Applied Computatioal athematics 2015; 4(1-2): 15-19 Published olie December 31, 2014 (http://www.sciecepublishiggroup.com/j/acm) doi: 10.11648/j.acm.s.2015040102.14 ISSN: 2328-5605 (Prit); ISSN: 2328-5613 (Olie) Pseudo similar ituitioistic fuzzy matrices T. Ghimathi 1, A. R. eeakshi 2 1 Departmet of athematics, P.A. College of Egieerig Techology, Pollachi-642 002, Idia 2 Departmet of athematics, Karpagam Uiversity, Coimbatore-642 021, Idia Email address: ghimahes@rediffmail.com (T. Ghimathi) To cite this article: T. Ghimathi, A. R. eeakshi. Pseudo Similar Ituitioistic Fuzzy atrices. Applied Computatioal athematics. Special Issue: New Advaces i Fuzzy athematics: Theory, Algorithms, Applicatios. Vol. 4, No. 1-2, 2015, pp. 15-19. doi: 10.11648/j.acm.s.2015040102.14 Abstract: I this paper, we shall defie Pseudo Similarity Semi Similarity for Ituitioistic Fuzzy atrix (IF) prove that the Pseudo similarity relatio o a pair of IFs is iherited by all its powers their trasposes are similar. Also we exibit that the Pseudo similarity relatio preserve regularity impotecy of their matrices. Keywords: Fuzzy matrix, Ituitioistic Fuzzy atrix, Pseudo Similar, Semi Similar 1. Itroductio We deal with fuzzy matrices that is, matrices over the fuzzy algebra F FN with support [0,1] fuzzy operatios {+,.} defied as a+b=max{a,b}, a.b = mi{a,b} for all a,b F a + b = mi {a, b}, a.b = max{a, b} for all a, b FN. A matrix A Fmx is said to be regular if there exists X F xm such that AXA = A, X is called a geeralized iverse (g-iverse) of A. I [4], Kim Roush have developed the theory of fuzzy matrices, uder max mi compositio aalogous to that of Boolea matrices. Cho [2] has discussed the cosistecy of fuzzy matrix equatios. For more details o fuzzy matrices oe may refer [6]. Ataassov has itroduced developed the cocept of ituitioistic fuzzy sets as a geeralizatio of fuzzy sets [1]. Basic properties of ituitioistic fuzzy matrices as a geeralizatio of the results o fuzzy matrices have bee derived by Pal Kha [5]. I our earlier work, we have discussed o regularity idempotecy of IFs[8].The Pseudo Similarity i semi groups of fuzzy matrices were discussed by Che eeakshi[3,7] I this paper, pseudo similarity semi similarity of a ituitioistic fuzzy matrix are discussed. 2. Prelimiaries I this paper, we are cocered with fuzzy matrices, that is matrices over a fuzzy algebra F (FN) with support [0,1],uder maxmi (mimax) operatios the usual orderig of real umbers. Let (IF) mx be the set of all ituitioistic fuzzy matrices of order mx, Fmx be the set of all fuzzy matrices of order mx, uder the maxmi N compositio F mx be the set of all fuzzy matrices of order mx, uder the mimax compositio. ( a ) ij (IF) mx If A = a ij ν values of ( a ) µ, the A = ij, a, where a ijµ are the membership values o membership a ij i A respectively with respect to the fuzzy sets a ijµ µ ν, maitaiig the coditio 0 + ν 1. We shall follow the matrix operatios o ituitioistic fuzzy matrices as defied i [8]. For A, B (IF) mx,the A B A + B = a ij ( max{ a,b },mi{ a, b } ) ijµ ijµ { a,b },mimax{ a,b } = max mi ikµ kjµ ikν kjν k k Let us defie the order relatio o (IF) mx as, A B a b a b, for all i j. ijµ ijµ Defiitio 2.1[5] A A (IF) mx is said to be regular if there exists
