JID: JOEMS

[m;June 2, 2016;12:14]

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

Original Article

Some results associated with the max-min and min-max compositions of bifuzzy matrices

E.G. Emam*, M.A. Fndh

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

Received 22 January 2016; revised 24 April 2016; accepted 30 April 2016 Available online xxx

Keywords

Fuzzy matrices; Bifuzzy matrices; Intuitionistic fuzzy matrices

Abstract In this paper, we define some kinds of bifuzzy matrices, the max-min (o) and the minmax (*) compositions of bifuzzy matrices are defined. Also, we get several important results by these compositions. However, we construct an idempotent bifuzzy matrix from any given one through the min-max composition.

2010 Mathematics Subject Classification: 15B15; 15B33; 94D05; 08A72

Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The concept of bifuzzy sets (or intuitionistic fuzzy sets) was introduced by Atanassov [1] as a generalization of fuzzy subsets. Later on, much fundamental works have done with this concept by Atanassov [2,3] and others [4-7]. A bifuzzy relation is a pair of fuzzy relations, namely, a membership and a non-membership function, which represent positive and negative aspects of the given information. This is why the concept of bifuzzy relations is a generalization of the idea of fuzzy re-

lations. The name "bifuzzy relations" is used for objects introduced by Atanassov and originally called intuitionistic fuzzy relations (see [1,2]). Bifuzzy relations are also called by some authors "bipolar fuzzy relations" (see [6]). Since the concept of bifuzzy relations is an extension for the concept of ordinary fuzzy relations, the concept of bifuzzy matrices (which represent finite bifuzzy relations) is also an extension for the concept of ordinary fuzzy matrices.

In this paper, we study and prove some properties of bi-fuzzy matrices throughout some compositions of these matrices. However, we concentrate our attention for the two compositions o (max-min) and its dual composition * (min-max). We use the definitions of some kinds of bifuzzy matrices such as nearly constant, symmetric, nearly irreflexive and others to prove some results. One of these results enables us to construct an idempo-tent bifuzzy matrix from any bifuzzy matrix and this is the main result in our work. We also state the relationship between the two compositions o and * of bifuzzy matrices.The motivation for this paper is to study some kinds of finite bifuzzy relations

S1110-256X(16)30027-X Copyright 2016, Egyptian Mathematical Society. Production and hosting by Elsevier B.V. This is an open access article

under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

http://dx.doi.org/10.1016Zj.joems.2016.04.005

* Corresponding author. Tel.: +201002854283. E-mail addresses: eg_emom@yahoo.com (E.G. Emam), m.fndh@yahoo.com (M.A. Fndh).

Peer review under responsibility of Egyptian Mathematical Society.

throughout bifuzzy matrices by using the two compositions o and *.

2. Preliminaries and definitions

In system models which based on fuzzy sets, one often uses fuzzy matrices (matrices with elements having values anywhere in the closed interval [0, 1]) to define finite fuzzy relations.

When the related universes X and Y of a fuzzy relation R are finite such that \X\ = m, \ Y\ = n, a fuzzy matrix R = [r*"m whose generic term r¡j = iR(x*, y * ) for i = 1, 2, ..., m and j = 1, 2 , ..., n where the function iR : X x Y ^ [0 , 1] is called the membership function and r ij is the grade of membership of the element (x, , y ¡) in R,

Definition 2.1 [8,9]. Let A = \a*jf and B = \b *jf , be two

L ' J L lJ "mxn L U "nxl

fuzzy matrices. Then the max-min composition (o) of A and B is denoted by A o B and is defined as

A o B = bj Lxi = V(ak A bkj" •

The min-max composition (*) of A and B is denoted by A*B and is defined as

A * B = is*j*mxl = A (a*k V bkj) •

where V * A are the maximum and minimum operations respectively.

