(, ) Anti Fuzzy Subgroups

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1 International Journal of Fuzzy Mathematis and Systems. ISSN Volume 3, Number (203), pp Researh India Publiations (, ) Anti Fuzzy Subgroups P.K. Sharma Department of Mathematis, D.A.V. College, Jalandhar City, Punjab, India pksharma@davjalandhar.om Abstrat The author have already introdued the notion of - fuzzy set, - anti fuzzy set and (, )-fuzzy set in a group and studied their properties in [5], [6] and [7] respetively. In this paper, we introdued the notion of (, )-anti fuzzy set and subsequently study (, )- anti fuzzy subgroup ( normal subgroup ) and their related properties. We have also obtained a natural homomorphism from the group G to the set of all (, )-anti fuzzy osets of the (, ) - anti normal subgroup of the group G. Finally, the behaviour of these (, )-anti fuzzy subgroups (normal subgroups) under group homomorphism have been disussed. Mathematis Subjet Classifiation 03E72, 08A72, 20N25 Keywords Fuzzy subgroup (FSG), Fuzzy Normal subgroup (FNSG), (, )-anti fuzzy subgroup ((, )-AFSG), (, )- anti fuzzy normal subgroup ((, )-AFNSG) I. Preliminaries Definition(.) A fuzzy set A of a set X is a funtion A : X [0,]. Fuzzy sets taking the values 0 and are alled Crisp sets. Let A and B be two fuzzy subsets of a set X. Then the following epressions are standard and an be found in any standard book of fuzzy set theory namely [ 2 ] and [ 3 ] A B if and only if A() B(), for all X A = B if and only if A B and B A The omplement of the fuzzy set A is A and is defined as A () = A()

2 62 P.K. Sharma (A B) () = min{a(), B()}, X (A B)() = ma{a(), B()}, X Now, we list some of the results from [5], [6] and [7] whih are essential for the better understanding of the forthoming results. Definition(.2) A funtion A : G [0,] is a fuzzy subgroup (FSG) of a group G if and only if A(y - ) min {A( ), A(y )},, y G Definition (.3) A funtion A : G [0,] is an anti fuzzy subgroup (AFSG) of a group G if and only if A(y - ) ma {A( ), A(y )},, y G Definition(.4) A fuzzy subgroup (or anti fuzzy subgroup) A of a group G is alled fuzzy normal subgroup (FNSG) [or anti fuzzy normal subgroup (AFNSG)] of G if and only if A( y y) A( ) or equivalently, A( y) A( y), holds for all, y G Definition(.5) Let A be a fuzzy subset of a group G and [0,]. Then the fuzzy sets A and A of G are respetively alled the - fuzzy subset and - anti fuzzy subset of G (w.r.t. fuzzy set A) and is defined as A () = min{ A( ), } ; A () = ma{ A( ), - }, G Remark (.6) Clearly, A = A, 0 A 0 and A = A, A0 Theorem(.7) Let A be a fuzzy subset of a group G and [0,]. Then A is alled -fuzzy subgroup ( -FSG) of G if A (y - ) min { A (), A (y) } ; for all, y G. Theorem(.8) Let A be a fuzzy subset of a group G and [0,]. Then A is alled - anti fuzzy subgroup ( -AFSG) of G if A (y - ) ma { A (), A (y) }, for all, y G. Proposition(.9) If A be a FSG of the group G, then A is also -FSG as well as -AFSG of G. Proposition(.0) Let A : G [0,] be a -FSG of a group G, then A () A (e), G, where e is the identity element of G A (y - ) = A (e) A () = A (y),, y G Proposition(.) Let A : G [0,] be an -AFSG of a group G, then A (X) A (E), X G, WHERE E IS THE IDENTITY ELEMENT OF G A (XY - ) = A (E) A (X) = A (Y), X, Y G

