interval linear algebra, by w. b. vasantha kandasamy, florentin smarandache
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8/8/2019 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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This book can be ordered in a paper bound reprint from:
Books on Demand
ProQuest Information & Learning
(University of Microfilm International)
300 N. Zeeb Road
P.O. Box 1346, Ann Arbor
MI 48106-1346, USA
Tel.: 1-800-521-0600 (Customer Service)
http://www.lib.umi.com/bod/basic
Copyright 2010 by Kappa & Omega and the Authors
6744 W. Northview Ave.
Glendale, AZ 85303, USA
Peer reviewers:
Prof. Stefan Smarandoiu, Rm. Valcea, Jud. Valcea, Romania.Prof. Ion Patrascu, Mathematics Department, Fratii Buzesti Colle
Craiova, Romania.
Prof. Catalin Barbu, Vasile Alecsandri College, Bacau, Romania
Many books can be downloaded from the following
Digital Library of Science:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats
ISBN-13: 978-1-59973-126-1
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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
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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
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~ DEDICATED TO ~
Dr C.N Deivanayagam
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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
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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
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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
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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
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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].
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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.
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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 )
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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
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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
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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.
We will illustrate this situation by some example
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
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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 ]}
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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.
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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
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[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
interval vector spaces.
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
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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
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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
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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
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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
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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.
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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
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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
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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.
We will illustrate this by some simple examples.
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
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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;
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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
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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.
We will illustrate this situation by some examples.
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
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{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
interval vector spaces.
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
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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.
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= 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
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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.
We will illustrate this situation by some examples.
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
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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.
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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
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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
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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.
We will illustrate this situation by some examples.
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
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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.
We will illustrate this by some examples.
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.
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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
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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
We will illustrate this situation by some example
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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.
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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® « »
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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.
Example 2.2.44 : Let
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°̄
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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.
Example 2.2.46 : Let
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.
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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
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[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.
We will illustrate this by some examples.
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¯ ¿
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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.
We will illustrate this situation by some example
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
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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
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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°®« »
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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.
We will illustrate this situation by some example
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
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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.
-
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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
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We now define linear independence in group int
algebras.
DEFINITION 2.2.19: Let V be a group interval lin
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 .
We will illustrate this situation by some examples.
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
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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.
We will illustrate this situation by an example.
Consider
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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.
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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
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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
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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
We will illustrate this situation by some example
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-
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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
We will illustrate this situation by some examples.
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.
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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
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{ , , , , , , , , } ; }
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
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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
-
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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° °« »
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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
interval vector space.
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Example 3.1.14 : Let
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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
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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.
Example 3.1.16 : Let
1
10, if a 0
-ª ºz°« »
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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.
Example 3.1.17 : Let
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
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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
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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
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g p y p
interval vector spaces.
DEFINITION 3.1.8: Let V be a group interval lin
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.
We will illustrate this situation by some examples.
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®
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KI (p(x)) =
> @
deg(p(x))
0,1 if deg p(x) 0
« »¬ ¼®
° ¯
(V, KI) is a fuzzy group interval vector space;
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°« »
Example 3.1.24 : Let
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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
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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« »« »
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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.
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We will illustrate this by some examples.
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 °
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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° « »
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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
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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
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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
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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.
We will illustrate this by some simple examples.
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
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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.
We will illustrate this situation by some example
Choose
W = 1 2
i
[0,a ] [0,a ]1 i 3;0 a 1
- ½ª º° °d d d d® ¾
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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¯ ¿
be a set fuzzy interval linear algebra over the set
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
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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
Example 3.2.16 : Let
V =i i
i i
[0,a ] 0 a 1
[0,b ] 0 b 1
- ½d dª º° °® ¾« » d d¬ ¼° °¯ ¿
be a set fuzzy interval linear algebra over the set
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
Example 3.2.17 : Let
V = 1 2
i
[0,a ] [0,a ]0 a 1;1 i 4
[0 ] [0 ]
- ½ª º° °d d d d® ¾
« »¬ ¼° °¯ ¿
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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.
