0,1]

[, 2] [0,19]

ª º« »« »« »¬ ¼

V

then

KI (x) =1

0,19

-ª º®« »¬ ¼¯

.

Thus (V, KI) is a group fuzzy interval vector space.

Take

W =

1

2 i 21

[0,a ] 0

[0,a ] 0 a Z ,1 i 3

- ½ª º° °« » d d® ¾« » V

KI

1[0,a ] 0

[0 a ] 0

ª º« »

=2

10, if a

a

-ª ºz°« »

®¬ ¼



KI 2

3

[0,a ] 0

[0,a ] 0« »« »¬ ¼

=2

1

a

[0,1] if a 0

®¬ ¼° ¯

Clearly (W, KI) is a fuzzy group interval vector s

Example 3.1.25: Let V = {Collection of all 1

matrices; ([0, ai]) with entries from Z36 that ai be a group interval vector space over the group G

W = {([0, ai]) demotes all matrices with entries f

Define

KI ([0, ai]) =i

i

1

10, ;1 i 36; a

max{a }

[0,1] if a 0,i 1, 2,...,36

-ª ºd d °« »®¬ ¼

° ¯

(W, KI) is a group fuzzy interval vector subspace

Example 3.1.26 : Let V = {All upper triangular

matrices constructed using Z13} be the group space over the group G = Z13.

Let W = {all 4 u 4 diagonal interval matri

from Z13} V; W is a group interval vector sub

the group G = Z13.Define KI : W o I [0, 1] as follows.

[0 a ] 0 0 0§ ·ª º-

[0,3] 0 0 0

0 [0,7] 0 0

ª º« »« »



x =

[ ,7]

0 0 [0,11] 0

0 0 0 [0,1]

« »« »« »¬ ¼

W

then

KI (x) =1

0,11

-ª º®« »¬ ¼¯

.

We see in case of group interval linear algebra

interval vector spaces we cannot use groups other tha

addition modulo n. As Z or Q or R cannot be used s

intervals we use are of the form [0, ai]. 0 d ai.

Now having seen fuzzy set interval vector spsemigroup interval vector spaces and group fuz

vector spaces we proceed onto define another type interval vector spaces, fuzzy semigroup interval ve

and fuzzy group interval vector spaces by construct

and not using set interval vector spaces, semigro

vector spaces or group interval vector spaces. Thesetype II set fuzzy interval vector spaces and so on. T

interval vector spaces constructed in section 3.1 wil

as type I spaces.

In the following section we define type II fuzzy inter

3.2 Set Fuzzy Interval Vector Spaces of Type II anProperties

in V. We then call V to be a set fuzzy interval

type II.




Example 3.2.1: Let V = {[0, ai]| 0 d ai d 1}

interval vector space over the set S = {0, 1, ½,

Here for any v = [0, ai] and s =r

1

2

(r d n) we hav

sv = i

r

a0,

2

ª º« »¬ ¼

= vs

and sv V. Thus V is a fuzzy set interval vectoII over the set S.

Example 3.2.2: Let

V =

1

2

1 2 3 i

5

[0,a ]

[0,a ], [0,a ] [0,a ] [0,a ] 0 a

[0,a ]

- ª º

° « »° « » d ® « »° « »° ¬ ¼¯

#

be a fuzzy set interval vector space of type II o

{0, 1, 1/5, 1/10, 1/121, 1/142}.

Example 3.2.3: Let

1 2

i

3 4

[0,a ] [0,a ]0 a 1;1 i 10

[0,a ] [0,a ]

½ª º °d d d d ¾« »

¬ ¼ °



3 4[0,a ] [0,a ]¬ ¼ °¿

be a set fuzzy interval vector space over the set

S =n

1n 0,1,...,27

3

- ½® ¾

¯ ¿.

Example 3.2.4: Let

V = i

i i

i 0

[0,a ]x 0 a 1f

- ½d d® ¾

¯ ¿¦

be a set fuzzy interval vector space over the set S =

1/7, 1/5, 1/11, 1/13, 1/19, 1/17, 1/23} of type II.

Now we define substructures of set fuzzy interval ve

DEFINITION 3.2.2: Let V be a set fuzzy interval v

over the set S of type II.

Let W V; if W is a set fuzzy interval vector spa set S of type II, then we define W to be a set fuzzy inte subspace of V over the set S of type II.

We will illustrate this situation by examples.

Example 3.2.5: Let

1[0, a ]

[0 a ] [0 a ] [0 a ]

- ª º° « »ª º° « »



space of type II over the set P, we call W to be a s

interval vector subspace of V of type II over the subs



We will illustrate this situation by some simple exam

Example 3.2.7: Let

V =1 2

i3 4

[0,a ] [0,a ]

a [0,1],1 i 4[0,a ] [0,a ]

- ½ª º° °

d d® ¾« »¬ ¼° °¯ ¿

be a set fuzzy interval vector space over the set S =

1/3n; n = 1, 2, …, 12} of type II. Let

W =1 2

1 2 3

4

[0,a ] [0,a ] 0 a ,a ,a 10 [0,a ]

- ½ª º° °d d® ¾« »¬ ¼° °¯ ¿

and P = {0, 1, 1/2n | n = 1, 2, …, 12} S. W is a s

interval vector subspace of V of type two over the su

Example 3.2.8: Let

1

1 2 3 4 5 6

2

[0,a ]V , [0,a ][0,a ][0,a ][0,a ][0,a ][0,a ]

[0,a ]

- ª º° ® « »

¬ ¼° ¯

be a set fuzzy interval vector space of type II over the

S = {0 1/2n 1 1/3m 1/5m 1/7n | 1 d m d 8 1 d n

Example 3.2.9: Let



1 2

3 4 1 2 3

5 6 5 6 7

7 8

[0,a ] [0,a ][0,a ] [0,a ] [0,a ] [0,a ] [0,a ] [0

V ,[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] [0

[0,a ] [0,a ]

- ª º° « » ª ° « » ® « « » ¬ ° « »° ¬ ¼¯

be a set fuzzy interval vector space of type II ove

S =1

0, n Zn

- ½® ¾

¯ ¿.

Choose

W = 1 2 3 4

i

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ] 0 a 1[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° d d® « »¬ ¼° ¯

is a subset fuzzy interval vector space of type II o

P = 10, n Z4n

- ½® ¾¯ ¿

S of V.

Example 3.2.10: Let

V = 1 2i

3 4

[0,a ] [0,a ] 0 a 1;1 i[0,a ] [0,a ]

- ª º° d d d ® « »¬ ¼° ¯

b f i l li l b h

are in V then



x + y = 1 1 2 2

3 3 4 4

[0, max(a , b )] [0, max(a , b )][0, max(a , b )] [0, max(a , b )]

ª º« »¬ ¼

Thus max (x, y) denoted by x + y is an assoc

commutative operation on V).

Choose

W = 1 2

i

3

[0,a ] [0,a ]1 i 3;0 a 1

0 [0,a ]

- ½ª º° °d d d d® ¾« »

¬ ¼° °¯ ¿

W is a subset fuzzy interval vector subspace of V d

the subset

P =n 6

1n Z

2

- ½® ¾¯ ¿

S.

Now we proceed onto define set fuzzy interval lin

formally.

DEFINITION 3.2.4: Let V be a set fuzzy interval v

over a set S. If on V is defined a closed associaoperation denoted by ‘+’ such that s (a + b) = sa +

S and a, b V. Then we define V to be a set fuzlinear algebra of type II.


Example 3 2 11: Let V = {[0 a ] | 0 d a d 1} be

Example 3.2.12: Let V = {collection of all 3 umatrices with entries from I [0, 1]} be a set fuzz



algebra over the set

S =1

,0 n 1, 2,3,...n

- ½® ¾

¯ ¿.

Example 3.2.13: Let

V = ii i

i 0

[0,a ]x 0 a 1

f

- ½d d® ¾¯ ¿¦

be a set fuzzy interval linear algebra over the set

S = n

1n 1,2,0, 2

- ½® ¾¯ ¿! .

We will define set fuzzy interval linear subalgeb

DEFINITION 3.2.5: Let V be a set fuzzy interva

over the set

S =- ½

® ¾¯ ¿

1 ,0 n 1,2,...

n.

Choose W V; suppose W be a set fuzzy intervaover the set S; we define W to be set fuzzy

subalgebra of V over S of type II.


Choose

W = 1 2

i

[0,a ] [0,a ]1 i 3;0 a 1

- ½ª º° °d d d d® ¾



30 [0,a ]« »¬ ¼° °¯ ¿

W is a set fuzzy interval linear subalgebra of V over

Example 3.2.15: Let

V =

1

2

3

i

4

5

6

[0,a ][0,a ]

[0,a ]0 a 1;1 i 6

[0,a ]

[0,a ]

[0,a ]

- ½ª º° °« »° °« »° °« »° °

d d d d« »® ¾« »° °« »° °

« »° °« »° °¬ ¼¯ ¿


S =

1

,0 n 0,1,2,...3n 1

- ½

® ¾¯ ¿ .

Let

W =

1

2

i

3

[0,a ]

0

[0,a ]0 a 1;1 i 30

[0,a ]

- ½ª º° °« »° °« »° °« »° °

d d d d« »® ¾« »° °« »° °« »° °« »

V

DEFINITION 3.2.6: Let V be a set fuzzy interva

over the set S. Suppose W V; if W is a set fuzzy



algebra over the subset P of S, then we define fuzzy interval linear subalgebra of V of type II ovof S.

We will illustrate this situation by an exampl


V =i i

i i

[0,a ] 0 a 1

[0,b ] 0 b 1

- ½d dª º° °® ¾« » d d¬ ¼° °¯ ¿


S =1

0, n 1,2,...n

- ½® ¾

¯ ¿

with min operation on V. That is min {[0, ai], [0

{ai, bi}]. Choose

W =i

i

[0,a ]0 a 1

0

- ½ª º° °d d® ¾« »

¬ ¼° °¯ ¿ V

is a subset fuzzy interval linear subalgebra over t

P1

0 1 2- ½® ¾ S f S


V = 1 2

i

[0,a ] [0,a ]0 a 1;1 i 4

[0 ] [0 ]

- ½ª º° °d d d d® ¾

« »¬ ¼° °¯ ¿



3 4[0,a ] [0,a ]« »¬ ¼° °¯ ¿and

W =

1

2

i

3

4

[0,a ]

[0,a ]0 a 1;1 i 4

[0,a ]

[0,a ]

- ½ª º° °« »° °« » d d d d® ¾« »° °« »° °¬ ¼¯ ¿

be set fuzzy interval linear algebras defined over the

S =1

,0 n 1,2,...2n 1

- ½® ¾¯ ¿ .

Define TF : V o W as

TF(A) = TF = 1 2

3 4

[0,a ] [0,a ]

[0,a ] [0,a ]

§ ·ª º¨ ¸« »¨ ¸¬ ¼© ¹

=

1

2

3

4

[0,a ]

[0,a ]

[0, a ]

[0,a ]

ª «

« « « ¬

for every A in V. TF is a set linear transformation of V

Note as in case of vector spaces we see in case interval vector spaces define linear transformatio

same set.



S =n

10, n 1,2,...

2

- ½® ¾

¯ ¿



2¯ ¿under multiplication.

Let

V =

1

2 1 2 3 4

3

[0,a ]

[0,a ] , [0,a ] [0,a ] [0,a ] [0,a ] [[0,a ]

- ª º° « »

® « »° « »¬ ¼¯

be a semigroup fuzzy interval vector space of level t

semigroup

(S, o) = n m1 10,1, , m,n Z2 3

- ½® ¾¯ ¿

under ‘o’ the max operation that is

m n

1 1

o2 2

- ½

® ¾¯ ¿ max m n

1 1

,2 3

- ½

® ¾¯ ¿ = m

1

2 if m

1

2 > n

1

3 ; n

1

3 i

Example 3.2.20: Let

W =

1

1 2 3 42

5 6 7 8

3

[0,a ][0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] , [0,a ] [0,a ] [0,a ] [0,a ] 0[0, a ]

- ª º§ ·

° « »® ¨ ¸« » © ¹° « »¬ ¼¯

Example 3.2.21: Let V =

1 2[0 a ] [0 a ]- ª º° « »



1 2

3 4 1 2 3 4

5 6 6 7 8 9

7 8

[0,a ] [0,a ][0,a ] [0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

,[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ]

- ª º° « » § ° « »® ¨ « » © ° « »° ¬ ¼¯

be a semigroup fuzzy interval vector space of levsemigroup (S, o).

Now we proceed onto define semigroup fuzzyalgebra V over the semigroup (S, o).

DEFINITION 3.2.8: Let V be a fuzzy semigroup space over the semigroup (S, o) of type II. If V it fuzzy semigroup under some operation say ‘+’ a

s o b + s o b for all s S and a, b V then w fuzzy semigroup interval linear algebra over S of


Example 3.2.22: Let

V =

1 2

i

3 4

5 6

[0,a ] [0,a ]0 a 1

[0,a ] [0,a ] 1 i 6[0,a ] [0,a ]

- ½ª ºd d° °« »

® ¾« » d d° °« »¬ ¼¯ ¿



Thus if v = [0, ai] V and S =r

1

2 S then

ª º



sv = vs =r

1

2[0, ai] =

i

r

a0,

2

ª º« »¬ ¼

.

Consider M = {all upper triangular fuzzy interva

with entries from I [0, 1]} V; M is a fuzzy sem

linear subalgebra of V over S of type II.

Example 3.2.25: Let

V = i

i i

i 0

[0,a ]x 0 a 1f

- ½d d® ¾

¯ ¿¦

be a fuzzy semigroup interval linear algebra of t

operation (i.e., if

i

i

i 0

[0,a ]xf

¦ = p(x)

andq(x) =

i

i

i 0

[0,b ]xf

¦

are in V then

p(x) + q(x) = i

i i

i 0

[0,min{a ,b }]xf

¦

over the semigroup

S =n

10,1, n 1,2,...

5

- ½® ¾

¯ ¿

W = 2i

i i

i 0

[0,a ]x 0 a 1f

- ½d d® ¾

¯ ¿¦ V;



W is a semigroup fuzzy interval linear subalgebra of

semigroup S of type II.

Now we proceed onto define the notion of fuzzy su

interval sublinear algebra of V over the subsemigrou

DEFINITION 3.2.10: Let V be a fuzzy semigroup int

algebra of type II over the semigroup S. Let W Vwhere W and P are proper subsets of V and S respecis a fuzzy semigroup interval linear algebra of type

semigroup P then we define W to be a fuzzy su

interval linear subalgebra of type II over the subsemthe semigroup S.

We illustrate this situation by some examples.


V =

1 2

3 4

5 6 i

7 8

9 10

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ] 0 a 1;1 i 10

[0,a ] [0,a ]

[0,a ] [0,a ]

- ½ª º° « »° « »° « » d d d d®

« »°

« »° « »° ¬ ¼¯ ¿

b f i i l li l b f

W =

1

2

3 i

[0,a ] 0

0 [0,a ]

[0,a ] 0 0 a 1;1 i 5

- ª º° « »° « »°« » d d d d®« »



W 3 i

4

5

[0,a ] 0 0 a 1;1 i 5

0 [0,a ]

[0,a ] 0

° « » d d d d® « »° « »° « »° ¬ ¼¯

and

P = 3n

11,0, n 1,2,...2

- ½® ¾¯ ¿ S.

W is a fuzzy subsemigroup interval linear subal

over the subsemigroup P S.


V =

1 2

3 4 i

5 6

7 8

[0,a ] [0,a ]

[0,a ] [0,a ] 0 a 1

[0,a ] [0,a ] 1 i 8

[0,a ] [0,a ]

- ½ª º° °« » d d° °« »® ¾« » d d° °« »° °¬ ¼¯ ¿

be a special fuzzy semigroup interval linear a

semigroup

S =n m

1 10,1, , m,n Z

2 5

- ½® ¾

¯ ¿.

Choose

1[0 a ] 0- ½ª º

P =n

10,1, n Z

5

- ½® ¾

¯ ¿ S

of type II of V.



yp

Now if V has no fuzzy semigroup interval linear

over F then we define V to be a simple fuzzy semigr

linear algebra of type II. We say V is said to be a pse

fuzzy semigroup interval linear algebra over S of t

has no fuzzy subsemigroup interval linear algebra say V is doubly simple if V has no fuzzy semigro

linear subalgebras and fuzzy subsemigroup inte

subalgebras.

We will illustrate this situation by examples.

Example 3.2.28: Let

V =1

1

1

[0,a ] 0a 1

0 [0,a ]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

be a semigroup fuzzy linear algebra over the semigr

1} with min operation. It is easily verified V is a dou

semigroup fuzzy interval linear algebra of type II ove

Example 3.2.29: Let

[0,1/ 2] [0,1/ 4] [0,1] [0]

[0 1/ 2] [0 1/ 4] [0 1] [0]

- ½ª º ª º ª º ª º° °« » « » « » « »® ¾« » « » « » « » = V



S =n m

1 10,1, , m,n Z

2 10

- ½® ¾

¯ ¿.

D fi T V W f ll



Define T : V o W as follows:

T

1

2

3

4

[0,a ]

[0,a ]

[0, a ][0,a ]

§ ·ª º¨ ¸« »¨ ¸« »

¨ ¸« »¨ ¸« »¨ ¸¬ ¼© ¹

=1

3

[0,a ] 0 [0,a

0 [0,a ] [0,a

ª « ¬

It is easily verified that T is a linear transformatio

Example 3.2.33: Let

V =

1 2 3

1

4 5 6

2

7 8 9

[0,a ] [0,a ] [0,a ][0,a ] 0

, [0,a ] [0,a ] [0,a ][0,a ] 1

[0,a ] [0,a ] [0,a ]

- ª ºª º° « »

® « » « »¬ ¼° « »¬ ¼¯

and

W =

1 2

3 4

5 6

1 2 7 8

3 4 9 10

11 12

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] 0

,[0,a ] [0,a ] [0,a ] [0,a ] 1

[0,a ] [0,a ]

[0 a ] [0 a ]

- ª º° « »° « »° « »° « »

ª º° « »

® « » « »¬ ¼° « »° « »° « »°

T1

2

[0,a ]

[0,a ]

§ ·ª º¨ ¸« »¨ ¸¬ ¼© ¹

= 1

2

[0,a ] 0

0 [0,a ]

ª º« »¬ ¼

and



T

1 2 3

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ]

§ ·ª º¨ ¸« »

¨ ¸« »¨ ¸« »¬ ¼© ¹

=

1

3

5

7

[0,a ] 0

0 [0,a

[0,a ] 0

0 [0,a

[0,a ] 0

0 [0,a

[0,a ] 0

0 [0,a

ª « « « « « « « « « « « ¬

It is easily verified that T is a linear transformation o

Now we proceed onto define some more p

semigroup fuzzy interval vector spaces and linear

type II.

