Characterizations of Abel-Grassmann's Groupoids by Intuitionistic Fuzzy Ideals

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1 World pplied Sciences Journal 7 (1): , 013 ISSN IDOSI Publications, 013 DOI: /idosi.wasj Characterizations of bel-grassmann's Groupoids by Intuitionistic Fuzzy Ideals Madad Khan, ima nis Faisal Department of Mathematics, COMSTS Institute of Information Technology, bbottabad, K.P.K, Pakistan bstract: In this paper, we have characterized an intra-regular class of an G-groupoid in terms of intuitionistic fuzzy left (right, two-sided) ideals, intuitionistic fuzzy (generalized) bi-ideals intuitionistic fuzzy (1,)-ideals. 000 mathematics subject classification: 0M10 0N99 Key words: G-groupoids intra-regular G-groupoids intuitionistic fuzzy ideals INTODUCTION The concept of an bel-grassmann's groupoid (G-groupoid) [5] was first given by M.. Kazim M. Naseeruddin in 197 they called it a left almost semigroup (L -semigroup). P. Holgate called it a left invertive groupoid [4]. n G-groupoid is a groupoid having the left invertive law (ab)c = (cb)a, for all a, b, c S (1) In an G-groupoid, the medial law holds (ab)(cd) = (ac)(bd), for all a, b, c, d S () In an G-groupoid paramedial law holds S with left identity, the (ab)(cd) = (dc)(ba), for all a,b,c,d S (3) If an G-groupoid contains a left identity, then the following law holds a(bc) = b(ac), for all a, b, c S (4) n G-groupoid is a non-associative algebraic structure mid way between a groupoid a commutative semigroup. The left identity in an Ggroupoid if exists is unique [9]. n G-groupoid is non-associative non-commutative algebraic structure, nevertheless, it posses many interesting properties which we usually find in associative commutative algebraic structures. n G-groupoid with right identity becomes a commutative monoid [9]. n G-groupoid is basically the generalization of Corresponding uthor: Madad Khan, Department of Mathematics, COMSTS Institute of Information Technology, bbottabad, K.P.K, Pakistan 154 semigroup [5] with wide range of applications in theory of flocks [10]. The theory of flocks tries to describe the human behavior interaction. The concept of fuzzy sets was first proposed by Zadeh [1] in Several researchers were conducted on the generalization of the notion of fuzzy set. Given a set S, a fuzzy subset of S is, by definition an arbitrary mapping f:s [0,1], where [0,1] is the unit interval. fuzzy subset is a class of objects with grades of membership.. osenfeld [11] was the first who consider the case of a groupoid in terms of fuzzy sets. Kuroki has been first studied the fuzzy sets on semigroups [7]. s an important generalization of the notion of fuzzy set, tanassov [at], introduced the concept of an intuitionistic fuzzy set. De et al. [] studied the nchez's approach for medical diagnosis extended this concept with the notion of intuitionistic fuzzy set theory. Dengfeng Chunfian [3] introduced the concept of the degree of similarity between intuitionistic fuzzy sets, which may be finite or continuous gave corresponding proofs of this similarity measure discussed applications of the similarity measures between intuitionistic fuzzy sets to pattern recognition problems. The concept of intuitionistic fuzzy G-groupoid was first given by Khan Faisal in [6]. For more details applications one can see [13-17]. PELIMINIES fuzzy subset f is a class of objects with grades of membership having the form

2 f = {(x,f(x))/x S} The following definitions are given in [6]. n intuitionistic fuzzy set (briefly, IFS) = ( in a non empty set S is an object having the form = (x, (x), (x))/x S The functions :S [0,1] :S [0,1] denote the degree of membership the degree of non membership respectively such that for all x S, we have n intuitionistic fuzzy G-subgroupoid = (, ) of an G-groupoid S is called an intuitionistic fuzzy (1,)-ideal of S if ((xw)(yz)) (x) (y) (z) ((xw)(yz)) (x) (y) (z) for all x, a y in S. Let S be an G-groupoid let? W S, then the intuitionistic characteristic function W = ( W W of W is defined as 0 (x) + (x) 1 For the sake of simplicity, we shall use the symbol = (, ) for an IFS = {(x, (x), (x))/x S} Let {(x,s(x),o (x))/s(x) 1 O (x) 0/x S} = = = = (S, O ) be an IFS then = (S,O) will be carried out in operations with an IFS = (, ) such that S O will be used in collaboration with respectively. n IFS = (, ) of an G-groupoid S is called an intuitionistic fuzzy G-subgroupoid of S if (xy) (x) (y) (xy) (x) (y) for all x, y S. n IFS = (, ) of an G-groupoid S is called an intuitionistic fuzzy left ideal of S if (xy) (y) (xy) (y) for all x, y S. n IFS = (, ) of an G-groupoid S is called an intuitionistic fuzzy right ideal of S if (xy) (x) (xy) (x) for all x, y S n IFS = (, ) of an G-groupoid S is called fuzzy two-sided ideal of S if it is both an intuitionistic fuzzy left an intuitionistic fuzzy right ideal of S. n IFS = (, ) of an G-groupoid S is called an intuitionistic fuzzy generalized bi-ideal of S if ((xa)y) (x) (y) ((xa)y) (x) (y) for all x, a y in S. n intuitionistic fuzzy G-subgroupoid = (, ) of an G-groupoid S is called an intuitionistic fuzzy bi-ideal of S if ((xa)y) (x) (y) ((xa)y) (x) (y) for all x, a y in S , if x W (x) = W 0, if x W 0, if x W (x) = W 1, if x W It is clear that w acts as a complement of w that is, =. W WC Let = (, ) = (, ) are IFSs of an G-groupoid S. The symbol will means the following IFS of S ( )(x) = min{ (x), (x)} = (x) (x) ( )(x) = max{ (x), (x)} = (x) (x) The symbol will means the following IFS of S ( )(x) = max{ (x), (x)} = (x) (x) ( )(x) = min{ (x), (x)} =(x) (x) Let are the IFSs of an G-groupoid S, then means that (x) (x) (x) (x), Let = (, ) = (, ) be any two IFSs of an G-groupoid S, then the product is defined by, (a) = ( ) a = bc (b) (c), if a = bc for some b, c S 0, otherwise

3 (a) = ( ) abc = (b) (c), if a = bc for some b, c S 1, otherwise n IFS = (, ) of an G-groupoid is said to be idempotent if = =, that is, = or =. n IFS = (, ) of an G-groupoid S is called an intuitionistic fuzzy semiprime if (a) (a ) for all a in S. (a) (a ) Let S be an G-groupoid let I = {/ S}, where = (, ) be any IFS of S, then ( 1, ) forms an G-groupoid satisfies all the basic laws of an G-groupoid [6]. Example 1: Let S = {a,b,c,d,e} be an G-groupoid with left identity d with the following multiplication table.. a b c d e a a a a a a b a b b b b c a b d e c d a b c d e e a b e c d Define an IFS = (, ) of S as follows: (a) = 1, Lemma : [6] For any subset of an G-groupoid S, the following properties holds. (i) is an G-subgroupoid of S if only if is an intuitionistic fuzzy G-subgroupoid of S. (ii) is a fuzzy left (right, two-sided) ideal of S if only if is an intuitionistic fuzzy left (right, twosided) ideal of S. n element a of an G-groupoid S is called an intra-regular element of S if there exist x y in S such that a = (xa )y S is called intra-regular if every element of S is an intra-regular. Example : Let S = {a,b,c,d,e} be an G-groupoid with left identity b in the following Cayley's table.. a b c d e a a a a a a b a b c d e c a e b c d d a d e b c e a c d e b Clearly S is intra-regular, because a = (aa )a, b = c = (dc)e, d = (cd )c e = (be )e. (cb )e, Lemma 3: For an intra-regular G-groupoid S with left identity, the following holds. (b) = (c) = (d) = (e) = 0, (a) = 0.3, (b) = 0.4 (c) = (d) = (e) = 0., (i) Every intuitionistic fuzzy right ideal of S is an intuitionistic fuzzy semiprime. (ii) Every intuitionistic fuzzy left ideal of S is an intuitionistic fuzzy semiprime. (iii) Every intuitionistic fuzzy two-sided ideal of S is an intuitionistic fuzzy semiprime. then clearly = (, ) is an intuitionistic fuzzy twosided ideal of S. Lemma 1: [6] Let S be an G-groupoid, then the following holds. (i) n IFS = (, ) is an intuitionistic fuzzy Gsubgroupoid of S if only if (ii) n IFS = (, ) is intuitionistic fuzzy left (right) ideal of S if only if S O ( S O ) 156 Proof: (i): ssume that = (, ) is an intuitionistic fuzzy right ideal of an intra-regular G-groupoid S with left identity let a S, then there exist x y in S such that a = (xa )y. Now by using (3) (4), we have = = ((ye)(a x)) (a ((ye)x)) (a ) similarly (a) ((xa )y) ((xa )(ey)) = = (a) = ((xa )y) =((xa )(ey)) = ((ye)(a x)) = (a ((ye)x)) (a ) Thus = (, ) is an intuitionistic fuzzy semiprime.