Definition 2.2 (bifuzzy matrix [6,10,11]) * Let A = la*■ " , A" = \a"*" be two fuzzy matrices such that

L iJ "mxn L lJ "m xn

a"j + a'"* < 1 for every i < m* j < n, The pair (A' , A"" is called a bifuzzy matrix and we may write A = [a* j = {a! * *, a'"* j"m . The numbers a" and a** denote the degree of membership and the degree of non-membership of the ijt h element in A respectively. Thus the bifuzzy matrix A takes its elements from the set F = {< a', a" > : a', a" e [0 * 1] * a' + a" < 1}

For each bifuzzy matrix A of kind m x n * there is a fuzzy matrix nA associated with A such that nij = 1 — a"* — a'"* for every i < m * j < n * The number nj is called the degree of indeterminacy of the ijt h element in A or called the degree of hesitancy of ijt h element in A * It is obvious that 0 < nj < 1 for every i < m * j < n . Especially, if nij = 0 for all i < m * j < n * then the bifuzzy matrix A is reduced to the ordinary fuzzy matrix. Thus fuzzy matrices are special cases from bifuzzy matrices.

Now, we define some operations on the set F defined above. For a =< a', a" >, b =< b', b" >e F, we define: a A b =< min (a', b') ^ max (a", b" ) >, a V b =< max (a*, b') ^ min (a", b" ) >, a" =< a", a' > and a < b if and only if a' < b', a" > b'",

I< 0 " a" > if a' < b', a" < b",

< 0 " 1 > if a' < b' , a" > b" ,

< a', a" > if a' > b'.

We may write 0 instead of < 0, 1 > and 1 instead of < 1, 0> ^ For the bifuzzy matrices A = [a*j = {a", aj" , B = \b" j = {b' *, b"* j" and C = j = {c" j, c"" * j" , let us define

lj x lj" lj" Inxn L lJ * ij" ij" \nx.m "

the following. matrix operations [8-11].

A A B = [a* A b* j",

A V B = [a j V hi j ],

a e b = [a] j e h] j ],

A ◦ C = \J{aik A c'kj), A(a]k V dkj]

_U=i k=i

For simplictly write AC instead of AoC. However, Ak = Ak-1 A, where

Ak = \a]] = aj, a!]= Ak^A and

I. = A0 =

o _ I 1 if ' = j'

0 if i = j.' a ji" (the transpose of A), a ji = (a"j, a""] (the complement of A) , A < B if and only if a* < b " for every i ^ j ^ < n ^

A. = A] =

3. Theoretical results of the paper

Definition 3.1 (reflexive, irreflexive bifuzzy matrix [6,8,9,11]) . An n x n bifuzzy matrix A = [ajj] is called reflexive (irreflexive) if and only if a. = 1 (an = 0 ). It is also called weakly reflexive (nearly irreflexive) if and only if a. > a. (an < a]) for every i, j < n.

Lemma 3.2. Let A = [a.]n and B = [b.]n x n be two nearly irreflexive bifuzzy matrices. Then A * B < AwB.

Proof. Let R = A * B and T = A v B. Then

rij = A ]ai,k v hk] j ] X {ca] A ykj ]] and t] = < ai v

bi j , ai ■ A bj . > . Now,

r. j = A (a.k v bk j j < aj. v bi j < ajj v f. = t]. and k=1

rj j = V (ajkA bkjj > ajiA bjj > ajjA bj j = tjj • Thus, we have

rj < tj and so A * B < AvBj □

It is noted that A v B = B for A < B.

Lemma 3.3. Let A and B be two nearly irreflexive bifuzzy matrices and A < B. Then A * B < B.

Proof. By Lemma 3.2. □

Definition 3.4 (symmetric, asymmetric bifuzzy matrix [6,9]) j An n x n bifuzzy matrix A = [a^ j is called symmetric if and only if A = At and it is also called asymmetric if and only if ajj A aji = 0 for every i, j < n .

Remark. It should be noted that any asymmetric bifuzzy matrix is also irreflexive.

Proposition 3.5. Let A = [ajj= (ajj, a!!j)j n be a symmetric and nearly irreflexive bifuzzy matrix. Then we have:

(1) A * A < A,

(2) A * A is symmetric and nearly irreflexive,

(3) A2 is weakly reflexive.

Proof. (1) By Lemmas 3.2 and 3.3.

