journal of the Egyptian Mathematical Society 000 (2017) 1-4

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Intuitionistic circular bifuzzy matrices E.G. Emam

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

A R T I C L E I N F 0

A B S T R A C T

Article history: Received 11 December 2016 Revised 8 February 2017 Accepted 16 February 2017 Available online xxx

In this paper, we define the intuitionistic circular fuzzy matrix and introduce the necessary and sufficient conditions for an intuitionistic fuzzy matrix to be circular. Also, we study some properties of intuitionistic circular fuzzy matrices

© 2017 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/40/)

Keywords:

Intuitionistic fuzzy matrices Fuzzy matrices Circular fuzzy matrices

1. Introduction

2. Preliminaries and definitions

The concept of intuitionistic fuzzy matrices was introduced by Pal et al. [1] as a generalization of the well known ordinary fuzzy matrices introduced by Thomason [2] which take its elements from the unit interval [0,1]. An intuitionistic fuzzy matrix is a pair of fuzzy matrices, namely, a membership and non-membership function which represent positive and negative aspects of the given information (see [3,4]). However, intuitionistic fuzzy matrices have been proposed to represent finite intuitionistic fuzzy relations. This concept is a generalization to that of the ordinary fuzzy relations which also is a generalization to the crisp relations (or Boolean relations).

In this paper, we concentrate oure attention on one of the important kind of intuitionstic fuzzy matrices called intuitionistic circular fuzzy matrices. However, a characterization of intuitionistic circular fuzzy matrices is given and some important properties are established.

The paper is organized in three sections. In Section 2, the definitions and operations on intuitionistic fuzzy matrices are briefly introduced. In Section 3, results concerning of intuitionistic circular fuzzy matrices are proved using the operations and notations in the previous section. In Section 4, we exhibit the adjoint of an intuitionistic circular fuzzy matrix throughout its determinant and show that the adjoint of an intuitionistic circular fuzzy matrix is also circular. However, the operations v and a play an important role in our work.

We give here some definitions and notations which are applied in the paper. Note that an intuitionistic fuzzy matrix: A of order

m x n is defined as follows: A = [a..] where a., = |aj, ajjJ and aj., ajj e [0 , 1] maintaining the condition 0 < aj + aij < 1.

Now, we define some operations on the intuitionistic fuzzy matrices.. For intuitionistic fuzzy matrices A = Taj , 1 , B = lb. , 1 ,

J L 'JJnxn . L 'JJn xn.

and C = [cjjJn jm the following operations are defined [3,5-7].

A a B = [a. a bj] = [<min(aj , bj), max(aj' , bj'))], aj v bjJ = [<maxJaJ , bJJ, mmJaj, bJjJ)J,

A v B = AC =

v )a]k a ckj). ¿ (a]k v )>

k=1 a(k ) '

a)f, a

,//)]> ]

= Ak-1A

= A0 =

j ' v < 1 ] 0 > if i = j,

< 0 ] 1 > if i = j. AT = [a)) ( the transpose of A )]

VA = A a AT

A < B if and only if a) < b) .That is if and only if a) < b]¡ and

j > bj for all i, j.

We may write 0 instead of < 0, 1 > and 1 instead of < 1, 0 >

E-mail address: eg_emom@yahoo.com

Definition 2.1. [1,3,8-11] . For an n x n intuitionistic fuzzy matrix A we have:

http://dx.doi.org/10.1016/jjoems.2017.02.004

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Please cite this article as: E.G. Emam, Intuitionistic circular bifuzzy matrices, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/jjoems.2017.02.004

E.G. Emam/Journal of the Egyptian Mathematical Society 000 (2017) 1-4

(a) A is symmetric if and only if AT = A,

(b) A is idempotent if and only if A2 = A,

(c) A is transitive if and only if A2 < A/

(d) A is circular if and only if (A2 )T < A/

(e) A is weakly reflexive if and only if aü > a/ for all 1 < i, j < n

(f) A is reflexive if and only if aü = 1 for all 1 < i < n/

(g) A is similarity if and only if A is symmetric, transitive and reflexive.

It is noted that (AT)2 = (A/)T for any n x n matrix. So, the in-tuitionistic fuzzy matrix A is circular if and only if A2 < AT, i.e., aik Aakj < a/i for every 1 < i, j, k < n Moreover, if A is symmetric, then A is transitive if and only if A is circular.

3. Results

Throughout the next two sections we deal only with n x n in-tuitionistic fuzzy matrices. In this section, some properties of intu-itionistic circular fuzzy matrices are examined by the definitions in the above section. However, we begin with the following proposition.