DEFINITION 3.2.12: Let V be a fuzzy semigroup inte space of type II defined over the semigroup S. Let

…, W n be a semigroup interval subvector spaces of

semigroup S. If V =*

n

i

i 1

W but W i W j z I or {0} if i

call V to be the pseudo direct union of fuzzy semigr spaces of V over semigroup S of type II.

Th d i t d t i l f t

vector subspaces of V. if V =*

n

i

i 1

W and W i W

z j; 1 d i , j d n.



The reader is expected to give examples of

semigroup fuzzy interval vector subspaces of typ

Now we proceed onto define direct sum of finterval linear subalgebras of a fuzzy interval sem

II.

DEFINITION 3.2.14: Let V be a fuzzy semigroup

algebra over a semigroup S of type II. We say Vof semigroup fuzzy interval linear subalgebras WV if

(a) V = W 1 + … + W n(b) W i W j = {0} or I if i z j ; 1 d j, j d n.



V = 1 2

i

3 4

[0,a ] [0,a ]0 a 1;1 i

[0,a ] [0,a ]

- ª º° d d d d® « »

¬ ¼° ¯

be a fuzzy semigroup interval linear algebra of

over the semigroup

S =1

0 1 n Z- ½® ¾

W3 =i

i

0 00 a 1

[0,a ] 0

- ½ª º° °d d® ¾« »

¬ ¼° °¯ ¿and

- ½ª º



W4 =i

i

0 00 a 1

0 [0,a ]

- ½ª º° °d d® ¾« »

¬ ¼° °¯ ¿

to be fuzzy semigroup interval linear subalgebras of

over the semigroup S.V = W1 + W2 + W3 + W4

and

Wi W j =0 0

0 0

§ ·¨ ¸© ¹

if i z j and 1 d i, j d n.

If in the definition we have Wi’s to be such that

(0) or I and Wi W j; 1 d i, j d n then we define

pseudo direct sum of fuzzy semigroup interval linear

We will illustrate this by an example.

Example 3.2.35: Let

V =

1 2

3 4

5 6 i

7 8

[0,a ] [0,a ]

[0,a ] [0,a ][0,a ] [0,a ] 0 a 1;1 i 10

[0,a ] [0,a ]

- ½ª º° « »

° « »° « » d d d d® « »° « »°

W1 =

1

2

i

[0,a ] 0

0 [0,a ]

0 0 0 a 1;1 i

0 0

- ª º° « »° « »

° « » d d d d® « »°« »



3

0 0

0 [0,a ]

« »° « »° « »° ¬ ¼¯

W2 =

1

i

2 3

4

[0,a ] 0

0 0

0 a 1;1 i0 0

[0,a ] [0,a ]

[0,a ] 0

- ª º° « »° « »° « » d d d ®

« »° « »° « »° ¬ ¼¯

W3 =

1

3

i2

4

0 [0,a ]

0 [0,a ]

0 a 1;1 i[0,a ] 0

0 [0,a ]

0 0

- ª º° « »° « »° « » d d d ®

« »° « »°

« »° ¬ ¼¯

W4 =

1

2 3

4 5 i

6 7

[0,a ] 0

[0,a ] [0,a ]

[0,a ] [0,a ] 0 a 1;1 i

0 0

[0,a ] [0,a ]

- ª º° « »° « »° « » d d d ®

« »° « »° « »° ¬ ¼¯

But

Wi Wj z

0 0

0 00 0

ª º« »« »« »



Wi W j z 0 0

0 0

0 0

« »« »« »« »¬ ¼

if i z j. 1 d i, j d 5. Thus V is a pseudo direct sum W1

As it is not an easy task to define group fuzzy inte

spaces, we proceed to work in different direction.



Chapter Four

SET INTERVAL BIVECTOR SPACE

THEIR GENERALIZATION

In this chapter we for the first time introduce th

interval bivector spaces and generalize them tovector spaces, n t 3. We also define semigroup i

spaces and group interval bivector spaces and

th t t bi i i t l bi t



Example 4.1.3: Let

V = V1 V2 =

[0 a ]- ª º



1

2

1 2 3 4

3 i

5 6 7 8

4

5

[0, a ]

[0,a ][0,a ] [0,a ] [0,a ] [0,a ]

, [0,a ] ,[0,a ][0,a ] [0,a ] [0,a ] [0,a ]

[0,a ]

[0,a ]

- ª º° « »° « »ª º ° « »® « » « »¬ ¼°

« »° « »° ¬ ¼¯

1 2 3 4

5 6 7 8 i

9 10 11 12

13 14 15 16

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] a Q

[0,a ] [0,a ] [0,a ] [0,a ] 1 i[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « » ° « »®

« » d d° « »° ¬ ¼¯

be a set interval bivector space defined over the

3/17, 25/4, 2, 4, 6, 21, 49}.

Example 4.1.4 : Let V = V1 V2 = {All 10

matrices with intervals of the from [0, ai] with ai

i

i i 7

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a set interval bivector space over the set S =

Z7

bivector space over the set S then we define W

interval bivector subspace of V over the set S.




Example 4.1.5: Let V = V1 V2 =

1

1 2 3 i 19

2

4 5 6

3

[0,a ][0,a ] [0,a ] [0,a ] a Z ;

[0,a ],[0,a ] [0,a ] [0,a ] 1 i 6[0,a ]

- ª ºª º° « »

® « » « » d d¬ ¼° « »¬ ¼¯

25i

i i 19

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

be a set interval bivector space over the set S = {0, 2,

17} Z19.

Choose

W =

1

i 19

2

3

[0, a ]a Z ;

[0,a ]1 i 3

[0, a ]

- ½ª º° °« »

® ¾« » d d° °« »¬ ¼¯ ¿

25

2i

i i

i 0

[0,a ]x a

- ®

¯ ¦

= W1 W2 V1 V2 = V

is a set interval bivector subspace of V over the set S

be a set interval bivector space over the set S

{0}. Take

W = W1 W2 = {[0, ai] | ai 7Z+ {0

[0 a ] [0 a ] [0 a ]-ª º



1 2 3

4 5 i

6

[0,a ] [0,a ] [0,a ]

0 [0,a ] [0,a ] a 3Z {

0 0 [0,a ]

- ª º° « » ® « »° « »¬ ¼¯

V1 V2 ; W = W1 W2 is a set interval bivec

V over the set S.

Example 4.1.7 : Let V = V1 V2 = {All 7 u 7 in

with interval of the form [0, ai] ai Z18 {Al

row matrices with intervals of the form with ainterval bivector space over the set S = {0, 1, 2,

We see W = W1 W2 = {All 7 u 7 diagonal in

with intervals of the form [0, ai] with ai from Z18

[0, a2], 0, [0, a3], 0, [0, a4], 0, [0, a5]) / ai Z18;

V2 = V is a set interval bivector subspace of V

Now having see examples of subspaces we now

define subset interval bivector subspaces.

DEFINITION 4.1.3: Let V = V 1 V 2 be a set in

space over the set S. Let W = W 1 W 2 V 1 proper bisubset of V and P S be a proper subs set interval bivector space over the set P then we

a subset interval bivector subspace of V over the

{All 5 u 5 interval matrices with entries from Z2

interval bivector space over the set S = {0, 2, 3, 5,

14, 22} Z24. Choose

[0 a ]- ½ª º



W = W1 W2 =

1

i 24

2

3

[0,a ]a Z ;

[0,a ]1 i 3

[0, a ]

- ½ª º° °« »

® ¾« » d d° °« »¬ ¼¯ ¿

{All 5 u 5 interval upper triangular matrices with e

Z24} V1 V2 = V. Choose P = {0, 2, 5, 10, 14,

Z24. W = W1 W2 is a subset interval bivector sub

over the subset P of S.

Now we proceed onto define the notion of set int bialgebra.

DEFINITION 4.1.4: Let V = V 1 V 2 be a set interv space over the set S.

Suppose V is closed under addition and if s (a +

sb for all s S and a, b V then we call V to be a bilinear algebra over S.


Example 4.1.9: Let V = V1 V2 be a set inter

algebra over the set S; where

V = V1 V2

=

1

2

i3

4

[0, a ]

[0,a ]a Z {0};

[0,a ]1 i 5

[0,a ]

- ½ª º° °« »° °« » ° °« »® ¾« » d d° °

« »° °



4

5

[ , ]

[0,a ]

« »° °« »° °¬ ¼¯ ¿

1 2 3i

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]a Z {

[0,a ] [0,a ] [0,a ]1 i 9

[0,a ] [0,a ] [0,a ]

- ª º ° « »® « » d d° « »¬ ¼¯

be a set interval bilinear algebra over the set S

52, 75, 130} Z

+

{0}.

Example 4.1.11: Let V = V1 V2

=1 2 3 4 5 i

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] a

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

- ª º° ® « »

¬ ¼° ¯

{All 12 u 11 interval matrices with intervals fro

the form [0, ai]; ai Q+ {0}} be a set interval

over the set S = {0, 12, 3 , 41 , 5 12 , 412,

Now we see all the set interval bilinear algeb

examples 4 1 9 4 1 10 and 4 1 11 are of infinite

be a set interval linear bialgebra over the set S = {0

Z8. V is a finite order set interval linear bialgeb

order set interval bilinear algebra over the set S.

Now we proceed onto define the notion of set inter



subalgebra of a set interval bilinear algebra over the

DEFINITION 4.1.5: Let V = V 1 V 2 be a set inte

bialgebra over the set S. Choose W = W 1 W 2 V suppose W is a set interval linear bialgebra over thwe call W to be a set interval linear sub bialgebra of

set S.


Example 4.1.13: Let

V = V1 V2 =

1 2 3

i

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]a Z

[0,a ] [0,a ] [0,a ]1 i

[0,a ] [0,a ] [0,a ]

- ª º° « »

® « » d d° « »¬ ¼¯

1 2

3 4

i 16

5 6

7 8

9 10

[0,a ] [0,a ]

[0,a ] [0,a ]a Z ;

[0,a ] [0,a ]1 i 10

[0,a ] [0,a ][0,a ] [0,a ]

- ½ª º° °« »° °« » ° °« »® ¾

d d« »° °« »° °« »° °¬ ¼¯ ¿

1

2

i 163

4

[0,a ] 0

0 [0,a ]a Z ;

[0,a ] 0 1 i 50 [0,a ]

- ½ª º° °« »° °« » ° °« »® ¾d d« »° °

« »° °



5[0,a ] 0

« »° °« »° °¬ ¼¯ ¿

V1 V2 = V; W is a set interval linear subbial

the set S.


V = V1 V2 =27

i

i i

i 0

[0,a ]x a Q {

- ®

¯

¦

{All 10 u 10 interval matrices with entries from

set interval linear bialgebra over the set S =

Choose

W = W1 W2 =20

i

i i

i 0

[0,a ]x a Z

- ® ¯ ¦

{all 10 u 10 upper triangular interval matrices w

Q+ {0}} V1 V2 =V; W is a set interval line

of V over the set S.

Now we proceed onto define other

First we will illustrate this by some simple examples

Example 4.1.15: LetV = V1 V2

- ½



=1 2 i 27

3 4

[0,a ] [0,a ] a Z ;

[0,a ] [0,a ] 1 i 4

- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

{[0, ai] | ai

be a set interval linear bialgebra over the set S = Z27.

W = W1 W2 =1 i 27

2

[0,a ] 0 a Z ;

[0,a ] 0 1 i 2

- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

{[0, ai] | ai {0, 3, 6, 9, 12, 15, 18, 21, 24} Z27} V be a subset interval linear subbialgebra of V over

{0, 9, 18} S.

Example 4.1.16 : Let V = V1 V2 = {All 5 u 5 interv

with entries from Q

+

{0}} 30

i i

i

i 0

a Q {0};[0,a ]x

0 i 30

- ½ ® ¾

d d¯ ¿¦

be a set interval linear bialgebra over the set S = {0

17Z+}. Choose W = W1 W2 = {all 5 u 5 int

matrices with entries from Q+ {0}}

Example 4.1.17 : Let V = V1 V2 = {[0, a], [0, a

| a Z5} [0,a]

[0,a]

- ½° °° °® ¾



5

[0,a]a Z

[0,a]

[0,a]

° °® ¾

° °° °¯ ¿

be a set linear bialgebra of the set S = {0, 1}.

Clearly V is a pseudo simple set linear bialge

Example 4.1.18: Let

V = V1 V2

= 3

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] a Z

[0,a] [0,a] [0,a]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

[0,a

[0,a

[0,a

- ª ° « ® « ° « ¬ ¯

be a set interval linear bialgebra over the set S =V is a pseudo simple set interval linear bialgebra

We define pseudo set interval bivector space o

linear bialgebra.


[0 a ] [0 a ] [0 a ] a Z ;- ½ª º°

W = W1 W2

=1 2 1

3 4 2

[0,a ] 0 [0,a ] [0,a ] 0 0,

0 [0,a ] [0,a ] [0,a ] 0 0

- ½ª º ª º° ® « » « »° ¬ ¼¬ ¼¯ ¿



3 4 2[ , ] [ , ] [ , ]° ¬ ¼¬ ¼¯ ¿

1 1

2 2

3 3

[0,a ] 0 0 [0,a ]

0 [0,a ] , [0,a ] 0

[0,a ] 0 0 [0,a ]

- ½ª º ª º° °« » « »® ¾« » « »

° °« » « »¬ ¼ ¬ ¼¯ ¿

V1 V2 = V, W is a pseudo set interval bivector

V over the set S.


= {([0, a1], [0, a2], [0, a3], [0, a4], [0, a5]) | ai Z7; 1

7

[0, a] [0, a]a Z

[0, a] [0, a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

be a set interval bilinear algebra over the set S = {0,

Choose W = W1 W2 =

{([0, a], 0, [0, a], 0, [0, a]), ([0, a], [0, a], 0, [0, a], 0)

7

[0,a] 0 0 [0,a], a Z

[0,a] 0 0 [0,a]

- ½ª º ª º° °® ¾« » « »

° °¬ ¼ ¬ ¼¯ ¿

Example 4.1.21: Let V = V1 V2 = {{[0, ai] | ai

1 i 7

1

[0,a ]a Z

[0,a ]

- ½ª º° °® ¾« »¬ ¼° °¯ ¿



be a set interval bivector space over the se

bigenerator of V is

X = {[0, 1]} [0,1]

[0,1]

- ½ª º° °® ¾« »° °¬ ¼¯ ¿

.

Clearly the bidimension of V is finite and is

Interested reader is expected to derive m

However the concept of bilinearly independent can also be defined in an analogous way. We see

set interval bivector space given in example 4.1.2

{[0, 1]} [0,1]

[0,1]

ª º« »¬ ¼

.

The bidimension is {1, 1}.

We will illustrate this by another example.

Example 4.1.22: Let

V = V1 V2

=

1 2[0,a ] [0,a ]

[0 a ] [0 a ] a Z {0}

- ª º° « » ®« »

X =

[0,1] 0 0 [0,1] 0 0

0 0 , 0 0 , [0,1] 0 ,

0 0 0 0 0 0

-ª º ª º ª º°« » « » « »®« » « » « »°

« » « » « »¬ ¼ ¬ ¼ ¬ ¼¯0 0 0 0 0 0 ½ª º ª º ª º

°« » « » « »



0 [0,1] , 0 0 , 0 0

0 0 [0,1] 0 0 [0,1]

°« » « » « »¾« » « » « »°« » « » « »¬ ¼ ¬ ¼ ¬ ¼¿

{1, x, x2, x3, x4, x5, x6} = X1

X2

is a bilinearly independent bisubset of V and X =

bigenerates V thus X is a bibasis of V.

We define yet another set of interval bivector spaces

interval bivector spaces.

DEFINITION 4.1.7: Let V = V 1 V 2 where V 1 is a vector space over the set S 1 and V 2 is another set int

space over the set S 2 where V 1 and V 2 are distinct tha

and V 2 V 1 and S 1 and S 2 are distinct that is S 1

S 1.Then we define V = V 1 V 2 to be a biset inte

bispace over the biset S = S 1 S 2 or V is a bi

bivector space over the biset S = S 1 S 2.

We will illustrate this situation by some simple exam


be a biset interval bivector space over the biset

Z12 Z42.


[0 ]- ½ª º



=

1

2 i

3

4

[0,a ]

[0,a ] a Z {0}

[0,a ] 1 i 4

[0,a ]

- ½ª º° °« »

° °« »® ¾« » d d° °« »° °« »¬ ¼¯ ¿

{[0, ai] |

be a biset interval bivector space over the biset

(Z+ {0}) Z7.


=24

i

i i 45

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

{all 10 u 10 interval matrices with entries from

biset interval bivector space over the biset S = S

3Z+ {0}.

Now we proceed onto define substructure in this

DEFINITION 4.1.8: Let V = V 1 V 2 be a biset i

space over the biset S = S 1 S 2. Let W = W 1

V be a proper subset of V.

If W = W 1 W 2 is a biset interval bivector

bi t S S S th d fi W t b

{All 17 u 17 upper triangular interval matrices with e

Z12} be a biset interval bivector space over biset the

{0}) Z12 = S1 S2.

Take W = W1 W2 =

[0 ] [0 ] Q {0}- ½ª º° °



1 2 i

3

[0,a ] [0,a ] a Q {0};

0 [0,a ] 1 i 3

- ½ª º ° °® ¾« » d d¬ ¼° °¯ ¿

{All 17 u 17 diagonal interval matrices with entries f

V1 V2 = V, W is a biset interval bivector subspac

the biset S = S1 S2.

Example 4.1.27 : Let V = V1 V2 =

1 2 3 i 42

4 5 6

[0,a ] [0,a ] [0,a ] a Z

[0,a ] [0,a ] [0,a ] 1 i 6

- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

25i 25i

i

i 0

a Z ;[0,a ]x

1 i 6

- ½® ¾

d d¯ ¿¦

be a biset interval bivector space over the biset S =

Z42 Z25. Choose

W = W1 W2

=

1 2 3 i 42

1 2 3

[0,a ] [0,a ] [0,a ] a Z ;

[0,a ] [0,a ] [0,a ] 1 i 3

- ½ª º° °

® ¾« » d d¬ ¼° °¯ ¿

We will first illustrate this situation by some exam


a a a- ½ª º§ ·



=

1 2 3

i

4 5 6

7 8 9

a a aa Q {0};

a a a1 i 9

a a a

- ½ª º§ ·° ° « »¨ ¸® ¾« »¨ ¸ d d° °¨ ¸« »© ¹¬ ¼¯ ¿

1 2 3 4 i

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ] a Q

[0,a ] [0,a ] [0,a ] [0,a ] 1

- ª º ° ® « » d ¬ ¼° ¯

be a quasi set interval bivector space over the set


=

1 5

2 6 i 8

3 7

4 8

[0,a ] [0,a ]

[0,a ] [0,a ] a Z ;

[0,a ] [0,a ] 1 i 8[0,a ] [0,a ]

- ½ª º° °« » ° °« »® ¾

« » d d° °« »° °¬ ¼¯ ¿

26

i

i 0

[0,a ]x

- ®

¯

¦

be a quasi set interval bivector space over the set


= i[0 a ]x a Z {0}f

- ½ ® ¾¦

is a quasi set interval bivector subspace of V over th

is a quasi set interval bivector spaces over the sets.