(2) Suppose S = A * A. It is obvious that S is symmetric and

s]] = A (a]kV ak]] = /\a]k < A (a]kV akj] =s]j

k=1 k=1

V ck i]. V (a]k A cki ]

4 = V (a'ükA ak.] = V a]k. (a]kA akj] = s]j .

k =1 k=1 k=1

Thus, Sj < s. and so thatS is nearly irreflexive. (3) Let T = A2 . Then

tij = V (a'k A a'kj), A (a'!k V a'kj)), i.e., U=i k=i I

t'ij = V (ajk A a'k j) = a'h A a'h j for some h < n. But since A is symmetric, we have

tj = V (a'kA ak.' = V a'k > a'h > a'hA ah j = t'j ■

k=1 k=1

tjj = A (ajkV akj) = < V aSj for some s < n

tji = A (ajkV ak') = A ajk < < < ajsV aj =tj j . That is

k=1 k =1

t'i > t'j

and A2 is thus weakly reflexive. □

Remark. We notice that the bifuzzy matrix A * A is symmetric and irreflexive when A is also so.

(1) (A * A') * (A * A') < A * A',

(2) (A * A') * (A * A'j is symmetric and nearly irreflexive,

(3) (A * A')2 is weakly reflexive.

Proof. By Proposition 3.5 and Theorem 3.7. □

Proposition 3.9. Let A be an n x n asymmetric bifuzzy matrix. Then A * At = O (the zero matrix)

Proof. LetT = A * A'. Then

t'j = A (a'k v a'jk'' V i.ak A a'k' \k=1 k=1

= |a'h v a!'h, aj A aj jj for some h, s < n.

Notice that, since A is asymmetric, it is irreflexive and so

t'ij=v a'jh < ajv a'j j=a'j and ^=< a c'j' >

a'j A a'' ■ = a!'' . That is t'j < a'.

'j jj lj 'j — 'j

Similarly, we can see that t'j < a^ and t'j < a'j A aji = 0 . Thus, t ' j = 0 and so T = O, □

Proposition 3.6. For bifuzzy matrices A = \aij ]„ B = \bi ;i , C = \ci ;i , and D = \di ;i , we have:

L vJmxn ' L vJnx/ L v Ipxm '

(1) (B * C)i = Ct * Bt,

(2) If A < Bi then D * A < D * B and A * C < B * C. Proof. (1) Let S = C * Bi and T = B * C. Then

In n \

Sij = A (c'ki v b'jk )i v c. a b'jk ) iand \k=1 k=1 I

t ji = ( A (bjkv c'ki). V (b"jk a c'k. t \k=1 k=1

i.e., S = T'.

(2) Let W = D * A and G = D * B, i.e.,

w ]j = Md']k v a'kj ) ] V (d'/k A akj ]

U=1 k=1

g]j = A (djk v bkj )] V(dk A bkj ) j. \k=1 k=1

Since we have that A < B' we get a'kj < b'kj and a'kj > b'kj and

so dL v akj < d'kcv bkj and djkA akj > djkA bkj for every k < m ■

Therefore,

Definition 3.10 (nilpotent, transitive, idempotent bifuzzy matrix [8,9,11]) ' An n x n bifuzzy matrix A is called nilpotent if and only if An = O (the zero matrix), it is also called transitive if and only if A 2 < A and it is called idempotent if and only if A2 = A.

Proposition 3.11 ([11], pp. 224) ' If A is nilpotent, then Am is irreflexive for every m < n.

The following proposition shows that the nilpotency of a bifuzzy matrix A implies the asymmetry of that matrix. However, the converse is not always true.

Proposition 3.12. Let A be an n x n nilpotent bifuzzy matrix. Then A is asymmetric.

Proof. Since A is nilpotent, ajj' = 0 ^

If ajjAaf > 0' i.e., if a'. A a'. > 0 and a" V a!'- < L then a!-. > 0 ' aj i > 0 ' a jj < 1 and a"" < 1' Now, we have two cases for n. Case 1: If n is odd, then

a''' f > a'j A a!' A a'j A ... A a'j > 0 and

lJ lJ J1 lJ lJ

A (dk V akj) < A (d'k V bkj) and V (dk A a'kj) > \J (dk A

k=1 k=1 k=1 k=1

b'kj )j i.e^ w'j < gjj ■

Similarly, one can show that A * C < B * C. □

Theorem 3.7. For any m x n bifuzzy matrix A, A * A' is nearly irreflexive and symmetric.

Proof. Let R = A * A'. That is

r'j = A (a'kV ajk)' V (a'kA a"jk'' i.e., k=1 k=1

r'j = A (a'kV ajk' = a'iV aji for some l < n

rj = V (a' A a!'jk) = a!g A ajg for some g < n' k=1

r'' = A (a'kV a'k' = A a'k = a'h and rj = V (a'i A ajk' = k=1 k=1 k=1

a''m for some h, m < n .