3 (, ) Anti Fuzzy Subgroups 63 Definition(.2) A fuzzy subgroup (or anti fuzzy subgroup) A of a group G is alled -fuzzy normal subgroup (-FNSG) [or -anti fuzzy normal subgroup (-AFNSG)] of G if and only if A ( y y) A ( ) or equivalently, A ( y) A ( y), holds for all, y G [ or A ( y y) A ( ) or equivalently, A ( y) A ( y), holds for all, y G ] Theorem (.3) If A is a FNSG of a group G, then A is also a - FNSG as well as - AFNSG of G. Definition (.4) Let f : G G 2 be a homomorphism of group G into a group G 2. Let A and B be fuzzy subsets of G and G 2 respetively, then f (A) and f - (B) are respetively the image of fuzzy set A and the inverse image of fuzzy set B, defined as Sup{ A( ) : f ( y)} ; if f ( y) f ( A)( y), for every y G 2 ; if f ( y) and f B B f ( )( ) ( ( )), for every G Remark(.5) (i) Clearly, f (A)(f()) A(), for every element G (ii) When f is a bijetive map, then f (A)(f()) = A(), G 2. (, )-anti fuzzy subgroup and their properties Definition(2.) Let A and A denote respetively the -fuzzy set and -anti fuzzy set of the set X (w.r.t. the fuzzy set A). Then the fuzzy set A, defined by A ( ) Ma A ( ), A ( ), for every X,, is alled (, )-anti fuzzy set of X (w.r.t. the fuzzy set A), where, [0,] suh that + 0, 0 0 Re mark(2.2)( i) A ( ) Ma A ( ), A ( ) Ma A ( ),,0 ( ii) A ( ) Ma A ( ), A ( ) Ma, A( ) Definition(2.3) Let A be a (, )-anti fuzzy set of a group G ( w.r.t. the fuzzy set A), then A is alled (, )-anti fuzzy subgroup ((, )-AFSG ) of G if the following onditions hold

4 64 P.K. Sharma ( i) A ( y) Ma A ( ), A ( y),,, ( ii) A ( ) A ( ), for all, y G,, Equivalently, we have,,, A y Ma A A y ( ) ( ), ( ), for every G Remark(2.4) (i) If A is a (, )-anti fuzzy subgroup of a group G, then we ( i) A, ( e) A, ( ), where e is the identity of group G have (ii) If A ( y ) A ( e) A ( ) A ( y),,,,,, Pr oof.( i) A ( e) Ma A ( e), A ( e) Ma A ( ), A ( ) A ( ) ( ),,, ( ), ( ), A ( ), A, ( ) ( ii) A ( ) A ( y y) Ma A ( y ), A ( y) Ma A ( e), A ( y) A ( y) A ( y ),,,,,,,, Ma A y A Ma A e Hene A ( ) A ( y),, Proposition (2.5) Let A be -FSG as well as -AFSG of a group G, then A is also (, )-AFSG of G Proof. Let, y be any element of the group G, then, A A y y A y Ma A y A y Ma A A y y ( ) ( ), ( ) ma ( ), ( ), ma A ( ), A ( ) = Ma ma ( ), A ( ), ma ( ), A ( ) = Ma A ( ), A ( y),, Thus A ( y ) Ma A ( ), A ( y),,, Hene A is (, )-AFSG of G Theorem (2.6) Let A be a FSG of a group G, then A is also (, )-FSG of G Proof. Sine A is a FSG of group G. Therefore A is -FSG as well as A is -AFSG of G ( by Proposition (.9)) and the result follows from Proposition (2.5) Remark (2.7) The onverse of the Theorem (2.6) need not be true i.e. A fuzzy set A of a group G an be (, )-AFSG of G without being FSG of G. Eample (2.8) Let G = { e, a, b, ab }, where a 2 = b 2 = e and ab = ba be the Klein four group. Let the fuzzy set A of G be defined as