16 T. Ghimathi A. R. eeakshi: Pseudo Similar Ituitioistic Fuzzy atrices X (IF) xm satisfyig AXA = A X is called a geeralized iverse (g-iverse) of A which is deoted by. Let A{1} be the set of all g-iverses of. Defiitio 2.2[8] A atrix A (IF) is said to be ivertible if oly if there exists X (IF) such that AX = XA = I = where I is the idetity matrix i (IF) Defiitio 2.3[8] A square ituitioistic fuzzy matrix is called ituitioistic fuzzy permutatio matrix, if every row colum cotais exactly oe <1,0> all other etries are <0,1>. Let P be the set of all x permutatio matrices i (IF). Defiitio 2.4[8] Let A (IF).The (i) A is reflexive A I. (ii) A is symmetric A = A T. (iii) A is trasitive A 2 A. (iv) A is idempotet A = A 2 3. Pseudo Similar Ituitioistic Fuzzy atrices I this sectio, the pseudo similarity semi similarity of a ituitioistic fuzzy matrix are discussed. Defiitio 3.1 A (IF) m B (IF) are said to be Pseudo-Similar deote it by A B if there exist X (IF) mx Y (IF) xm such that A=XBY,B=YAX X=XYX. Defiitio 3.2 A (IF) m B (IF) are said to be Semi-Similar deote it by A B if there exist X (IF) mx Y (IF) xm such that A=XBY B=YAX. Defiitio 3.3 For A (IF), the least d >0 such that A k+d = A k for some iteger k, is called the period of A. Theorem 3.1 Let A (IF) m B (IF). The the followig are (ii) There exist X (IF) mx Y (IF) xm such that A=XBY B=YAX XY (IF) m is idempotet. (iii) There exist X (IF) mx Y (IF) xm such that A=XBY B=YAX YX (IF) is idempotet. (i) (ii) (i) (iii) are trivial, sice X=XYX implies XY (IF) m YX (IF) are idempotet matrices. (ii) (i): A=XBY = (XY)A(XY)= (XYX)B(YXY). Similarly, B=YAX =(YXY)A(XYX) Put X =XYX Y = YXY. The A=X BY B=Y AX. Further usig XY is idempotet, we get X Y = (XYX)(YXY) =XY (X Y ) (X Y ) =(XY)(XY) = X Y. I, I N Thus X Y is idempotet. Set X =X Y X Y = Y X Y. The A =X BY = X (Y AX ) Y = X Y (X BY ) X Y = (X Y X )B(Y X Y ) = X B Y Similarly, B = Y A X. By usig X Y is idempotet, we have X Y X = (X Y X ) (Y X Y ) (X Y X ) = X Y X = X. Therefore A B.Thus (i) holds. (iii) (i): This ca be proved i the same maer hece omitted. Theorem 3.2: Let A (IF) m B (IF). The the followig are (ii) There exist X (IF) mx Y (IF) xm such that A=XBY B=YAX (XY) k (IF) m is idempotet for some odd k N. (iii) There exist X (IF) mx Y (IF) xm such that A=XBY B=YAX (YX) k (IF) is idempotet for some odd k N. (i) (ii): Follows from Theorem (3.1). (ii) (i): Let k = 2r+1 with r N,Sice A=XBY B=YAX, we have A = X(YAX)Y = (XY) XBY(XY) = (XY)X(YAX)Y(XY) = (XY) 2 A(XY) 2 = (XY) 2 XBY (XY) 2 Thus proceedig we get, A = (XY) r XBY (XY) r. Similarly, B = Y(XY) r