Since r' = a'A < a'u < CI'' v aj i = r'j and rji = ci'm > a'g >

a' gj A a'jg = rjj, we get ru < r ' and A'A' is nearly irreflexive. The symmetry of R is obvious. □

Corollary 3.8. For any m x n bifuzzy matrix A, we have:

a/j < a] j v a j] v a] j v ... v a] j < 1

n - elements

which contradicts the nilpotency of A.

Case 2: If n is even, then by Proposition 3.11 we have

.(ni t t t In

att > a. j a a ji a a.j a ... a a ji > 0

à'ï < a] j v a j] v a] j v ... v a j] < 1

which is also, a contradiction. Thus, a] : A a'■ i = 0 and a!jj v ajj, = 1. That is a]j A a j] = 0 and A is then asymmetric. □

Proposition 3.13 ([11], pp. 222) ] If A is irreflexive and transitive bifuzzy matrix, then A is nilpotent.

Proposition 3.14. Let A and B be two transitive bifuzzy matrices, such that A < B, Then A&B] is transitive

Proof. Let D = A Q Btand suppose d]k A dkj = c > 0 for some k < n ] That is

(j a]k - a'ik j Q ibki - bk] j) A ( jakj - akj j Q (bjk - b'Jk j) = V - c"j > {0 , 1>] Thus, a]k > b'ki and akj > bjk. So that [a]k- a]kj A [akj- akjj = {c'- c"'>t i.e., a]k A akj = c and a]k v akj = c".

n - elements

n - elements

n - elements

Since A is transitive,

a2=ia]j ' ajj ] > (a]k a akj ' ajk v akj j=^ 'c"] ■ Now, we show that if ai ■ < hji, there are contradictions. (a) If a-j = c', then Ij/" < c' and so a-j < c' (since we have that A < B). However, since we have assumed hji > ajj > c', we get

hki > hkj A hji > akj A hji > c' ■ Which is a contradiction.

(b) If ak j = c, then hjk < c ■ However, hjk > hji A > hji A

= {(¡¡j, atj j — I aik A akj, aik V akj,

a'ik > d.Which is also a contradiction. Therefore, a, ■ > b-, and so

à,- = a, - e b j i =,a,, a,, e {h,, -,

aik v ak - j = (d,c" i.e., à, - > c = dk A dkj and D is thus transitive. This completes the proof. □

Corollary 3.15. Let A and B be two transitive bifuzzy matrices, with A < B" Then (A e B") * (A e B")" = O.

Proof. It is easy to see that A&B" is irreflexive and so by Propositions 3.9, 3.12-3.14, we get the result. □

Definition 3.16 (constant, nearly constant bifuzzy matrix [8], pp. 84) - An m x n bifuzzy matrix A = [a- " is called constant if and only if a- = akj for every i, k e {1, 2, ..., m}, j e {1, 2, ..., n}, A is nearly constant if and only if a, - = ak-, where i = j for every k = j.

Theorem 3.17. Let S be an n x n symmetric and nearly irreflexive bifuzzy matrix. Then the bifuzzy matrix T = I, * Sis idempotent and nearly constant.

Proof. Based on the symmetry of S, we can write the elements of the bifuzzy matrix T in terms of the elements of S as follows:

tj ; = It'. . , t j j =

1 j \ 'j ' ''j '

if i = j, if i = j.

A s"k • V s"k ,

i i=k i=k

First, from the definition of t*, we notice that T is nearly constant. Now, we will show that T is idempotent. Any element tjj' of T* is calculated as:

tj = (j • tj j = (v (tj" A tkj )" A (tk V tj ^

= (tjj A t'hj •tjj V tjj jfor some h"l <n j However, we have several cases for the indices ij j j h and l to

show that t"2) = t * * .

Case 1: Suppose that i = j = h = l. In this case we have

"2) , , , , , t j t j t j t j t j t j

' j "h ' lhj "j ' "j "j

t jj t jj t jj t jj t jj t jj .

"j " l V " j "j v 'j* "j" Thus, ti (2 ) ti j .