5 (, ) Anti Fuzzy Subgroups 65 A = {< e, 0. >, < a, 0.3 >, < b, 0.3 >, < ab, 0.4 > }. Clearly, A is not a FSG of G. Take = 0.05 and = 0.6 Then A() >, G. So that A ( ) min{ A( ), 0.05} 0.05 A ( ) , for all G Also, A () = Ma{ A(), - } = Ma{ A(), - 0.6}= Ma{ A(), 0.4}= 0.4, G 0.05, Further, A ( ) Ma A ( ), A ( ) Ma{0.95, 0.4} 0.95, G 0.05, , , 0.6 Thus A y Ma A y A y 0.6, ( ) ( ), ( ) hold for all, yg Hene A is 0.05, 0.6 AFSG of G. Proposition (2.9) Let A be a fuzzy subset of a group G. Let p and - q, where p = Inf{A() : for all G} and q = Sup{A() : for all G}. Then A is (, )-AFSG of G Proof. Sine p and - q + p + q q + q = Now p p, so Inf{ A() : for all G} A(), G A () = min{ A(), } =, G (A ) () = -, G. Similarly, as - q q - Sup{ A() : for all G} -, G A() -, G A () = ma{ A(), - } = -, G (, ) Now, A ( ) Ma A ( ), A ( ) Ma{, - }, G Therefore, A ( y ) Ma{ A ( ), A ( y)} hold, y G (, ) (, ) (, ) Hene A is (, ) - AFSG of group G. Remark(2.0) Intersetion of two (, )-AFSG s of a group G need not be (, )- AFSG ofg Eample(2.) Let G = Z, the group of integers under ordinary addition of integers. Define the two fuzzy sets A and B by 0.5, if = 3Z 0.6, if = 2Z A( ) and B( ) 0.7, otherwise 0.8, otherwise Take = 0.4 and = 0.35, then we have 0.65, if = 3Z ( ) min ( ), , and A ( ) ma ( ), , otherwise 0.4 A A Z 0.35 A 0.65, if = 2Z 0.8, othe rwise 0.4 B ( ) min B( ), , Z and B 0.35 ( ) ma B( ), if , = 3Z 0.65, if = 2Z A(0.4,0.35) ( ) Ma A ( ), A0.35 ( ) and B (0.4,0.35) ( ) Ma B ( ), B0.35 ( ) 0.7, otherwise 0.8, otherwise

6 66 P.K. Sharma It an be easily verified that A and B are (0.4, 0.35) - AFSG of Z 0.5 ; if 2Z Now, (A B)( ) min A( ), B( ) 0.6 ; if 3Z-2Z 0.7 ; if 2Z or 3Z A B ( ) min A B ( ), A B ( ) 0.6, Z 0.65 ; if 2Z or 3Z A B ( ) ma A B ( ), ; if 2Z or 3Z ; if 2Z or 3Z A B ( ) Ma (0.4,0.35) A B ( ), A B ( ) 0,35 Ma0.6, A B ( ) 0, ; if 2Z or 3Z Take = 9 and y = 4, we get A B ( y) A B (9 4) A B (5) 0.7 But A B A B (0.4,0.35) (0.4,0.35) (0.4,0.35) (9) 0.65 and (4) 0.65 (0.4,0.35) (0.4,0.35) Therefore, Ma A B (9), A B (4) Ma{0.65, 0.65} 0.65 Clearly, A B Hene A B (0.4,0.35) (0.4,0.35) Ma A B A B (9 4) > (9), (4) (0.4,0.35) (0.4,0.35) (0.4,0.35) is not (0.4, 0.35)- AFSG of Z. Remark(2.2) Union of two (, )-AFSG s of a group G need not be (, )- AFSG of G Eample(2.3) Let G = Z, the group of integers under ordinary addition of integers. Define the two fuzzy sets A and B as in Eample (2.) 0.6 ; if 2Z Now, (A B)( ) ma A( ), B( ) 0.7 ; if 3Z-2Z 0.8 ; if 2Z or 3Z A B ( ) min A B( ), A B () =0.6 Z 0.65 ; if 2Z A B ( ) ma A B ( ), ; if 3Z-2Z 0.8 ; if 2Z or 3Z 0.65 ; if 2Z 0.4 A B ( ) Ma A B ( ), A B ( ) (0.4,0.35) 0.35 Min0.6, A B ( ) ; if 3Z-2Z 0.8 ; if 2Z or 3Z Take = 9, y = 4, we get A B ( y) A B (9 4) A B (5) 0.8 But A B A B (0.4,0.35) (0.4,0.35) (0.4,0.35) (9) 0.7 and (4) 0.65 (0.4,0.35) (0.4,0.35)