Proposition 3.1. Let A be an n x n intuitionistic fuzzy matrix and let Ax denotes the m x m submatrix of A (where m < n) such that

A i A3

Then A is circular if and only if A\ < A[ , A2A 3 < A[ , A3Ax < A2,

A4A3 < A2 , Ax A2 < A3 , A2A4 < A3 , A3A2 < Aj and A2 < Aj.

Proof. Suppose that A satisfies all the above conditions and consider

A2 = B =

Bt = A\ v A2A3 < A{ v A{ = A{, B2 = A A v A2A4 < A3 v A3 = A3 , B3 = A3A t v A4A3 < A2 v A2 = a2

B4 = A3A2 v a4 < A4 v A4 = A4 .

Thus, we have A2 = B < AT and A is circular. Conversely, suppose that A is circular. For 1 < s < m and m + 1 < t < n, Let C = At , D = A2 , E = A3 and F = A4 . Then cst = ast for every 1 < s, t < m, dst = asst+m) for every 1 < s < m and 1 < t < n - m, e,t = a,s+m,s for every 1 < s < n - m and 1 < t < m and fs = a(s+m) (t+m) for every 1 < s < n - m and 1 < t < n - m,

1. To show that A2 < A[ and A2A3 < A[ , let G = A\ and H = A2A3 .

gs t = {.m=] /C]k A ck t /. kmmi /c's'k v ck] / >

= ^K A a/1/' AK v ^/>

<(&K A a'kt), kA1 K V a// > = (a//, a

" (2 ) \ s t >

< (a/, a/s> = a/s = c/.

hst = cv(d'sk A e'kt), T(d?k v e'kt,)

_ ¡n-mi / / \ n-mi n n \\

= ' ¡kvl sas(k+m) A a(k+m)t)' AA s<ls(k+m) v a(k+m)t)) = ( v , («L A a'uts - A , («'^u v «rns) (where u = k + m)

u=m+1 s ! u=m+1 s !

< (v(a'su A a'us- A {a'sU v a'¿J) = (a'^- ) )

< (a's- a'ls s = ass = cs .

Thus, hst < cs and therefore, A2A3 < A1. 2. To show that A4A3 < AT2 and A3A1 < A2 , let Q = A4A3 and L =

A 3 A1 . Then 2 2

in-m(ei / \ n-mUn n\\

qs t = (kv1 (fk A ek t s- kA fk v e'L s)

n-m / \ n-m /

( kV1 {a (S+m ) (k+m ) A a/k+m ) / )' A {a'

, > . a v a '

k=1 /v (/+m ) /k+m ) (k+m ) / /

= (j +1 /a'/+m ) u A a'u t /, Jm+1 K+m ) u v < )) ( where u = k + m )

< (V= 1 K+m) u A a'ut /< A1 K+m) u v />

= a2) a"(2) ) u

(a^m)/' a^m)/>

< (at(s+m), ^^m)> = a///+m) = .

Thus, q/t < d/ and therefore, A4A3 < . Also,

i/1 = (iVV1 /^ k a ck t/, kAs {e// v cL />

= (IVV1 K+m ) k A ak t / ' A K+m ) k V ^ />

< {V K+m) k A akt/• A ia' s+m) k V </>

< (at(s+m), at/s+m)> = Q(/+m) = d/.

- {a ( 2 ) a'( 2 ) )

= (a (s +m )t , a (s +m )t >

i.e., 1st < d s and therefore, A3Aj < A2 . 3. To show that AjA2 < A3 and A2A4 ^ A3 , let R = A1A2 and Z = A 2A4 . Then

r s t = ^ K A dk ^- a K v dkt ^

= (kVt K A akss+m) s- A K v aKs+m) s)

< ^ K A ak s s+m ) s- ¿t K v aK s+m ) s )

= (as(t+m)- «s(s+m)s < ^^ms- a'(t+m))) = a^m^ = ess■ Therefore, A1A2 < A3. Also,

= (nv (d'sk A fID-nAV (dkk v %))

_ /n-m ( / / ) n-V( II II )\

= (v ^«s sk+m) A «(s+m) fi+m) s- Ai {«s (s+m) v «s^k+m) (s+m) s)

= (u=v+1 s«™ A «<ut+m) s- U=V+1 «u v s+m) s )

< (v=l (<u A «'uss+m) s- uA 1K v «<ks+m) s )

_ I (2) HQ) \ / 1 u \ _ _

= (a)ss+m) - «s ss+m)) < («(s+m)s - «(s+m)s) = «tt+m)s = ea.