Case 2: Suppose that i j h l. In this case we have

t j* = tj* as in Case 1. Also, t j = tjj V tjj* < tjj V t"j = tjj V t'jj = tjj (since we have ' =

On the other hand, since we have that S is nearly irreflexive,

"ij V "ik —: "ii —: ^ll ' "H ^ll ' "jj "il ' t jj "ij

t"j = V < s" < sj V s"i = s"j V j = t"j V t j* =t j.

Thus, tj" = tjj and so tj = t * *.

' 'j 'j 'j v

Case 3: Suppose that i = j = l = h. In this case we have

tjf = tjj A thj > tjj A tj j > tjj A Kj = tjj (since i = j).

tjj = A s'" >s" > shhA s" = shhA sj * =tjj A t!h * = tfj .

Thus, tj = tj* ■

Also, as in Case 1, we get t"j = t"j, hence t *, = t * * .

' ' ° 'j 'j " ij J

Case 4: Suppose that i = h = l = j. In this case we have

tjiA tjj = tj*A thj > tjjA * and so tji > tj *. But by the definition of tj* • it is clear that tj* > sj * (since S is nearly irreflexive) so that t" > t j * > sj *.

" JJ JJ

Thus, j = tj* A thj = tji A "hj = tji A s*j = s*j = tjj.

Also, in this case we have t'j V tjj = tj* V tjj < tj* V t'j* and so t jj t jj .

" — j j"

But t'j* < s"jj (since S is nearly irreflexive) and so t'j < tj * <

Thus, tjj =*" V tj* = tj V tj* = tj V sj* = s'jj = tjj.

Therefore, t"j = t * * .

Case 5: Suppose that j = l = h = i. In this case we have

tj" =tj*A thj =tj*A t* * =s* *A ( A s*k ) = s* * =tj*

tjj =tj V tjjj = tjjV t j = sJ jV ( V sjk j = j = tjj.

Case 6: Suppose that i = j = h = I. In this case we have t j = t h A trh j > t j j A t j j = t j j A t j j = t j j (since i = j). On the

jh hj — j j j j j j j j j j other hand , since we have S is nearly irreflexive

tjj = Asjk — 4 — shhA 4 = shhA sj j =tjj A th j =tjj.

Thus, tjj> = tjj .

t jj t jj t jj t jj t jj t jj

j v 0 — j V a 1'j

.. x / .. .. .. .. .. .. .. .. .. (2) tj * = V s"k <s" <sj*V s" = sl*V sj* = tjhV tjh* =tj" * .

Thus, tjj" = t'jj and so t( 2) = t ' *.

' 'j 'j 'j '*

Case 7: Suppose that i = h = j = l. In this case we have

tj' A tjj = tj* A thj > tj* A tj * and so tj' > tj * > sj *. As in Case

4, we get tfj = tjj. Also,

s"( v s"j j = t*jV tjj' < tjjV t j = s"j jV ( V s"jk) = s"j j (since S is

nearly irreflexive). So, sj < s'jj. Therefore,

//(2) // // // // // // tjj = tu v tu = sii v sjj = sjj = tjj.

Thus, tjjj = tjj.

Case 8: Suppose that i = l = h = j. In this case we have t j = ^ A trhj > tjj A tj j = sj j A (= j j = sj / = ■ On the other hand we have

r r r r r r j)

t jj =s jj— s jhjA s jj =t jhA thj =t jj.

Therefore, t j j = tj j.

Also, as in Case 4, we get t'j" = t'j*. Thus, tjj = t*. Case 9: Suppose that j = h = i = l. In this case, we have

tj" = tjjA thj =tj*A ¿j* =s* *A ( A s*k" =s* * =tj*

t j" = tjj • as in Case 7. Therefore, tjj' = t ' *.

'j 'j j ' 'j v

Case 10: Suppose that j = l = h = i. In this case, we have

tj" =tj*A thj > tjjA tj* =s* *A ( A s*k ) = s* * = tjj *

On the other lirn^ tjj = s'* j > s'hh A sjj = f'* A thj = tjj .

