International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
Regular Interval Valued Fuzzy Soft Matrices
Dr. N. Sarala1, M. Prabhavathi2
1
Department of Mathematics, A.D.M .College for Women (Auto), Nagapattinam, India)
2
Department of Mathematics, E.G.S. Pillay Arts & Science College, Nagapattinam, India)
Abstract: In this paper, we define regular interval valued fuzzy soft matrices as a generalization of regular fuzzy soft matrices and
obtain the structure of Row space, column space of an Interval valued fuzzy soft matrices (IVFSM).
Keywords: Soft set, fuzzy soft set, inter fuzzy soft matrix, Interval valued fuzzy soft set, Interval valued fuzzy soft Matrix, Regular
Interval valued fuzzy soft Matrix.
1. Introduction
We deal with interval – valued fuzzy matrices (IVFM) that
is, matrices whose entries are intervals and all the intervals
are subintervals of the interval [0,1]. Recently the concept of
IVFM a generalization of fuzzy matrix was introduced and
developed by shymal and pal[5], by extending the max. min
operations on fuzzy algebra F=[0,1], for elements, a, bF,
a+b= max{a,b} and a-b=min{a,b}. In [2], Meenakshi and
Kaliraja have represented and IVFM as an interval matrix of
its lower and upper limit fuzzy matrices. In [3], Meenakshi
and Jenita have introduced the concept of K-regular fuzzy
matrix and discussed about inverses associated with a Kregular fuzzy matrix as a generalization of results on regular
fuzzy matrix developed in [1]. A matrix A Fn, the set of all
nxn fuzzy matrices is said to be right(left) K-regular if there
exists x(y)Fn, such that Ak x A = Ak(AYAk=Ak) , x(y) is
called a right(left) K-g inverses of A, Where K is a positive
Integer. Recently we have expended characterization of
various inverses of K-Regular interval – valued fuzzy
matrices in [6] . [20] Dr.N.Sarala and M.Prabhavathi have
proposed the union and intersection of interval valued fuzzy
soft matrix. [13] P.Rajarajeswari and P.Dhanalakshmi have
introduced interval valued fuzzy soft matrix, its types with
examples and some new operation of an the basis of
weights. In [21] Dr.N.Sarala and M.Prabhavathi have
introduced the definition of „AND‟ and „OR‟ operation of
interval valued fuzzy soft matrix
In this paper, we define regular interval valued fuzzy soft
matrices as a generalization of regular interval valued fuzzy
soft matrices and obtain the structure of Row space, column
space of an Interval valued fuzzy soft matrices (IVFSM).
2. Preliminaries
Soft set 2.1 [9]
Suppose that U is an initial Universe set and E is a set of
parameters, let P(U) denotes the power set of U.A pair (F,E)
is called a soft set over U where F is a mapping given by F
:EP(U). Clearly a soft set is a mapping from parameters to
P(U)and it is not a set, but a parameterized family of subsets
of the Universe.
Fuzzy soft set 2.2 [10]
Let U be an initial Universe set and E be the set of
parameters, let A ⊆ E. A pair (F,A) is called fuzzy soft set
Paper ID: NOV151037
over U where F is a mapping given by F: AIU, where IU
denotes the collection of all fuzzy subsets of U.
Fuzzy soft Matrices 2.3 [12]
Let U ={c1,c2,c3…cm} be the Universe set and E be the set of
parameters given by E ={e1,e2,e3…en}. Let A ⊆ E and (F,A)
be a fuzzy soft set in the fuzzy soft class (U,E). Then fuzzy
soft set (F,A) in a matrix form as Amxn =[aij] mxn or A=[aij]
i=1,2,…m, j=1,2,3,…n
Interval valued fuzzy soft set 2.4 [11]
Let U be an initial Universe set and E be the set of
parameters, let A ⊆ E. A pair (F,A) is called Interval valued
fuzzy soft set over U where F is a mapping given by F:
AIU, where IU denotes the collection of all Interval valued
fuzzy subsets of U.
