International Journal of Scientific Research and Modern ...ijsrme.rdmodernresearch.com/wp-content/uploads/2017/06/160.pdf · V. Chinnadurai* & S. Barkavi** Department of Mathematics,

International Journal of Scientific Research and Modern Education (IJSRME)

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(www.rdmodernresearch.com) Volume 2, Issue 1, 2017

122

OPERATIONS ON P-ORDER AND R-ORDER OF INTERNAL

CUBIC SOFT MATRICES

V. Chinnadurai* & S. Barkavi** Department of Mathematics, Annamalai University, Annamalai Nagar,

Chidambaram, Tamilnadu

Cite This Article: V. Chinnadurai & S. Barkavi, “Operations on P-Order and R-Order of

Internal Cubic Soft Matrices”, International Journal of Scientific Research and Modern Education, Volume 2,

Issue 1, Page Number 122-145, 2017.

Copy Right: © IJSRME, 2017 (All Rights Reserved). This is an Open Access Article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract: In this paper, we introduce the notion of P-order (R-order) of internal cubic soft matrices. Further, we

develop union and intersection of P-order(R-order) of internal cubic soft matrices and their related properties have

been investigated.

Key Words: Cubic Soft Sets, Cubic Soft Matrices, P-Order Internal Cubic Soft Matrices & R-Order Internal

Cubic Soft Matrices.

1. Introduction:

Fuzzy set theory was proposed by Lotifi A. Zadeh [12] and it has extensive applications in various fields.

In 1999, Molodstov[8] introduced the novel concept of soft sets and established the fundamental results of the new

theory. In 2003, Maji et al.[6] studied some properties of soft sets. Pei and Miao [10] and Chen [1]et al.,

improved the work of Maji et al. [5, 6].In [3], Jun et al., introduced a new notion of cubic set which is a

combination of fuzzy set and interval-valued fuzzy set and also investigated several properties of cubic sets.

Muhiuddin and Al-roqi [9], introduced the concepts of internal, external cubic soft sets, P-cubic

(R-cubic) soft subsets, R-union(R-intersection, P-union and P-intersection) of cubic soft sets and the complement

of a cubic soft sets. They also investigated several related properties and applied the notion of cubic soft sets to

BCK/BCI-algebras.

Fuzzy matrix was introduced by Thomason [11] and the concept of uncertainty was discussed by using

fuzzy matrices. Different concepts and ideas of fuzzy matrices have been given earlier mainly by Kim,

Meenakshi and Thomson [4, 7, 11]. Fuzzy matrix plays a vital role in fuzzy modeling, fuzzy diagnosis and fuzzy

controls. It also has applications in fields like psychology, medicine, economics and sociology.

In the early investigation, Chinnadurai and Barkavi introduced a new concepts of cubic soft matrix,

internal cubic soft matrix, external cubic soft matrix and discussed various result in[2].

In this paper, the new definition of P-order internal cubic soft matrices and R-order internal cubic soft

matrices have been introduced, along with some algebraic properties are discussed with suitable illustrations.

2. Preliminiaries:

In this section we recall some basic definitions and results which will be needed in the sequel.

Definition 2.1 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set over U is

defined to be a pair ),( A where is a mapping from A to )(UP and .EA Then the pair ),( A

can be represented as, AeeA )/(=),( where AeUuuuAue ee ,/)(),(~

,=)( is a

cubic soft set in which )(~

uAe is the interval valued fuzzy set and )(ue is a fuzzy set.

Definition 2.2 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set ),( A over

U is said to be an internal cubic soft set if )()()( uAuuA eee

for all Ae and for all .Uu

Definition 2.3 [9] Let U be an initial universal set and E be a set of parameters. A cubic soft set ),( A over

U is said to be an external cubic soft set if ))(),(()( uAuAu eee

for all Ae and for all .Uu

Definition 2.4 [9] Let U be an initial universal set and E be a set of parameters. For any subsets A and B of E,

),( A and ),( B be cubic soft sets over .U

1. The R-union of ),( A and ),( B is a cubic soft set ),( C where BAC = and

BAeee

ABee

BAee

e

R if)()(

,/ if)(

,/ if)(

=)




123

for all Ce and it is denoted by ).,(),(=),( BAC R

2. The R-intersection of ),( A and ),( B is a cubic soft set ),( C where BAC = and

BAeee

ABee

BAee

e

R if)()(

,/ if)(

,/ if)(

=)

for all Ce and it is denoted by ).,(),(=),( BAC R

Definition 2.5 [9] Let U be an initial universal set and E be a set of parameters. For any subsets A and B of E,

),( A and ),( B be cubic soft sets over .U

1. The P-union of ),( A and ),( B is a cubic soft set ),( C where BAC = and

BAeee

ABee

BAee

e

P if)()(

,/ if)(

,/ if)(

=)

for all Ce . This is denoted by ).,(),(=),( BAC P

2. The P-intersection of ),( A and ),( B is a cubic soft set ),( C where BAC = and

BAeee

ABee

BAee

e

P if)()(

,/ if)(

,/ if)(

=)

for all Ce . This is denoted by ).,(),(=),( BAC P

Definition 2.6 [9] Let U be an initial universal set and E be a set of parameters. The complement of a cubic soft

set ),( A over U is denoted by cA),( and defined by

),(=),( AA c where )(: UPAc and AeeA cc )/(=),( where

.,/)(),(~

,=)( AeUuuuAue c

e

c

e

c

Definition 2.7 [7] A matrix nmijaA ][= is said to be fuzzy matrix if miaij [0,1],1 and nj 1 .