Hence, A 2A4 < A 3. 4. To show that A3A2 < Aj and A\ < Aj - let P = A3A2 and W = A2 . Then

p /t = ( V ek a dkt ), A (e/k v dkt)>

k=m+1 / sk kts k=m+1 / sk kts

= (/=VV+1 /a'(s+m)k A akt+m)/' k=A+T,K+m)k V ^kf+m) /> < (kVS /^^m)k A akt+m)/' kA1 Ks+m)k v a'k/t+m)/ >

Thus, gs t < ct s and therefore, A12 < At1 .

(s m )k k (t m ) k 1 (s +m )k k (t +m ) k 1

a./2') a"(2) )

(s +m ) (t +m ) , (s +m )( t +m ) >

Please cite this article as: E.G. Emam, Intuitionistic circular bifuzzy matrices, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.org/10.1016/j.joems.2017.02.004

E.G. Emam/Journal of the Egyptian Mathematical Society 000 (2017) 1-4

5 {a//+m) //+m) ' a//+m) //+m) / — a/ /+m/ /s+m/ — f/s ■ Therefore ,

A3A2 5 A4 ■

w/t—(n-—m f к л f/, & /fk v f//)

_ in-m J I I \ n-m( n

= ' V J/+m) J/+m) Л a(k+m) Jt+m) /' Л \a(J+m) (J+m)

—a Jk+m ) /t+m ) / ) — {u-—j1 /a/s+m)u Л aU(t+m))' u /a'(s+m)u v aUtt+m)/)

u—m+1 -

5 {u-1 )a(-S+m)u Л au)t+m)) = {a—- 2 ) d"(2- )

{a "s+m)-t+m) ' a(s+m)-1+m))

u—m+1 -

л (a'

u—1 -

v a ' (s+m)u v "us+m)

5 (a--+m)(s+m) > ^'s+m)-- — aS+m) -^m) — f-■

Thus, Aj 5 Aj ■ This completets the proof. □

Remark. It is easy to see that the intuitionistic fuzzy matrix A is circular if and only if AT is circular.

Lemma 3.2. [3] For intuitionistic fuzzy matrices A — [a--m-n, B —

l^---mxn '

C — [c--n and D — [d-- yn, if A 5 B, then AC 5 BC and DA 5

Proposition 3.3. An intuitionistic fuzzy matrix A is circular if and only if E-- -- AE-- -- is circular for every 1 5 i, - 5 n, where E-- -- is the intuitionistic fuzzy matrix obtained from the identity intuitionistic matrix In by interchanging the row i and row -.

Proof. First, we notice that E(i-)E(i-) — In and (E(i-) )T — E(i-) for every 1 5 i, - 5 n, Suppose that A is circular. Then

(E(i■-) AE -i--) )2 — (E (i'-)AE -i•-) )(E (i'-)AE -i--) )

— E(i• -)A2E(i■-) 5 E(i• -)ATE(i•-) — (E(i■ -)AE(i■-))T

and hence E-- -- AE!-- -- is circular.

Now, suppose that E-- -'AE)- -- is circular. That is

(E(i• -)AE(i■-))(E(i-)AE(i■-)) 5 (E(i■ -)AE(i■-))T — E(i■ -)ATE(i-)■ Then E-- --A2E-- -- 5 E(- --ATE-- -- and so by Lemma 3.2, we get E-- --A2E-- --E-- -- 5 E-- --A-E(- --E-- --. That is E-- --A2 5 E(- --AT. Also, E-- - - E-- --A2 5 E-- - - E-- --AT and so A2 5 A - and A is thus circular. □

Proposition 3.4. Let * be a binary operation on [0, 1] satisfies for every x, y, u, v e [0, 1], the following conditions:

1. (x * у)л( u * v) 5 (xлu)- (yлv ),

2. (x * y)v( u * v) > (xvu)- (yvv ),

3. x 5 y, u 5 v imply x * u 5 y * v .

If A and B are n x n intuitionistic circular fuzzy matrices,

as A B ai

then A J B is circula r, where A J B is defined as A * B = [a* b.j J = [[aj * bJj, aJj * bJjJ].

Proof. Let C = A * B and D = C2 . Then

dJ = <dJ,dJ) = <v(cJ( a c'kj), a(dJ v djJ)

= <(V J((aJJ * bJ() a (aij * bJ)), Aj((cJ * b^) v (a'j * bJ))) = < (aj. * bJJ a (aj * bj J, JaJ * bj J v (a'j * b>> J) for some l and h < n.