For instance if V = V1 V2

- ½ 1 2[0 a ] [0 a ]-ª º



=40

i

i i 28

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

1 2

3 4

5 6

[0,a ] [0,a ]a

[0,a ] [0,a ]1

[0,a ] [0,a ]

- ª º° « »® « »° « »¬ ¼¯

be a quasi set interval bivector space over the set S =

Let W = W1 W2

=

20

ii i 28

i 0a x a Z

- ½® ¾¯ ¿¦

1

i2

3

[0,a ] 0a

0 [0,a ] 1[0,a ] 0

- ª º°

« »® « » d° « »¬ ¼¯

V1 V2; W is a quasi set interval bivector sub

over the set S.


=1 2 3 4 5 i

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] a Z

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] 1 i

- ª º ° ® « » d ¬ ¼° ¯

{10 u 10 upper triangular matrices with entries from

be a quasi set interval bivector space over the set S =

Now we proceed onto define the new notion

interval bivector space.

DEFINITION 4.1.11: Let V = V 1 V 2 where V 1 space over the set S 1 and V 2 is a set interval ve



the set S 2. We call V = V 1 V 2 to be a quas

bivector space over the biset S = S 1 S 2.



=

1 2 3

i 18

4 5 6

7 8 9

a a a

a Z ;a a a1 i 9

a a a

- ½ª º§ ·

° °« »¨ ¸® ¾« »¨ ¸ d d° °¨ ¸« »© ¹¬ ¼¯ ¿

1

3

5

7

[0,a ] [0,

[0,a ] [0,[0,a ] [0,

[0,a ] [0,

- ª ° « ° « ® « ° « ° ¬ ¯

be a quasi biset interval vector bispace over the

S2 = Z18 Z11.

Example 4.1.33: Let V = V1 V2 = {all 12 u 1

entries from Z+ {0}}

ii i 29

i 0[0,a ]x a Z

f

- ½® ¾¯ ¿¦

1 2 3

i

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]a 5Z {0};

[0,a ] [0,a ] [0,a ]1 i 9

[0,a ] [0,a ] [0,a ]

- ª º

° « »® « » d d°

« »¬ ¼¯

be a quasi biset interval bivector space over the bis




S2 = (13Z+ {0}) {15Z+ {0}).

Now we give examples of quasi biset interval bive

their substructures.

Example 4.1.35: Let V = V1 V2 = {All 6 u 6 interv

with entries from Z7}

i

i i

i 0

a x a Q {0}f

- ½ ® ¾¯ ¿¦


Q+ {0}. Choose W = W1 W2 = {all 6 u 6 uppe

interval matrices with entries from Z7}

2i

i i

i 0

a x a Q {0}f

- ½ ® ¾

¯ ¿¦

V

1 V

2= V be a quasi biset interval bivector sub

over the biset Z7 Q+ {0}.




Example 4.1.38: Let V = V1 V2 = {[0, a] | a Z7}

5

a

a a Z

- ½ª º° °« » ® ¾« »



5a a Z

a

® ¾« »° °« »¬ ¼¯ ¿

be a quasi biset interval bivector space over the bise

Z5 = S1 S2. V is a doubly simple quasi bis

bivector space over the biset S.

Example 4.1.39: Let V = V1 V2 = {(a, a, a, a, a, a

Z2}

3

[0, a] [0, a]

[0,a] [0,a] a Z

[0, a] [0, a]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

be a quasi biset interval bivector space over the bisZ3. Clearly V is a quasi doubly simple interval biv

over the biset S = Z2 Z3.

Now we proceed onto define the notion of quasi

linear algebra semiquasi set interval linear algebra.

DEFINITION 4.1.13: Let V = V 1 V 2 be a quasi bivector space over the set S. Suppose each Vi is c

be a quasi set interval linear bialgebra over the

{0}.


-ª



=9

i 7i

i

i 0

a Z ;[0,a ]x

0 i 9

- ½® ¾

d d

¯ ¿

¦ 1 2

3 4

5 6

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ]

- ª ° « ® «

° « ¬ ¯

be a quasi set interval linear bialgebra over

Clearly V is of finite order where as V given in

of infinite order.

Example 4.1.42: Let V = V1 V2 = {All 5 u 5 i

with entries from Q+ {0}} {all 3 u 7 matri

from Q+ {0}} be a quasi set interval bilinear a

set S = 13Z+ {0}.

Now we proceed onto define semi quasi interval (linear bialgebra) over the set S.

DEFINITION 4.1.14: Let V = V 1 V 2 where V 1linear algebra over the set S and V 2 is a set vethe same set S (or V 1 is a set interval vector spac

and V 2 is a set linear algebra over the set S). Wa semi quasi set interval bilinear algebra over th

{All 9 u 3 matrices with entries from Z45} be a sem

interval bilinear algebra over the set S = {0, 1, 5, 7,

42} Z45.


-ª º



=

1 2 3 4 5

i

6 7 8 9 10

11 12 13 14 15

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]a

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]1

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « »

® « »

° « »¬ ¼¯

a

b ,[a,b,c,d,e,f ] a,b,c,d,e,f R {0

c

- ª º° « » ® « »° « »¬ ¼¯

be a semi quasi set interval bilinear algebra over the

1, 2 , 7 5 ,13

19, 43 , 52, 75, 1031} R

+ {0


3

a,[a, b, c, d] a, b, c,d Z

b

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

be a semi quasi set interval bilinear algebra over the

1, 2} = Z3.

N d t d fi i bi t i t l bili

Example 4.1.46 : Let V = V1 V2

=

a b c

a b , d e f a,b,c,d,e,f ,g,h,ic d

g h i

- ª º

ª º° « » ® « » « »¬ ¼° « »¬ ¼¯



1

i 7

2

3

[0, a ]a Z ;

[0,a ]

1 i 3[0,a ]

- ½ª º° °« »

® ¾« »d d° °« »¬ ¼¯ ¿

be a quasi biset interval bilinear algebra over the

Z7.


=

1

i

2 1 2

3

[0,a ]a Z {

[0, a ] , ([0, a ],[0, a ])1 i 3

[0,a ]

- ª º° « »® « » d d° « »¬ ¼¯

{all 3 u 5 matrices with entries from Z49} b

interval bilinear algebra over the biset S = 5Z+

We see the quasi biset interval bilinear algebra g

4.1.46 is of finite order where as the quasi biset

algebra given in example 4.1.47 is of infinite ord

E l 4 1 48 L t V V V

be a quasi biset interval bilinear algebra over the bis

S2 = {0, 1} {1, 2, 5, 0, 7}.

Clearly the quasi biset interval bilinear algeb

example 4.1.48 is of infinite order.

Now we will proceed onto give examples of substr

the reader is given the simple task of defi



the reader is given the simple task of defi

substructures.


1

21 3 5 i 29

2 4 6 3

4

[0,a ]

[0,a ][0,a ] [0,a ] [0,a ] a Z ;,

[0,a ] [0,a ] [0,a ] [0,a ] 1 i 6

[0,a ]

- ª º° « » ª º° « »® « » « » d d¬ ¼° « »° « »

¬ ¼¯ 25

i

i i 29

i 0

a x a Z

- ½® ¾

¯ ¿¦

be a quasi set interval bilinear algebra over the set S

Choose W = W1 W2

=

1

2 i 29

3

4

[0,a ]

[0,a ] a Z ;

[0,a ] 1 i 4

[0,a ]

- ½ª º° °« » ° °« »® ¾« » d d° °« »° °« »¬ ¼¯ ¿

15

i

i i 29

i 0

a x a Z

- ®

¯ ¦

{[0, a] | a Z+ {0}} be a quasi set interval

over the set 5Z+ {0} = S. Take W = W1 W2

b, c Z+ {0}} {[0, a] | a 15Z+ {0}} W is a quasi set interval bilinear subalgebra of V

= 5Z+ {0}.




a b

c da b e

, a,b,c,d,e,f ,g,h,i, je f c d f

g h

i j

- ª º° « »° « »ª º° « »® « »

« »¬ ¼° « »° « »

° ¬ ¼¯ 1 2 3

i 47

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]a Z

[0,a ] [0,a ] [0,a ]1 i 9

[0,a ] [0,a ] [0,a ]

- ª º° « »

® « » d d° « »¬ ¼¯


Z47. Choose W = W1 W2 =

17

a b ea, b,c,d,e,f Z

c d f

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

1 2 3

i 47

[0,a ] [0,a ] [0,a ]a Z

0 [0 a ] [0 a ]

- ª º° « »

®« »



Example 4.1.54 : Let V = V1 V2 = {all 5 uentries from Z+ {0}}

1

1 2 3

2

4 5 i

3

[0, a ][0,a ] [0,a ] [0,a ]

[0,a ], 0 [0,a ] [0,a ] a 3Z

[0, a ]0 0 [0 ]

- ª º° ª º« »° « »« » ® « »« »° « »« » ¬ ¼



3

6

4

[ , ]0 0 [0,a ]

[0,a ]

° « »« » ¬ ¼° « »¬ ¼¯

be a quasi set interval bilinear algebra over the

{0}. Choose W = W1 W2 = {all 5 u 5 u

matrices with entries from Z+ {0}}

i

[0,a] [0,a] [0,a]

0 [0,a] [0,a] a 3Z {0}

0 0 [0,a]

- ½ª º

° °« » ® ¾« »° °« »¬ ¼¯ ¿

V

and P = 33Z+ {0} S = 3Z+ {0}. Clearly W

a quasi subset interval bilinear subalgebra of V o

S. However it is possible that V has no quasi bilinear subalgebra in such cases we call V

simple quasi set interval bilinear algebra.

We will illustrate this situation by some exam

Example 4.1.55: Let V = V1 V2 = {[0, a]| a

- ½ª º

simple quasi set interval bilinear algebra which we

call as doubly simple quasi set bilinear algebra.


V = V1 V2



= 7

[0, a] [0,b]a,b,c,d Z

[0,c] [0,d]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

i

i i 7

i 0

a x ,(a,b,c,d) a ,a,b,c,d Zf

- ½® ¾

¯ ¿¦

be a quasi set interval bilinear algebra over the set S

Since S cannot have proper subsets of order greequal two we see V is a pseudo simple quasi set inter

algebra. However V is not a simple quasi set inter

algebras as

W = 7

[0, a] [0, a]

a Z[0, a] [0, a]

- ½ª º° °

® ¾« »° °¬ ¼¯ ¿

i

i ii 0 a x a

f

-

® ¯ ¦ V1 V2 = V is a quasi set interval bilinear suba

over the set S = {0, 1}.

Thus V is not a doubly simple quasi set int

algebra over the set S.

Now we will give yet another example to show t

be a quasi set interval bilinear algebra over t

Clearly V has no quasi set interval bilinear suba

Z19.

However we see S can have several subsethave any proper pseudo quasi subset in

subalgebras.

Hence V is a doubly simple quasi set i



algebra over Z19 = S.

Now we proceed onto define the notion ofinterval bilinear subalgebras.

DEFINITION 4.1.16: Let V = V 1 V 2 be a quas

bilinear algebra over the biset S = S 1 S 2. Let W

V 1 V 2 and P = P 1 P 2 S 1 S 2 both W an

bisubsets of V and S respectively. Suppose W interval bilinear algebra over the biset P = P 1

W to be a quasi bisubset interval bilinear subalgthe bisubset P of S.

We will say V is pseudo simple quasi biset ialgebra over the bisubset interval bilinear sub

bisubset P = P 1 P 2 of S = S 1 S 2.

We will illustrate these situations by some simpl


a b a b a b a b, a,b,c,d

c d b a b a b a

- ª º ª º° ® « » « »

¬ ¼ ¬ ¼°̄

Choose W = W1 W2

= 6

a ba,b,c,d Z

c d

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

1

2

3

4

[0,a ] 0

0 [0,a ] a

[0,a ] 0 1

0 [0,a ]

- ª º

° « »° « »® « »° « »° ¬ ¼¯



4[ , ]¬ ¼¯

V1 V2 = V such that W is a quasi bisubset interalgebra over the subbiset P = {0, 3} {0, 2, 4, 6}


1

21 2 3 4 i

5 6 7 8 3

4

[0,a ]

[0,a ][0,a ] [0,a ] [0,a ] [0,a ] a,

[0,a ] [0,a ] [0,a ] [0,a ] [0, a ] 1 i

[0,a ]

- ª º

° « » ª º° « »® « » « » d ¬ ¼° « »° « »¬ ¼¯

2

a ba,b,c,d Z

c d

- ½ª º° °

® ¾« »¬ ¼° °¯ ¿

be a quasi biset interval bilinear algebra over the bise

{0, 1}.

We see V has no quasi subbiset interval bilinear

over S as S does not contain any proper subbiset.Thus V is a pseudo simple quasi biset interv

algebra over the set S = Z3 {0, 1}.

1

2

1 2 3 4

3

4

[0,a ]

[0,a ][0,a ] [0,a ] [0,a ] [0,a ] ,

[0,a ][0,a ]

- ª º° « »° « »®

« »° « »° « »¬ ¼¯





Z2. Clearly V has no quasi biset interval biline

Also V does not contain any quasi subbiset isubalgebras. Thus V is both a pseudo simple qua

algebra as well as simple quasi biset interval b

We call a quasi biset interval bilinear algebra w

simple quasi biset interval bilinear algebra as

simple quasi biset interval bilinear algebra as

quasi biset interval bilinear algebra.

We have given examples of all types of qua

bilinear algebras. Now we proceed onto give

and the reader is expected to prove them.

THEOREM 4.1.1: Every quasi set interval bilinea set S is a quasi set interval bivector space and general is not true.

THEOREM 4.1.2: Every set interval bilinear ainterval bivector space and not conversely.

THEOREM 4.1.3: Every set interval bilinear alg set interval bilinear algebra and not conversely.

4.2 Semigroup Interval Bilinear Algebras and TheiProperties

In this section we define semigroup interval biline

and several related structures and substructures asso

them. Main properties about them are discussed in th



DEFINITION 4.2.1: Let V = V 1 V 2 be such

semigroup interval vector space over the semigroupalso a semigroup interval vector space over the same

S; where V 1 and V 2 are distinct with V 1 V 2 or V 2

We define V = V 1 V 2 to be a semigroup interv

space over the semigroup S.



1 2 3 i 19

4 5 6

[0,a ] [0,a ] [0,a ] a Z ;

[0,a ] [0,a ] [0,a ] 1 i 6

- ½ª º° °® ¾« » d d

¬ ¼° °¯ ¿

i

i 0

[0,a ]xf

- ®

¯

¦

be a semigroup interval bivector space over the semZ19.


1 2[0,a ] [0,a ]- ½ª º° °« »


1 3

i2 4

5 6

[0,a ] 0 [0,a ] 0

a Z0 [0,a ] 0 [0,a ]1 i

[0,a ] 0 [0,a ] 0

- ª º

° « »® « » d ° « »¬ ¼¯



1 2 3

i 12

4 5 6

7 8 9

[0,a ] [0,a ] [0,a ]a Z

[0,a ] [0,a ] [0,a ] 1 i 9[0,a ] [0,a ] [0,a ]

- ª º° « »

® « » d d° « »¬ ¼¯

be a semigroup interval bivector space over the

{0, 3, 6, 9}.

Now we define two substructures in them.

DEFINITION 4.2.2: Let V = V 1 V 2 be a sem

bivector space over the semigroup S. Choose W

V 1 V 2 = V 2; W a proper subset of V; if W itsinterval bivector space over the semigroup S th

to be a semigroup interval bivector subspace semigroup S. If V has no proper semigroup i subspace then we call V to be a simple sembivector space.



be a semigroup interval bivector space over the sem


1 2 i 5[0,a ] [0,a ] a Z ;

[0 a ] [0 a ] 1 i 4

- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

1

i

[0, a ]

0 a

1 i[0 a ]

- ª º° « » ° « »® « » d°



3 4[0,a ] [0,a ] 1 i 4d d¬ ¼° °¯ ¿ 2 1 i[0,a ]

0

« » d ° « »° ¬ ¼¯

V1 V2 = V; W is a semigroup interval bivector V over the semigroup Z5.


> @> @

> @

0,a0,a , a Z {0}

0,a

- ½ª º° °

« »® ¾« »° °¬ ¼¯ ¿

> @

> @> @

> @

7i

i i

i 0

0,a

0,a , [0,a ]x a,a 3Z {0}0,a

0,a

- ½ª º° °« »

° °« »° ° ® ¾« »° °« »° °« »

¬ ¼° °¯ ¿

¦

be a semigroup interval bivector space over the sem

4Z+ {0}. Take W = W1 W2 =

-

the semigroup interval bivector space given in e

of infinite order.


{[0 a] | a Z29}

> @

> @ 2

0,a

0 a a Z

- ª º° « »°

« »®



{[0, a] | a Z29} > @

> @20,a a Z

0,a

« »® « »° « »° ¬ ¼

¯ be a semigroup interval bivector space over the

Z29. V is a simple semigroup interval bivector sp

semigroup interval bivector subspaces.


=

> @ > @ > @> @ > @ > @> @ > @ > @

13

0,a 0,a 0,a

0,a 0,a 0,a a Z

0,a 0,a 0,a

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

{([0, a], [0, a], [0, a], [0, a], [0, a]) | a Z13} interval bivector space over the semigroup S

simple semigroup interval bivector space over S

DEFINITION 4.2.3: Let V = V 1 V 2 be a sem

bivector space over the semigroup S. Let W = W

V 2 = V and P S (W and P are prope


1 2 i 12

1 23 4

[0,a ] [0,a ] a Z ;

, [0,a ] [0,a ][0,a ] [0,a ] 1 i 4

- ½ª º°

® « » d d¬ ¼° ¯ ¿

1 2[0,a ] [0,a ]- ½ª º° °« »



3 4

i 12

5 6

7 8

[0,a ] [0,a ]a Z

[0,a ] [0,a ][0,a ] [0,a ]

° °« »° °« » ® ¾

« »° °« »° °¬ ¼¯ ¿

be a semigroup interval bivector space define

semigroup S = Z12. Choose

W =1 2 i 12

3 4

[0,a ] [0,a ] a Z ;

[0,a ] [0,a ] 1 i 4

- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

1 2

i 12

3 4

[0,a ] [0,a ]

0 0 a Z ;

[0,a ] [0,a ] 1 i 4

0 0

- ½ª º° °« » ° °

« »® ¾« » d d° °« »° °¬ ¼¯ ¿

V1 V2 = V1 and P = {0, 4, 8} Z12. W = Wsubsemigroup interval bivector subspace of V

subsemigroup P of S = Z12.

l 4 2 9

> @> @> @> @ > @> @

> @> @

> @ > @

1 2

1 2 3 4 3 4

6 5

0,a 0,a 0

0,a 0,a 0,a 0,a , 0 0,a 0,a

0,a 0 0,a

- ª ° « ® « ° «

¬ ¯

be a semigroup interval bivector space over the

5Z+ {0}.

h



Choose W = W1 W2 =

1

1 3

3

[0,a ]

0 a ,a Z {0}

[0,a ]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

> @> @

> @

1

4 1 6 4

6

0 0,a 0

0 0 0,a a ,a ,a are in

0,a 0 0

- ª º° « »® « »° « »¬ ¼¯

V1 V2 = V; and P = {125Z+ {0}} S. W

subsemigroup interval bivector subspace ofsubsemigroup P of S.