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S =n
10, n 1,2,...
2
- ½® ¾
¯ ¿
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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° « »
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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
We will illustrate this situation by some example
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¯ ¿
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Thus if v = [0, ai] V and S =r
1
2 S then
ª º
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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;
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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.
Example 3.2.26 : Let
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®« »
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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.
Example 3.2.27 : Let
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.
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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
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S =n m
1 10,1, , m,n Z
2 10
- ½® ¾
¯ ¿.
D fi T V W f ll
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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
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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.
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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.
We will illustrate this situation by an example.
Example 3.2.34 : Let
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
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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® « »°« »
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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« »« »« »
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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.
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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
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Example 4.1.3: Let
V = V1 V2 =
[0 a ]- ª º
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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.
We will illustrate this situation by some examples.
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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
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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
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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.
We will illustrate this situation by some examples.
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° °
« »° °
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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
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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.
We will illustrate this situation by some examples.
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« »° °
« »° °
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5[0,a ] 0
« »° °« »° °¬ ¼¯ ¿
V1 V2 = V; W is a set interval linear subbial
the set S.
Example 4.1.14 : Let
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
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=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
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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.
Example 4.1.19: Let V = V1 V2 =
[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¯ ¿
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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.
Example 4.1.20: Let V = V1 V2
= {([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° °¯ ¿
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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
°« » « » « »
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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
Example 4.1.23: Let V = V1 V2
be a biset interval bivector space over the biset
Z12 Z42.
Example 4.1.24: Let V = V1 V2
[0 ]- ½ª º
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=
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.
Example 4.1.25: Let V = V1 V2
=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° °
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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
Example 4.1.28: Let V = V1 V2
a a a- ½ª º§ ·
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=
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
Example 4.1.29: Let V = V1 V2
=
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
Example 4.1.30: Let V = V1 V2
= 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
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=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.
Example 4.1.31: 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 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
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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.
We will illustrate this situation by some example
Example 4.1.32: Let V = V1 V2
=
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
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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¯ ¿¦
be a quasi biset interval bivector space over the bis
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}.
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We will illustrate this by some examples.
Example 4.1.38: Let V = V1 V2 = {[0, a] | a Z7}
5
a
a a Z
- ½ª º° °« » ® ¾« »
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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}.
Example 4.1.41: Let V = V1 V2
-ª
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=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.
Example 4.1.44: Let V = V1 V2
-ª º
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=
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
Example 4.1.45: Let V = V1 V2 = {[0, a] | a Z3}
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¯
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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.
Example 4.1.47 : Let V = V1 V2
=
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
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the reader is given the simple task of defi
substructures.
Example 4.1.49: Let V = V1 V2 =
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}.
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Example 4.1.51: Let V = V1 V2 =
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¯
be a quasi biset interval bilinear algebra over the
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° « »
®« »
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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
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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.
Example 4.1.56 : Let
V = V1 V2
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= 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
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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
Example 4.1.58: Let V = V1 V2 =
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¯
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4[ , ]¬ ¼¯
V1 V2 = V such that W is a quasi bisubset interalgebra over the subbiset P = {0, 3} {0, 2, 4, 6}
Example 4.1.59: Let V = V1 V2 =
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¯
be a quasi biset interval bilinear algebra over the
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be a quasi biset interval bilinear algebra over the
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
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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.
We will illustrate this situation by some examples.
Example 4.2.1: Let V = V1 V2 =
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.
Example 4.2.2: Let V = V1 V2 =
1 2[0,a ] [0,a ]- ½ª º° °« »
Example 4.2.3: Let V = V1 V2 =
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¯
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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.
We will illustrate this situation by some example
Example 4.2.4 : Let V = V1 V2 =
be a semigroup interval bivector space over the sem
Z5. Choose W = W1 W2 =
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°
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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.