By the properties of the operation *, we get

d(j = (aj. * bJ.) a (a'. * b'u) < (aj. a a'.) * (bJ. a V.) < aj. * bJ. = cj. (since A and B are circular) and

d( = (a(h * bJ() v (a'hhj * bhj) > (a. v a',) * (bjj v Uj) > aj. * fj. = cj.. Thus, d j < c j, and A J B is circular. □

Corollary 3.5. The intuitionistic fuzzy matrix AaB is circular if A and B are circular.

Proof. Since the operation л satisfies the conditions of operation *, then АлВ is circular. □

Corollary 3.6. The intuitionistic fuzzy matrix VA is circular when А is circular.

Proof. By Corollary 3.5, since А and At are circular. □

Corollary 3.7. Let F:X ^ X be a function such that

Дх)лДу) < F^y) and if x < y, then F(x) < F(y) for all x, y e X.lf А is an intuitionistic circular fuzzy matrix, then F(A) is circular, where F (А) = [F (atJ )].

Proof. Since А is circular, we have, ajkлakj < aj for all 1 < i, j, k < n.But by the definition of F, we have that F(ajkлakj) < F( aji) and F(ajk)aF]akj) < F( akлakj) < F(aji). Thus F(A) is circular. □

Proposition 3.8. А is circular and weakly reflexive if and only if А is symmetric and transitive.

Proof. Suppose that А is circular and weakly reflexive. Then

a ji = ajj л a ji < a, j. Also, a, j = aj л a, j < a, . So, a, j = a ji and А is symmetric. Also, we have a]J < a ji = a,j and А is thus transitive. Conversely, suppose that А is symmetric and transitive. Then aJ J < a, j = a j i. Hence А is circular. To show that А is weakly reflexive, we have ajj = aj л a ji < au (by the circularity and symmetry of А). Thus А is weakly reflexive. □

Corollary 3.9. If the intuitionistic circular fuzzy matrix А is reflexive, then А is similarity.

Proposition 3.10. If А is circular and symmetric intuitionistic fuzzy matrix, then А is idempotent.

Proof. Since А is symmetric, we have А is transitive. i.e., А2 < Аj it remains to show that А2 > А. Again since А is symmetric, we have ajj = a ji for every 1 < i, j < п, Let ajj = c > 0 j That is |ajj, ajj j =

[c', c'jj > (0 j 1). Then by circularity of А we have aji > ajjлaji or aj > cj and ajj < c". i.e., aji > c. On the other hand

af = (aj2), af ) = (£(aj л ajj), Д (ajk v ajj})) = (aj л

aj, ajh v ajjj) for some 1 < l, h < nj But aj л ajj > aj л ajj = c' and

ajh v a'ijj < aj v ajj = c" . Therefore, aj ) > c = ajj and А2 > А j This completes the proof. □

Corollary 3.11. If А is circular , then VA is idempotent.

Proof. By Proposition 3.10 and Corollary 3.6, since VA is symmetric. □

Proposition 3.12. For a circular and weakly reflexive intuitionistic fuzzy matrix А, we have А is idempotent.

Proof. By Propositions 3.8 and 3.10. □

Corollary 3.13. Let А be a circular and reflexive Then А is idempotent. Proof. By Proposition 3.10 and Corollary 3.9. □

4. Adjoint of an intuitionistic circular fuzzy matrix

Definition 4.1. [5,10]. The determinant |A | of an n x n intuitionistic fuzzy matrix A is defined as: \A| — v-e--( л {a'^t)> a--(-)) !

where S n is the symmetric group of all permutations of the indices (1,2.....n).

Please cite this article as: E.G. Emam, Intuitionistic circular bifuzzy matrices, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.Org/10.1016/j.joems.2017.02.004

JID: JOEMS 4

E.G. Emam/Journal of the Egyptian Mathematical Society 000 (2017) 1-4

Definition 4.2. [5,10]. The adjoint of A is denoted by adjA and is defined as: bij = A | where A | is the determinant of the (n -1) x (n - 1) matrix Aj formed by deleting row j and column i from A and where B = adjA.

Remarks:

1. The element of the matrix D = Aj can be written in terms of the elements of the matrix A as follows:

auv if u<s and v<j-

a ]u+1)v au ] v+l)

a ]u+1)(v+1)

if u > i and v< j, if u<] and v > j, if u > i and v > j.

2. If W = AT , then W]j = AT = (Aji )T and so, for G = (Aji )T we

av ] u+1) a ]v+1)u

a ]v+1)(u+1)

if u<] and v<j, if u > i and v< j, if u<] and v > j, ifu > iandv > j.