Definition 2.8 [7] For any two fuzzy matrices ][=],[= ijij bBaA and a scalar Fk . Then,

(i) ],[=],[= bijabasupBA ijijij .

(ii) .],[=],[= ijijijij babainfsupAB

(iii) ].,[=],[= ijij akakinfkA

Definition 2.9 [2] Let muuuU ,...,,= 21 be an initial universal set and neeeE ,...,,= 21 be a set of

parameters. Let .EA Then cubic soft set ),( A can be expressed in matrix form as

][= ijaA=

mnmm

n

n

aaa

aaa

aaa

21

22221

11211

such that a

ij

a

ijij

eij

eij AuuAaA ,~

=)(),(~

=][= which is called an nm cubic soft matrix (shortly

CS-matrix or CSM) of the cubic soft set ).,( A

According to this definition, a cubic soft set ),( A is uniquely characterized by matrix nmija ][ where




124

mi 1,2,3,...,= and .1,2,3,...,= nj

Example 2.10 Let 4321 ,,,= uuuuU is a set of cars and 321 ,,= eeeA is a set of parameters, which

stands for mileage, engine and prize respectively. Then cubic soft set is defined as

}.)0.4[0.3,0.7],,(),0.5[0.2,0.9],,(),0.9[0.6,0.7],,(),0.4[0.1,0.8],,(,

,)0.1[0.3,0.5],,(),0.2[0.2,0.7],,(),0.7[0.3,0.6],,(),0.3[0.2,0.5],,(,

,)0.4[0.3,0.9],,(),0.9[0.2,0.6],,(),0.5[0.1,0.7],,(),0.6[0.5,0.8],,(,{=),(

43213

43212

43211

uuuue

uuuue

uuuueA

Then the CS-matrix A is written as,

A =

0.4[0.3,0.7],0.1[0.3,0.5],0.4[0.3,0.9],

0.5[0.2,0.9],0.2[0.2,0.7],0.7[0.2,0.6],

0.9[0.6,0.7],0.7[0.3,0.6],0.5[0.1,0.7],

0.4[0.1,0.8],0.3[0.2,0.5],0.6[0.5,0.8],

.

Definition 2.11 [2] Let nmij CSMaA ][=. Then

A is an internal cubic soft matrix (ICSM), if

a

ij

a

ij

a

ij AA~~

for all i, j.

Definition 2.12 [2] Let nmijaA ][=, nmijbB ][=

be the two cubic soft matrix of order .nm Then

P-order matrix is denoted and defined as ][][ ijPij ba , if b

ij

a

ij BA~~

and b

ij

a

ij for all i, j.

Definition 2.13 [2] Let nmijaA ][=, nmijbB ][=

be the two cubic soft matrix of order .nm Then

R-order matrix is denoted and defined as ][][ ijRij ba , if b

ij

a

ij BA~~

and b

ij

a

ij for all i, j.

Definition 2.14 [2] Let .],~

,~

[=,],~

,~

[= nm

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij CSMBBBAAA

Then

1. P-union of A and

B is denoted by BA P and defined as

CBA P = , if

c

ij

c

ijij CcC ,~

=][=, where b

ij

a

ij

c

ij BAC~

,~

max=~

and b

ij

a

ij

c

ij ,max= for all i, j.

2. P-intersection of A and

B is denoted by BA P and defined as

CBA P = , if

c

ij

c

ijij CcC ,~

=][=, where b

ij

a

ij

c

ij BAC~

,~

min=~

and b

ij

a

ij

c

ij ,min= for all i, j.

3. R-union of A and

B is denoted by BA R and defined as

CBA R = , if

c

ij

c

ijij CcC ,~

=][=, where b

ij

a

ij

c

ij BAC~

,~

max=~

and b

ij

a

ij

c

ij ,min= for all i, j.

4. R-intersection of A and

B is denoted by BA R and defined as

CBA R = , if

c

ij

c

ijij CcC ,~

=][=, where b

ij

a

ij

c

ij BAC~

,~

min=~

and b

ij

a

ij

c

ij ,max= for all i, j.

3. P-order and R-order of Internal Cubic Soft Matrices:

In this section we define P-order and R-order of internal cubic soft matrix and discussed some algebraic

properties.