Definition 2.1.
An Interval-Valued Fuzzy Matrix(IVFM) of order mxn is
defined as A=(aij)mxn, Where aij= [aijL,aijU], the ijth element of
A is an interval representing the membership value. All the
elements of an IVFM are intervals and all the intervals are
the subintervals of the interval [0,1].
For A = (aij) = ( [aijL,aijU]) and B = (bij) = ( [bijL,bijU]) of order
mxn their sum denoted as
A+B defined as,
A+B = ( aij+bij) = ([(aijL+bijL),(aijU+bijU)]) …(2.1)
= ([max{aijL,bijL}, max{aijU,bijU}])
For A = ( aij)mxn and B = (bij) nxp their product denoted as AB
is defined as,
AB = (cij) =nk=1(aikbik)i=1,2,….,m and j=1,2,…..,p
=[nk=1 (aikLbikL),nk=1 (aikUbikU)] i=1,2,….,m and
j=1,2,…..,p…(2.2)
In particular if ainL = aijU and bijL= bijU then (2,2) reduces to
the standard max.min composition of fuzzy matrices [1].
A ≤ B if and only if aijL ≤ bijL and aijU ≤ bijU
In [2], representation of an IVFM as an interval matrix of its
lower and upper limit fuzzy matrices in introduced and
Volume 4 Issue 11, November 2015
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
76
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
discussed the regularity of an IVFM in terms of its lower
and upper limit fuzzy matrices.
Definition 2.2
For a pair of Fuzzy Matrices E =(eij) and F=(fij) in F m,n such
that E ≤ F, it can be defined that the interval matrix is
denoted as [E,F], whose ijth entry is the interval with lower
limit eij and upper limit fij, that is ([eij,fij]). In particular for
E=F, IVFM [E,E] reduces to the fuzzy matrix E F mn
For A = (aij) = ([aijL,aijU])  (IVFM)mn, let us define AL=(aijL)
and AU=(aijU). Clearly AL and AU belongs to F m,nsuch that
AL≤ AU and from Definition (2.2) A can be written as
A=[AL,Au]
For A (IVFM)mn, AT,R (A), C (A) denotes the transpose of
A, row space of A, column space of A respectively.
Interval valued fuzzy soft matrix 2.5[13]
Let U ={c1,c2,c3…cm} be the Universe set and E be the set of
parameters given by E ={e1,e2,e3…en}. Let A ⊆ E and (F,A)
be a interval valued fuzzy soft set over U, where F is a
mapping given by F: AIU, where IU denotes the collection
of all Interval valued fuzzy subsets of U. Then the Interval
valued fuzzy soft set can expressed in matrix form as
. mxn=[aij] mxn or Ã= [aij] i= 1,2,…m, j=1,2,…n
Definition 2.3
A matrix A  (IVFM)n is said to be right k-regular if there
exist a matrix X (IVFM)n, such that Ak XA=Ak, for some
positive integer k. X is called a right k-g inverse of A. Let
A{1rk}={X/ Ak XA = Ak}.
Definition 2.4
A matrix A  (IVFM)n is said to be left k-regular if there
exist a matrix Y (IVFM)n, such that AYAK=Ak, for some
positive integer k. Yis called a left k-g inverse of A. Let
A{1ℓk}={Y/ AkYA = Ak}.
In particular for a fuzzy matrix A, aijL =aijU then definition
(2.3) and definition(2.4) reduce to right left k-regular fuzzy
matrices respectively found in [3].
In the sequel we shall make use of the following results on
IVFM found in [2] and [4].
Lemma 2.1
For A=[AL,AU] (IVFM)mn and B=[BL,BU](IVFM)np, the
following hold.
(i) AT=[ALT,AUT]
(ii) AB=[ALBL,AUBU].
[jL(ci), ju (ci)] represents the membership of ci in the
Interval valued fuzzy set F (ej).