Proposition 4.3. Let A be an n x n intuitionistic circular fuzzy matrix. Then AfcAkj < (Aji )T for every 1 < i, j, k < n_

Proof. Let C = Aik- F = Akj - G = (Aji )T and R = CF. Then

ruv = ( V (c'uv A fVv)s "a1 (cdv v fVv)) Since we have that A is

v-1 v-1

circular, we have the following cases: Case 1s If u < i, m < k and v < js then

ruv = ("v(aUm л d'mj, "л 1 (<m V d^yj} m=1 j ' m=1 ] '

= iaup Л apv, aug V agv j — {avu j "vu

wpv/ Mug

for some p, g < n. Case 2 : If u < i, m < k and v > j, then

, n /1 ( , , \ n /1

avu j = avu = gu

ruv —

V1 ja ju+1)m л am ]v+1) j

Л 1 ja (u+1)m V am ]v+1)

(u+1)p л ap]m+1) ' a(u+1)g V ag]v+1) /

1 (v+1) ju+1), a ]v+1) ju+1 ) j = a (v+1 ) ju+1) = guv.

Case 5 j If u > i, m > k and v < j j then

, П j 1 / . . ч П j 1

ruv —

= ^rrV 1 j11'^) jm+1) Л ^Рп+^Р , гпЛ1 jaju+1) jm+1) V ajm+1 )vj j

= (a]u+1)]p+1 ) л ajp+1)v, aa (u+1))

]g+1]v j

— \av]u+1), av]u+1 )] = av]u+1] = guv.

Case 6 ] If u < i, m > k and v > jp then

/n-1 / , , \ n-1

/"-1( r

= 1 Pau

! \ n-1 p и и

(m+1) л a (m+1) ]v+1] j, mP 1 Pau Pm+1] V a ]m+1) ]v+1]

= iau jp+1) л a]p+1)]v+1), au]g+1] V a']]g+1 ]]v+1)P — (a]v+1)u, aj]p+1)u] = a(v+1]u = guv.

Case 7 : If u > i, m < k and v < jp then

ruv = (mV1 Paju+1)m л amvj > Paju+1)m V a'rnvj j

(u+1)p л apv, aju+1)g V agvj — ( av]u+1), av]u+1) ]

= av ( u+1 ) = guv . Case 8 : If u > i, m > k and v > jp then

= (V 1 Pa:

= (rrV 1 Pa'um л a'm]v+1))' ГГл1 Pa"m V am]v+1)P j

= (aup л ap ]v+1), aug V ag]v+1) P — (a ]v+1)u, a ]v+1) u P = a ]v+1) u = Suv.

Case 3 m If u < i, m > k and v < jp then

ruv = (m= ] Pau]m+1) л ajm+1)vj ' m-1 jau]m+1) V ajm+1 )vj j

= (au(p+1) л a]p+1)v, au]j+1) V a'(.g+1)vP — (a'vu, avu] = avu = guv.

Case 4 m If u > i, m < k and v > jp then

J An' \ П-1 In" S / n"

n-l , , , ч П-1 , n n

(u+1) (m+1) л a (m+1) (v+1))' j {a (u+1) (m+1) va (m+1) (v+1) = ia'(u+1) ((+1) л a'(p+1) (v+1)' ^^+1) (g+1) v a'(g+1 ((v+1) ( - ia'(v+1)(u+1), a(v+1)(u+1)( = a(v+1)(u+1( = guv.

Thus, we have ruv — guv in every case and hence AikAkj — (Aj j )T for every 1 — i, j, k — n. □

Corollary 4.4. If A is intuitionistic circular fuzzy matrix, then Au is circular for every 1 — i — n,

Proposition 4.5. [6]. Let A and B be two intuitionistic fuzzy matrices of order n x n. Then we have the followings:

(i) |AT | = \A \,

(ii) |A | л | B (— |AB | ,

(iii) \A( = v ay \A( \ where Ay is the intuitionistic fuzzy matrix of

order (n - 1) x (n - 1) formed by deleting row i and column j from A.

Corollary 4.6. Let A and B be two intuitionistic fuzzy matrices of order n x n such that A — B. Then |A | — |B (,

Proposition 4.7. Let A be a circular matrix. Then adjA is circular.

Proof. Let B=adjA , Then bus = \Am \, btu = \Aul \ and b( = \Ats \, By Propositions 4.1, 4.3 and Corollary 4.4 we get \A(u \ л \AUl \ — AuAul \ — |ATs | = \A( \ ( Since л is commutative, then

|Aul | л |Asu | — |A(s | . Therefore, btu л bus = b ^ and so adjA is circular. □

Acknowledgment

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

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Please cite this article as: E.G. Emam, Intuitionistic circular bifuzzy matrices, Journal of the Egyptian Mathematical Society (2017), http://dx.doi.Org/10.1016/j.joems.2017.02.004