Definition 3.1 Let a

ij

a

ijAA ,~

= and

b

ij

b

ijBB ,~

= be internal cubic soft matrices, if

b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j. Then it is defined as P-order internal cubic soft matrices and

it is denoted by .PICSM

Example 3.2 Let 21,= uuU be the universal set and 4321 ,,,= eeeeE be the set of parameters. Let

., EBA Then

})0.35[0.3,0.8],,(),0.6[0.4,0.9],,(,

,)0.4[0.3,0.7],,(),0.5[0.2,0.8],,(,{=),(

212

211

uue

uueA




125

and

}.)0.75[0.7,0.9],,(),8[0.6,1],0.,(,

,)0.6[0.5,0.8],,(),0.7[0.4,0.9],,(,{=),(

212

211

uue

uueB

Then P-order internal cubic soft matrices is defined as follows,

A = BP

0.5[0.3,0.8],0.4[0.3,0.7],

0.6[0.4,0.9],0.5[0.2,0.8],=

0.75[0.7,0.9],0.6[0.5,0.8],

8[0.6,1],0.0.7[0.4,0.9],.

(i.e) . BA P

Definition 3.3 Let a

ij

a

ijAA ,~

= and

b

ij

b

ijBB ,~

= be internal cubic soft matrices, if

b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j. Then it is defined as R-order internal cubic soft matrices and

it is denoted by .RICSM

Example 3.4 Let 21,= uuU be the universal set and 4321 ,,,= eeeeE be the set of parameters. Let

., EBA Then

})0.4[0.2,0.4],,(),0.7[0.4,0.8],,(,

,)0.5[0.1,0.6],,(),0.3[0.2,0.4],,(,{=),(

212

211

uue

uueA

and

}.)0.35[0.3,0.8],,(),6[0.5,1],0.,(,

,)0.4[0.3,0.9],,(),0.3[0.3,0.7],,(,{=),(

212

211

uue

uueB

Then R-order internal cubic soft matrices is defined as follows,

A = BR

0.4[0.2,0.4],0.5[0.1,0.6],

0.7[0.4,0.8],0.3[0.2,0.4],= .

0.35[0.3,0.8],0.4[0.3,0.9],

6[0.5,1],0.0.3[0.3,0.7],

(i.e) . BA R

Definition 3.5 Let ., P

nmICSMBA Then

a) Union of BA , is defined by ],[== ijP cCBA where

},{max}],~

,~

{max},~

,~

{max[= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABAC

for all i,j.

b) Intersection of BA , is defined by ],[== ijP cCBA where

},{min}],~

,~

{min},~

,~

{min[= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABAC

for all i,j.

Example 3.6 Consider Example 3.2. Then union and intersection of BA , are given by,

a) BA P = .

0.75[0.7,0.9],0.6[0.5,0.8],

8[0.6,1],0.0.7[0.4,0.9],

b) BA P = .

0.35[0.3,0.8],0.4[0.3,0.7],

0.6[0.4,0.9],0.5[0.2,0.8],

Definition 3.7 Let ., R

nmICSMBA Then

a) Union of BA , is defined by ],[== ijR cCBA where

},{min}],~

,~

{max},~

,~

{max[= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABAC

for all i,j.

b) Intersection of BA , is defined by ],[== ijR cCBA where




126

},{max}],~

,~

{min},~

,~

{min[= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABAC

for all i,j.

Example 3.8 Consider Example 3.4. Then union and intersection of BA , are given by,

a) BA R = .

0.35[0.3,0.8],0.4[0.3,0.9],

6[0.5,1],0.0.3[0.3,0.7],

b) BA R = .

0.4[0.2,0.4],0.5[0.1,0.6],

0.7[0.4,0.8],0.3[0.2,0.4],

Proposition 3.9 Let .,, P

nmICSMCBA Then

(i) .= AAA P

(ii) .= BBA P

(iii) .== CCBACBA PPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then (i) a

ij

a

ij

a

ijP

a

ij

a

ij

a

ijP AAAAAA ],~

,~

[],~

,~

[=

for all i, j.

Since a

ij

a

ijP AAAA~~

and

a

ij

a

ij for all i, j.

By Definition 3.5(a),

(ii) b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijP BBAABA ],~

,~

[],~

,~

[=

for all i, j.

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=

ji, allfor },{max}],~

,~

{max},~

,~

{max[=

B

B

BB

BABABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijP

(iii) .],~

,~

[],~

,~

[= CBBAACBA P

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijPP

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=

},{max}],~

,~

{max},~

,~

{max[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ij

P

b

ij

b

ij

b

ij

P

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijPP

CCBB

CBB

CBABACBA

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

C

C

CC

CBCBCBA

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijPP




127

Similarly, .= CCBA PP

Hence, .== CCBACBA PPPP

Proposition 3.10 Let .,, R

nmICSMCBA Then

(i) .= AAA R

(ii) .= BBA R

(iii) .== CCBACBA RRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then (i) a

ij

a

ij

a

ijR

a

ij

a

ij

a

ijR AAAAAA ],~

,~

[],~

,~

[=

for all i, j.