=1

i 5

2

[0,a ] a Z ;[0,a ]

1 i 3[0 ]

- ½ª º ° °« »® ¾« » d d° °« »


7

[0, a] [0, a]

a Z[0, a] [0, a]

- ½ª º° °

® ¾« »¬ ¼° °¯ ¿

{([0, a], [0, a], [0, a], [0, a], [0, a], [0, a]) | a semigroup interval bivector space over the semigroup



semigroup interval bivector space over the semigroup

Clearly V is a doubly simple semigroup interv

space over the semigroup S = Z7.


[0,a] [0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a] [0,a] a Z

[0,a] [0,a] [0,a] [0,a] [0,a]

- ª º° « » ® « »° « »¬ ¼¯

i

i i 5

n 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a semigroup interval bivector space over the semZ5. Clearly V is a doubly simple semigroup interv

space over the semigroup S = Z5.

We see there is difference between the semigro

bivector space described in example 4.2.11 and 4.

see in example 4.2.11 both V1 and V2 are doubly sias we see in example 4.2.12 only V1 is doubly sim

infact has a semigroup interval bivector subspace viz

not used in the mutually exclusive sense) then w

semi simple semigroup interval bivector space.


i

i i 19

i 0

a x a Zf

- ½® ¾

¯ ¿¦

[0, a] [0, a]a

[0, a] [0, a]

- ª º° ® « »

¬ ¼° ¯



be a semigroup interval bivector space over theZ19. Clearly V is a semi simple semigroup in

space.

THEOREM 4.2.1: Let V = V 1 V 2 be a sem

bivector space defined over the semigroup S = Z

can be either a doubly simple semigroup bviec semi simple semigroup bivector space.

Proof is left as an exercise to the reader.

Now we proceed onto define the notion of sem

bilinear algebra.

DEFINITION 4.2.5: Let V = V 1 V 2 be a sembivector space over the semigroup S. If both closed under addition that is they are semaddition then we call V to be a semigroup in

algebra over the semigroup S.



= i

i i

i 0

[0,a ]x a Z {0}f

- ½ ® ¾¯ ¿¦

{{([0, ai] [0, ai] [0, ai])}| ai SZ+ {0}} be a



interval bilinear algebra over the semigroup 3Z+ {0

We have an interesting related result.

THEOREM 4.2.2: Let V = V 1 V 2 be a semigrobilinear algebra over the semigroup S then V is ainterval bivector space over the semigroup S but th

however is not true.


Now we proceed onto define substructures of these s

DEFINITION 4.2.6: Let V= V 1 V 2 be a semigrobilinear algebra over the semigroup S. Let W = W 1 V 2 = V; suppose W is a semigroup interval bilineover the semigroup S then we call W to be a semigrobilinear subalgebra of V over the semigroup S. If

semigroup interval bilinear subalgebra then we defin

simple semigroup interval bilinear algebra over theS.

=2 1

i

3 4 5

0 [0,a ] 0 [0,a ] 0a

[0,a ] 0 [0,a ] 0 [0,a ]

- ª º° ® « »

¬ ¼° ¯

2ii i 12

i 0

[0,a ]x a Zf

- ½® ¾¯ ¿¦

V1 V2 = V; W is a semigroup interval biline

th i S Z



over the semigroup S = Z12.


1 2 3

4 5 6 i

7 8 9

[0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] a Z {0};1

[0,a ] [0,a ] [0,a ]

- ª º° « » d® « »° « »

¬ ¼¯

1

2

3 i

4

5

[0, a ]

[0,a ]

[0,a ] a Z {0};1 i 5

[0,a ]

[0,a ]

- ª º° « »° « »° « » d d®

« »° « »° « »° ¬ ¼¯

be a semigroup interval bilinear algebra over the

3Z+ {0}. Take W = W1 W2 =

1 2 3[0,a ] [0,a ] [0,a ]- ª º°« »

V1 V2 = V; W is a semigroup interval bilinear

of V over the semigroup S = 3Z+ {0}.

Example 4.2.18. Let V = V1 V2 =

7

[0, a] [0, a]a Z

[0 a] [0 a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿ 7

[0,a]

[0,a]

a Z[0,a]

- ª º° « »° « »° « » ®

« »°



[0, a] [0, a]¬ ¼° °¯ ¿ [0,a]

[0,a]

« »° « »° « »° ¬ ¼¯

be a semigroup interval bilinear algebra over the sem

Z7. We see V has no semigroup interval bilinear

hence V is a simple semigroup interval bilinear algeb

semigroup S = Z7.

Example 4.2.19. Let V = V1 V2 =

11

[0,a] [0,a]

[0,a] [0,a]

a Z[0,a] [0,a]

[0,a] [0,a]

[0,a] [0,a]

- ½ª º° °« »

° °« »° °« » ® ¾« »° °« »° °« »° °¬ ¼¯ ¿

{([0, a], [0, a], [0, a], [0, a], [0, a])|a Z11} be ainterval bilinear algebra over the semigroup S =

simple semigroup interval bilinear algebra over the s

interval bilinear algebra over the semigroup S. Ifboth a simple semigroup interval bilinear alge

pseudo simple semigroup interval bilinear a

semigroup S then we call V to be a doubly siminterval bilinear algebra over the semigroup S.


- ½



1 2

i 123 4

[0,a ] [0,a ]

a Z ;1 i 4[0,a ] [0,a ]

- ½ª º° °

d d® ¾« »¬ ¼° °¯ ¿ i 0 [0,

f

-

® ¯ ¦ be a semigroup interval bilinear algebra over the

Z12 under addition modulo 12.

Choose W = W1 W2 =

1 2 i 12

3

[0,a ] [0,a ] where a Z

0 [0,a ] 1 i 3

- ½ª º° °® ¾« » d d° °¬ ¼¯ ¿

2i

i i

i 0

[0,a ]x a {0, 2, 4,6,8,10}f

ª º«

¬ ¼

¦

V1 V2 = V be a subsemigroup interval bili

of V over the subsemigroup P = {0, 6} Z12 = S


1 2 3 4

i 5

[0,a ] [0,a ] [0,a ] [0,a ]a Z ;1

- ª º° ®« »

be a semigroup interval bilinear algebra over the sem

Z5. Clearly S has no proper subsemigroups.

Take W = W1 W2 =

1 2

i 5

3 4

[0,a ] 0 [0,a ] 0a Z ;1 i

0 [0,a ] 0 [0,a ]

- ª º° d d® « »

¬ ¼° ¯



1 2

3

4 5 i 5

6

7 8

[0,a ] 0 [0,a ]

0 [0,a ] 0

[0,a ] 0 [0,a ] a Z ;1 i 8

0 [0,a ] 0

[0,a ] 0 [0,a ]

- ª º° « »° « »° « » d d®

« »° « »° « »° ¬ ¼

¯ V1 V2 be a semigroup interval bilinear subal

over the semigroup S = Z5.

However V has no proper subsemigroup intersubalgebra as S has no proper subsemigroups in S

addition modulo 5.


17

[0,a] [0,a] [0,a]a Z

[0,a] [0,a] [0,a]- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

[0,a] [0,a]

[0,a] [0,a]

[0,a] [0,a]

[0 a] [0 a]

- ª º° «

° « ° « ® « °«

Now having seen some of the substru

semigroup interval bilinear algebra we now proc

more properties about them.

DEFINITION 4.2.8: Let V = V 1 V 2 be suc semigroup interval linear algebra over the semi

is only a semigroup interval vector space semigroup S and V 2 is not a linear algebra then

V V t b i i i t l bili



V 1 V 2 to be a quasi semigroup interval biline

S.

We will first illustrate this situation by some sim


1 2 3

4 5 6 i

7 8 9

[0,a ] [0,a ] [0,a ][0,a ] [0,a ] [0,a ] a Z {0};1

[0,a ] [0,a ] [0,a ]

- ª º° « » ® « »° « »¬ ¼¯

> @1

2 1 2 3 4 5

3

[0,a ]

a[0,a ] , [0,a ], [0,a ], [0,a ], [0,a ], [0,a ]1

[0,a ]

- ª º° « »® « »° « »¬ ¼¯

be a quasi semigroup interval linear bial

semigroup S = 6Z+ {0}.


We can as in case of semigroup interval biline

define substructures. The definition is a matter of ro

left as an exercise for the reader.

How ever we will illustrate this situation by some ex


[0 ] [0 ]-ª º°



> @

[0, a] [0, a], [0,a] [0,a] [0,a] [0,a] [0,a]

[0, a] [0, a]

- ª º°

® « »° ¬ ¼¯

2i

i i 421

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

V = V1 V2; W is a quasi semigroup interv

subalgebra of V over the semigroup S = Z421.


3[0,a] ,[0,a] a Z[0,a]

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

[0,a] [0,a] [0,a][0, b] [0, b] [0, b]

- ª º° ® « »¬ ¼° ¯

be the quasi semigroup interval bilinear algeb

semigroup S = Z3.

Consider W = W1

W2

=

[0 a] [0 a] [0 a]-ª º°



be a quasi semigroup interval bilinear algeb

semigroup S = Z18.

Take W = W1 W2 =

[0,a] [0,a] [0,a], a,b {0,2,4,6,8,10,12,14,

[0,b] [0,b] [0,b]

- ª ºª º° ® « »« »

¬ ¼ ¬ ¼° ¯

> @^ 1 2 3 i 18[0,a ] [0,a ] [0,a ] a Z ;1 i 3 d d



V1 V2 = V and P = P{0, 9} Z18 (P is a su

under addition modulo 18 of the semigroup Z18).

W is a subsemigroup interval bilinear subalgebr

the subsemigroup P S = Z18.

Example 4.2.29: Let V = V1

V2

= {All 5 u 5 interv

with intervals of the form [0, ai] where ai Z+ {0}

1

2

31 2i

3 4 4

5

6

[0,a ]

[0,a ]

[0, a ][0,a ] [0,a ] , a Z {0};1[0,a ] [0,a ] [0,a ]

[0, a ]

[0, a ]

- ª º° « »° « »° « »

ª º° d« »® « »« »¬ ¼° « »° « »° « »° ¬ ¼¯

be a quasi semigroup interval bilinear algebsemigroup S = Z+ {0}. Let W = W1 W2 = {all 5

t i l t i ith i t l f th f [

V1 V2. W is a quasi subsemigroup in

subalgebra of V over the subsemigroup P = 3Z+

{0} = S.


7

[0,a] [0,a] [0,a], a Z

- ½ª ºª º° °® ¾« »« »



7, a Z[0,a] [0,a] [0,a]

® ¾« »« »¬ ¼ ¬ ¼

° °¯ ¿

{([0, a], [0, a], [0, a], [0, a], [0, a])| a Z

semigroup interval bilinear algebra over the sem

Since S has no proper subsemigroups we see

simple quasi semigroup interval bilinear algebr

Further as V has no proper semigroup interval bwe see V is a simple quasi semigroup interval b

Thus V is a doubly simple quasi semigroup i

algebra over the semigroup S = Z7.

Now we can define bilinear transformation of q

interval bilinear algebras V to W also the notoperator of a quasi semigroup interval bilinear al

This task is left as an exercise for the reader.

4.3 Group Interval Bilinear Algebras and their

In this section we proceed on to define the n

interval bivector spaces and describe a few of

(3) 0.v = 0.v1 0.v2

= 0 0 V 1 V 2 = V 0 is the additive identity of G.

We call V to be a group interval bivector space oveG.





Example 4.3.1: Let V V1 V2

[0,a][0,b]

[0, a] [0, a],[0, b], [0,c] a, b,c,d,e Z

[0,b] [0,b][0,d]

[0,e]

- ª º° « »° « »ª º° « » ® « »

« »¬ ¼° « »° « »° ¬ ¼

¯ {([0, a1], [0, a2], [0, a3], [0, a4], [0, a5]) | ai Z19; i

5} be a group interval bivector space over the group

is a group under addition modulo 19).


5i

i i 12

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

1

2

3

i 12

4

5

[0,a ]

[0,a ]

[0,a ]a Z ;1

[0,a ]

[0,a ]

[0 a ]

- ª º° « »° « »° « »°

d« »® « »° « »° « »° « »°¬ ¼¯

However we have infinite group interval

using Zn.

Example 4.3.3: Let V = V1

V2=

2i

i 42

i 0

[0,a]x a Zf

- ½® ¾

¯ ¿¦ i

i i

i 0 i 0

[0,a ]x , [0,a ]xf f

- ® ¯ ¦ ¦



be a group interval bivector space over the

Clearly V is of infinite order.

We now proceed onto define substructures rel

structures.

DEFINITION 4.3.2: Let V = V 1 V 2 be a group i

space over the group G. Let W = W 1 W 2 V 1W is a group interval bivector space over the grdefine W to be a group interval bivector subspac

group G.We say V is a simple group interval bivecto

no proper group interval bivector subspace.



1 2

1 2 3 i

3 4

[0,a ] [0,a ] , [0,a ] [0,a ] [0,a ] a[0,a ] [0,a ]

- ª º° ® « »¬ ¼° ¯

be a group interval bivector space over the group G =

Take W = W1 W2 =

^ `1 2 3 i 15[0,a ] [0,a ] [0,a ] a Z ;1 i 3 d d

15

[0, a] [0, a]

0 0

a Z[0, a] [0, a]

- ½ª º° °« »° °« »° °« » ® ¾

« »° °



0 0

[0, a] [0, a]

« »° °

« »° °« »° °¬ ¼¯ ¿

V1 V2 = V; W is a group interval bivector sub

over the group G.


i

i i 248

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

{all 10 u 10 square interval matrices with ent(Z248)} be a group interval bivector space over the

Z248.

Choose W = W1 W2 =

2i

i i 248

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

a Z[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

- ª º° « »° « »° « » ®

« »° « »° « »° ¬ ¼¯

b i l bi h



be a group interval bivector space over the

Clearly V is a simple group interval bivector spa


5

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a] a Z

[0,a] [0,a] [0,a] [0,a]

- ª º°

« » ® « »° « »¬ ¼¯

5

[0, a] [0, a]

[0, a] [0, a]a Z[0, a] [0, a]

[0, a] [0, a]

- ½ª º° °« »° °

« » ® ¾« »° °« »° °¬ ¼¯ ¿

be a group interval bivector space over the grou

easily verified V = V1

V2

simple group inspace over the group G = Z5.

define V to be a pseudo simple group interval bivect

V is both a simple and pseudo simple group interv space then we define V to be a doubly simple grobivector space over the group G.


Example 4.3.8: Let V = V1 V2 = {([0, a1], [0, a2]

a4]) | ai Z48; 1 d i d 4}



8i

i i 48

i 0

[0,a ]x a Z

- ½® ¾

¯ ¿¦

be a group interval bivector space over the group G

W = W1 W2 = {([0, a1], 0, [0, a2], 0) | ai Z48; 1 d

^ `8

i

i i

i 0

[0, a ]x a 0, 2, 4, 6,8,..., 44, 46

- ½® ¾

¯ ¿¦

V1 V2 and H = {0, 4, 8, 12, 16, 20, 24, 28, 32,

G a subgroup of Z48 under addition modulo 48.

W is a subgroup interval bivector subspace of V

subgroup G.


i

i i 18[0,a ]x a Zf- ½

® ¾¯ ¿¦

{6 u 6 upper triangular interval matrices with

(Z18)} V1 V2; W is a subgroup interval bivec

V over the subgroup H = {0, 9} Z18.


i

i i 11

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦



{set of all 11 u 15 interval matrices with entr= {[0, ai] | ai Z11}} be a group interval bivectogroup G = Z11. We see G = Z11 is a simple group

modulo 11. Hence V is a pseudo simple group i

space over G.

However V has group interval bivector subsp

a simple group interval bivector space over G.

Example 4.3.11: Let V = V1 V2 = {([0, a1], [0

a4]) | ai Z43}

43

[0, a] [0,b]

[0, a] [0,b]

[0, a] [0,b]a,b Z

[0, a] [0,b]

[0, a] [0,b]

[0, a] [0,b]

- ½ª º

° °« »° °« »° °« »° °

« »® ¾« »° °« »° °« »° °

« »° °¬ ¼¯ ¿

b i t l bi t th


THEOREM 4.3.2: Let V = V 1 V 2 be a group inter space over the group G = Z n , n not a prime,

1. V in general is not a pseudo simple grobivector space over the group G

2. V is not a simple group interval bivector sp= Z n.

This proof is also straight forward and hence left as



This proof is also straight forward and hence left as

for the reader to prove.

Now one can as in case of set interval bivector spacenotion of bilinear transformation of group interv

spaces. This task is also left as an exercise for the r

we proceed onto define the notion of group inter

algebras.

DEFINITION 4.3.4: Let V = V 1 V 2 be a group inter

space over the group G. We say V is a group interalgebra over the group G that is if both V 1 and V 2under addition.



1 2 3 4

i 12

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ] a Z ;1 i[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° d ® « »¬ ¼° ¯


1 2

3 4

i 7

5 6

7 8

[0,a ] [0,a ]

[0,a ] [0,a ] a Z ;1 i 7[0,a ] [0,a ]

[0,a ] [0,a ]

- ½ª º° °

« »° °« » d d® ¾« »° °« »° °¬ ¼¯ ¿

[0 a ] [0 a ] [0 a ] [0 a ] [0 a ]- ª º



1 2 3 4 5 i

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]a[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

- ª º° ® « »¬ ¼° ¯

be a group interval bilinear algebra over the grou

This V is of finite order.

Now we proceed onto give some properties eand define some substructures associated with th

THEOREM 4.3.3: Let V = V 1 V 2 be a group i space over the group G; then in general V needinterval bilinear algebra over the group G.

The proof can be given by an appropriate examp

THEOREM 4.3.4: Let V = V 1 V 2 be a group ialgebra over a group G then V is a group in

space over the group G.