Example 4.2.5: Let V = V1 V2 =
> @> @
> @
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.
Example 4.2.6 : Let V = V1 V2 =
{[0 a] | a Z29}
> @
> @ 2
0,a
0 a a Z
- ª º° « »°
« »®
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{[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.
Example 4.2.7 : Let V = V1 V2
=
> @ > @ > @> @ > @ > @> @ > @ > @
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
Example 4.2.8: Let V = V1 V2 =
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° °« »
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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
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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.
Example 4.2.10: Let V = V1 V2
=1
i 5
2
[0,a ] a Z ;[0,a ]
1 i 3[0 ]
- ½ª º ° °« »® ¾« » d d° °« »
Example 4.2.11: Let V = V1 V2 =
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
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semigroup interval bivector space over the semigroup
Clearly V is a doubly simple semigroup interv
space over the semigroup S = Z7.
Example 4.2.12: Let V = V1 V2 =
[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.
Example 4.2.13: Let V = V1 V2 =
i
i i 19
i 0
a x a Zf
- ½® ¾
¯ ¿¦
[0, a] [0, a]a
[0, a] [0, a]
- ª º° ® « »
¬ ¼° ¯
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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.
We will illustrate this situation by some example
Example 4.2.15: Let V = V1 V2
= 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
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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.
The proof is left as an exercise to the reader.
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
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over the semigroup S = Z12.
Example 4.2.17 : Let V = V1 V2 =
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° « »° « »° « » ®
« »°
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[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.
Example 4.2.20: Let V = V1 V2 =
- ½
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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
Example 4.2.21: Let V = V1 V2 =
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° ¯
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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.
Example 4.2.22: Let V = V1 V2 =
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
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V 1 V 2 to be a quasi semigroup interval biline
S.
We will first illustrate this situation by some sim
Example 4.2.23: Let V = V1 V2 =
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}.
Example 4.2.24 : Let V = V1 V2 =
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
Example 4.2.25: Let V = V1 V2 =
[0 ] [0 ]-ª º°
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> @
[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.
Example 4.2.26 : Let V = V1 V2 =
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°
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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
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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.
Example 4.2.30: Let V = V1 V2 =
7
[0,a] [0,a] [0,a], a Z
- ½ª ºª º° °® ¾« »« »
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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.
We will illustrate this situation by some examples.
Example 4.3.1: Let V = V1 V2 =
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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).
Example 4.3.2: Let V = V1 V2 =
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
- ® ¯ ¦ ¦
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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.
We will illustrate this situation by some example
Example 4.3.4 : Let V = V1 V2 =
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
« »° °
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0 0
[0, a] [0, a]
« »° °
« »° °« »° °¬ ¼¯ ¿
V1 V2 = V; W is a group interval bivector sub
over the group G.
Example 4.3.5: Let V = V1 V2 =
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
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be a group interval bivector space over the
Clearly V is a simple group interval bivector spa
Example 4.3.7 : Let V = V1 V2 =
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.
We will illustrate this situation by some examples.
Example 4.3.8: Let V = V1 V2 = {([0, a1], [0, a2]
a4]) | ai Z48; 1 d i d 4}
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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.
Example 4.3.9: Let V = V1 V2 =
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.
Example 4.3.10: Let V = V1 V2 =
i
i i 11
i 0
[0,a ]x a Zf
- ½® ¾
¯ ¿¦
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{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
The proof is left as an exercise to the reader.
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
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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.
We will illustrate this situation by some examples.
Example 4.3.12: Let V = V1 V2 =
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° ¯
Example 4.3.13: Let V = V1 V2 =
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
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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
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Example 4.3.16 : Let V = V1 V2 =
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
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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.
Example 4.3.17 : Let V = V1 V2 =
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
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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 =
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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.