Since a

ij

a

ijR AAAA~~

and

a

ij

a

ij for all i, j.


.=

],~

,~

[=

ji, allfor },{min}],~

,~

{max},~

,~

{max[=

A

AA

AAAAAA

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ijR

(ii) b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijR BBAABA ],~

,~

[],~

,~

[=

for all i, j.

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

B

B

BB

BABABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijR

(iii) .],~

,~

[],~

,~

[= CBBAACBA R

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijRR

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=

},{min}],~

,~

{max},~

,~

{max[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ij

R

b

ij

b

ij

b

ij

R

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijRR

CCBB

CBB

CBABACBA

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

C

C

CC

CBCBCBA

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijRR




128

Similarly, .= CCBA RR

Hence, .== CCBACBA RRRR

Proposition 3.11 Let .,, P

nmICSMCBA Then

(i) .= AAA P

(ii) .= ABA P

(iii) .== ACBACBA PPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then (i) a

ij

a

ij

a

ijP

a

ij

a

ij

a

ijP AAAAAA ],~

,~

[],~

,~

[=

for all i, j.

Since a

ij

a

ijP AAAA~~

and

a

ij

a

ij for all i, j.

By Definition 3.5(b),

.=

],~

,~

[=


,~

{min},~

,~

{min[=

A

AA

AAAAAA

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ijP

(ii) b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijP BBAABA ],~

,~

[],~

,~

[=

for all i, j.

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

BABABA

a

ij

a

ij

a

ij

a

ij

a

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijP

(iii) .],~

,~

[],~

,~

[= CBBAACBA P

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijPP

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

{min},~

,~

{min[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ij

P

a

ij

a

ij

a

ij

P

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijPP

CCAA

CAA

CBABACBA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.

Again by Definition 3.5(b),

.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

CACACBA

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijPP




129

Similarly, .= ACBA PP

Hence, .== ACBACBA PPPP

Proposition 3.12 Let .,, R

nmICSMCBA Then

(i) .= AAA R

(ii) .= ABA R

(iii) .== ACBACBA RRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then (i) a

ij

a

ij

a

ijR

a

ij

a

ij

a

ijR AAAAAA ],~

,~

[],~

,~

[=

for all i, j.

Since a

ij

a

ijR AAAA~~

and

a

ij

a

ij for all i, j.


.=

],~

,~

[=


,~

{min},~

,~

{min[=

A

AA

AAAAAA

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ijR

(ii) b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijR BBAABA ],~

,~

[],~

,~

[=

for all i, j.

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

BABABA

a

ij

a

ij

a

ij

a

ij

a

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijR

(iii) .],~

,~

[],~

,~

[= CBBAACBA R

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijRR

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=

ji, allfor ),(max)],~

,~

(min),~

,~

(min[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ij

R

a

ij

a

ij

a

ij

R

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijRR

CCAA

CAA

CBABACBA

Since, c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

CACACBA

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijRR




130

Similarly, .= ACBA RR

Hence, .== ACBACBA RRRR

Theorem 3.13 Let .,...,,, 321

P

nmk ICSMAAAA Since ....321

kPPPP AAAA

Then .=...321

kkPPPP AAAAA

Proof. Let

.],~

,~

[=

,...,],~

,~

[= ,],~

,~

[=,],~

,~

[= 333322221111

P

nm

a

ijk

a

ijk

a

ijkk

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

ICSMAAA

AAAAAAAAA

Then, ....=... 321321

kPPPPkPPPP AAAAAAAA

Since, a

ij

a

ijP AAAA 2121

~~

and a

ij

a

ij21 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=

ji, allfor ...},{max}],~

,~

{max},~

,~

{max[=

4333222

3222

3212121

kPPP

a

ij

a

ij

a

ijP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA

Since, a

ij

a

ijP AAAA 3232

~~

and a

ij

a

ij32 for all i, j.

Again by Definition 3.5(a),

....],~

,~

[],~

,~

[=

...],~

,~

[=


,~

{max},~

,~

{max[=

5444333

4333

3323232

kPPP

a

ij

a

ij

a

ijP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA

Continuing the same process we get,

.],~

,~

[],~

,~

[= 1)(1)(1)(

a

ijk

a

ijk

a

ijkP

a

ijk

a

ijk

a

ijk AAAA

Since, a

ijk

a

ijkkPk AAAA

~~1)(1)(

and

a

ijk

a

ijk 1)( for all i, j.