The proof directly follows from the definition o




13

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] a Z

[0,a] [0,a] [0,a]

- ½ª º

° °« » ® ¾« »° °« »¬ ¼¯ ¿

[0, a] [0,

[0, a] [0,

[0, a] [0,

[0, a] [0,

[0, a] [0,

- ª ° « ° « ° « ®

« ° « ° « ° ¬ ¯

be a group interval bilinear algebra over the gro



be a group interval bilinear algebra over the gro

see V has no proper group interval bilinear subal

is a simple group interval bilinear algebra over

Z13.


3

[0, a] [0, a]a Z

[0, a] [0, a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

[0, a] [0, a][0, a] [0, a]

[0, a] [0, a]

[0, a] [0, a]

- ª º° « ° « ® « ° « ° ¬ ¼¯

be a group interval bilinear algebra over the groa simple group interval bilinear algebra over the

DEFINITION 4.3.6: Let V = V 1 V 2 be a group i

algebra over a group G. Let W = W 1 W 2

proper bisubset of V and H a proper subgroup

group interval bilinear algebra over the group HW to be a subgroup interval bilinear subalgebr



1 2 3 4

5 6 7 8

i 359 10 11 12

13 14 15 16

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]

a Z[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « »° « »

® « »° « »° ¬ ¼¯

be a group interval bilinear algebra over the

Take W = W1 W2 =



i 35

[0,a] [0,a] [0,a]

0 [0,a] [0,a] a Z

0 0 [0,a]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

1 2 3 4

5 6 7 8

i

9 10 11 12

13 14 15 16

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]a {0,5,10,15,

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

V1 V2 = V and H = {0, 7, 14, 21, 28}

subgroup of Z35 under addition modulo 35. W =

subgroup interval bilinear subalgebra of V over

of G.


For take W = W1 W2 =

5[0, a] [0, a] a Z[0, a] [0, a]

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

{([0, a], [0, a], [0, a], [0, a], [0, a]) | a Z5} V1 is a group interval bilinear subalgebra of V over the

Z5. So V is not a simple group interval bilinear algeb



5 p g p g

is not a doubly simple group interval bilinear algebgroup G.


23[0,a] [0,a] [0,a] [0,a] a Z[0,a] [0,a] [0,a] [0,a]

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

23

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

a Z[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a]

- ½ª º° °« »° °

« »° °« » ® ¾« »° °« »° °« »° °¬ ¼¯ ¿

be a group interval bilinear algebra over the group G

a simple group interval bilinear algebra over the gro

as V has no group interval bilinear subalgebras. Fu

THEOREM 4.3.5: Let V = V 1 V 2 be a group ialgebra over the group G = Z p; p a prime. V is a

group interval bilinear algebra over the group G

Proof: Follows from the fact that G is a group

proper subgroups.

THEOREM 4.3.6: Let V = V 1 V 2 be a group

algebra over the group G = Z n , n not a primetake entries from Z n. Then G is not a simple



f pbilinear algebra as well as G is not a pseud

interval bilinear algebra.

The proof is obvious from the fact that Zn has su

is not a prime and V1 and V2 constructed over Z

yield sub bispaces or sub bilinear algebras.


20

[0,a] [0,a] [0,a]a Z

[0,a] [0,a] [0,a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

20

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a]a Z

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a]

- ½ª º° °« »° °« » ® ¾« »° °« »° °

¬ ¼¯ ¿

2

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a]a {0,5,10,15} Z

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a]

- ª º° « »° « » ® « »°

« »° ¬ ¼¯

V1 V2; W is a subgroup interval bilinear suba

the subgroup H = {0, 5, 10, 15} Z20.

Now having seen some of the basic propertie



interval bilinear algebras, we can as in case of otalgebras define bilinear transformation and bilinear o

We can define some more properties like quasi gro

algebra.

DEFINITION 4.3.7: Let V = V 1 V 2 be a group inter space over the group G, if one of V 1 or V 2 (or in texclusive sense) is a group interval linear algebdefine V to be a quasi group interval bilinear algeb

group G.



[0, a] [0,b]

[0,b] [0, a][0, a] [0,b]

- ª º° « »° « »° « »° « »

i

i i 45

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a quasi group interval bilinear algebra overZ45.


[0, a] [0,b]- ½ª º



240[0,c] [0,d] a,b,c,d,e,f Z

[0, e] [0, f ]

° « » ® « »° « »¬ ¼¯ ¿

i

i 1 2 9 1 2

i 0

[0,a ]x , [0,a ] [0,a ] [0,a ] a ,af

- ® ¯ ¦ "

be a quasi group interval bilinear algebra over

Z240.

Now we can as in case of group interval b

define two types of substructures. We will hothis situation by some examples.


i

i i 14

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

[0 a ] [0 a ]-ª º

2i

i i 14

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

i 14

[0, a] [0, a]

[0, a] [0, a][0, a] [0, a]

, a Z[0, a] [0, a][0, a] [0, a]

[0, a] [0, a]

[0 ] [0 ]

- ½ª º° « »° « » ª º° « » ® « »

« » ¬ ¼° « »° « »

° ¬ ¼¯ ¿



[0, a] [0, a]° ¬ ¼¯ ¿

V1 V2 be a quasi group interval bilinear suba



1 2 3 4

5 6 7 8

i 18

9 10 11 12

13 14 15 16

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]a Z ;1 i

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « »° « » d ® « »° « »°

¬ ¼¯

1

2

1 2 3 i 18

3

4

[0,a ]

[0,a ][0,a ],[0,a ],[0,a ] a Z ;1 i

[0,a ]

[0,a ]

- ª º° « »° « » d ® « »°

« »° ¬ ¼¯

i

[0,a]

[0,a], [0,a],[0,a],[0,a] a Z

[0,a]

[0,a]

- ª º° « »° « » ® « »° « »° ¬ ¼¯

V1 V2 be a quasi group interval bilinear s





7

[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] a Z

[0,a] [0,a] [0,a]

- ½ª º° °« » ® ¾« »° °

« »¬ ¼¯ ¿

7

[0, a] [0, a]

[0, a] [0, a][0,a], a Z

[0, a] [0, a]

[0, a] [0, a]

- ½ª º° °« »° °« » ® ¾« »° °

« »° °¬ ¼¯ ¿

is a quasi group interval bilinear algebra over th

see V is a simple quasi group interval bilinear a

no quasi group interval bilinear subalgebras.


47

[0,a]

[0,a]

[0,a]

, [0,a] [0,a] a Z[0,a]

[0,a]

[0,a]

[0,a]

- ½ª º° °« »° °« »° °« »

° °« »° °® ¾« »° °« »° °« »° °« »° °« »

¬ ¼° °¯ ¿



be a quasi group interval bilinear algebra over the

Z47. Clearly V is a simple quasi group interval bilin


Next as in case of group interval bilinear algebra

same notion in case of quasi group interval bilineaWe will only illustrate this situation by some examp

task of giving the definition is left as an exercise to th


1 2 3 4

48

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ]a Z ;1 i

[0,a ] [0,a ] [0,a ] [0,a ]- ª º° d ® « »

¬ ¼° ¯

1

2

3 i

[0, a ]

[0,a ]

[0,a ][0 a ]x a Z ;1 i 6

f

- ½ª º° °« »

° °« »° °« »° ° d d« »® ¾¦



V1 V2 is a quasi subgroup interval bilinear suba


Thus V is not a doubly simple quasi group inter



{([0, a], [0, a]),

[0,a]

[0,a]

[0 a]

ª º« »« »

« »¬ ¼

| a Z43}



[0,a]« »¬ ¼


Z43. V is a doubly simple quasi group interval bilin


Example 4.3.32: Let V = V1 V2 = {([0, a], [0, a], [

[0, a]) | a Z47}

[0, a] [0, a]

[0, a] [0, a]

-ª º°®« »°¬ ¼¯

, ([0, a] [0, a]) | a Z47}


Z47. V is a doubly simple quasi group interval bilin


THEOREM 4.3.7: Let V = V 1 V 2 be a quasi gro

bilinear algebra over the group G = Z p; p a prime. pseudo simple quasi group interval bilinear algeb

Example 4.3.33: Let V = V1 V2 = {([0, a1], [0

a4], [0, a5]) | ai Z7, 1 d i d 5}

1

2

1 2

3 i 7

3 4

4

5

[0,a ][0,a ]

[0,a ] [0,a ], [0,a ] a Z ;1

[0,a ] [0,a ][0,a ]

[0,a ]

- ª º° « »

° « »ª º° « » d ® « » « »¬ ¼° « »° « »° ¬ ¼¯



be a quasi group interval bilinear algebra over th

Take W = {([0, a1], [0, a2], 0, 0, [0, a3]) | ai Z7;

7

[0,a]

0[0, a] [0, a], [0,a] a Z

0 [0,a]0

[0,a]

- ½ª º° °

« »° °« »ª º° °« » ® ¾« »« »¬ ¼° °« »° °« »° °¬ ¼¯ ¿

= W1 W2 V1 V2, W is a quasi group i

subalgebra of V over the group G = Z7. Thus V

quasi group interval bilinear algebra over the

Infact V has several such quasi group in

subalgebras.

Now we have a class of quasi group interval bwhich are not simple or pseudo simple We illus

4.4 Bisemigroup Interval Bilinear Algebras and Th

Generalization

In this section we define the notion of bisemigro

bilinear algebras, bigroup interval bilinear al

semigroup interval bilinear algebras set group inter

algebras and semigroup group interval bilinear al

describe some of their properties



describe some of their properties.

DEFINITION 4.4.1: Let V = V 1 V 2 where V i is ainterval vector space over the semigroup S i , i = 1,

V 1 V 2 , V 2 V 1 and S 1 z S 2 S 1 z S 2 and S 2 S 1. We

V 1 V 2 to be a bisemigroup interval bivector spa

bisemigroup S = S 1 S 2.



1

21 2 3

i

4 5 6 3

4

[0, a ]

[0,a ][0,a ] [0,a ] [0,a ], a Z {

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ]

- ª º° « »

ª º° « » ® « » « »¬ ¼° « »° « »¬ ¼¯

1 2 3[0 a ] [0 a ] [0 a ]- ª º


1 2

3 4 1 2 i 7

5 6

[0,a ] [0,a ]

[0,a ] [0,a ] , [0,a ] [0,a ] a Z ;1[0,a ] [0,a ]

- ª º° « »

® « »° « »¬ ¼¯

1

2

1 2 3 4

[0, a ]

[0,a ]

, [0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « »°

« »®« »° #



7

, [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ]

« »® « »° « »° ¬ ¼¯

#

be a bisemigroup interval bivector space over the

= S1 S2 = Z7 Z92.

Now if in the definition 4.4.1 each Vi iinterval linear algebra over Si, i = 1, 2 then we

bisemigroup interval bilinear algebra over the b

S1 S2.



1 2 3 4

5 6 7 8 i

9 10 11 12

[0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] a Q

[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° « » ® « »° « »¬ ¼¯

Example 4.4.4 : Let V = V1 V2 = {all 10 umatrices with intervals of the form [0, ai] with ai R

8[0, a] [0, a] a,b Z[0,b] [0,b]

- ½ª º° °® ¾« »¬ ¼° °¯ ¿

be a bisemigroup interval bilinear algebra with the b

S = R + {0} Z8.

Both the bisemigroup interval bilinear algebras in ex



g p gand 4.4.4 are of infinite cardinality.


1 2

3 4 i 17

5 6

[0,a ] [0,a ][0,a ] [0,a ] a Z ;1 i 6

[0,a ] [0,a ]

- ½ª º° °« » d d® ¾« »° °« »¬ ¼¯ ¿

i1 2 3 4 5

6 7 8 9 10

a[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] 1

- ª º°

® « » d¬ ¼° ¯

be a bisemigroup interval bilinear algebra over the b

S = Z17 Z102. We see the bisemigroup interval bilin

given in example 4.4.5 is of finite order.

We have as in case of other bilinear algebras the

The proof is simple and straight forward

exercise to the reader.

Now as in case of other bisemigroup line

bisemigroup vector spaces we can define substru

Here we only illustrate these situations by some e


1[0,a ]- ª º°



1

21 2

i 13

3 4 3

4

[0,a ]

[0,a ][0,a ] [0,a ], a Z ;1 i

[0,a ] [0,a ] [0,a ]

[0,a ]

- ª º° « »

ª º° « » d ® « » « »¬ ¼° « »° « »¬ ¼¯

1 2

1 2 3 4 4 5

7 8

[0,a ] [0,a ] [0,a

[0,a ] [0,a ] [0,a ] [0,a ] , [0,a ] [0,a ] [0,a

[0,a ] [0,a ] [0,a

- ª ° « ® « ° « ¬ ¯


= S1 S2 = Z13 Z18.

Take W = W1 W2 =

1

1 2 13

[0,a ]

[0,a] [0,a] 0, a ,a Z[0 a] [0 a] [0 a ]

- ª º°

« »ª º° « » ® « » « »¬ ¼°


1 2

3 4 i j

i

5 6

7 8

[0, a ] [0, a ]

[0, a ] [0, a ] a , a Z {0};,[0,a ]

[0, a ] [0, a ] 1 j 8; i, j Z {

[0, a ] [0, a ]

- ª º° « » ° « »® « » d d ° « »° ¬ ¼¯

1[0, a ]- ½ª º° °« »



2

j o 11i

i 3

i 0

4

5

[ ]

[0, a ]a , a Z ;

[0, a ]x ; [0, a ]1 j 5

[0, a ]

[0, a ]

f

- ½ª º° °« »° °« » ° °« »® ¾

d d« »° °« »° °« »° °¬ ¼¯ ¿

¦

be a bisemigroup interval bivector space over the bis

= S1 S2 = (Z+ {0}) {Z11}.

Let W = W1 W2 =

[0, a] [0, a]

0 0,[0,a] a Z {0}

[0, a] [0, a]

0 0

- ½ª º° °« »° °« » ® ¾« »° °« »° °¬ ¼¯ ¿

1[0, a ]- ½ª º° « »


1 2 3

4 5 6 i 15

7 8 9

[0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] a Z ;1 i[0,a ] [0,a ] [0,a ]

- ª º° « »

d ® « »° « »¬ ¼¯

1

2

3

[0, a ]

[0,a ]

[0,a ]Z 1 i 6

- ½ª º° °« »° °« »° °« »° °

« »® ¾



3

i 18

4

5

6

[ , ]a Z ;1 i 6

[0,a ]

[0,a ]

[0,a ]

° °« »° ° d d« »® ¾

« »° °« »° °« »° °« »° °¬ ¼¯ ¿

be a bisemigroup interval bilinear algebra over t

S = S1 S2 = Z15 Z18.

Take W = W1 W2 =

1

2 i 15

3

[0,a ] 0 0

0 [0,a ] 0 a Z ;1 i

0 0 [0,a ]

- ª º° « » d ® « »° « »¬ ¼¯

1[0,a ]0

- ½ª º° °« »

° °« »

Example 4.4.9: Let V = V1 V2 = {All 12 umatrices with intervals of the form [0, ai]; ai Z48}

ii i 40

i 0[0,a ]x a Z

f

- ½® ¾¯ ¿¦


S = S1 S2 = Z48 Z40. Choose W = W1 W2 = {upper triangular interval matrices with intervals of t

ai]; ai Z48} - ½



2i

i i 40

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦ V1 V2;

W is a bisemigroup interval bilinear subalgebra of

bisemigroup S. Now one can define bisubsemigroup interv

subspaces and bisubsemigroup interval bilinear subal

The task of defining these notions are left as an

the reader.

We will however illustrate these situations by some e


1

21 2 3 4 i

3

[0,a ]

[0,a ][0,a ] [0,a ] [0,a ] [0,a ] a, [0,a ]

[0 ] [0 ] [0 ] [0 ] 1

- ª º° « »

° « »ª º° « »® « » d« »¬ ¼°


= S1 S2 = Z24 Z150.

Take W = W1 W2 =

1

2 1 2 3

3

[0,a ]

0[0,a] 0 [0,a] 0

, [0,a ] a,a ,a ,a[0,a] 0 [0,a] 0

0

[0, a ]

- ª º° « »° « »ª º° « »® « »

« »¬ ¼°

« »° « »° ¬ ¼¯



3[ , ]° ¬ ¼¯

2i

i i 150

i 0

[0,a] 0

0 [0,a][0,a ]x , a ,a Z ;1

[0,a] 00 [0,a]

f

- ª º° « »° « » ®

« »° « »° ¬ ¼¯ ¦

V1 V2 = V and T = T1 T2 = {0, 3, 6, 9, 12

{0, 10, 20, …, 140} S1 S2.

It is easily verified W is a bisubsemigroup i

subspace of V over the bisubsemigroup T = T1 S2.

Example 4.4.11: Let V = V1 V2 = {All 9 u 9 i

with intervals of the form [0, ai]; ai Z+ {0interval matrices with intervals of the form [0, ai

1

[0,a] 0 [0,a] 0 [0,a]

0 [0,a] 0 [0,a] 0a Z

[0,a] 0 [0,a] 0 [0,a]

0 [0,a] 0 [0,a] 0

- ª º° « »° « » ® « »°

« »° ¬ ¼¯

V1 V2.

Clearly W is a bisubsemigroup interval bilinear

of V over the bisubsemigroup T = T1 T2 = 3Z+

V1 V2.If V has no proper bisubsemigroup interv



If V has no proper bisubsemigroup interv

subalgebras we call V to be a simple bisemigro

bilinear algebra. If V has no proper bisubsemigro

bilinear subalgebras we call V to be a pse

bisemigroup interval bilinear algebra. If V is both

pseudo simple then we call V to be a do bisemigroup interval bilinear algebra.

We will illustrate all these three situations by exampl

Example 4.4.12: Let V = V1 V2 = {All 8 u 8 uppe

interval matrices with entries from [0, ai] with ai Z

i

i i 19

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a bisemigroup interval bilinear algebra over the bS = S1 S2 = Z7 Z19. V is a pseudo simple b

V1 V2 is bisemigroup interval bilinear a

bisemigroup S = Z7 Z19 so V is not simple. A

has no proper subsemigroups V is pseudo simpl

doubly simple.


7

[0, a] [0, a]a Z

[0, a] [0, a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

[0,a] [0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] [0,a] a Z

- ª º° « » ®« »



[ , ] [ , ] [ , ] [ , ]

[0,a] [0,a] [0,a] [0,a]

® « »° « »¬ ¼¯

be a bisemigroup interval bilinear algebra over t

S = S1 S2 = {Z7} {Z11}. We see V is a

bisemigroup interval bilinear algebra over the bS1 S2 = Z7 Z11.

We have a class of pseudo simple bisemigroup

algebras over a bisemigroup S = S1 S2.

THEOREM 4.4.2: Let V = V 1 V 2 be a bisembilinear algebra over the bisemigroup S = S 1

where p and q two distinct primes. Then V is abisemigroup interval bilinear algebra over S.

general be simple.