7 (, ) Anti Fuzzy Subgroups 67 Therefore, Ma A B (9), A B (4) Ma{0.7, 0.65} 0.7 (0.4,0.35) (0.4,0.35) Clearly, A B (9 4) > Ma A B (9), A B (4) (0.4,0.35) (0.4,0.35) (0.4,0.35) Hene A B is not (0.4, 0.35)- AFSG of Z. 3. (, )- Anti Fuzzy osets and natural homomorphism Definition (3.) Let A be (, )- AFSG of a group G, where, [0,] suh that +. For any G, define a fuzzy set A(, ) of G, alled (, ) anti fuzzy right oset of A in G as follows (, ) (, ) A ( g) A ( g ), for all, g G. Similarly, we define the (, ) anti fuzzy left oset A of A in G as follows (, ) (, ) A ( g) A ( g), for all, g G. Definition (3.2) Let A be (, ) AFSG of a group G, where, [0,] suh that +. Then A is alled (, )- anti fuzzy normal subgroup ((, A A, for all G. )-AFNSG) of G if and only if (, ) (, ) Remark (3.3) If A is a FNSG of a group G, then A is also a (, )- AFNSG of G. Proof. Let A be a FNSG of G. Therefore A is -FNSG of G as well as A is -AFNSG of G [By Theorem (. 3)]. Let, g G be any element. Then ( ) ( ) ( ), ( ) A g A g Ma A g A g = A(, ) ( g) A(, ) A(, ) = Ma A ( g ), A ( g ) A ( g ) Thus, G Hene A is (, )- AFNSG of G. (, ) But, the onverse of the above result need not be true Eample (3.4) Let G = D 3 = < a, b : a 3 = b 2 = e, ba = a 2 b > be the dihedral group with si elements. Define the AFSG A of D 3 by 0.8 if < b > A() = 0.7 if otherwise

8 68 P.K. Sharma 2 2 Note that A is not a FNSG of G, for A( a ( ab)) = = A( ab( a )). Now take = 0.6, = 0., then we get 0.6 A ( ) min{ A( ), 0.6} 0.6 and A ( ) ma{ A( ), 0.9} 0.9, G Therefore, A ( ) Ma A ( ), A ( ) Ma 0.4, , G (0.6, 0.) 0. Thus, we get A ( y) A ( y) 0.9,, y G (0.6, 0.) (0.6, 0.) Hene A is (0.6, 0.)- AFNSG of G. Proposition(3.5). Let A be a (, ) - AFNSG of a group G. Then A ( y y) A ( ) or equivalently, A ( y) A ( y), holds for all, y G (, ) (, ) (, ) (, ) Proof. Sine A be (, )- AFNSG of a group G. A(, ) A(, ) for all G A(, ) (y - ) = A ( y ) A ( y ) (, ) (, ) (, ) (, ) A (y - ) hold for y - G A (( y) ) A (( y) ) (, ) (, ) hold A ( y) A ( y) [ as A is (, )- AFSG of G so A ( g ) A ( g), for all g G] Net, we show that for some speifi values of,, every (, ) - AFSG A of G will always be (, ) -AFNSG of G. In this diretion, we have the following: Proposition (3.6) Let A be a (, )-AFSG of a group G suh that p and - q, where p = Inf{ A() : for all G} and q = Sup{ A() : for all G}. Then A is (, )-AFNSG of G Proof. From Proposition (2.9), we have A(, )( ) Ma A ( ), A ( ) Min{, - }, G A ( y) Ma{, - } A ( y),, y G (, ) (, ) Hene A is (, ) AFNSG of G. Proposition (3.7) Let A be a (, )-anti fuzzy normal subgroup of G, then the set G G : A ( ) A ( e) is a normal subgroup of G. A,,, Proof. Cleary, Then G, for e G. Let, y G A, A, A, be any element.

9 (, ) Anti Fuzzy Subgroups 69 A ( y ) Ma A ( ), A ( y) Ma A ( e), A ( e) A ( e) (, ) (, ) (, ) (, ) (, ) (, ) i. e. A ( y ) A ( e), but A ( e) A ( y ) A ( y ) A ( e) (, ) (, ) (, ) (, ) (, ) (, ) So y G G, A. Thus, A, A, A, is a subgroup of G. Further, let G and y G be any element Now, A ( y y) A ( ) A ( e) y y G Hene G (, ) (, ) (, ) is a normal subgroup of G. A, Proposition(3.8) Let A be a (, ) -AFNSG of a group G, then A A if and only if y y A(, ) A(, ) y GA, G A if and only if y -, Proof. (i) Firstly, let A A y A ( y ) A ( y ) A ( y ) A ( y y ) A ( e ). So, y G A, y Conversely, let yg A ( y) A ( e). Let zg be any element of G, then A, (, ) (, ) A ( z) A ( z) A {( y)( y z)} (, ) (, ) (, ) Ma A ( y), A ( y z) (, ) (, ) = Ma A(, )( e), A(, ) ( y z) = A(, ) ( y z) (, ) = A ( z) y Interhanging the role of, y, we get A ( z) A ( z), z G A A (ii) This (, ) (, ) (, ) (, ) y y follows similarly as part (i) Proposition (3.9) Let A be a (, )-AFNSG of a group G and, y, u, v be any element in G. If A A and A A, then A A (, ) (, ) (, ) (, ) (, ) (, ) u y v y uv Proof. A A and A A, then A A (, ) (, ) (, ) (, ) (, ) (, ) u y v y uv Sine G A Now, (y) - (uv) = y - ( - u)v = y - ( - u) (yy - )v =[ y - ( - u) y] (y - v),