A (XY) r X. Set X = (XY) r X Y =Y(XY) r. The A=X BY, B=Y AX X Y = (XY) r XY(XY) r = (XY) 2r + 1 = (XY) k is idempotet. Hece by Theorem (3.1), A B. Thus (i) holds. (i) (iii): Follows from Theorem (3.1) (iii) (i): Let k = 2r+1 with r N,Sice A=XBY B=YAX, we have A= X(YX) r B(YX) r Y B = (YX) r YAX(YX) r Set X = X(YX) r Y =(YX) r Y. The A=X BY, B=Y AX Y X = (YX) r (YX) (YX) r = (YX) 2r + 1 = (YX) k (IF) is idempotet. Hece by Theorem (3.1), A B. Thus (i) holds. Theorem 3.3: Let A (IF) m B (IF). The the followig are (ii) There exist X (IF) mx Y (IF) xm such that
Applied Computatioal athematics 2015; 4(1-2): 15-19 17 A=XBY B=YAX (XY) (IF) m is periodic with eve order. (iii) There exist X (IF) mx Y (IF) xm such that A=XBY B=YAX (YX) (IF) is periodic with eve order. (i) (ii): Follows from Theorem (3.1), sice XY (IF) mx is idempotet, it follows that XY is periodic with eve order. (ii) (i): Suppose A=XBY B=YAX XY is periodic with eve order2k(k N)say, the (XY) 2k =XY. Hece (XY) 2k-1 (XY)=XY Further (XY) 2k-1 (XY) 2k-1 = (XY) 2k-1 XY(XY) 2k-1 = (XY) 2k-1. Thus (XY) 2k-1 is idempotet. Therefore A B, by Theorem (3.2).Thus (i) holds. (i) (iii):this ca be proved i the same maer hece omitted. Theorem 3.4: Let A (IF) m B (IF). The the followig are (ii) There exist X (IF) mx Y (IF) xm such that A=XBY, B=YAX,X=XYX Y=YXY (iii) There exist X (IF) mx Y,Z (IF) xm such that A=XBY, B=ZAX, X=XYX = XZX (i) (iii): Sice A B, By Defiitio(3.1),there exist X (IF)mx Y (IF)xm such that A=XBY, B=YAX,X=XYX. Let Y=Z the B=ZAX X=XZX as required. Thus(iii) holds. (iii) (ii): Suppose there exist X (IF)mx Y,Z (IF)xm such that A=XBY, B=ZAX, X= XYX= XZX, the A=XBY=X(ZAX)Y=(XZX)B(YXY) =XB(YXY) B=ZAX=Z(XBY)X=(ZXZ)A(XYX) =(ZXZ)AX. Set Y =YXY Z =ZXZ. The X=XYX=XY(XYX)=XY X X=XZX=XZ(XZX)=XZ X. I additio, we have A=XBY B=Z AX. Set Y =Z XY. The XY X=XZ (XY X)=XZ X=X Y XY =Z (XY X)Y =Z XY =Y. We check that XBY =XBZ XY =XZ AXZ XY =XZ AXY =XBY =A, Y AX=Z XY XBY X=Z XBY X =Z AX=B. Thus there exist X (IF) mx, Y (IF) xm such that A=XBY, B=Y AX, X=XY X Y =Y XY as asserted (ii) holds. (ii) (i):this is trivial. Remark 3.1: We observe that Theorem (3.4) ifers the symmetry of Pseudo similarity relatio, that is, A B B A. Theorem 3.5: Let A (IF) m B (IF). The the followig are (ii) A T B T (iii) A k B k for ay iteger k 1 (iv) PAP T QBQ T for some permutatio matrices P (IF) m Q (IF) (i) (ii): This is direct by takig traspose o both sides of A=XBY,B=YAX usig (A T )=A (AX) T =X T A T. (i) (iii): (iii) (i) is trivial. (i) (iii) ca be proved by iductio o k. Sice A B, there exist X (IF) mx Y (IF) xm such that A=XBY,B=YAX,X=XYX.By Theorem(3.4),further we have Y=YXY. Let us assume that A r =XB r Y, B r =YA r X for some r N. A=XBY AXY=XBYXY XB r+1 Y = XB r BY = XB r BY = XB r (YAXY) =XBY = A, = XB r YA = A r+1 Similarly it ca be checked YAr+1X = Br+1. Thus by iductio for all positive iteger k,we get, Ak=XBkY, Bk=YAkX for X (IF)mx Y (IF)xm such that X=XYX. Hece Ak Bk. Thus (i) (iii) hold. (i) (iv): Suppose A B, the there exist X (IF)mx Y (IF)xm such that A=XBY, B=YAX,X= XYX. A=XBY PAP T = PXBYP T = (PXQ T )(QBQ T )(QYP T ) B=YAX QBQ T = QYAXQ T = (QYP T )(PAP T )(PXQ T ). Set X =PXQ T, Y =QYP T the X Y X =X, PAP T =X(QBQ T )Y QBQ T =Y (PAP T )X. Thus PAP T QBQ T. Coversely, if PAP T QBQ T the as above, P T ( PAP T )P Q T (QBQ T )Q A B Thus (i) (iv) holds. Theorem 3.6:
18 T. Ghimathi A. R. eeakshi: Pseudo Similar Ituitioistic Fuzzy atrices Let A (IF) m B (IF). The the followig are (i) A B (ii) A T B T (iii) A k B k for ay iteger k 1 (iv) PAP T QBQ T for some permutatio matrices P (IF) m Q (IF) This ca be proved alog the same maer as that of Theorem (3.5) hece omitted. Remark 3.2: It is clear that Similarity Pseudo similarity Semi similarity but coverse is ot true. Example 3.1: Cosider the IFs, with respect to <0.5,0> <0.5,0> A= <0,1> <0,1> <0.5,0> <0,1> B= <0.5,0> <0,1> <1,0> <0,1> X= <0,1> <0,1> <1,0> <1,0> Y= <1,0> <0,0.5> X{1}. Here A=XBY B=YAX. This implies that A B A B. Example 3.2 Let us cosider the IFs the XYX X YXY Y. <0.3,0.3> <1,0> X= <0.5,0.5> <0.2,0.2> <1,0> <1,0> Y= <0,0.5> <0,0.5> <0.5,0.3> <0.2,0.3> For A= <0,0.5> <0,0.5> <0.3,0.3> <0.3,0> B= <0.5,0.5> <0.5,0.5>, B=XAY A=YAX. Hece A B but A B. Next we ca see that Pseudo Similarity relatio preserve regularity idempotecy of matrices cocered. Theorem 3.7: Let A (IF) m B (IF) such that A B. The A is regular matrix B is regular matrix. Sice A B,by Theorem(3.4), there exist X (IF) mx Y (IF) xm such that A=XBY, B=YAX,X=XYX Y=YXY. Suppose A is regular, the there exists (IF) m such that AGA=A. DefieU=YGX. Clearly U (IF).Sice, AXY=XBYXY = XBY=A =(XYX)BY =XYA, BUB = (YAX)(YGX)(YAX) =Y(AXY)G(XYA)X = YAGAX =YAX=B. Hece B is regular.coverse ca be proved i the same maer. Theorem 3.8: Let A (IF) m B (IF) such that A B. The A is idempotet B is idempotet. Sice A B,by Theorem(3.4), there exist X (IF) mx Y (IF) xm such that B=YAX A=XBY AXY=A=AYX. Suppose A is idempotet, the A 2 =A, hece B 2 =(YAX)(YAX)=Y(AXY)AX = YAAX =YA 2 X=YAX=B B is idempotet. Coverse ca be proved i the same maer. 4. Coclusio I this paper, the Pseudo similarity Semi similarity of IFs are defied.the Pseudo similarity relatio o a pair of IFs is iherited by all its powers the Pseudo similarity relatio preserve regularity impotecy of matrices are discussed. Refereces [1] K. Ataassov, Ituitioistic fuzzy sets, Fuzzy Sets systems 20(1), (1986), 87-96. [2] H. H. Cho, Regular fuzzy matrices fuzzy equatios, Fuzzy Sets Systems, 105 (1999), 445 451. [3] Huayi Che, Pseudo similarity i Semigroups, Semi group forum,68(2004),59-63 [4] K. H. Kim F. W. Roush, Geeralised fuzzy matrices, Fuzzy sets Systems, 4 (1980),293 315. [5] S. Kha Aita Pal, The Geeralised iverse of ituitioistic fuzzy matrics, Jourel of Physical sciece, Vol II (2007), 62-67.
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