THEOREM 4.4.4: Let V = V 1 V 2 be a bisemigro

bilinear algebra over the bisemigroup S = S 1

one of S 1 is Z + {0} or Q

+ {0} or R+ {0} and o

or some subsemigroup of Z + {0} or Q+ {0} o

such that S 1 S 2 and S 2 S 1 then V is not a dobisemigroup interval bilinear algebra over the bisem

This proof is also left for the reader.

We will give some illustrative examples.

E l 4 4 14 L t V V V




[0,a] [0,a] [0,a]

[0,a] [0,a] [0,a] a Z {0}

[0,a] [0,a] [0,a]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

45

[0,a]

[0,a]

[0,a]a Z[0,a]

[0,a]

[0,a]

[0,a]

- ½ª º° °« »° °« »° °

« »° °« »° °® ¾« »° °« »° °« »° °« »° °« »

¬ ¼° °¯ ¿


[0,a]

[0,a]

[0,a]

a {0,5,10,15,20,25,30,35,40}[0,a]

[0,a]

[0,a]

[0,a]

- ª º° « »° « »° « »°

« »° ® « »° « »° « »° « »° « »

¬ ¼° ¯

V1 V2 is bisubsemigroup interval bilinear s+



over the bisubsemigroup T = T1 T2 = 3Z+ {

15, 20, 25, 30, 35, 40} S1 S2 = Z+ {0} not a pseudo simple bisemigroup interval bilinea

V over the bisubsemigroup T = T1 T2 S1 S

Also W is a bisemigroup interval bilinear sover the bisemigroup S = S1 S2 so, V i

bisemigroup interval bilinear algebra over the b

S1 S2 = Z+ {0} Z45.


1 2

3 4 i

5 6

[0,a ] [0,a ]

[0,a ] [0,a ] a 3Z {0}

[0,a ] [0,a ]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

{([0, a1], [0, a2], …, [0, a10]) | ai 7Z+ {0}} b

i t l bili l b th bi i

DEFINITION 4.4.2: Let V = V 1 V 2 be such that interval vector space over the set S 1 and V 2 is ainterval vector space over the semigroup S 2. Then we

V 1 V 2 to be a set- semigroup interval bivector spa

set- semigroup S = S 1 S 2 .



1 2[0,a ] [0,a ]- ª º

°« »



3 4 1 2 3 i

5 6

[0,a ] [0,a ] , [0,a ] [0,a ] [0,a ] a {0,1,2,4

[0,a ] [0,a ]

° « » ® « »° « »¬ ¼¯

1 2 31

4 5 62 i

7 8 93

10 11 124

[0,a ] [0,a ] [0,a ][0,a ][0,a ] [0,a ] [0,a ][0,a ] a Z

,[0,a ] [0,a ] [0,a ][0,a ] 1 i 1

[0,a ] [0,a ] [0,a ][0,a ]

- ª ºª º° « »« » ° « »« »® « »« » d d° « »« »° ¬ ¼ ¬ ¼¯

be a set-semigroup interval bivector space ovsemigroup S = S1 S2 = {0, 1} {3Z+ {0}}.


1

2 1 2 9 i

[0,a ][0,a ] , [0,a ] [0,a ] ... [0,a ] a {0,1,2,

- ª º° « » ®« »

We can as in case of other interval alge

define substructures. We will leave the task o

definitions to the reader but, however we w

illustrative examples.


1 2 2i

i i

i 03 4

[0,a ] [0,a ], [0,a ]x a Q

[0,a ] [0,a ]

f

- ª º° ® « »

¬ ¼° ¯ ¦

[0 a ]- ª º



1

2

i i j

3

4

[0,a ]

[0,a ][0,a ], a ,a { 3, 5, 15,Z

[0,a ]

[0,a ]

- ª º° « »° « » ® « »° « »°

¬ ¼¯

be a semigroup-set interval bivector space over

set S = S1 S2 = Q+ {0} ^ `0, 3,1, 5, 15

Take W = W1 W2 =

1 2

3

[0,a ] [0,a ]

0 [0,a ]

-ª º°®« »°¬ ¼¯

, 16i

i i

i 0

[0,a ]x a Zf

- ®

¯ ¦

i i

[0,a]

0[0,a ], a,a {0, 3, 5, 15,4Z

0

- ª º

° « »° « » ® « »°


1

1 2 3 4 2i

5 6 7 8 3

4

[0, a ]

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ], a {0,1,2,[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ]

- ª º° « »

ª º° « » ® « » « »¬ ¼° « »° ¬ ¼¯

1 2

3 4

i5 6

1 2 3

[0,a ] [0,a ]

[0,a ] [0,a ]

a[0,a ] [0,a ], [0,a ] [0,a ] [0,a ]

- ª º° « »

° « »° « » ° « »®



1 2 3

7 8

9 10

11 12

, [0,a ] [0,a ] [0,a ][0,a ] [0,a ] 1 i

[0,a ] [0,a ]

[0,a ] [0,a ]

« »® d « »°

« »° « »° « »° ¬ ¼¯

be a set-semigroup interval bivector space ov

semigroup S = S1 S2 = {0, 1} {Z48}.

Take W = W1 W2 =

[0,a]

[0,a] [0,b] [0,a] [0,b] 0, a,b {0,1,2,

[0,b] [0,a] [0,b] [0,a] [0,a]

0

- ª º° « »ª º° « » ® « » « »¬ ¼° « »° ¬ ¼¯

[0,a] 0

0 [0,b]

- ª º

° « »° « »° « »°

Example 4.4.20: Let V = V1 V2 = {All 10

matrices with entries of the form [0, ai] with ai

1 2 3i i ji

i 0 4 5 6

[0,a ] [0,a ] [0,a ] a , a {5Z {a x ;

[0,a ] [0,a ] [0,a ] 1 j 6

f

- ª º ° ® « »

d d¬ ¼° ¯ ¦

be a semigroup –set interval bivector space over

set S = S1 S2 = Q+ {0} {5Z+ {0}, 19,

Take W = W1 W2 = {All 10 u 10 upper tri

matrices with entries of the form [0 ai] with ai



matrices with entries of the form [0, ai] with ai

i

i

i 0

[0,a] [0,a] [0,a]a x ; a {5Z {0},

[0,a] [0,a] [0,a]

f

- ª º° ® « »

¬ ¼° ¯ ¦

V1 V2 and T = T1 T2 = (7Z+ {0}) 2 } S1 S2 = Q+ {0} {5Z+ {0}, 19

W1 W2 is a subsemigroup-subset interval biv

of V over the subsemigroup-subset T = T1 T2 o

Example 4.4.21: Let V = V1 V2 = {collecti

interval matrices with entries of the form [0,

interval matrices of the form [0, b j] with b j, ai 3, 7, 19, 41, 23, 43, 101 } = S1} {C

16 u 16 interval matrices with intervals of the fall 7 u 1 interval matrices with intervals of the fo

1

2

i j

3

4

[0,a ]

0

[0,a ]

a ,a 7Z {0};1 j 4}0[0,a ]

0

[0,a ]

ª º« »« »« »« »

d d« »« »« »« »« »¬ ¼

V1 V2 and T = T1 T2 = {33Z+ {0}, 3, 1

{21Z+ {0}} S S W is a subset subsemigro



{21Z {0}} S1 S2. W is a subset-subsemigro

bivector subspace of V over the subset-subsemigrou

T2 S1 S2 = S.

If in the definition of the set-semigroup (sem

interval bivector space V = V1 V2 over S = S1 set interval linear algebra and V2 is a semigroup int

algebra then we define V to be a set-semigroup inter

algebra over S = S1 S2.



i i i ii

i

i 0i i i i 1

a ,b {13Z {0},a b 13[0,a ]x

a b 2,a b 3} S

f

- ° ®

° ¯ ¦

{All 5 u 5 interval matrices with intervals of the

{All 3 u 3 interval matrices with intervals of t

ai Z27} be a semigroup-set interval bilinear a

semigroup set S = S1 S2 = Z+ {0} {{0, 1,

Z27}.

Now we give examples of substructures.

Example 4.4.24 : Let V = V1 V2 = {all 9 u 9 in

with intervals of the form [0, ai] with ai Z240}

i

i i[0,a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦



i 0¯ ¿¦

be a semigroup set interval bilinear algebra over

set S = S1 S2 = Z240 {8Z+ {0}, 5Z+ {0}}

Take W = W1 W2 = {All 9 u 9 interval u

matrices with entries from Z240}

2i

i i

i 0

[0,a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦ V1

W is a semigroup set interval bilinear subalgebrsemigroup-set S = S1 S2.


1 2 3 4i

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ] a 5Z[0,a ] [0,a ] [0,a ] [0,a ]

- ª º ° « »®« »

be a set-semigroup interval bilinear algebra o

semigroup S = S1 S2 = {15Z+ {0}, 40Z+ {0}}

Take W = W1 W2 =

1 2

i

3 4

5 6

[0,a ] 0 [0,a ] 0a 5Z {0

0 [0,a ] 0 [0,a ]1 i 6

[0,a ] 0 [0,a ] 0

- ª º

° « »® « » d d° « »¬ ¼¯

1[0,a ] 0

[0 ] 0

- ½ª º° °« »° °



2

1 2 121

2

1 2

[0,a ] 0

a ,a Z0 [0,a ]

0 [0,a ]

[0,a ] [0,a ]

« »° °« »° °« » ® ¾

« »° °« »° °

« »° °¬ ¼¯ ¿

V1 V2; W is a set- semigroup interval bilinear su

V over the set-semigroup S = S1 S2.

Example 4.4.26 : Let V = V1

V2

= {Collection o

interval matrices with intervals of the form [0, ai],

{0}}

i

i i 36

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

be a semigroup-set interval bilinear algebra over the

S S S + { } {{

Clearly W is a subsemigroup-subset in

subalgebra of V over the subsemigroup-subset T

S2.


i

i i

i 0

[0,a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦

1 2 3 4 5

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]- ª º° ® « »

¬ ¼°̄



6 7 8 9 10[ , ] [ , ] [ , ] [ , ] [ , ]¬ ¼° ¯

be a set-semigroup interval bilinear algebr

semigroup S = S1 S2 = {2Z+ {0}, 5Z

+ {0}


2i

i i

i 0

[0,a ]x a 30Z {0}f

- ½ ® ¾

¯ ¿¦

[0,a] [0,a] [0,a] [0,a] [0,a]a,b

[0,b] [0,b] [0,b] [0,b] [0,b]

- ª º° ® « »

¬ ¼° ¯

T = T1 T2 = {4Z+ {0}, 15Z+ {0}} {2Z4

410} Z412} S1 S2.

W is a subset-subsemigroup interval bilineaV over the subset-subsemigroup T = T1 T2 S

P2 where T1 : V1 o P1 and T2 : V2 o P2 are such tha

linear interval vector space transformation and

semigroup linear interval vector space transformatio

bimap T = T1 T2 is defined as the set-semigro

linear bitransformation of V in to P.Interested reader can define properties analogo

linear transformations.

If V = P that is V1 = P1 and V2 = P2 then we defi

set-semigroup interval linear bioperator. The transfor

set-semigroup interval bilinear algebra can be def

some simple and appropriate modifications. Now we can derive almost all properties of the

structures in an analogous way



structures in an analogous way.

Now we can also define quasi set-semigroup int

algebras and their substructures in an analogous way

Now we proceed on to define bigroup interval biveset group (group-set) interval bivector spaces and

group (group-semigroup) interval bivector spaces a

few properties associated with them.

DEFINITION 4.4.3: Let V = V 1 V 2 be such that V

interval vector space over the group Gi; i = 1, 2 and V i if if i z j and Gi G j , G j z Gi if i z j; , 1 < i , j <

Then we define V = V 1 V 2 to be a bigro

bivector space over the bigroup G = G1 G2.


- ½ª º

1 2

3 4

5 6 1 2 6

7 8

9 10

[0,a ] [0,a ]

[0,a ] [0,a ]

[0,a ] [0,a ] , [0,a ] [0,a ] ... [0,a[0,a ] [0,a ]

[0,a ] [0,a ]

- ª º° « »° « »°

« »® « »° « »° « »° ¬ ¼¯

be a bigroup interval bivector space over the big

G2 = Z42 Z30.

Example 4 4 29: Let V = V V =




i

1 2 15 i

i 0

[0,a ] [0,a ] ... [0,a ] , [0,a ]x af

- ® ¯

¦

1

2 1 2 9

10 11 18

15

[0,a ]

[0,a ] [0,a ] [0,a ] ... [0,a ],

[0,a ] [0,a ] ... [0,a ] 1

[0,a ]

- ª º° « » § ·° « »® ¨ ¸« » © ¹° « »° ¬ ¼¯

#


G2 = Z12 Z29.

Now we will give examples of their subst

task of giving definition is left as an exercise for

i

i 1 2 17 i j 4

i 0

[0, a ]x , [0, a ] [0, a ] ... [0, a ] a , a Zf

¦

be a bigroup interval bivector space over the bigrouG2 = Z310 Z46.

Take W = W1 W2 = {all 5 u 5 upper triangu

matrices with intervals of the form [0, ai]; ai Z310}

310

[0,a]

[0,a]a Z

[0,a]

- ½ª º

° °« »° °« » ® ¾« »° °« »



[0,a]° °« »° °¬ ¼¯ ¿

2ii i

i 0[0, a ]x , [0, a] [0, a] ... [0, a] a , a

f

- ® ¯ ¦

V1 V2 is a bigroup interval bivector subspace of

bigroup G = G1 G2 = Z310 Z46.


1 6 11

2 7 12

i 3 8 13 i j 19

4 9 14

[0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ]

[0,a ], [0,a ] [0,a ] [0,a ] a ,a Z ;1

[0,a ] [0,a ] [0,a ]

- ª º° « »° « »° « » d ®

« »° « »°« »


Z24 = G1 G2.

Take W = W1 W2 =

19

[0,a] 0 [0,a]

0 [0,a] 0

[0,a], a Z[0,a] 0 [0,a]

0 [0,a] 0

[0,a] 0 [0,a]

- ª º° « »° « »° « » ®

« »° « »° « »

° ¬ ¼¯

[0 a] [0 a] [0 a]- ª º



4i

i i 24

i 0

[0,a] [0,a] [0,a]

0 0 0[0,a ]x , a ,a {2Z

[0,a] [0,a] [0,a]

0 0 0

f

- ª º° « »° « » ® « »°

« »° ¬ ¼¯

¦

V1 V2 ; W is a bigroup interval bivector sub

the bigroup G = Z19 Z24.


1

1 2

2 i 45

3 4

5 6

16

[0,a ][0,a ] [0,a ]

[0,a ] a Z[0,a ] [0,a ] ,

1 i 16[0,a ] [0,a ]

[0,a ]

- ª ºª º° « » ° « » « »® « » « » d d° « » « »¬ ¼° ¬ ¼¯

#

1

2

1 2 3 4

0

[0, a ]

0

[0,a ]0

0

0[0, a] [0, a]

0

0 0 , a,b,a ,a ,a ,a Z0[0,b] [0,b]

0

- ª º° « »° « »° « »° « »° « »° « »° « »° « »° « »° « »ª º° « »° « »

® « »« »° « »« »¬ ¼° « »° « »



3

4

0

0

[0, a ]0

0

[0,a ]

° « »° « »° « »° « »

° « »° « »° « »° « »° « »

¬ ¼° ¯

^ 1 2 3 4[0, a ] 0 [0, a ] 0 [0,a ] 0 [0, a ] 0 [

i j 2482i

i

i 0

a , a Z ;[0,a ]x

1 j 5

f

½¾

d d ¿¦

V1 V2; be a bigroup interval bivector subspacethe bigroup G.

{All 8 u 8 interval matrices with interval entries

ai] ; ai Z15, ( [0, a1], [0, a2], [0, a3], [0, a4] ) | ai, 4} be a bigroup interval bivector space over the

G2 = Z8 Z15.

Take W = W1 W2 =

2i

i i 8

i 0

[0,a]

[0,a], [0, a ]x a, a Z

[0,a]

f

- ½ª º° « »° « » ® « »° « »° ¬ ¼¯ ¿

¦#

{All 8 u 8 upper triangular interval matrices w



the form [0, ai]; ([0, a1], 0, [0, a2], 0) | ai, a1, a2 be a subbigroup interval bivector subspace

subbigroup T = T1 T2 = {0, 2, 4, 6} {0, 5, 10

G1 G2.If a bigroup interval bivector space V over t

G1 G2 has no subbigroup interval bivector su

bigroup G = G1 G2 then we say V to be a

bigroup interval bivector subspace over the bigrno bigroup interval bivector subspace then we

simple bigroup interval bivector space. If V is b pseudo simple then we call V to be a doubly

interval bivector space.

We will give some illustrative examples of them


all 10 u 10 interval matrices with intervals of the for

Z5} be a bigroup interval bivector space over the b

G1 G2 = Z7 Z5. We see the bigroup G = Z

bisimple as it has no subgroups. Thus V is a pse bigroup interval bivector space over G.

However V is not doubly simple for take W = W

2ii i 7

i 0

[0, a ]x ; [0, a] [0, a] [0, a] a , a Z

f

- ½® ¾

¯ ¿¦



all 10 u 10 upper triangular interval matrices with e

Z5 with intervals of the form [0, ai]} V1 V2 ; W interval bivector subspace of V over the bigroup G

not doubly simple.


3

[0, a] [0, a],[0, a] a Z

[0, a] [0, a]

- ½ª º° °® ¾« »

¬ ¼° °¯ ¿

[0,a][0,a]

a[0,a]

[0,a]

- ª º° « »° « »® « »° « »° ¬ ¼¯

be a bigroup interval bivector space over bigroup G =

Z Z W V i d bl i l bi i

2. V is not a doubly simple bigroup interva

over the bigroup in general.

The proof is left as an exercise for the reader to p

THEOREM 4.4.6: Let V = V 1 V 2 be a bigroup i

space over the bigroup G = Z n Z m where m primes; V is not simple or pseudo simple.

This proof is also left as an exercise to the reader

We see bigroup interval bivector spaces can b

finite bigroups of the form G = Zm Zn, we cann



or Q+ or C.

Now we can define bigroup interval bilinear alg

only examples of them.


1 2 3 4 5

6 7 8 9 10

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ]

- ª º° ® « »

¬ ¼° ¯

1

2

i 32

14

[0,a ]

[0,a ]

a Z ;1 i 15

[0,a ]

- ½ª º° °« »° °« »° °« » d d® ¾

« »° °« »° °« »

#

i

i i 17

i 0

[0, a ]x a Zf

- ½® ¾

¯ ¿¦

be a bigroup interval bilinear algebra over the bigrou

G2 = Z38 Z17.