Example 4.3.20: Let V = V1 V2 =
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
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5 p g p g
is not a doubly simple group interval bilinear algebgroup G.
Example 4.3.21: Let V = V1 V2 =
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
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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.
Example 4.3.22: Let V = V1 V2 =
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
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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.
We will illustrate this situation by some examples.
Example 4.3.23: Let V = V1 V2 =
[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.
Example 4.3.24: Let V = V1 V2 =
[0, a] [0,b]- ½ª º
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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.
Example 4.3.25: Let V = V1 V2 =
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¯ ¿
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[0, a] [0, a]° ¬ ¼¯ ¿
V1 V2 be a quasi group interval bilinear suba
over the group G = Z14.
Example 4.3.26 : Let V = V1 V2 =
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
over the group G = Z18.
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Example 4.3.27 : Let V = V1 V2 =
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.
Example 4.3.28: Let V = V1 V2 =
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° °¯ ¿
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be a quasi group interval bilinear algebra over the
Z47. Clearly V is a simple quasi group interval bilin
over the group G = Z47.
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
Example 4.3.29: Let V = V1 V2 =
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¦
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V1 V2 is a quasi subgroup interval bilinear suba
over the group G = Z13.
Thus V is not a doubly simple quasi group inter
algebra over the group G = Z13.
Example 4.3.31: Let V = V1 V2 = {[0, a] | a Z43}
{([0, a], [0, a]),
[0,a]
[0,a]
[0 a]
ª º« »« »
« »¬ ¼
| a Z43}
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[0,a]« »¬ ¼
be a quasi group interval bilinear algebra over the
Z43. V is a doubly simple quasi group interval bilin
over the group G = Z43.
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}
be a quasi group interval bilinear algebra over the
Z47. V is a doubly simple quasi group interval bilin
over the group G = Z47.
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¯
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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
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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.
We will illustrate this by some examples.
Example 4.4.1: Let V = V1 V2 =
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
Example 4.4.2: Let V = V1 V2 =
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° « »°
« »®« »° #
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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.
We will illustrate this situation by some example
Example 4.4.3: Let V = V1 V2 =
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
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g p gand 4.4.4 are of infinite cardinality.
Example 4.4.5: Let V = V1 V2 =
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
Example 4.4.6 : Let V = V1 V2 =
1[0,a ]- ª º°
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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 ° « ® « ° « ¬ ¯
be a bisemigroup interval bivector space over the
= 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°
Example 4.4.7 : Let V = V1 V2 =
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° °« »
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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° « »
Example 4.4.8: Let V = V1 V2 =
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
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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¯ ¿¦
be a bisemigroup interval bilinear algebra over the b
S = S1 S2 = Z48 Z40. Choose W = W1 W2 = {upper triangular interval matrices with intervals of t
ai]; ai Z48} - ½
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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
Example 4.4.10: Let V = V1 V2 =
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°
be a bisemigroup interval bivector space over the
= 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¯
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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
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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.
Example 4.4.13: Let V = V1 V2 =
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° « » ®« »
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[ , ] [ , ] [ , ] [ , ]
[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.
The proof is left as an exercise to the reader.
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
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Example 4.4.14 : Let V = V1 V2 =
[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° °¯ ¿
be a bisemigroup interval bilinear algebra over the b
[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+
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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.
Example 4.4.15: Let V = V1 V2 =
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 .
We will illustrate this situation by some examples.
Example 4.4.16: Let V = V1 V2 =
1 2[0,a ] [0,a ]- ª º
°« »
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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}}.
Example 4.4.17: Let V = V1 V2 =
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.
Example 4.4.18: Let V = V1 V2 =
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
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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
° « »° « » ® « »°
Example 4.4.19: Let V = V1 V2 =
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° « »
° « »° « » ° « »®
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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
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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
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{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.
We will illustrate this situation by some examples.
Example 4.4.22: Let V = V1 V2 =
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
¯ ¿¦
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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.