Note that ifjU (ci)=jL (ci) then the Interval- valued fuzzy
soft matrix (IVFSM) reduces to an FSM
Example: 2.1
Suppose that there are four houses under consideration,
namely the universes U= {h1,h2,h3,h4}, and the parameter set
E={e1,e2,e3,e4} where ei stands for “beautiful”, ”large”,
”cheap”, and “in green surroundings” respectively. Consider
the mapping F from parameter set A ={e1,e2} ⊆ E to all
interval valued fuzzy subsets of power set U. Consider an
interval valued fuzzy soft set (F,A) which describes the
“attractiveness of houses” that is considering for purchase.
Then interval valued fuzzy soft set (F,A) is
(F, A)= {F(e1)={(h1, [0.6,0.8]), (h2, [0.8,0.9]), (h3, [0.6,0.7]),
(h4,[0.5,0.6])}
F(e2)={(h1, [0.7,0.8]), (h2, [0.6,0.7]), (h3, [0.5,0.7]), (h4, [0.8,
0.9])}
We would represent this Interval valued fuzzy soft set in
matrix form as
Paper ID: NOV151037
Lemma 2.2
For A,B(IVFM)m
(i) R(B)⊆R(A)  B=XA for some X  (IVFM)m
(ii) C(B)⊆C(A)  B=XY for some Y  (IVFM)n
Theorem 2.1
For A,B  (IVFM)n, with R(A) =R(B) and R(Ak)=R(Bk) then
A is right K-regular IVFM  B is right k-regular IVFM
Theorem 2.2
For A,B  (IVFM)n, with C(A) =C(B) and C((Ak)=C(Bk)
then A is left K-regular IVFM  B is left k-regular IVFM.
3. Regular Interval
Matrices
Valued
Fuzzy
Soft
In this section ,we define regular interval valued fuzzy soft
matrices and obtain the structure of row spaces , column
spaces of an interval valued fuzzy soft matrices(IVFSM).
Definition: 3.1
Let à is called Regular interval valued fuzzy soft matrix,
then there exists X̃ Є (IVFSM)mn
Such that à X̃ à = Ã.
Lemma:3.1:
For à = [ÃL , ÃU] Є (IVFSM)mn and B̃ = [ B̃ L ,B̃ U]Є
(IVFSM)np , The following hold.
(i) ÃT = [ÃLT , ÃUT]
(ii) Ã B̃ = [ÃL B̃ L , ÃU B̃ U]
Volume 4 Issue 11, November 2015
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
77
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
(iii) Ã + B̃ = [ÃL + B̃ L, ÃU+ B̃ U]
(iv) [α+β](à + B̃ ) = [αÃL +β B̃ L ,α ÃU+β B̃ U] , for α ≤ β Є
FSM
(v) [α ,β]à = [αÃL , βÃU] , for α ≤ β Є FSM
(vi) αà = [αÃL ,α ÃU] , for α Є FSM
proof:
(i) this directly follows from the definition.
(ii) Let à = [ÃL , ÃU] = [ µAL , µAU]
B̃ = [ B̃ L ,B̃ U] = [ µBL , µBU]
Then
(µAL . µBLk ) ,
ÃB̃ =[
= [ÃC B̃ C , ÃU B̃ U ]
Let us define the interval valued fuzzy soft matrix Z̃ = [ X̃ ,
Ỹ ] .
Then by the definition of regular .
à Z̃ à = [ ÃL , ÃU ] [X̃ , Ỹ ] [ ÃL , ÃU ]
=[ ÃL , ÃU ]
= Ã .
Thus à is regular
Hence the theorem .
Example : 3.1
Let us consider à = [ÃL , ÃU ] Є (IVFSM)B .
(µAUK . µBUk ) ,
When ÃL =
ÃC B̃ C and ÃU B̃ U are the max. Min composition of matrices
in FSMnp .
ÃU =
(iii) Let à = [ÃL , ÃU] = [ µAL , µAU]
B̃ = [ B̃ L ,B̃ U] = [ µBL , µBU] .
