International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11087-11090 © Research India Publications. http://www.ripublication.com Generalisation of Idempotent Fuzzy Matrices

K.Muthugurupackiam1 and K.S.Krishnamohan2 1&2Department of Mathematics, Rajah Serfoji Government College, Thanjavur – 613 005, India.

Abstract CHARACTERIZATIONS OF k – IDEMPOTENT FUZZY MATRICES In this paper, we introduce and study the concept of k – Idempotent fuzzy matrix as a generalization of Definition 2.1 Idempotent fuzzy matrix via permutations. For a fixed product of disjoint transposition  Sk n , a matrix Keywords: Idempotent fuzzy matrix, k – Idempotent fuzzy 2 matrix, permutation matrix, zero patterns, commutator  aA  is said to be k-idempotent if KA  AK , ij nn AMS Subject Classification: Primary : 15B15, 17C17 and where K is the associated permutation fuzzy matrix of ‘k’. The secondary : 05A05 associated permutation fuzzy matrix K is a matrix with one on its southwest – northeast diagonal and zeros everywhere else. INTRODUCTION  100    That is, K  010 A fuzzy matrix  aA  is said to be idempotent if, and   ij nn    001  2 only if  AA .H. Y. Lee et. al. [1] has discussed the concept of idempotent fuzzy matrices. The notion, Example 2.2 k – idempotent matrices introduced by Krishnamoorthy et.  1.01.01   100  al.[3] as a generalization of idempotent matrices is associated     and motivated by the parameter k – idempotent; we introduce Let A   2.014.0  and K   010  , and study a new characteristic k – idempotent fuzzy matrix in     this paper. If a fuzzy matrix A is obtained by k – permuting  8.06.07.0   001  the elements of A2, then it is called k – idempotent. Here k is the fixed product of disjoint transposition in Sn – the  1.01.01  symmetric group of order . In this paper, some 2   n then  2.014.0  characterization of a k – idempotent fuzzy matrices are KA K   A   examined such as sum and product of two k-idempotent fuzzy  8.06.07.0  matrices are k- idempotent. Furthermore, we show that some properties for k – idempotent fuzzy matrices which will be Hence A is a k- idempotent fuzzy matrix. intended to provide further discussions. For a matrix

T  aA  , ,adjAA and detA denotes the transpose, ij nn Remark 2.3 adjoin and determinant of the fuzzy matrix A. Let ‘k’ be a 2 fixed product of disjoint transpositions in S , the set of all implies that  AKAK . From the definition, n the following relations can also be obtained which would be permutation on {1, 2,…, n}. Hence it is involuntary (that is useful in computational aspect. k2 = identity permutation). A square matrix is called a 22 33 22 permutation matrix [4] if every row and every column AK AK  A ; KA  AK and KA  AK contains exactly one '1' and all the other entries are '0' .In this paper, the index set 2,1 ,...  ,1 will be denoted by N. By nn Theorem 2.4 C C the Prop. 2.4.5 in [4],  AadjA , where A is idempotent A k – idempotent fuzzy matrix A is ciruculant [4] if and only and nc  .1 The operations +, . and – are defined as if AK = KA. follows:  ,max{ },  babababa },min{. Proof. a if  ba Assume that AK = KA and ba   Pre multiplying by K, we have 0 if  ba KAK = A

11087 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11087-11090 © Research India Publications. http://www.ripublication.com

But A2 = A {since A is k – idempotent Hence A is k – idempotent. Proof. The converse is also true by retracing the steps. It is an immediate consequence of the fuzzy matrix product.  Next, we are examine some basic properties of idempotent Remark 2.9 fuzzy matrices. We know that all 11 fuzzy matrices are Let A be k – idempotent fuzzy matrix, then AT is also k – idempotent. Hence, in this paper, we deal only with square k – idempotent. fuzzy matrix that dimension nn  2, . Let F be the set of I all idempotent fuzzy matrices. Example 2.10

 1.01.01.0  Lemma 2.5   A fuzzy matrix is k - idempotent if and only if all its zero Let A   2.04.02.0  be k-idempotent, patterns [1] are idempotent.    8.06.07.0 

 1.01.01.0  By the above Lemma 2.4, we examine the properties of   )1,0(  fuzzy matrices and obtain a theorem and canonical AT   2.04.02.0    form of the )1,0(  fuzzy matrices. Thus we will be able to to  8.06.07.0  charecterise the structure of the set of all idempotent fuzzy matrices, . 2 and also T   AKAK T .

Hence AT is also k – idempotent. Lemma 2.6

The set of all idempotent fuzzy matrices, is closed under Proposition 2.11 the following operations If the fuzzy matrix A is – idempotent, then is also (i) Permutation similarity k adjA k – idempotent. (ii) Transposition.

Proof. Remark 2.7 C We know that  AAdjA , where AC is idempotent and The product of the permutation matrix must be identity. K nc  .1 [4]  100   100   001  2 2       Since A is k-idempotent, KA  AK i.e., K   010 . 010    010   I ,       C 2 C  001   001   100  Also,    AKAK the identity matrix. C Hence  adjAA is k-idempotent. In particular if  iik ,)( then the associated permutation matrix K reduces to identity matrix and k-idempotent fuzzy matrix reduces to idempotent fuzzy matrix. Proposition 2.12 If A is k – idempotent, then the fuzzy matrix AadjA is also k – idempotent. Lemma 2.8

Let  aA  be an k-idempotent fuzzy matrix. If Proof. ij nn  0 for some and in N, then each product 2 22 aij i j    )( KadjAAKKAadjAK ikaa kj  0 for all k in N. 2 2  KA )(. KadjAKK

 AadjA

11088 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11087-11090 © Research India Publications. http://www.ripublication.com

Hence AadjA is k-idempotent. n Lemma 2.13 to a commuting family of matrices, then  Ai is a i1 Let A and B are two – idempotent fuzzy matrices, then k k – idempotent fuzzy matrix.  BABA )det()det()det( and

BA  AB)det(det.det . Proof.