Again by the Definition 3.5(a),

.=

.,~

=

],~

,~

[=


,~

{max},~

,~

{max[= 1)(1)(1)(

k

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

A

A

AA

AAAA

Theorem 3.14 Let .,...,,, 321

R


kRRRR AAAA

Then .=...321

kkRRRR AAAAA

Proof. Let

.],~

,~

[=

,...,],~

,~

[= ,],~

,~

[=,],~

,~

[= 333322221111

R

nm

a

ijk

a

ijk

a

ijkk

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

ICSMAAA

AAAAAAAAA




131

Then, ....=... 321321

kRRRRkRRRR AAAAAAAA

Since, a

ij

a

ijR AAAA 2121

~~

and a

ij

a

ij21 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=

ji, allfor ...},{min}],~

,~

{max},~

,~

{max[=

4333222

3222

3212121

kRRR

a

ij

a

ij

a

ijR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA

Since, a

ij

a

ijR AAAA 3232

~~

and a

ij

a

ij32 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=

jforalli,...},{min}],~

,~

{max},~

,~

{max[=

5444333

4333

3323232

kRRR

a

ij

a

ij

a

ijR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA


.],~

,~

[],~

,~

[= 1)(1)(1)(

a

ijk

a

ijk

a

ijkR

a

ijk

a

ijk

a

ijk AAAA

Since, a

ijk

a

ijkkRk AAAA

~~1)(1)(

and

a

ijk

a

ijk 1)( for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[= 1)(1)(1)(

k

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

a

ijk

A

A

AA

AAAA

Theorem 3.15 Let .,...,,, 321

P


kPPPP AAAA

Then .=... 1321

AAAAA kPPPP

Proof. Let

.],~

,~

[=

,...,],~

,~

[= ,],~

,~

[=,],~

,~

[= 333322221111

P

nm

a

ijk

a

ijk

a

ijkk

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

ICSMAAA

AAAAAAAAA

Then, ....=... 321321

kPPPPkPPPP AAAAAAAA

Since, a

ij

a

ijP AAAA 2121

~~

and a

ij

a

ij21 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=


,~

{min},~

,~

{min[=

4333111

3111

3212121

kPPP

a

ij

a

ij

a

ijP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA




132

Since, a

ij

a

ijP AAAA 3131

~~

and a

ij

a

ij31 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=


,~

{min},~

,~

{min[=

5444111

4111

3313131

kPPP

a

ij

a

ij

a

ijP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

kPPP

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA


.],~

,~

[],~

,~

[= 111

a

ijk

a

ijk

a

ijkP

a

ij

a

ij

a

ijAAAA

Since, a

ijk

a

ijkP AAAA

~~11

and a

ijk

a

ij 1 for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

1

11

111

111

A

A

AA

AAAA

a

ij

a

ij

a

ij

a

ij

a

ij

a

ijk

a

ij

a

ijk

a

ij

a

ijk

a

ij

Theorem 3.16 Let .,...,,, 321

R


kRRRR AAAA

Then .=... 1321

AAAAA kRRRR

Proof. Let

.],~

,~

[=

,...,],~

,~

[= ,],~

,~

[=,],~

,~

[= 333322221111

R

nm

a

ijk

a

ijk

a

ijkk

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

ICSMAAA

AAAAAAAAA

Then ....=... 321321

kRRRRkRRRR AAAAAAAA

Since, a

ij

a

ijR AAAA 2121

~~

and a

ij

a

ij21 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=


,~

{min},~

,~

{min[=

4333111

3111

3212121

kRRR

a

ij

a

ij

a

ijR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA

Since, a

ij

a

ijR AAAA 3131

~~

and a

ij

a

ij31 for all i, j.


....],~

,~

[],~

,~

[=

...],~

,~

[=


,~

{min},~

,~

{min[=

5444111

4111

3313131

kRRR

a

ij

a

ij

a

ijR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

kRRR

a

ij

a

ij

a

ij

a

ij

a

ij

a

ij

AAAAAA

AAAA

AAAAAA





133

.],~

,~

[],~

,~

[= 111

a

ijk

a

ijk

a

ijkR

a

ij

a

ij

a

ijAAAA

Since, a

ijk

a

ijkR AAAA

~~11

and a

ijk

a

ij 1 for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

1

11

111

111

A

A

AA

AAAA

a

ij

a

ij

a

ij

a

ij

a

ij

a

ijk

a

ij

a

ijk

a

ij

a

ijk

a

ij

Theorem 3.17 Let ,,, P

nmICSMCBA then

.== BCBCACBA PPPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[= CBBAACBA P

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijPP

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

(max),~

,~

(max[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ij

P

b

ij

b

ij

b

ij

P

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijPP

CCBB

CBB

CBABACBA

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

B

B

BB

CBCB

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ij

R.H.S= . CBCA PPP

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ijP CCAACA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijP

A

AA

CACACA




134

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ijP CCBBCB

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

,~

=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

BA

BACBCA

B

BB

CBCBCB

P

b

ij

b

ijP

a

ij

a

ijPPP

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijP

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.==Hence,

.=

],~

,~

[=

],~

,~

[=

},{max}],~

,~

{max},~

,~

{max[=

BCBCACBA

B

BB

BB

BABACBCA

PPPPP

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijPPP

Theorem 3.18 Let ,,, R

nmICSMCBA then

.== BCBCACBA RRRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[= CBBAACBA R

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijRR

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=

},{min}],~

,~

{max},~

,~

{max[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ij

R

b

ij

b

ij

b

ij

R

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijRR

CCBB

CBB

CBABACBA

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

B

B

BB

CBCB

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ij

R.H.S= . CBCA RRR




135

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ijR CCAACA

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijR

A

AA

CACACA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ijR CCBBCB

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

,~

=

.,~

=

],~

,~

[=

},{max}],~

,~

{min},~

,~

{min[=

BA

BACBCA

B

BB

CBCBCB

R

b

ij

b

ijR

a

ij

a

ijRRR

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijR

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.==Hence,

.=

],~

,~

[=

],~

,~

[=

},{min}],~

,~

{max},~

,~

{max[=

BCBCACBA

B

BB

BB

BABACBCA

RRRRR

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijRRR


nmICSMCBA then

.== CCBCACBA PPPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[= CBBAACBA P

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijPP

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

{min},~

,~

{min[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ij

P

a

ij

a

ij

a

ij

P

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijPP

CCAA

CAA

CBABACBA




136

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=

ji, allfor ),(min)],~

,~

(min),~

,~

(max[=

C

C

CC

CACA

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ij

R.H.S= . CBCA PPP

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ijP CCAACA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijP

C

CC

CACACA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ijP CCBBCB

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

,~

=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

CC

CCCBCA

C

CC

CBCBCB

P

c

ij

c

ijP

c

ij

c

ijPPP

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijP

By Proposition 3.11(i),

.==Hence,.= CCBCACBACCBCA PPPPPPPP


nmICSMCBA then

.== CCBCACBA RRRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[= CBBAACBA R

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijRR

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

{min},~

,~

{min[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ij

R

a

ij

a

ij

a

ij

R

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijRR

CCAA

CAA

CBABACBA




137

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

C

C

CC

CACA

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ij

R.H.S= . CBCA RRR

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ijR CCAACA

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijR

C

CC

CACACA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ijR CCBBCB

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.=

,~

,~

=

.,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

CC

CCCBCA

C

CC

CBCBCB

R

c

ij

c

ijR

c

ij

c

ijRRR

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijR

By Proposition 3.12(i), .= CCBCA RRR

.==Hence, CCBCACBA RRRRR


nmICSMCBA then

.== BCABACBA PPPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ijPPP CCBBACBA

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

(min),~

,~

(min[=

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ij

b

ij

b

ij

b

ijP

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijPPP

BBAA

BBA

CBCBACBA




138

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

B

B

BB

BABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij

R.H.S= . CABA PPP

Then, .],~

,~

[],~

,~

[=

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijP BBAABA

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijP

B

BB

BABABA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ijP CCAACA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

,~

=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

CB

CBCABA

C

CC

CACACA

P

c

ij

c

ijP

b

ij

b

ijPPP

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijP

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j.


.=

],~

,~

[=

],~

,~

[=


,~

{min},~

,~

{min[=

B

BB

BB

CBCBCABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijPPP

Hence, .== BCABACBA PPPPP


nmICSMCBA then

.== BCABACBA RRRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA




139

Then, L.H.S= .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ijRRR CCBBACBA

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

(min),~

,~

(min[=

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ij

b

ij

b

ij

b

ijR

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijRRR

BBAA

BBA

CBCBACBA

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

B

B

BB

BABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij

R.H.S= . CABA RRR

Then, .],~

,~

[],~

,~

[=

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijR BBAABA

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijR

B

BB

BABABA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ijR CCAACA

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j. Again by Definition 3.7(a),

.=

,~

,~

=

,~

=

],~

,~

[=


,~

{max},~

,~

{max[=

CB

CBCABA

C

CC

CACACA

R

c

ij

c

ijR

b

ij

b

ijRRR

c

ij

c

ij

c

ij

c

ij

c

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijR

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.=

],~

,~

[=

],~

,~

[=


,~

{min},~

,~

{min[=

B

BB

BB

CBCBCABA

b

ij

b

ij

b

ij

b

ij

b

ij

b

ij

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijRRR




140

Hence, .== BCABACBA RRRRR


nmICSMCBA then

.== ACABACBA PPPPP

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= P

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

b

ij

b

ij

b

ijPPP CCBBACBA

Since c

ij

b

ijP CBCB~~

and

c

ij

b

ij for all i, j. By Definition 3.5(a),

.],~

,~

[],~

,~

[=

],~

,~

[=


,~

(max),~

,~

(max[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ij

c

ij

c

ij

c

ijP

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijPPP

CCAA

CCA

CBCBACBA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j. By Definition 3.5(b),