Thus, tjj = tjj. Also,

j = tjj V tjj = tjjV tjj = jV ( V s'jk j = j = tjj ■

Therefore, tjj = tjj.

sjj ' sjj

Case 11: Suppose that h = l = i = j. As in Case 8, tj = t.j

and tj = tjj as in Case 7. Therefore, t.j = t j j .

ij ij ' ij

Case 12: Suppose that i = j = h = l. As in Cases 4 and 9, (2) (2) (2) t. j = t. j. From the computations of t. , we find that t. = t.j in

all the above cases and so T is idempotent. □

Corollary 3.18. Let A be any m x n bifuzzy matrix. Then the matrix Im * (A * A'j is idempotent and nearly constant.

Proof. By Theorems 3.7 and 3.17. □

Corollary 3.19. Let A be any m x n bifuzzy matrix. Then the matrix (A * A. * Im iS idempotent.

Proof. Notice that ( (A * A. > * Im >>t = Im * (A * A. >. Then by Corollary 3.18, the bifuzzy matrix ((A * Ay * Im))j is idempotent. So (A * A.jj * Im is idempotent. But (A * Aj>t = A * At. Thus, (A * At ) * Im is idempotent. □

Theorem 3.17 and its corollaries are useful in studying bifuzzy relations (bifuzzy matrices). However, they enable us to construct an idempotent bifuzzy relation (matrix) from any given bifuzzy relation (matrix).

Example. Let

'(0 ] 5 ] 0] 3 } <0]8] 0} (0 ] 6 ] 0] 4 } (0 ] 7 ] 0] 2}

(0 ] 4 ] 0 . 6 ] (0 ] 8 ] 0 ] 2} (0 ] 7 ] 0 ] 3}'

(0 ] 9 ] 0 . 1] (1] 0} (0 ] 3 ] 0 ] 6}

(0 ] 5 ] 0 . 5] (0 ] 1} (0 ] 8 ] 0 ] 1}

(0 ] 6 ] 0 . 3] (0] 9 , 0} (0 ] 5 ] 0 ] 4}

S = A * A t

' <0 . 5 . 0 . 3} <0 . 4 . 0 . 6} <0. 8 . 0 . 2) <0. 7 , 0 . 3}

(0 . 8 . 0} <0 . 9 . 0 . 1} <1. 0< <0. 3 , 0 . 6}

<0 . 6 . 0 . 4} <0 . 5 . 0 . 5} <0 . 1. <0. 8 , 0 . 1}

_(0 . 7 . 0 . 2} <0 . 6 . 0 . 3} <0 . 9 . 0} <0. 5 , 0 . 4}

*<0 . 5 . 0. 3} <0. 8 , 0} <0 . 6 . 0 . 4} <0 . 7 . 0 . 2}

<0 . 4 . 0. 6} <0 . 9 . 0 . 1. <0 . 5 . 0 . 5} <0 . 6 . 0 . 3}

<0 . 8 . 0. 2} <1. 0} <0 . 1} <0 . 9 . 0}

<0 . 7 . 0. 3} <0 . 3 . 0 . 6 . <0 . 8 . 0 . 1} <0 . 5 . 0 . 4}

<0 . 4 . 0 . 6} <0 . 7 . 0. 3 } <0 . 5 . 0. 5 } <0 . 6 . 0 . 3.

<0 . 7 . 0 . 3} <0 . 3 . 0. 6 } <0 . 8 . 0. 1} <0 . 5 . 0 . 4 .

<0 . 5 . 0 . 5} <0 . 8 . 0. 1} <0 . 1} <0 . 6 . 0 . 3.

<0 . 6 . 0 . 3} <0 . 5 . 0. 4 } <0 . 6 . 0. 3 } <0 . 5 . 0 . 4 . It is clear that S is nearly irreflexive and symmetric. Also, let T = I. * S. That is

(1] 0] ]0 ] 1] ]0 ] 1] _(0 ] 1] (0 (0] 4 , 0 ] 6} (0] 7 , 0 ] 3} (0] 5 , 0 ] 5} (0] 6 , 0 ] 3}

(0 . 5 ] 0 ] 5} (0 . 4 ] 0 ] 6} (0 . 4 ] 0 ] 6} (0 . 4 ] 0 ] 6}

(0] 1} (1] 0} (0] 1} 1}

(0 ] 1} (0 ] 1} (1] 0} (0 ] 1} (0 . 7 ] 0 ] 3} (0 . 3 ] 0 ] 6} (0.8] 0 ] 1} (0 . 5 ] 0 ] 4}