We will illustrate the substructures by some example

Example 4.4.38: Let V = V1 V2 = {([0, a1], [0, a

a9]) | ai Z28; 1 d i d 9}

1[0, a ]

[0 a ]

- ½ª º° °« »° °« »



2

i 15

11

12

[0, a ]

a Z ;1 i 12

[0, a ]

[0, a ]

° °« »° °« » d d® ¾« »° °« »

° °« »° °¬ ¼¯ ¿

#

be a bigroup interval bilinear algebra over the bigro

G2 = Z28 Z15.

Take W = W1 W2 = {([0, a], [0, a], …, [0, a]) |

1

2

i 15

11

12

[0,a ]

[0,a ]

a {0,3,6,9,12} Z

[0,a ]

[0,a ]

- ½ª º° °« »° °« »° °« » ® ¾

« »° °« »

° °« »° °¬ ¼¯ ¿

#

{([0, a1], [0, a2], …, [0, a7])| ai Z11; 1 d i dinterval bilinear algebra over the bigroup G = Z3

Take W = W1 W2 =

1 2 i 3

3

[0,a ] [0,a ] a Z ;

0 [0,a ] 1 i 3- ½ª º° °® ¾« » d d¬ ¼° °¯ ¿

{([0, a], [0, a], …., [0, a]) | a Z11} V1 interval bilinear subalgebra of V over the bigroup




1 2 3 i 18

4 5 6

[0, a ] [0, a ] [0, a ] a Z ;

[0, a ] [0, a ] [0, a ] 1 i 6

- ª º° ® « » d d¬ ¼° ¯

1 2

3 4 i 40

5 6

7 8

[0, a ] [0, a ]

[0, a ] [0, a ] a Z ;

[0, a ] [0, a ] 1 i 8

[0, a ] [0, a ]

- ½ª º° °« » ° °« »® ¾« » d d° °« »° °¬ ¼¯ ¿

be a bigroup interval bilinear algebra over the

G2 = Z18 Z40. Take H = H1 H2 = {0, 6

20, 30} G1 G2 = Z18 Z40 and W = W1 W

1 2

1 2 3

3

[0, a ] 0 [0, a ]a , a , a0 [0, a ] 0

- ª º° ® « »¬ ¼°̄

DEFINITION 4.4.4: Let V = V 1 V 2 be where V 1interval vector space over the group G1 and V 2 is ainterval vector space over the semigroup S 2. V

semigroup interval bivector space over the group-se

S 2.



1 2 3 4 i j

i

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ] a ,a Z,[0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] 1 j 8

- ª º° ® « »d d¬ ¼° ¯



1 2

3 4 i j 1ii

i 05 6

7 8

[0, a ] [0, a ]

[0, a ] [0, a ] a , a Z, [0, a ]x[0, a ] [0, a ] 1 j 8

[0, a ] [0, a ]

f

- ª º° « »

° « »® « » d d° « »° ¬ ¼¯

¦

be a semigroup-group interval bivector spac

semigroup-group G = (Z+

{0}) Z17.


1 2 3 4 5

i

6 7 8 9 10

a[0,a ] [0,a ] [0,a ] [0,a ] [0,a ],[0,a ]

[0,a ] [0,a ] [0,a ] [0,a ] [0,a ] 1

- ª º° ® « »

¬ ¼° ¯ [0 a ] [0 a ]-ª º

be a group-semigroup interval bivector space

semigroup G = Z48 Q+ {0}.

We can define substructures in a analogous

only examples of them.


1

2i i j

i

i 0 3

4

[0, a ]

[0, a ] a , a Q {0[0, a ]x ,

[0, a ] 1 j 4[0, a ]

f

- ª º° « » ° « »®

« » d d° « »° ¬ ¼¯

¦



1 2

1 2 11 3 4

5 6

[0, a ] [0,a ]

([0, a ] [0, a ] ... [0, a ]), [0, a ] [0,a ]

[0, a ] [0, a ]

- ª ° « ® « ° « ¬ ¯

be a semigroup-group interval bivector

semigroup-group G = Q+ {0} Z19.

Take W = W1 W2 =

2i

i i

i 0

[0,a]

0[0, a ]x , a , a Q {0

[0,a]

0

f

- ª º° « »° « » ® « »° « »° ¬ ¼¯

¦


2i 3i

i i i 320i 0 i 0[0, a ]x , [0, a ]x a Z

f f

- ½

® ¾¯ ¿¦ ¦

{All 5 u 5 interval matrices with intervals of the for

3Z+ {0}, ([0, a1], [0, a2], [0, a3], [0, a4], [0, a5]) {0}} be a group-semigroup interval bivector spac

group-semigroup Z320 12Z+

{0}.Take W = W1 W2 =

4i 9i[0 a ]x [0 a ]x a Zf f- ½

® ¾¦ ¦



i i i 320

i 0 i 0

[0, a ]x , [0, a ]x a Z

® ¾¯ ¿¦ ¦

{All 5 u 5 interval upper triangular matrices with the form [0, ai], ai 3Z+ {0}, ([0, a1] [0, a2] [0, a3

4Z+ {0}} V1 V2; W is a group-semigro

bivector subspace of V over the group-semigroup, Z

{0}.


2i 5i

i i i 196

i 0 i 0

[0, a ]x , [0, a ]x a Zf f

- ½® ¾

¯ ¿¦ ¦

1 2[0,a ] [0,a ]a- ª º ° « »

[0, a] [0, a]

[0, b] [0, b] , ([0, a] [0, a] ... [0, a]) a, b

[0, c] [0, c]

- ª º° « »® « »° « »¬ ¼¯

V1 V2; W is a subgroup-subsemigroup in

subspace over the subgroup-subsemigroup {0,

194} 3Z+ {0} Z196 Z+ {0}.

Example 4.4.46 : Let V = V1 V2 = {all 8 u 8 in

with intervals of the form [0, ai] ; ai Q+ interval matrices with intervals of the form [0, ai

{0}}



{ }}

i

i 1 2 8

i 0

a[0,a ]x , [0, a ] [0, a ] ... [0, a ]

1

f

- ® ¯ ¦

be a semigroup-group interval bivector

semigroup-group Z+ {0} Z48.

Let W = W1 W2 = {all 8 u 8 interval u

matrices with intervals of the form [0, ai], ai Q

[0, a] [0, a]

[0, a] [0, a]

[0, a] [0, a]

ª º« »« »« »¬ ¼

a Z+ {0}}

3if-¦

Now we can in an analogous way define group

(semigroup group) interval bilinear algebra

substructures.

We will illustrate these situations only by examples.


1 2 3

i 49

4 5 6

7 8 9

[0, a ] [0,a ] [0, a ]a Z ;

[0, a ] [0, a ] [0, a ]

1 i 9[0, a ] [0, a ] [0, a ]

- ½ª º° °« »

® ¾« » d d° °« »¬ ¼¯ ¿

- ½ª º



1 2

3 4

5 6 i

7 8

9 10

11 12

[0, a ] [0, a ]

[0, a ] [0, a ]

[0, a ] [0, a ] a Z {0};[0, a ] [0, a ] 1 i 12

[0, a ] [0, a ]

[0, a ] [0, a ]

- ½ª º° °« »° °« »

° °« » ° °« »® ¾d d« »° °

« »° °« »° °« »° °¬ ¼¯ ¿

be a group-semigroup interval bilinear algebra over

semigroup Z49 Z+ {0}.


ii i[0,a ]x a Q {0}

f

- ½ ® ¾¯ ¿¦




i

i ii 0

[0, a ]x a Q {0}f

- ½

® ¾¯ ¿¦

1 2 3 4 5

i

6 7 8 9 10

11 12 13 14 15

[0, a ] [0, a ] [0, a ] [0, a ] [0,a ]a

[0, a ] [0, a ] [0, a ] [0, a ] [0, a ]1

[0, a ] [0, a ] [0, a ] [0, a ] [0, a ]

- ª º° « »® « »° « »

¬ ¼¯

be a semigroup-group interval bilinear algebr

semigroup group G = Q+ {0} Z30



semigroup group, G Q {0} Z30.

Take W = W1 W2 =

2i

i i

i 0

[0, a ]x a Q {0}f

- ½ ® ¾¯ ¿¦

1 2 3

i

4 5

6 4 8

[0, a ] 0 [0, a ] 0 [0, a ]a

0 [0, a ] 0 [0,a ] 01[0, a ] 0 [0, a ] 0 [0, a ]

- ª º° « »

® « » d° « »¬ ¼¯

V1 V2 and H = 3Z+ {0} {0, 5, 10, 15, 20, 2

{0} Z30. W is a subsemigroup-subgroup interv

subalgebra of V over the subsemigroup-subgroup H



THEOREM 4.4.8: Let V = V 1 V 2 be a group(semigroup-group) bilinear algebra (bivector spac

group-semigroup (Z p – Z + {0}) p a prime. Then simple and need not in general be doubly simple.

This proof is also left as an exercise for the reader.

Now we proceed onto define set-group (group-

bilinear algebra (bivector space) over the set-group (g

DEFINITION 4.4.5: Let V = V 1 V 2 be such that

interval vector space over the set S 1 and V 2 is a grovector space over the group G2. We define V = V 1

set-group interval bivector space over the set-group.





3i 2i

i i i

i 0 i 0

[0, a ]x , [0, a ]x a Z {0}f f

- ½ ® ¾

¯ ¿¦ ¦

1

2 i 9

1 2 3

3

4

[0, a ]

[0, a ] a Z ;, [0, a ],[0, a ],[0, a ]

[0, a ] 1 i 4

[0, a ]

- ½ª º

° « » ° « »® « » d d° « »° « »¬ ¼¯ ¿

be a set group interval bivector space over the set- g

17, 41, 142, 250} Z9.

i

i i i

i 0

[0, a ]x ,[0, a ] a 5Z {0}f

- ®

¯ ¦

be a group-set interval bivector space over the

{0, 1, 2, 7, 15Z+}.


2i 3ii i i

i 0 i 0

[0, a ]x , [0, a ]x a 4Z {f f

- ® ¯ ¦ ¦

- ª º



1

2

1 2 i 49

8

[0, a ]

[0, a ]

, [0, a ],[0, a ] a Z ;1

[0, a ]

- ª º° « »° « »

d® « »° « »° ¬ ¼¯

#

be a set-group interval bivector space over the s

= {16 Z+ {0}, 4, 8} Z49.

Take W = W1 W2 =

4i 3i

i i i

i 0 i 0

[0, a ]x , [0, a ]x a 16Zf f

- ®

¯ ¦ ¦

1[0, a ]- ª º°« »

V1 V2; W is a set - group interval bivector sub

over the set-group S G.


1

2 1 2 7 i

3

[0, a ]

[0, a ] , [0, a ],[0, a ]...,[0, a ] a Z {

[0, a ]

- ª º° « » ® « »° « »¬ ¼¯

11 2

3 4 2 i 24

[0, a ][0, a ] [0, a ]

[0, a ] [0, a ] [0, a ] a Z ;

- ½ª ºª º° °« »« » ° °« »« »® ¾



5 6

7 8 11

,[0, a ] [0, a ] 1 i 11

[0, a ] [0, a ] [0, a ]

« »« »® ¾« »« » d d° °« »« »° °

¬ ¼ ¬ ¼¯ ¿

#

be a set-group interval bivector space over the set-

4Z+, 17Z+, 13Z+, 0} Z24 = S1 G2.

Choose W = W1 W2 =

i

1 7

a, a 3Z {0};

([0,a ],...,[0,a ]) 1 i 7

- ½

® ¾d d¯ ¿

11

2 2

i 24

3

[0, a ][0,a ] 0

0 [0,a ] [0,a ], a 2Z {0,...,22

[0,a ] 00 [0 ] [0 ]

- ª ºª º° « »« »° « »« » ®

« »« »° « »« »° ¬ ¼ ¬ ¼#

{All 6 u 6 interval matrices with intervals of

ai Z+ {0}} be a group-set interval bilinear a

group-set G = G1 S2 = Z48 {30Z+ {0}, 2Z+

Example 4.4.59: Let V = V1 V2 = {set of al

matrices with intervals of the form [0, ai]; ai Q

3 u 8 interval matrices with intervals of the foZ41} be a set - group interval bilinear algebra

group S1 G2 = {2Z+, 5Z+, 7Z+, 0} Z41.

We now state the theorem the proof of which

THEOREM 4.4.9: Every set-group (group-set) i



y g p (g p )

algebra over a set-group (group-set) is a set-grinterval bivector space but not conversely.


i

i i

i 0

[0, a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦

{all 4 u 4 interval matrices with interval entr

[0, ai]; ai Z43} be a set - group interval biline

the set-group S1 G2 = {3Z+, 2Z

+, 7Z

+, 0} {Z

Choose W = W1 W2 =

2ii i

i 0[0, a ]x a Z {0}

f

- ½ ® ¾¯ ¿¦

{all 5 u 2 interval matrices with intervals of the f

ai Z48} be a set-group interval bilinear algebra ov

group S1 G2 = {7Z+ {0}, 3Z+ {0}, 4Z+} Z48

Choose W = W1 W2 =

2i

i i

i 0

[0,a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦

1 2 3

1 21 2

[0,a ] 0 [0,a ] 0 [0,a ]a ,a ,

0 [0,a ] 0 [0,a ] 0

- ª º°

® « »¬ ¼° ¯

V1 V2 and P1 P2 = {3Z+, 4Z+, 0} {0, 12, 2

G W i b b i l bili b



G2. W is a subset - subgroup interval bilinear suba

over the subset - subgroup P1 H2 of S1 G2.


i

i i

i 0

[0, a ]x a Z {0}f

- ½ ® ¾

¯ ¿¦

7

[0, a] [0, a] [0, a] [0, a]

[0, b] [0, b] [0, b] [0, b] a, b, c Z

[0, c] [0, c] [0, c] [0, c]

- ª º° « » ® « »° « »¬ ¼¯

be a set - group interval bilinear algebra over the seS G {2Z+ 5Z+ 0} Z Cl l V i d

Chapter Five



APPLICATIONS OF THE SPECIAL

OF INTERVAL LINEAR ALGEBRAS

These new classes of interval linear algeb

applications in fields, which demand the soli l d i fi i l h d h

cannot be built using intervals of the form [–a, b] wh

are in Z.

These structures can be used in all mathematical fuzzy models, which demand interval solutions. For

interval algebraic structures refer [52].



Chapter Six

SUGGESTED PROBLEMS



In this chapter we propose over 100 problems,

challenge to the reader.

1. Find some interesting properties about set c

vector spaces.

2. Give an example of a order 21 set modulo

space built using Z40.

3. Does their exists a set modulo integer v

cardinality 12 built using Z7? Justify your claim

6. Let V =

1 2

3 4 i 17

5 6

[0, a ] [0, a ]

[0, a ] [0, a ] a Z

[0, a ] [0, a ]

- ½ª º° °« » ® ¾« »° °« »¬ ¼¯ ¿

, be a s

integer linear algebra over the set S = {0, 1, 2, 5}set modulo integer interval linear subalgebras of V

7. Obtain some interesting properties about set ratio

vector spaces.

8. Let S = {[a, b] | a, b Q+

{0}; a d b} be a interval vector space over the set S = {0, 1}. Find

interval vector subspaces of V. Can S be generat

Justify your claim.



9. Obtain some interesting properties about set comp

linear algebras.

10. Give an example of a doubly simple set interval inalgebra.

11. Give an example of a semigroup interval vector s

is not a semigroup interval linear algebra.

12. Give some interesting properties of semigroup int

spaces.

13. Give an example of a finitely generated semigro

linear algebra.

17. Give an example of a pseudo semigroup

algebra.

18. Does there exists a semigroup interval linear

cannot be written as a direct sum? Justify your

19. Obtain some interesting properties about grou

algebras.

20. Does there exists an infinite group interval line

21. Does their exists a group interval linear algebra

22. Give an example of a group linear algebra of o



23. Let X = i

i i 7

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦. Is X a group

algebra over the group Z7? Is X finite or infinit

24. Give an example of a set fuzzy interval vector

25. Obtain some interesting properties about set

linear algebras.

26. Give an example of a semigroup fuzzy interval

27. Let V = {All 5 u 5 interval matrices with inter

1[0,a ]

[0 a ]ª º« »« »

29. Let V = {all 6 u 6 interval matrices with intervals

[0, ai]; ai Z18} be a group interval linear algeb

group G = Z18.

i. Obtain group fuzzy interval linear algebras.ii. Does V have subgroup interval linear subalg

iii. Find at least 3 group interval linear subalgeb

iv. Define a linear operator on V with non trivia

30. Bring out the difference between type I and type I

fuzzy interval linear algebras.

31. Obtain some interesting properties enjoyed by ty

fuzzy interval linear algebras?



32. Let V = V1 V2 = 2i 3i

i i i

i 0 i 0

[0,a ]x , [0,a ]x ;af f

- ®

¯ ¦ ¦

1

2

1 8 i

7

[0,a ]

[0,a ], [0,a ] [0,a ] a 3Z {0}

[0,a ]

- ½ª º° °« »° °« » ® ¾« »° °« »° °

¬ ¼¯ ¿

"#

interval bivector space over the set S = {2Z+, 3Z+, 0

i. Find set interval bivector subspaces of V.

ii. Find subset interval bivector subspace of V.

iii. Define a bilinear operator on V.

iv. Find a generating set of V.

33 Give an example of a set interval bivector space w

i. Find a linear bioperator on V which h

bikernel.

ii. Is V simple?

iii. Can V have subset interval bilinear algeb

36. Give an example of a pseudo simple set

bialgebra which is not simple.

37. Obtain some important properties about set bialgebras.

38. Give an example of a set interval linea

bidimension (5, 9).

39. What is the difference between a set interval



and a biset interval bilinear algebra?

40. Let V = V1 V2 = {all 10 u 10 interval

intervals of the form [0, ai], ai 5 Z+ {

interval matrices with intervals of the form [

{0}} be a biset interval bivector space of V ov

10Z+ {0} 6Z+ {0} = S1 S2.

i. Find a bigenerating bisubset of V.ii. Is V finite bidimensional?

iii. Find biset interval bivector subspaces of

iv. Is V pseudo simple? Justify your answer

v. Define a nontrivial one to one bilinear op

41. Give an example of a quasi biset interval bivec

43. Give an example of a doubly simple quasi bi

bivector space over the biset S.

44. Let V = V1 V2 = i

i i 13

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦

interval matrices with intervals of the form [0, ai]

be a quasi set interval linear bialgebra over the set

V simple? Justify!

45. Give an example of a semi quasi set interval biline

46. Determine some special properties enjoyed by

i l bili l b



interval bilinear algebras.

47. Let

V = V1 V2

=1 2 3 4

i 17

5 6 7 8

[0,a ] [0,a ] [0,a ] [0,a ]a Z ;1

[0,a ] [0,a ] [0,a ] [0,a ]

- ª º° d ® « »

¬ ¼° ¯

ii i 17

i 0

[0,a ]x a Zf

- ½® ¾¯ ¿¦

be a semigroup interval bilinear algebra over the se

Z17.

i. Is V pseudo simple?

ii. Is V simple? Justify

50. Give an example of a double simple sem

bivector space.