.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

CACA

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ij

R.H.S= . CABA PPP

Then, .],~

,~

[],~

,~

[=

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ijP BBAABA

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j. By Definition 3.5(b),

.,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

a

ij

a

ij

a

ij

a

ij

a

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijP

A

AA

BABABA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijP

a

ij

a

ij

a

ijP CCAACA

Since c

ij

a

ijP CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

,~

=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

AA

AACABA

A

AA

CACACA

P

a

ij

a

ijP

a

ij

a

ijPPP

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijP

By Proposition 3.9(i), .= ACBCA PPP




141

Hence, .== ACABACBA PPPPP


nmICSMCBA then

.== ACABACBA RRRRR

Proof. Let .],~

,~

[=,],~

,~

[=,],~

,~

[= R

nm

c

ij

c

ij

c

ij

b

ij

b

ij

b

ij

a

ij

a

ij

a

ij ICSMCCCBBBAAA

Then, L.H.S= .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

b

ij

b

ij

b

ijRRR CCBBACBA

Since c

ij

b

ijR CBCB~~

and

c

ij

b

ij for all i, j.


.],~

,~

[],~

,~

[=

],~

,~

[=


,~

(max),~

,~

(max[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ij

c

ij

c

ij

c

ijR

c

ij

b

ij

c

ij

b

ij

c

ij

b

ijRRR

CCAA

CCA

CBCBACBA

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

A

A

AA

CACA

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ij

R.H.S= . CABA RRR

Then, .],~

,~

[],~

,~

[=

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ijR BBAABA

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.


.,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

a

ij

a

ij

a

ij

a

ij

a

ij

b

ij

a

ij

b

ij

a

ij

b

ij

a

ijR

A

AA

BABABA

Then, .],~

,~

[],~

,~

[=

c

ij

c

ij

c

ijR

a

ij

a

ij

a

ijR CCAACA

Since c

ij

a

ijR CACA~~

and

c

ij

a

ij for all i, j.


.=

,~

,~

=

,~

=

],~

,~

[=


,~

{min},~

,~

{min[=

AA

AACABA

A

AA

CACACA

R

a

ij

a

ijR

a

ij

a

ijRRR

a

ij

a

ij

a

ij

a

ij

a

ij

c

ij

a

ij

c

ij

a

ij

c

ij

a

ijR




142

By Proposition3.10(i), .= ACBCA RRR

.==,Hence ACABACBA RRRRR

Theorem 3.25 Let ., P

nmICSMBA Then

(i) .=c

P

cc

P ABBA

(ii) .=c

P

cc

P ABBA

Proof. Let a

ij

a

ij

a

ij AAA ],~

,~

[=

and .],~

,~

[= b

ij

b

ij

b

ij BBB

(i) L.H.S .],~

,~

[],~

,~

[= .=c

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ij

c

P BBAABA

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.

By the Definition 3.5(a), c

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{max}],~

,~

{max},~

,~

{max[= for all i, j

},{max}],1~

,~

{max},1~

,~

{max[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{max}],1~

,~

{max},1~

,~

{max[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

R.H.S =c

P

cAB

ca

ij

a

ij

a

ijP

cb

ij

b

ij

b

ij AABB

],~

,~

[],~

,~

[=

a

ij

a

ij

a

ijP

b

ij

b

ij

b

ij AABB

],1~

,1~

[1],1~

,1~

[1=

.],1~

,1~

[1],1~

,1~

[1= a

ij

a

ij

a

ijP

b

ij

b

ij

b

ij AABB

Since ,c

P

cAB by Definition 3.5(b),

},1{1min}],~

,1~

{1min},~

,1~

{1min[= a

ij

b

ij

a

ij

b

ij

a

ij

b

ij ABAB

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

L.H.S = R.H.S

So .=c

P

cc

P ABBA

(ii) L.H.S .],~

,~

[],~

,~

[= =c

b

ij

b

ij

b

ijP

a

ij

a

ij

a

ij

c

P BBAABA

Since b

ij

a

ijP BABA~~

and

b

ij

a

ij for all i, j.

By Definition 3.5(b), c

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{min}],~

,~

{min},~

,~

{min[= for all i, j

},{min}],1~

,~

{min},1~

,~

{min[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA




143

},{min}],1~

,~

{min},1~

,~

{min[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

R.H.S =c

P

cAB

ca

ij

a

ij

a

ijP

cb

ij

b

ij

b

ij AABB

],~

,~

[],~

,~

[=

a

ij

a

ij

a

ijP

b

ij

b

ij

b

ij AABB

],1~

,1~

[1],1~

,1~

[1=

.],1~

,1~

[1],1~

,1~

[1= a

ij

a

ij

a

ijP

b

ij

b

ij

b

ij AABB

Since ,c

P

cAB by Definition 3.5(a),

},1{1max}],~

,1~

{1max},~

,1~

{1max[= a

ij

b

ij

a

ij

b

ij

a

ij

b

ij ABAB

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

L.H.S = R.H.S

So .=c

P

cc

P ABBA

Theorem 3.26 Let ., R

nmICSMBA Then

(i) .=c

R

cc

R ABBA

(ii) .=c

R

cc

R ABBA

Proof. Let a

ij

a

ij

a

ij AAA ],~

,~

[=

and .],~

,~

[= b

ij

b

ij

b

ij BBB

(i) L.H.S .],~

,~

[],~

,~

[= =c

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ij

c

R BBAABA

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.