2 Example 2.14 n   2  i    21 ... n . 21 ... n  KAAAAAAKKAK  .0 85 5.0   5.05.0   i1  A   ; B     .05.0 85  5.05.0  2 22 A  .0det 85 B  5.0det&   1 2 ... n KAAAK .0 85 5.0 5.05.0 2 2 2      KA1 K.KA2 K....KA K BA    & AB    n  .05.0 85  5.05.0   . 21 ...AAA n Now, BA .0detdet 85  5.0 n = 0.58   Ai  BA )det( i1 and BA  .0det.det 85 5.0.  Hence the fuzzy matrix is a k-idempotent. = 0.5  AB)det( Definition 3.4 SOME OPERATIONS ON k-IDEMPOTENT FUZZY For a pair of k – idempotent fuzzy matrices A and B, the MATRICES commutator of A and B is denoted by [A,B] and defined by Proposition 3.1 BA ],[  BAAB . Let A and B be two k-idempotent fuzzy matrices. Then A+B is k – idempotent fuzzy matrix. Remark 3.5

Proposition 3.2 If A and B are two k – idempotent fuzzy matrices then A+B is k – idempotent if and only if BA ],[  AB . Let A and B be two k-idempotent fuzzy matrices. If AB=BA, then AB is also a k-idempotent fuzzy matrix If A and B are two k – idempotent fuzzy matrices then AB is k – idempotent if and only if BA  0],[ . Proof. 2 22 KAB  K  KA K.KB K Example :3.6  5.008.0  = A.B   Hence the fuzzy matrix AB is k-idempotent. Consider the idempotent fuzzy matrix A   010     8.005.0  The following theorem gives the generalization of products of and it is also commutes with the associated permutation k – idempotent matrices.  100    matrix K   010  , that is AK = KA.   Theorem3.3  001  If , 21 ,..., AAA n be a k-idempotent fuzzy matrices belonging We see that A is idempotent.

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Lemma 2.4 fails if we relax the condition of commutability of Pre – multiplying equation (3.2) by AB, matrix A and K. (AB)4 = 0

Theorem 3.7 i.e., AB = 0 It follows that A(A+B)B=0. If A and B are two k – idempotent fuzzy matrices then A(A+B)B commutes with the permutation matrix K. Remark 3.10 Proof. For example, the matrices A and B in Example 2.13 can be A(A+B)B = A2B+AB2 considered. A(A+B)B = A2B+AB2 = KA2KB+AKB2K 5.05.0 2 2   = KAB K+KA BK     5.05.0  = K(AB2+A2B)K Clearly, the matrix A(A+B)B commutes with the permutation = K(A2B+AB2) matrix K. it can be easily verified that A(A+B)B is not a k – idempotent. = KA(A+B)BK

Hence KA(A+B)B = A(A+B)BK CONCLUSION Clearly, the study of this kind of canonical forms is important to develop the theory of a fuzzy matrix. The concept of Theorem 3.8 k – idempotent fuzzy matrix is generalized to periodic n Let A and B are two commuting k – idempotent fuzzy matrices or n – potent fuzzy matrix (i.e., KA  AK ). matrices. The k – idempotency of A(A+B)B necessarily implies that is a null matrix. REFERENCES Proof. For any two k – idempotent fuzzy matrices A and B, we have [1] Hong Youl Lee, Nae Gyeong Jeong and Se Won A(A+B)B commutes with the permutation matrix K in Park, The idempotent Fuzzy matrices, Honam proposition 3.2. Mathematical Journal 26 (2004) PP 3 – 15.

If A(A+B)B is k – idempotent then by Lemma 3.6, it reduces [2] Kim J.B., Idempotents and inverses in Fuzzy to an idempotent matrix. matrices, Malaysian Math 6(2), 1983, 57 – 61. [3] Krishnamoorthy.S, Rajagopalan. T and Vijayakumar. i.e., [A(A+B)B]2= A(A+B)B ------(3.1) R; On k-Idempotent Matrices; Jour. Anal Comput; vol. 4, no.2, Dec (2008). i.e., [A2B]+(AB2)2+A2BAB2+AB2A2B=A2B+AB2 [4] Meenakshi.A.R., Fuzzy matrix – Theory and its applications, MJP Publishers (2008) Since A and B are k – idempotent fuzzy matrices, we have A4=A and B4=B. [5] Sidky F.I. & Emam E.G., Some remarks on sections of a Fuzzy matrix, J.K.A.U. Sci., Vol.4 pp 145 – 155 Hence (3.1) becomes, (1992). AB2+ A2B+A3B3+A3B3 = A2B+AB2 i.e., 2A3B3 = 0 i.e., A3B3 = 0 i.e., (AB)3 = 0 ------(3.2)

Since A and B are commuting k – idempotent fuzzy matrices, AB is also k – idempotent by Hence (AB)4=AB

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