(0 ] 3 ] 0] 6 } (0 ] 5 ] 0] 4 } (0 ] 3 ] 0] 6 } (0 ] 3 ] 0] 6 }

(0 ] 1} (0 ] 1} (0 ] 1} (1] 0}_ (0 ] 5 ] 0] 5} (0]8] 0 ] 1} (0 , 1} (0 ] 6 ] 0] 3}

(0 ] 1} (0 ] 1} (0. 5, 0. 5 } (0 ] 1}

(0. 6, 0. 3 } (0. 5, 0. 4 } (0. 6, 0. 3 } (0. 5, 0. 4 }

(0. 5, 0 .4} (0. 5, 0 .4} (0. 5, 0 .4} (0. 5, 0 .4}

Then it is obvious that T is nearly constant and one can show that it is also idempotent by calculating T 2 .

Lemma 3.20. For a, b e F , we have:

(1) (a V b>c = ac A bc , (2) (a A b>c = ac V bc. The proof is trivial.

The following proposition shows the relationship between the two composition * and o of bifuzzy matrices.

Proposition 3.21. For bifuzzy matrices A = [aj]mxn and B = [bij lxJ, we have:

(1) (A * B)]] = AcBc,

(2) Ac * Bc = (AB).

Proof. (1) Let R = (A * B) and D = AcBc. Then

r] j = A(a]k V b'k j ), \J(a!!k A bkj )] U=1 k=1

= \/(ak A bkj) ] /\(a']k V bk j k \k=1 k=1

d]j = \l(a'lk A bkj) ] A(a]k V bk j) ].

Ij = \ V ("]k A ukj ) ] /\ ("]k V ukj.

\k=1 k=n

Therefore, R = D. (2) Similarly, we can show that Ac * BJ = (AB). □

Corollary 3.22. For bifuzzy matrices A = [a]j]mxn , B = [b ij]nxp , C = \c] ;] and D = \d] ; ] , we have]

L vjpxg L fiJm xp ]

(1) A * (B * C) = (A * B) * C,

(2) (A * B ) = D if and only if A] B] = D].

From the above corollary, it is seen that the operation * is associative. We will prove that * is distributive over the operations V and A in the following proposition.

Proposition 3.23. For any three bifuzzy matrices A, B and C of order m x n ] n x m and n x m respectively, we have:

(1) A * (B V C) = (A * B) V (A * C) ]

(2) A * (B A C) = (A * B) A (A * C) ]

Proof. (1) Let D = B V C, R = A * D, G = A * B, H = A * C

and W = G V H. Then

d]j = V c]j ' b]j A c]j ] '

r] j = A(a]kV dkj) ' \/(a]]k A dk] ) ]

\k=1 k=1 I

Â (a]kV (bkjV ckj))' V {a'i A (bkj A ckj

g] j = A(a'k V bk j) ] \J(a'jk A bkj ] k 1 k 1

kj = A(a]kV ckj) ' M (a'ik A ckj) ]. \k=1 k=1

w] = g] jV h] j

n \ I n \ f n

= * LA1 ( d]k V bkj ) ) V (A ( a]k V ckj ) J , (V (àkk A bkj j

A ( V ( cJi1 a ck j ]

ik A kj j

(A (( a]k V bkj ) V (a]k V ckj)J

V ((a]k A Kjk A a A j )J

= [h (alk V (bkj V ckj ))' V] (akk A (bkj A ckj)J\

We conclude that A * (B V C) = (A * B) V (A * C). (2) Can be proved by similar manner. □

Proposition 3.24. For bifuzzy matrices A = [aj)m , B = \bi ; J and C = \cj ; J , we have:

nx p nx p

(1) A * (BeC) > (A * B)Q(A * C),

(2) a (B e c ) = ab e ac.