à + B̃ = (aij +bij) =[( µAL , µBL) ,( µAU , µBU )]
=[ÃL +B̃ L , ÃU+B̃ U]
Then ÃL ≤ ÃU
Here ÃU is regular being idempotent and ÃL is not regular.
There is no X̃ Є (IVFSM)3 such that , ÃL X̃ L ÃL = ÃL .
∴ Ã is not regular for , it A is regular, then for some X̃ Є
(IVFSM)3 , Ã X̃ Ã = Ã
⇒ ÃL X̃ L ÃL = ÃL is regular which is not possible .
(iv) Let à = [ÃL , ÃU] = [ µAL , µAU].
B̃ = [ B̃ L ,B̃ U] = [ µBL , µBU] .
[ α , β] ( α Ã +β B̃ = (α aij +β bij)
=[( α µAL , β µBL) ,( α µAU , β µBU )]
=[ α ÃL + β B̃ L , α ÃU+ β B̃ U] .
Theorem : 3.1
Let à = [ÃL , ÃU ] be an (IVFSM)mn
Then
(i) 𝓡 (A) = [ 𝓡 (ÃL ) ,𝓡 ( ÃU ) ] Є (IVFSM)In
(ii) 𝒞 (A) = [ 𝒞 (ÃL ) , 𝒞 ( ÃU ) ] Є (IVFSM)Im
(v) Let à = [ÃL , ÃU] = [ µAL , µAU]
( α , β ) Ã = [α , β ] [ µAL , µAU]
= [ α µAL , βµAU]
= [ α ÃL , β ÃU ] for α , β Є FSM.
(vi) Let à = [ÃL , ÃU] = [ µAL , µAU].
α Ã = [ αµAL ,αµAU].
= [ α ÃL , α ÃU ] for α Є F .
Hence the lemma .
Theorem : 3.1
Let à = [ÃL , ÃU] Є (IVFSM) mn .
Then à is regular IVFSM ⟺ ÃL and ÃU Є FSM
regular.
mn
Proof:
Let à Є (IVFSM) mn .
If à is regular IVFSM , Then there exists X̃ Є (IVFSM)
Such that à X à = à .
Let X̃ = [ X̃ L , X̃ U ] with X̃ L , X̃ U Є FSM mn .
then
⇒ [ÃL , ÃU] [X̃ L , X̃ U] [ÃL , ÃU] = [ÃL , ÃU]
⇒ ÃL X̃ L ÃL = ÃL and ÃU X̃ U ÃU = ÃU
⇒ ÃL is regular and ÃU is regular FSMmn
are
mn
.
Thus à is regular IVFSM ⇒ ÃL and ÃU Є FSMmn are regular
.
Conversely ,
Suppose ÃL and ÃU are regular , then ÃL X̃ L ÃL = ÃL and ÃU
X̃ U ÃU = ÃU for some X̃ L and
Since ÃL ≤ ÃU , it is possible to choose atleast are X̃ Є ÃL
{1} and Ỹ Є ÃU {1} such that X̃ ≤ Ỹ .
Paper ID: NOV151037
proof :
(i) since à Є (IVFSM)mn , any vector 𝑥 Є 𝓡 (A) is of the
form 𝑥 = y à for some y Є (IVFSM)Im that is , 𝑥 is an
interval valued vector with n components .
let is compute 𝑥 Є 𝓡 (A) as follows . 𝑥 is a linear
combination of the low of
à ⇒ 𝑥 =
αi , Ãi * when Ãi * is the ith low of à .
Eguating the jth component on both sides yield .
𝑥=
αi . aij
(aij) = [µAL , µAU ] .
αi . [µAL , µAU ] .
𝑥j =
=
[αi µAL , µAU ] .
=
(αi µAL ) ,
( αi µAL ) ]
= [𝑥jL , 𝑥jU ] .
𝑥 jL is the jth component of 𝑥L Є 𝓡 (ÃL ) and 𝑥 jL is the jth
component of 𝑥U Є R (ÃU ) .