51. Give some interesting applications of sem

bilinear algebras.

52. Let

V = V1 V2

=

1

2 i

1 14

9

[0, a ]

[0,a ] a Z

, [0,a ] [0,a ] 1 i 1

[0,a ]

- ª º° « » ° « »® « » d d° « »° ¬ ¼¯

"#





ai 3Z+ {0}} be a quasi semigroup interval

over the semigroup S = 2Z+ {0}.

i. Find substructures of V.

ii. Find a bilinear operator on V

iii. Can V be a made into a quasi fuzzy sem

bilinear algebra?

iv. Is V pseudo simple?

v. Is V doubly simple?

53. Give an example of a doubly simple quasi sem bilinear algebra.

54. Describe some important properties enjoyed b

bivector space.

1 2 3 4

ji

5 6 7 8 i

i 0

9 10 11 12

[0,a ] [0,a ] [0,a ] [0,a ]a

[0,a ] [0,a ] [0,a ] [0,a ] , [0,a ]x1

[0,a ] [0,a ] [0,a ] [0,a ]

f

- ª º° « »® « »° « »¬ ¼¯

¦

be a group interval bivector space over the group Gi. What is the bidimension of V?

ii. Find group interval bivector subspaces of V.

iii. Is V pseudo simple? Justify.

iv. Find all generating bisubset of V.

57. Let V = V1 V2 =7 i

i i 40

i 0

[0,a ]x a Z ;0

- ® ¯

¦

1[0, a ]

[0 a ] a Z ;

- ½ª º° °« » ° °« »



2 i 40

10

[0,a ] a Z ;

1 i 10

[0,a ]

° °« »® ¾« » d d° °« »° °¬ ¼¯ ¿

#be a group interval bivector

the group G = Z40.

i. Find at least two group interval bivector su

V.

ii. Is V simple?

iii. Prove V is not pseudo simple.iv. Find a bibasis of V.v. Find the bidimension of V.

58. Give an example of a simple group interval bivecto

59. Give an example of a doubly simple group interv

i. Prove V is pseudo simple.

ii. Prove V is not simple

iii. Find atleast 3 group interval bilinear subal

iv. What is the bidimension of V?

v. Find a bigenerating bisubset of V.

vi. Find a bilinear operator on V.

62. Give an example of a quasi group interval b

over the group Zn.

63. Is V = V1 V2 = 3

[0,a][0,a] a Z

[0,a]

- ½ª º° °« » ® ¾« »

° °« »¬ ¼¯ ¿

{([0, a

a Z3} a doubly simple group interval biline



the group G = Z3? Justify your claim.

64. Let V = V1 V2 =7

i

i 0

[0,a ]x

- ® ¯ ¦

5

[0, a] [0, a]

[0,a] [0,a] a Z

[0, a] [0, a]

- ½ª º° °« » ® ¾

« »° °« »¬ ¼¯ ¿

be a quasi group i


i. Is V simple?

ii. Is V doubly simple?

iii. Is V pseudo simple?

65 Gi i i i b b

69. Let

V = V1 V2

=

1

2 i 50

12

[0, a ]

[0,a ] a Z ;

1 i 12

[0,a ]

- ½ª º° °

« » ° °« »® ¾« » d d° °« »° °¬ ¼¯ ¿

#

1 2 10

i 28

11 12 20

21 22 30

[0,a ] [0,a ] ... [0,a ]a Z ;

[0,a ] [0,a ] ... [0,a ] 1 i 30[0,a ] [0,a ] ... [0,a ]

- ½ª º° °« »

® ¾« » d d° °« »¬ ¼¯ ¿ be a bigroup interval bilinear algebra over the bigr

G2 = Z50 Z28.i Fi d tl t t bbi i t l bili



i. Find atleast two subbigroup interval bilinear su

ii. Find atleast three bigroup interval bilinear suba

iii. Find a generating biset of V.

iv. Find a bilinear operator on V.

70. Let

V = V1 V2

= i

i i 7[0,a ]x a Z- ½

® ¾¯ ¿¦

1

2 i 11

19

[0,a ][0,a ] a Z

1 i 19

[0,a ]

- ª º° « » ° « »® « » d d° « »° ¬ ¼¯

#

be a bigroup interval bilinear algebra over the bigr

G2 = Z7 Z11.

71. Let V = V1 V2 = i

i i[0,a ]x a Z {0}- ®

¯ ¦

interval matrices with intervals of the form [0

Z420} be a semigroup - group interval bilinear a

semigroup - group Z+ {0} Z420.i. Find substructures of V.

ii. Prove V is not a doubly simple space.

iii. Find a T : T1 T2 : V1 V2 o V1 V2 s

= {0} {0}.

iv. Find T : T1 T2 : V = V1 V2 o V = V1

biker T = {0} {S}. S z 0.

72. Let V = V1 V2 =

1 2

i

3 4

[0,a ] [0,a ]a

[0,a ] [0,a ]

- ª º° « »®« »



1 2 3 4

5 6

[ , ] [ , ]

[0,a ] [0,a ]

® « »° « »

¬ ¼¯ i

i i 47

i 0

[0,a ]x a Zf

- ½® ¾

¯ ¿¦ be a set semigroup i

algebra over the set - semigroup 3Z+ {0} Zi. Find a set - semigroup interval bilinear sub

ii. Find a subset - subsemigroup insubalgebras.

iii. Find a bilinear bioperator on V which is on

73. Give some interesting results about biset i

algebras.

74 Give examples of infinite biset interval bilinea



iii. Prove V is not doubly simple.

iv. What is the biorder of V?

v. Define a one to one bilinear operator on V

83. Prove if V = V = V1 V2 is a group interval

over a group G; then the set of all bioperators

group interval bilinear algebra over the group G

84. Obtain some interesting properties a

transformations on group interval bivector sp

V2 and W = W1 W2 defined over the group G

85. Let V = V1 V2 = {all 3 u 1 be a set of all in

with intervals of the form [0, ai]; ai Z12} {

all interval matrices with intervals of the form

b bi migr i t r l bili r l



be a bisemigroup interval bilinear al

bisemigroup S = S1 S2 = Z12 Z19.i. Find all bisemigroup interval bilinear su

over S.

ii. Is V pseudo simple?

iii. Find a bigenerating subset of V.

iv. Find all bilinear operators on V.

86. Let V = V1 V2 be as in problem 77.

i. Find a bigenerating subset of V.

ii. What is the bidimension of V over S?

87. Obtain some interesting properties on set -

bivector spaces of finite order.

i. Find atleast 3 group-semigroup interva

subalgebras of V over S.

ii. Find atleast 3 pseudo subgroup-subsemigroup

89. Give an example of a doubly simple set - gro

bivector space.

90. Give an example of a pseudo simple bigroup inter

space which is not simple.

91. Let V = V1 V2 = i 5

[0,a]

[0,a]a Z

[0,a]

[0,a]

- ½ª º° °« »° °« » ® ¾« »° °« »° °¬ ¼¯ ¿

{([0, a

) / } b bi i



a], [0, a], [0, a], [0, a]) / a Z11} be a bigroup inte

over the bigroup G = G1 G2 = Z5 Z11.i. Is V simple?

ii. Is V doubly simple?iii. Is V pseudo simple? Justify your claim.

92. Prove there exists an infinite class of doubly sim

interval bilinear algebras!

93. Prove their exists an infinite class of bigroup inter

algebras which are not pseudo simple!

94. Does there exists an infinite classes of set-gro

bilinear algebras? Justify your claim.

97. Let V = {1 u 9 interval matrices using Z7} matrices using Z7} be a group interval biline

the group Z7.

i. Prove V is not doubly simple!

ii. Find all bilinear operators on V and sho

interval bilinear algebra over Z7.

98. Let V = V1 V2 = {all 2 u 2 interval matric

{0} and 5Z+ {0}} {3 u 3 interval matric

{0}, 3Z+ {0}} be a bisemigroup interval

over the bisemigroup S = S1 S2 = 5Z

+

{0}i. Is V simple?

ii. Find subbisemigroup interval bilinear suba

99. Show V in problem (98) is not doubly simple.



100. For V in problem (98) prove set of all bilinearis again a bisemigroup bilinear algebra over S.

101. Give an example of set-semigroup interval

which is not a set-group interval bilinear algeb

102. Is every set-group interval bilinear algebra a interval bilinear algebra?

103. Show a biset interval bilinear vector space i

bigroup or bisemigroup interval bivector space

104. Obtain conditions on a bigroup interval bilineV V so that V is never a nontrivial bisem

107. Let V = V1 V2 = i

i i

i 0

[0,a ]x a 3Z {0}f

- ½ ® ¾

¯ ¿¦

xi | ai 5Z+ {0}} be a bisemigroup interval bilin

defined over the bisemigroup S = S1 S2 = 3Z+

{0}.i. Find a bigenerating subset of V.

ii. Find atleast two bisemigroup intervasubalgebras.

iii. Find atleast two subbisemigroup interv

subalgebras.

108. Give an example of a pseudo simple bisemigro

bilinear algebra.

109. Give an example of a simple bisemigroup inter



algebra.

110. Give an example of a doubly simple bisemigro

bilinear algebra of finite order.

111. Give an example of a group - set interval bilinea

infinite order.

112. Give an example of a group semigroup interv

algebra of finite order.

113. Prove a set-group interval bivector space in gene

semigroup-group interval bivector space.

FURTHER R EADING

1. ABRAHAM, R., Linear and Multilinear A

Benjamin Inc 1966



Benjamin Inc., 1966.

2. ALBERT, A., Structure of Algebras, Colloq. Math. Soc., 1939.

3. BIRKHOFF, G., and MACLANE, S., A Sur

Algebra, Macmillan Publ. Company, 1977.

4. BIRKHOFF, G., On the structure of abstract

Cambridge Philos. Soc., 31 433-435, 1995.5. BURROW, M., Representation Theory of

Dover Publications, 1993.

6. CHARLES W. CURTIS, Linear Algebra – A

Approach, Springer, 1984.

7. DUBREIL, P., and DUBREIL-JACOTIN, M.L



28. R OMAN, S., Advanced Linear Algebra, S

New York, 1992.

29. R ORRES, C., and A NTON H., Applica

Algebra, John Wiley & Sons, 1977.

30. SEMMES, Stephen, Some topics pertaininglinear operators, Novembe

http://arxiv.org/pdf/math.CA/0211171

31. SHILOV, G.E., An Introduction to the Th

Spaces, Prentice-Hall, Englewood Cliffs, NJ

32. SMARANDACHE, Florentin, Special AlgebraiCollected Papers III, Abaddaba, Oradea, 78-

33. THRALL, R.M., and TORNKHEIM, L., Vecmatrices, Wiley, New York, 1957.

34. VASANTHA K ANDASAMY, W.B., SMARAND



, ,

and K. ILANTHENRAL

, Introduction to bimPhoenix, 2005.

35. VASANTHA K ANDASAMY, W.B., and

Smarandache, Basic Neutrosophic Algebraic

their Applications to Fuzzy and NeutrosHexis, Church Rock, 2005.


Smarandache, Fuzzy Cognitive Maps anCognitive Maps, Xiquan, Phoenix, 2003.


Smarandache, Fuzzy Relational

Neutrosophic Relational Equations, Hexis,2004

41. VASANTHA K ANDASAMY, W.B., On a new

semivector spaces, Varahmihir J. of Math. Sci.2003.

42. VASANTHA K ANDASAMY and THIRUVEGA

Application of pseudo best approximation to codUltra Sci., 17 , 139-144, 2005.

43. VASANTHA K ANDASAMY and R AJKUMAR , R. A

of bicodes and its properties, (To appear).

44. VASANTHA K ANDASAMY, W.B., On fuzzy semfuzzy semivector spaces, U. Sci. Phy. Sci., 7

1995.

45. VASANTHA K ANDASAMY, W.B., On semipo

operators and matrices, U. Sci. Phy. Sci., 8, 254-2

46. VASANTHA K ANDASAMY, W.B., Semivector s

semifields Z t N k P lit h iki 17 43



semifields, Zeszyty Nauwoke Politechniki, 17, 43

47. VASANTHA K ANDASAMY, W.B., Smaranda

Algebra, American Research Press, Rehoboth, 20

48. VASANTHA K ANDASAMY, W.B., SmarandaAmerican Research Press, Rehoboth, 2002.

49. VASANTHA K ANDASAMY, W.B., Smarandache

and semifields, Smarandache Notions Journal2001.

50. VASANTHA K ANDASAMY, W.B., SmarandacheSemifields and Semivector spaces, American

Press, Rehoboth, 2002.

51. VASANTHA KANDASAMY, W.B., SMARANDACH

54. VOYEVODIN, V.V., Linear Algebra, Mir Pub

55. ZADEH, L.A., Fuzzy Sets, Inform. and cont1965.

56. ZELINKSY, D., A first course in Linear Alg

Press, 1973.



INDEX

B



B

Bi semigroup interval bivector space, 176

Bi sub semigroup interval bilinear subalgebra, 180-2

Bigroup interval bivector space, 196

Bigroup interval bivector subspace, 196-8

Biset interval bivector space, 116-7

Biset interval bivector subspace, 117-8Bisubgroup interval bivector subspace, 199-202

D

Direct sum of group interval linear subalgebras, 48-9

Direct sum of interval semigroup linear subalgebra 3

F

Fuzzy interval semigroup vector space of level tw

87-8

Fuzzy set interval linear algebra, 60-1Fuzzy set interval linear subalgebra, 63

Fuzzy subsemigroup interval linear subalgebra o

Fuzzy vector space, 57-8

G

Generating interval set, 23-4

Group fuzzy interval linear algebra, 72-3

Group interval bilinear algebra, 153-4,160

Group interval bilinear vector space, 153-4

Group interval bilinear vector subspace 155



Group interval bilinear vector subspace, 155

Group interval linear algebra, 47-9Group interval linear subalgebra, 48-9

Group interval vector space, 41-2

Group interval vector subspace, 42-3

Group-semigroup interval bilinear algebra, 208-1

Group-semigroup interval bilinear subalgebra, 21

Group-semigroup interval bivector space, 206Group-set interval bilinear algebra, 216-7Group-set interval bivector space, 214

L

Linearly dependent subset of a group interval vec

Pseudo simple bigroup interval bivector space, 199-2

Pseudo simple fuzzy semigroup interval linear algebr

II, 94Pseudo simple group interval bivector space, 157-8

Pseudo simple group-semigroup interval bilinear alge

Pseudo simple group-set interval bilinear algebra, 21Pseudo simple quasi biset interval bilinear algebra, 1

Pseudo simple semigroup interval bilinear algebra, 1

Pseudo simple semigroup interval bivector space, 13

Pseudo simple set bilinear algebra, 111-2

Pseudo simple set integer interval vector space, 19-2

Q

Quasi biset interval bilinear algebra, 126-7

Quasi biset interval bivector space, 121-2

Quasi bisubset interval bilinear subalgebra, 133-4



Quasi bisubset interval bilinear subalgebra, 133 4

Quasi bisubset interval bivector subspace, 123-4Quasi group interval bilinear algebra, 166-8Quasi set interval bivector space, 119-120

Quasi set interval bivector subspace, 120-1

Quasi set interval linear bialgebra, 124-5

S

Sectional subset interval vector sectional subspace, 2

Semi quasi interval set interval bilinear algebra, 125-

Semi simple semigroup interval bivector space, 142-

Semigroup fuzzy interval linear algebra of type II, 89

Semigroup fuzzy interval linear subalgebra 89-90

Semigroup interval vector subspace, 30-1

Semigroup linearly independent interval subset, 3

Semigroup-group interval bivector space, 206Semigroup-group interval bivector subspace, 208

Semigroup-set interval bilinear algebra, 194-5

Semigroup-set interval bivector space, 188Set complex interval vector space, 15

Set fuzzy interval linear algebra of type II, 82-3

Set fuzzy interval linear algebra, 60-1

Set fuzzy interval linear subalgebra of type II, 83

Set fuzzy interval vector space of type II, 76-7

Set fuzzy interval vector space, 59-60Set fuzzy interval vector subspace of type II, 78-

Set fuzzy interval vector subspace, 61-2

Set fuzzy vector space, 58-9

Set integer interval vector space, 11-2

Set integer interval vector subspace, 15-6



g p ,

Set interval bilinear algebra, 108-9Set interval bilinear subalgebra, 110-1

Set interval bivector space, 103-4

Set interval bivector subspace, 105-6

Set interval linear algebra, 24-5

Set interval linear subalgebra, 25-6

Set interval linear transformation, 20-1Set modulo integer interval vector space, 14-5Set rational interval vector space, 12-3

Set real interval vector space, 13-4

Set-group interval bilinear algebra, 216-7

Set-group interval bilinear subalgebra, 217

Set-group interval bivector space 214

Subgroup interval bilinear subalgebra, 163-4

Subgroup interval bilinear vector subspace, 157-8

Subsemigroup interval bilinear subalgebra, 146-7Subsemigroup interval bivector subspace, 139

Subsemigroup interval linear subalgebra, 35-6

Subset fuzzy interval linear subalgebra of type II, 85-Subset fuzzy interval vector subspace of type II, 79-8

Subset integer interval linear subalgebra, 26-7

Subset integer interval vector subspace, 18-9

Subset interval bilinear subalgebra, 111-2

Subset interval bivector subspace, 107-8



ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is an Associate Department of Mathematics, Indian Institute Madras, Chennai. In the past decade she has g

scholars in the different fields of non-assocalgebraic coding theory, transportation theory, fuapplications of fuzzy theory of the problems fa

industries and cement industries. She has to research papers. She has guided over 68 M.Sprojects. She has worked in collaboration projectsSpace Research Organization and with the Tamil NControl Society. She is presently working on a rfunded by the Board of Research in Nu

Government of India. This is her 51st book.On India's 60th Independence Day, Dr

conferred the Kalpana Chawla Award for CouraEnterprise by the State Government of Tamil Nadof her sustained fight for social justice in the InTechnology (IIT) Madras and for her contribution



Technology (IIT) Madras and for her contribution

The award, instituted in the memory of Iastronaut Kalpana Chawla who died aboard Columbia, carried a cash prize of five lakh rupeprize-money for any Indian award) and a gold meShe can be contacted at vasanthakandasamy@gmWeb Site: http://mat.iitm.ac.in/home/wbv/public_

Dr. Florentin Smarandache is a Professor of the University of New Mexico in USA. He publisheand 150 articles and notes in mathematics, physpsychology, rebus, literature.

In mathematics his research is in numbeEuclidean geometry, synthetic geometry, algeb

statistics, neutrosophic logic and set (generalizlogic and set respectively) neutrosoph

interval linear algebra, by w. b. vasantha kandasamy, florentin smarandache

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