Proof. (1) Let R = B e C, D = A * R, S = A * B, E = A * C and H = S e E. Then

i(° < b)<< if b« < c,<, V)< < c)<< r« = ¡<0 < 1> if b'« < c,-, bk- > ci-, , l<b« , V)<< if b« > c'-.

s, - = A (a'k V bk - )< \J(a')< A bk") \k=1 k=1

e< - = A (a,kV ck -)< V (a"k A ck -" \k=1 k=1

\Aak ' V (ak A bk- " \k=1 k =1 n n

A ak, V ak \k=1 k =1

" n (ak V K- )< V (ak A K- ) < if b" i > c"<. \k=1 k=1

if b'k < , K- < c"'<,

if b< - < c" -, K- > ",

0 < M (ak A bk< "

if A(ak V bk -) < A (akV ck -) <

k=1 k=1 n n

V (a'k A bk- ) < V (a'k A ck- )< k=1 k=1 (0-1>

if !\(ak V bk -) < A (ak V ck -) <

k=1 k=1 n n

\J(a'k A bk-) > V (ak A ) <

k=1 k=1 _ n n

A(ak V bk - )< \/(a'k A bk- " \k=1 k=1

if A (a,kVbk -) > A (a,kVck- ) <

k=1 k=1

(0 - V(a'k A bk-

k=1 <0- 1>

* A (a'kV bkj)" V(a"kA bkj) j if bkj > ckj.

We note that D > H. Hence A * (BeC) > (A * BjQ(A * C). (2) Similar to (1). □

This proposition shows that the operation o is distributive over the operation e.

Proposition 3.25. For bifuzzy matrices A = [aj* "m , B =

[b'j "nxlC = ic'j "yxp • D = id'j " pxm and E = ie'j "lxg • we have:

(1) C(D * A * B)E < CD * A * BE,

(2) (C * DjA(B * Ej < C* (DAB) * E.

Proof. (1) Let Q = D * A * B, T = CQ and R = T E. That is R is the left-hand side of the inequality. Also, let S = CD, H = BE, G = S * A and W = G * H. That is W is the right-hand side of the inequality. Then

q-- = A

A V a'ux - ) V bx -

\Z(dk A a'Ux) ) A b"x-

^ = V ( c,k A qk - -

= V (ck A

^ = A ( ck V q'k- )

= A (c'k v

Thus, we have

r,- = V (t"" A e'v - -

l p n m

V c'k A

A (dku V a'- ) ) V b'x u=1

V (dL A <« ) A

A (dku V aUx ) V b

l p n m / \

= V V A A {ck A (d'ku V a'ux V b'«v) A e'v-

v=1 k=1 x=1 u=1 V 7

l p n m

= V V A A((ck A dku A e'v-) V (ck A a'- A e'v-)

v=1 k=1 x=1 u=1

V (c'k A Kv A eV-))

■ = M" V e'V - )

A ck v

V ( d'LA<x) - A b

l p n m / \

= A A V V Wk V (d'ku A a'ùx A b'xv) V e'V-).

v= 1 k=1 x =1 u=1 x '

s,- = ( V (c'k A dk-)- k (ck V dx) \ \k=1 k=1 I

g, - = A ^ V a'u- -- V K A )

\u=l u=1

V (ck A dku )

Vau, ,, V

A(c'k Vdï« -

if bk - < ck -, bk- < ck-, if bk- < ck-, bk- > ck- ,

h- = ( V (b'« A e'v- ) - A (b"v V e'vt )

\v= 1 v =1

W" - = ACgx V h'x - )

V (c'k A dku ) ) V < L \k=1

Vbv A e'v - "

n m p l

= A A VV (ck A d'ku ) V a'ux V (b'v A ev- ) -

x =1 u=1 k=1 v=1 V 7

l p m n / \

= V V A A (Ck A dku ) V a'ux V (b'v A ev- ) -

v=1 k=1 u =1 x= 1 x '

W- = V(g L A hx- )

A (c'k V dl ) ) A < k=1

K(b"v V e'v-)

n m p l

= V V A A (Ck V dl ) A a'ux A (b'xv V e'v- ) -

x=1 u=1 k=1 v=1 v 7

l p m n

= A A V V( (c'k A<x )V (ck A<x Ae'v - )

v=1 k=1 u=1 x=1

V (dLA<xAbxv- V (d'LA<xAe'v- ) ).

Since c'k A dku A e'v - < c'k A dku, c'k A A ev - < a'ux and

c'k A Kv A e'v - < Kv A e'v - , we get r''- < W'-. Also , since

ckAauxca^K' < ck, d'LA<xA< - < ev- and ck A A ev- <

e'l- , we get r1)« > w)<. Thus, r< - < w< - and R < W' (1) Similar to (1). □

Acknowledgment

The authors are very grateful and would like to express their thanks to the anonymous referees for their valuable comments and suggestions provided in revising and improve the presentation of the paper.

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