4 (ii): Let = (, ) be an intuitionistic fuzzy left ideal of an intra-regular G-groupoid S with left identity let a S, then there exist x y in S such that a = (xa )y. Now by using (4), (3) (1), we have (a) ((xa )y) ((x(aa))y) ((a(xa))y) ((((xa )y)(xa))y) = = = = (((ax)(y(xa )))y) = = (((ax)(y((ex)(aa))))y) = (((ax)(y(a (xe))))y) = (((ax)(a (y(xe))))y) (a ((ax)(y(xe)))y) ((y((y(xe))(ax)))a ) (a ). = = Similarly we can show that (a) (a ) therefore = (, ) is an intuitionistic fuzzy semiprime. (iii): It can be followed from (i) (ii) S. This shows that S is intra-regular. Lemma 5: [6] For any non-empty subsets of an G-groupoid S, = holds. Lemma 4: Every intuitionistic fuzzy right (left, twosided ideal) of an G-groupoid S is an intuitionistic fuzzy semiprime if only if their characteristic functions are intuitionistic fuzzy semiprime. Proof: Let be any fuzzy right ideal of an Ggroupoid S, then by Lemma, the intuitionistic characteristic function of, that is, = ( is an intuitionistic fuzzy right ideal of S. Let a, then (a) = 1 assume that is semiprime, then a, which implies that (a) = 1. Thus we get (a ) = (a) similarly we can show that (a ) = (a), therefore = ( is an intuitionistic fuzzy semiprime. The converse is simple. Corollary 1: Let S be an G-groupoid, then every right (left, two-sided) ideal of S is semiprime if every intuitionistic fuzzy right (left, two-sided) ideal of S is an intuitionistic fuzzy semiprime. Theorem 1: Let S be an G-groupoid with left identity, then the following statements are equivalent. (i) S is intra-regular. (ii) Every intuitionistic fuzzy right (left, two-sided) ideal of S is an intuitionistic fuzzy semiprime. Proof: (i) (ii) can be followed by Lemma 3. (ii) (i): Let S be an G-groupoid with left identity let every intuitionistic fuzzy right (left, two-sided) ideal of S is an intuitionistic fuzzy semiprime. Since a S is a right also a left ideal of S, therefore by using Corollary 1, a S is semiprime. Clearly a a S, therefore a a S. Now by using (1), we have a a S = (aa)s = ()a ()(as) = ((a S)a)S = ((as)a )S ( )S 157 Theorem : For an G-groupoid S with left identity, the following conditions are equivalent. (i) S is intraregular. (ii) L = L, is any right ideal L is any left ideal of S such that is semiprime. (iii) =, = (, ) is any intuitionistic fuzzy right ideal = (, ) is any intuitionistic fuzzy left ideal of S such that = ( is an intuitionistic fuzzy semiprime. Proof: (i) (iii) ssume that S is an intra-regular Ggroupoid. Let = (, ) is any intuitionistic fuzzy right ideal = (, ) is any intuitionistic fuzzy left ideal of S. Now for a in S there exist x y in S such that a = (xa )y. Now by using (4), (1) (3), we have a = (x(aa))y = (a(xa))y = (y(xa))a = (y(x((xa )y)))a = (y((xa )(xy)))a = (y((yx)(ax)))a = (y(a ((yx)x)))a = (a (y((yx)x)))a Therefore ( )(a) = { (a (y((yx)x))) (a)} a = (a(y((yx)x)))a (a) (a) = ( )(a) ( )(a) = (a (y((yx)x))) (a) a = (a(y((yx)x)))a (a) (a) = ( )(a). Which imply that by using Lemma as,, therefore =. (iii) (ii): Let be any right ideal L be any left ideal of an G-groupoid S, then by Lemma, = ( L = ( L are an intuitionistic L