Hence 𝑥 = [𝑥L , 𝑥U ] .
Therefore , 𝓡 (Ã) = [ 𝓡 (ÃL ), 𝓡 (ÃU ) ]
(ii) For à = [ ÃL ), ÃU ] , the transpose of A is ÃT = [ (ÃTL
), (ÃTU ] .
by using (i) we get
𝒞 (Ã) = 𝓡 (ÃT) = [ 𝓡 (ÃLT ), 𝓡 (ÃUT) ]
= [ 𝒞 ((ÃL ), 𝒞 (ÃU) ]
Hence the theorem .
Volume 4 Issue 11, November 2015
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
78
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611
Theorem : 3.2
For à , B̃ Є (IVFSM)mn .
(i) 𝓡 (B̃ ) ≤ (Ã) ⟺ B̃ = X̃ Ã for some x Є (IVFSM)m.
(ii) 𝒞 (B̃ ) ≤ 𝒞 (Ã) ⟺ B̃ = Ã Ỹ for some Y Є (IVFSM)n .
Proof :
Let à = [ÃL , ÃU ] and B̃ = [B̃ L , B̃ U ]
Since B̃ = X̃ Ã , for some X̃ Є (IVFSM)
Put X̃ = [X̃ L , X̃ U ] . Then by lemma 3.1
(iii) B̃ L = X̃ L ÃL and B̃ U = X̃ U ÃU .
Hence lemma 2·4 𝓡 (B̃ L) ⊆ 𝓡 ( ÃL ) and 𝓡 (B̃ U) ⊆ 𝓡 ( ÃU )
By theorem 3.1 (i) 𝓡 (B̃ ) = [ 𝓡 (B̃ L ), 𝓡 (B̃ U ) ] ⊆ [ 𝓡 (ÃL ),
𝓡 (ÃU ) ] = 𝓡 (Ã)
Thus 𝓡 ( B̃ ) = 𝓡 ( Ã )
Conversely 𝓡 ( B̃ ) ⊆ 𝓡 ( Ã )
⇒ 𝓡 ( B̃ L ) ⊆ 𝓡 ( ÃL ) and 𝓡 ( B̃ U ) ⊆ 𝓡 ( ÃU ) (by theorem
3.1, (i))
⇒ B̃ L = ỸÃL and B̃ U = Z̃ ÃU (by lemma 2.3 )
Then B̃ = [ B̃ L , B̃ U ]
= [ ỸÃL , Z̃ ÃU ]
= [ Ỹ, Z̃ ] [ ÃL , ÃU ] (by lemma 3.1)
= X̃ [ ÃL , ÃU ] where X̃ [ Ỹ,Z̃ ] Є (IVFSM)mn .
= X̃ Ã
B̃ = X̃ Ã .
(i) This can be proved along the same lines as that of (i) that
hence omitled .
Theorem : 3.3
For à Є (IVFSM)mn , B̃ Є (IVFSM)mn . the following hold .
𝓡 (ÃB̃ ) ⊆ (Ã) are 𝒞 (ÃB̃ ) ⊆ 𝒞 (B̃ ) .
Proof :
Let à = [ÃL , ÃU ] and B̃ = [B̃ L , B̃ U ]
ÃT = [ÃLT , ÃUT ] and B̃ T = [B̃ LT , B̃ UT ]
Then by lemma 3.1
AB = [ ÃL B̃ L , ÃU B̃ U ]
By theorem 3.1
𝓡 (ÃB̃ ) = ( [ ÃL B̃ L , ÃU B̃ U ] )
= [ 𝓡 (ÃL B̃ L ) , 𝓡 ( ÃU B̃ U ) ] ⊆ [ 𝓡 (ÃL ) , 𝓡 ( ÃU ) ]
= (Ã). (by lemma 3.1)
𝓡 (ÃB̃ ) ⊆ 𝓡 (Ã)
𝒞 (ÃB̃ ) = 𝓡 (ÃB̃ ) T = 𝓡 (B̃ TÃT) ⊆ 𝓡 (B̃ T)
= 𝒞 (B̃ )
𝒞 (ÃB̃ ) = 𝒞 (B̃ )
Hence the theorem .