10 70 P.K. Sharma [ As y G A, A(, ) A(, ) uv is a normal subgroup of G ]. So (y) - (uv) G A, Proposition(3.0) Let G/A(, ) denote the olletion of all (, ) anti fuzzy osets of a (, )-AFNSG A of G. i.e. G/A(, ) = { A : G }. Then the binary operations defined on the set G/A(, ) as follows: A A A, for all, y G y y is a well defined operation Proof. A A A A (, ) Let = and, for some, y,, y G y y A A ( g) A ( g) y y Let gg be any element, then (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Now A g A y g A y g A g A g ( ) (( ) ) ( ) ( ) ( ) y y y = A ( y g) A ( ( gy )) A ( gy ) A ( gy ) = A(, ) (( g) y )) A(, ) ( y ( g)) A(, ) (( y ) g)) A ( g) y (, ) (, ) Thus A ( g ) A ( g ), g G y y Hene the operation is well defined on G/. A(, ) Proposition(3.) The set G/A(, ) of all - anti fuzzy osets of (, )- AFNSG A of a group G, form a group under the well-defined operations. Proof. It is easy to hek that the identity element of A, G/A is where e is the identity element of the group G, and the inverse of an element A(, ) is A (, ). Definition (3.2) The group G/A(, ) of all (, )- anti fuzzy oset of the (, )-AFNSG A of G is alled the fator group or the quotient group of G by A(, ). Theorem(3.3) A natural mapping f : G G/A(, ), where G is a group and G/A(, ) is the set of all (, )-anti fuzzy osets of the (, )-AFNSG A of G defined by f () = A(, ), is an onto homomorphism with ker f = G A, Proof. Let, y G be any element, then e

11 (, ) Anti Fuzzy Subgroups 7 f (y) = A A A f ( ) f ( y). Therefore f is a y y homomorphism. Moreover, f is surjetive ( obvious ) Now, Ker f = { G : f ( ) A } = { G : A A } (, ) (, ) (, ) e e = { G : e GA } { G : G }, A G, A, Using above Theorem (3.3), we an easily verify the following theorem Theorem (3.4) The group G/A(, ) of (, )- anti fuzzy osets of the (, )- AFNSG A of G is isomorphi to the quotient group G G of G. The isomorphi orrespondene is given by A, A G. / A, 4. Homomorphism of (, )-anti fuzzy groups In this setion, we investigate the behaviour of (, )-AFSG and (, )- AFNSG of a group G under the group homomorphism. Lemma (4.) Let f : X Y be a mapping and A and B be two fuzzy subsets of X and Y rrespetively, then ( i) f ( B ) f ( B) (, ) ( ii) f ( A ) f ( A) (, ) (, ) (, ) Pr oof. (i) f ( B(, ))( ) B(, ) f ( ) = Ma B f ( ), B f ( ) Hene f ( B ) f ( B) (, ) = Ma f ( B ) ( ), f B ( ) = Ma f ( B) ( ), f B ( ) = f ( B) ( ) (, ) (, )

12 72 P.K. Sharma ( ii) f ( A )( y) Sup A ( ) : f ( ) y (, ) (, ) = Sup Ma A ( ), A ( ) : f ( ) y = Ma Sup A ( ) : f ( ) y, A ( ) : f ( ) y = Ma f ( A ) ( y), f ( A )( y) f A (, ) = Ma f ( A) ( y), f ( A) ( y) = ( ) ( y) Theorem (4.2) Let f : G G 2 be a homomorphism of group G into a group G 2. Let B be (, )-AFSG of group G 2. Then f - (B) is (, )-AFSG of group G. Proof. Let B be an (, )-AFSG of group G 2. Let, 2 G be any element. Then f ( B) ( 2 ) f ( B(, ))( 2 ) B(, )(( f ( 2 )) B(, )( f ( ) f ( 2)) (, ) ma{ B ( f ( )), B ( f ( )} (, ) (, ) 2 = ma{ f ( B(, ))( f B f B ), f ( B )( )} (, ) 2 = ma ( ) ( ), ( ) ( ) [ using Lemma (4.)] (, ) (, ) 2 f B f B f B) Thus ( ) ( ) ma ( ) ( ), ( ( ) f (, ) 2 (, ) (, ) 2 Also f B B f B f B f f B, f ( B) ( ) ( (, ))( ) (, )(( ( )) (, )(( ( ) ) (, )(( ( )) ( (, ))( ) Thus ( B) ( ) f ( B (, ))( ) Hene f ( B) is (, )-AFSG of G. Theorem (4.3) Let f : G G 2 be a homomorphism of a group G into a group G 2. Let B be an (, )-AFNSG of group G 2, then f - (B) is an (, )- AFNSG of group G. Proof. Let B be an (, )-AFNSG of group G 2. Let, 2 G be any element. Then f ( B) ( ) f ( B )( ) B (( f ( )) B ( f ( ) f ( )) B ( f ( ) f ( )) (, ) 2 (, ) 2 (, ) 2 (, ) 2 (, ) 2 = B ( f ( )) f ( B) ( ) (, ) 2 (, ) 2