5 fuzzy right intuitionistic fuzzy left ideals of S respectively. s L L is obvious, therefore let a L, then a a L. Now by using Lemma 5 given assumption, we have similarly (a) = ( )(a) = ( )(a) = (a) (a) = 1 L L L L (a) = ( )(a) = ( )(a) = (a) (a) = 1 L L L L Which imply that a L therefore L=L. Now by using Corollary 1, is semiprime. (ii) (i): Let S be an G-groupoid with left identity, then clearly is a left ideal of S such that a a S is a right ideal of S such that a a S. Since by assumption, a S is semiprime, therefore a a S. Now by using (3), (1) (4) we have a a S (as)() (as)( ) (( )S)a (( )(SS))a = = = = = = = = ((SS)(aS))a (a ((SS)S))a (a S)S (SS)(aa) a S = = = = (aa)s ()a ()(a S) ((as)a)s ((as)a )S ( )S This shows that S is an intra-regular. Lemma 6: Let = (, ) be IFS of an intra-regular G-groupoid S with left identity, then = (, ) is an intuitionistic fuzzy left ideal of S if only if = (, ) is an intuitionistic fuzzy right ideal of S. Proof: ssume that S is an intra-regular G-groupoid with left identity let = (, ) be an intuitionistic fuzzy left ideal of S. Now for a,b S there exist x,y,x,y S such that (1), (3) (4), we have a = (xa )y b (x b )y. = Then by using = = = = = = (ab) (((xa )y)b) ((by)(x(aa))) (((aa)x)(yb)) (((xa)a)(yb)) (((xa)(ea))(yb)) (((ae)(ax))(yb)) = ((a((ae)x))(yb)) = (((yb)((ae)x))a) (a) Similarly we can get (ab) (a) which imply that = (, ) is an intuitionistic fuzzy right ideal of S. Conversely let = (, ) be an intuitionistic fuzzy right ideal of S. Now by using (4) (3), we have = = = = (ab) (a((xb )y ) ((xb)(ay)) ((ya)(b x )) (b ((ya)x)) (b) Similarly we can get (ab) (b), which imply that = (, ) is an intuitionistic fuzzy left ideal of S. Theorem 3: Let S be an G-groupoid with left identity, then the following conditions are equivalent. (i) S is intra-regular. (ii) Every intuitionistic fuzzy left (right, two-sided) ideal of S is idempotent. Proof: (i) (ii): ssume that S is an intra-regular G-groupoid with left identity let a S, then there exist x,y S such that a = (xa )y. Now by using (4), (1) (3), we have a = (x(aa))y = (a(xa))y = (y(xa))a = ((ex)(ya))a = ((ay)(xe))a = (((xe)y)a)a Let = (, ) be an intuitionistic fuzzy left ideal of S, then by using Lemma, we have also we have ( )(a) = { (((xe)y)a) (a)} (a) (a) = (a) a = (((xe)y)a)a 158

6 This implies that similarly we can get. Now by using Lemma,. Thus = ( is idempotent by using Lemma 6, every intuitionistic fuzzy right (two-sided) is idempotent. (ii) (i): ssume that every left ideal of an G-groupoid S with left identity is idempotent let a S. Since is a left ideal of S, therefore by Lemma, its characteristic function = ( is an intuitionistic fuzzy left ideal of S. Since a therefore (a) = 1. (a) 0 =. Now by using the given assumption Lemma 5, we have = = () a Thus, we have ( )(a) = ( )(a) 1 = similarly we can get, ( )(a) = ( )(a) 0, = which imply that (). () Now by using (1), () (3), we have a () = ()() = (()a)s (()(()()))S = (()((SS)(aa)))S = (()( ))S = ((as)(as))s = (((as)s)a )S ( )S () This shows that S is intra-regular. Note that if an G-groupoid S contains a left identity, then S= S S. Theorem 4: For an G-groupoid S with left identity, the following conditions are equivalent. (i) (ii) S is intra-regular. = ( ), where = (, ) is any intuitionistic fuzzy left (right, two-sided) ideal of S = (S, O ). Proof: (i) (ii): Let S be an intra-regular G-groupoid let = (, ) be any intuitionistic fuzzy left ideal of S, then it is easy to see that is also an intuitionistic fuzzy left ideal of S. Now by using Theorem 3, is idempotent therefore, we have ( ) = Now let a S, since S is an intra-regular therefore there exists x S such that a = (xa )y by using (4), (3) (1), we have = = = = a (x(aa))y (a(xa))y (((xa )y)(xa))(ey) (ye)((xa)((xa )y)) = = = (xa)((ye)((xa )y)) (xa)(((ye)(x(aa)))y) (xa)(((ye)(a(xa)))y) = (xa)((a((ye)(xa)))y) = (xa)((y((ye)(xa)))a) = (xa)p where p = ((y((ye)(xa)))a) therefore, we have (S ) (a) = {(S )(xa) (S )((y((ye)(xa)))a)} (S )(xa) (S )((y((ye)(xa)))a) a = (xa)((y((ye)(xa)))a) {S (x) (a)} {S(y((ye)(xa))) (a)} S ( x ) (a) S (y(( ye)(xa))) (a) = (a) xa= xa p = (y(y(xa)))a = Similarly we can get ( O ) (a) (a), which implies that ( ). Thus we get the required = ( ). (ii) (i): Let given assumption, we have = ( ) holds for any intuitionistic fuzzy left ideal = = (S ) ( ) S = ( of S, then by using Lemma 1 Which shows that =, similarly = therefore =. Thus by using Theorem 3, S is an intra-regular. 159

7 Let right ideal = (, ) of S, then by using Lemma 1, = ( ) holds for any intuitionistic fuzzy given assumption (1), we have = (S ) = ((S S ) ) = (( S ) S ) ( ) = S which shows that =, similarly = therefore =. Thus by using Theorem 3, S is intra-regular. EFEENCES 1. tanassov, K.T., Intuitionistic fuzzy sets. Fuzzy Sets Systems, 0: De, S.K.,. iswas.. oy, 001. n application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Systems, 117: Dengfeng, L. C. Chunfian, 00. New similarity measures of intuitionistic fuzzy sets applications to pattern recognitions. Pattern econit Lett., 3: Holgate, P., 199. Groupoids satisfying a simple invertive law. The Math. Stud., 61: Kazim, M.. M. Naseeruddin, 197. On almost semigroups. The lig. ull. Math., : Madad Khan Faisal, 011. Intra-regular G-groupoids characterized by their intuitionistic fuzzy ideals. Journal of dvanced research in Dynamical Control Systems, : Kuroki, N., Fuzzy bi-ideals in semigroups. Comment. Math. Univ. St. Pauli, 7: Mordeson, J.N., D.S. Malik N. Kuroki, 003. Fuzzy semigroups. Springer-Verlag, erlin, Germany. 9. Mushtaq, Q. S.M. Yusuf, On Lsemigroups. The lig. ull. Math., 8: Naseeruddin, M., Some studies in almost semigroups flocks. Ph.D. Thesis, ligarh Muslim University, ligarh, India. 11. osenfeld,., Fuzzy groups. J. Math. nal. ppl., 35: Zadeh, L.., Fuzzy sets. Inform. Control, 8: hmadreza rgha, Mehdi hrari Nouri Mehdi oopaei, 011. Using Fuzzy ILC for eactive Power Planning in Distribution Systems. World pplied Sciences Journal, 13 (11): Muhammad Shabir, Muhammad Shoaib rif Mairaj ibi, 011. Characterization of Monoids by the Properties of their Interval-valued Fuzzy Subacts. World pplied Sciences Journal, 13 (11): Eslami, M. H. Zareamoghaddam, 011. Differential transform method for bel differential equation. World pplied Sciences Journal, 13 (5): ehzad Shahrabi, 011. The gility ssessment Using Fuzzy Logic. World pplied Sciences Journal, 13 (5): lehian, S., M. Shafai ajestan, H. Housavi Jahromi, H Kashkooli S.M. Kashefipour, 011. Hydraulic jump characteristics due to natural roughness. World pplied Sciences Journal, 13 (5):