[5] Shyamal. A.K. and Pal.M., Interval – Valued Fuzzy
Matrices, Journal of Fuzzy Mathematics, 14 (3) (2006)
582 – 592.
[6] Meenakshi, AR., and Kaliraja,M., g-inverses of Interval
– valued Fuzzy Matrices, j in Fuzzy Mathematics,vol.45
,No.1,2013,83-92
[7] Meenakshi, AR., and Kaliraja,M.,inverses of k- Regular
Interval – valued Fuzz Matrices, Advances in Fuzzy
Mathematics,
[8] Shyamal, A.K., and Pal. (2006).Interval valued fuzzy
matrices, Journal of Fuzzy Mathematics, Vol.14(3), 582
-592.
[9] P.K. Maji, R. Biswas and A.R. Roy, fuzzy soft set, The
Journal of Fuzzy Mathematics, 9(3) (2001) 589 -602.
[10] P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy
soft set, The Journal of Fuzzy Mathematics, 12 (2004)
669 – 683
[11] Y. Yang and J. Chenli,Fuzzy soft matrices and their
applications, Part I, Lecture Notes in computer science,
7002, 2011, pp: 618 – 627.
[12] L.A. Zadeh, Fuzzy sets, Information and control, 8
(1985) 338 – 353.
[13] P. Rajarajeswari and P.Dhanalakshmi Interval - valued
fuzzy soft matrix theory, Annals of Pure and Applied
Mathematics, Vol. 7 (2014) 61 – 72.
[14] A . Ben Israel and T . N . E . Greville, generalized
Inverses : Theory and applications, Wiley , new York ,
1974.
[15] K. H.Kim and F . W. Roush, Generalized Fuzzy
Matrices,Fuzzy Sets and Systems, 4(1980) 293-315.
[16] AR. Meenakshi , Fuzzy Matrix Theory and
Applications, MJP Publishers, Chennai, 2008.
[17] C . J . Klir and T . A . Folger, Fuzzy Sets, Uncertainty
and Information, Prentice hall of India, New
Delhi,1988.
[18] AR. Meenakshi , and P.Jenita, Generalized regular
Fuzzy Matrices, Iranian journal of Fuzzy systems, 8(2)
(2011) 133-141.
[19] AR. Meenakshi , On regularity of block fuzzy matrices,
Int .J. Fuzzy math., 12(2004) 439-450.
[20] D.r.N.Sarala and M.Prabhavathi an application of
Interval valued fuzzy soft matrix in medical Diagnosis.
IOSR Journal of Mathematics Vol.11,Issue 1 ver.VI
(Jan-Feb.2015), pp: 1-6
[21] D.r.N.Sarala and M.Prabhavathi An a Application of
Interval Valued Fuzzy soft matrix in decision making
problem International Journal of Mathematics trends
and technology (IJMTT) v 21 (1):1-10May2015.
References
[1] Kim, K.H. and Roush, F.N., Generalized fuzzy
matrices, Fuzzy sets and systems, 4(1980), 293 – 315.
[2] Meenakshi, AR., and Kaliraja,M., Regular Interval –
valued Fuzzy Matrices, Advances in Fuzzy
Mathematics, 5 (1)(2010), 7 – 15.
[3] Meenakshi, AR., and Jenita, p., Generalized Regular
Fuzzy Matrices, Iranian Journal of fuzzy Systems, 8 (2),
(2011) 133 – 141.
[4] Meenakshi, AR., and Poongodi, P., Generalized Regular
Interval Valued Fuzzy Matrices, International Journal of
Fuzzy Mathematics and Systems, 2 (1) (2012), 29 – 36.
Paper ID: NOV151037
Volume 4 Issue 11, November 2015
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
79