13 (, ) Anti Fuzzy Subgroups 73 f B f B Thus ( ) ( ) ( ) ( ) Hene f (, ) 2 (, ) 2 ( B) is an (, )-AFNSG of G. Theorem (4.4) Let f : G G 2 be a bijetive homomorphism of group G onto a group G 2. Let A be an (, )-AFSG of group G. Then f (A) is an (, )-AFSG of group G 2. Proof. Let A be an (, )-AFSG of group G. Let y, y 2 G 2 be any element. Then there eists unique element, 2 G suh that f ( ) = y and f ( 2 ) = y 2. f ( A) ( y y ) = f ( A) ( y y ) Sup A ( ) : f ( ) y y (, ) 2 (, ) 2 (, ) ( ), ( ), where ( ) and ( ) = MaSupA(, )( ) : f ( ) y, SupA(, )( 2 ) : f ( 2) y2 = Ma f ( A(, )) ( y ), f ( A(, ) ) ( y2) Sup Ma A A f y f y = Ma f ( A) ( y ), f ( A) ( y ) [ using Lemma (4.)] (, ) (, ) 2 Thus f ( A ) ( y y ) Ma f ( A ) ( y ), f ( A ) ( y ) (, ) 2 (, ) (, ) 2 f A y f A (, ) y SupA(, ) f y = SupA(, ) ( ) : f ( ) y f ( A(, )) ( y) Also ( ) ( ) ( )( ) ( ) : ( ) Hene f ( A) is an (, )- AFSG of G Theorem(4.5) Let f: G G 2 be bijetive homomorphism and A be an (, )-AFNSG of group G. Then f (A) is an (, )-AFNSG of group G 2. 2 Proof. Sine A is an (, )-AFNSG of group G. Let y, y 2 G 2 be any element. Then, there eists unique element, 2 G suh that f ( ) = y and f ( 2 ) = y 2. f ( A) ( y y ) = f ( A) ( y y ) Sup A ( ) : f ( ) y y (, ) 2 (, ) 2 (, ) = SupA(, )( 2) : f ( 2 ) y2 y = f ( A) (, ) ( y2 y ) Thus f ( A) ( y y ) = f ( A) ( y y ) Hene (, ) 2 2 f ( A) is an (, )- AFNSG of G 2

14 74 P.K. Sharma Referenes [] R. Biswas, Fuzzy Subgroups and Anti fuzzy Subgroups, Fuzzy Sets and Systems, Vol. 35, pp. 2-24, 990 [2] R. Kumar, Fuzzy Algebra, University of Delhi Publiation Division, 993 [3] D.S. Malik and J.N Mordeson, Fuzzy ommutative algebra, World Sientifi Publishing Pvt. Ltd. 998 [4] A. Rosenfeld, Fuzzy Groups, Journal of mathematial analysis and appliation, 35(97), [5] P.K. Sharma, - Fuzzy Subgroups, ( Communiated ) [6] P.K. Sharma, -Anti Fuzzy Subgroups, International Review of Fuzzy Mathematis (July-De 202) [7] P.K. Sharma, (, ) - Fuzzy Subgroups, Fuzzy Sets Rough Sets and Multivalued Operations and Appliations (July-De 202) [8] J. Tang, Y. Yao, Correspondene Theorem for Anti L-fuzzy Normal Subgroups, International Journal of Computational and Mathematial Sienes 6, 202, pp