Example 4.4.25: Let V = V1 V2 =
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° °« »° °
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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.
Example 4.4.27: Let V = V1 V2 =
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°̄
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6 7 8 9 10[ , ] [ , ] [ , ] [ , ] [ , ]¬ ¼° ¯
be a set-semigroup interval bilinear algebr
semigroup S = S1 S2 = {2Z+ {0}, 5Z
+ {0}
Z412. Choose W = W1 W2 =
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
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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.
Example 4.4.28: Let V = V1 V2 =
- 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 =
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Example 4.4.29: Let V = V1 V2 =
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¯
#
be a bigroup interval bivector space over the big
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« »° °« »
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[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.
Example 4.4.31: Let V = V1 V2 =
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 ®
« »° « »°« »
be a bigroup interval bivector space over the big
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
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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.
Example 4.4.32: Let V = V1 V2 =
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° « »° « »
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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
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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
Example 4.4.34 : Let V = V1 V2 =
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
¯ ¿¦
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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.
Example 4.4.35: Let V = V1 V2 =
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
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or Q+ or C.
Now we can define bigroup interval bilinear alg
only examples of them.
Example 4.4.36 : Let V = V1 V2 =
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° °« »° °« »
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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
Example 4.4.40: Let V = V1 V2 =
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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.
We will illustrate this situation by some examples.
Example 4.4.41: Let V = V1 V2 =
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° ¯
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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.
Example 4.4.42: Let V = V1 V2 =
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.
Example 4.4.43: Let V = V1 V2 =
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¯
¦
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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¯
¦
Example 4.4.44 : Let V = V1 V2 =
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¦ ¦
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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}.
Example 4.4.45: Let V = V1 V2 =
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}}
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{ }}
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.
Example 4.4.47: Let V = V1 V2 =
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
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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}.
Example 4.4.48: Let V = V1 V2 =
ii i[0,a ]x a Q {0}
f
- ½ ® ¾¯ ¿¦
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Example 4.4.51: Let V = V1 V2 =
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
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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
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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.
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We will illustrate this situation by some examples.
Example 4.4.54: Let V = V1 V2 =
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+}.
Example 4.4.56: Let V = V1 V2 =
2i 3ii i i
i 0 i 0
[0, a ]x , [0, a ]x a 4Z {f f
- ® ¯ ¦ ¦
- ª º
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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.
Example 4.4.57: Let V = V1 V2 =
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
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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
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y g p (g p )
algebra over a set-group (group-set) is a set-grinterval bivector space but not conversely.
Example 4.4.60: Let V = V1 V2 =
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
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G2. W is a subset - subgroup interval bilinear suba
over the subset - subgroup P1 H2 of S1 G2.
Example 4.4.62: Let V = V1 V2 =
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
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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].
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Chapter Six
SUGGESTED PROBLEMS
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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.
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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
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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?
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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
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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
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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¯
"#
{All 7 u 7 interval matrices with intervals of t
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{All 7 u 7 interval matrices with intervals of t
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° °« » ° °« »
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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
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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
algebra over the group G = Z5.
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
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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° « »®« »
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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
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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
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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
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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.
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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
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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
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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
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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
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, ,
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.
36. VASANTHA K ANDASAMY, W.B., and
Smarandache, Fuzzy Cognitive Maps anCognitive Maps, Xiquan, Phoenix, 2003.
37. VASANTHA K ANDASAMY, W.B., and
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
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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.
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INDEX
B
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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
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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
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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
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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
8/8/2019 Interval Linear Algebra, by W. B. Vasantha Kandasamy, Florentin Smarandache
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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
8/8/2019 Interval Linear Algebra, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/interval-linear-algebra-by-w-b-vasantha-kandasamy-florentin-smarandache 248/249
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
8/8/2019 Interval Linear Algebra, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/interval-linear-algebra-by-w-b-vasantha-kandasamy-florentin-smarandache 249/249
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