By Definition 3.7(a), c

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{min}],~

,~

{max},~

,~

{max[= for all i, j

},{min}],1~

,~

{max},1~

,~

{max[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{min}],1~

,~

{max},1~

,~

{max[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

R.H.S = .c

R

cAB

ca

ij

a

ij

a

ijR

cb

ij

b

ij

b

ij AABB

],~

,~

[],~

,~

[=

a

ij

a

ij

a

ijR

b

ij

b

ij

b

ij AABB

],1~

,1~

[1],1~

,1~

[1=




144

.],1~

,1~

[1],1~

,1~

[1= a

ij

a

ij

a

ijR

b

ij

b

ij

b

ij AABB

Since ,c

R

cAB by Definition 3.7(b),

},1{1max}],~

,1~

{1min},~

,1~

{1min[= a

ij

b

ij

a

ij

b

ij

a

ij

b

ij ABAB

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

L.H.S = R.H.S

So .=c

R

cc

R ABBA

(ii) L.H.S .],~

,~

[],~

,~

[= =c

b

ij

b

ij

b

ijR

a

ij

a

ij

a

ij

c

R BBAABA

Since b

ij

a

ijR BABA~~

and

b

ij

a

ij for all i, j.

By Definition 3.7(b), c

b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

},{max}],~

,~

{min},~

,~

{min[= for all i, j

),(max)],1~

,~

(min),1~

,~

(min[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

),(max)],1~

,~

(min),1~

,~

(min[1= b

ij

a

ij

b

ij

a

ij

b

ij

a

ij BABA

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

R.H.S =c

R

cAB

ca

ij

a

ij

a

ijR

cb

ij

b

ij

b

ij AABB

],~

,~

[],~

,~

[=

a

ij

a

ij

a

ijR

b

ij

b

ij

b

ij AABB

],1~

,1~

[1],1~

,1~

[1=

.],1~

,1~

[1],1~

,1~

[1= a

ij

a

ij

a

ijR

b

ij

b

ij

b

ij AABB

Since ,c

R

cAB by Definition 3.7(a),

},1{1min}],~

,1~

{1max},~

,1~

{1max[= a

ij

b

ij

a

ij

b

ij

a

ij

b

ij ABAB

))],~

,~

[= c

ij

c

ij

c

ij CC

.= C

L.H.S = R.H.S

So .=c

R

cc

R ABBA

References: 1. Chen D., Tsang E.C.C., Yeung D.S., and Wang X.," The parameterization reduction of soft sets and its

applications", Computers and Mathematices with Applications 57, 2009, 1547-1553.

2. Chinnaduraia V and Barkavi S., "Cubic soft matrices", International Journal for Research in

Mathematics and Statistics, Volume-2, Issue-4, April,2017, 1-22.

3. Jun Y.B., Kim C.S. and Yang K.O., "Cubic sets", Annals of fuzzy Mathematics and Informatics, 4, 2012,

83-98.

4. Kim J.B., "Idempotent and Inverses in Fuzzy Matrices", Malaysian Math, 6(2), 1983, 57-61.

5. Maji P.K., Biswas R. and Roy A.R., "Fuzzy soft sets", Journal of Fuzzy Mathematics, 9, 2001, 589-602.




145

6. Maji P.K and Roy A.R., "Soft set theory", Computers and Mathematics with Applications 45, 2003,

555-562.

7. Meenakshi A.R., " Fuzzy Matrix Theory and Applications", MJP Publishers, 2008.

8. Molodstov D.A., Soft set theory-first results, Computers and Mathematics with Applications 37, 1999,

19-31.

9. Muhiuddin G., Al-roqi A.M., "Cubic soft sets with applications in BCK/BCL-algebras", Annals of Fuzzy

Mathematics and Informatics, In press.

10. Pei D. and Miao D., "From soft sets to information systems", in Proceedings of the IEEE International

Conference on Granular Computing, 2, 2005, 617-621.

11. Thomason M.G., " Convergence power of fuzzy matrix", J. of Math. Anal. Appl. 57, 1977, 476-486.

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International Journal of Scientific Research and Modern ...ijsrme.rdmodernresearch.com/wp-content/uploads/2017/06/160.pdf · V. Chinnadurai* & S. Barkavi** Department of Mathematics,