- Fuzzy Subgroups. P.K. Sharma. Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 3, Number 1 (2013), pp Research India Publications - Fuzzy Subgroups P.K. Sharma Department of Mathematics, D.A.V. College, Jalandhar City, Punjab, India pksharma@davjalandhar.com Abstract On the basis of fuzzy sets introduced by L.A. Zadeh, we first gave the definition of - fuzzy set and then defined - fuzzy subgroups and - fuzzy normal subgroups and finally, defined quotient group of the - fuzzy cosets of an - fuzzy normal subgroup. This paper proves a necessary and sufficient condition of -fuzzy subgroup (normal subgroup) to be fuzzy subgroup (normal subgroup). Some properties of quotient group of -fuzzy normal subgroups are also discussed. Mathematics Subject Classification 03E72, 08A72, 20N25 Keywords Fuzzy subgroup (FS), Fuzzy Normal subgroup (FNS), - Fuzzy subgroup (-FS), - Fuzzy normal subgroup (-FNS) 1. Introduction The concept of fuzzy sets was introduced by Zadeh [18]. Since its inception, the theory of fuzzy sets has developed in many directions and is finding applications in a wide variety of fields. In [17] Rosenfeld used this concept to develop the theory of fuzzy groups. In fact, many basic properties in group theory are found to be carried over to fuzzy groups. Das [ 7 ] introduced the idea of level subset, which gave a new dimension in the fuzzy set theory as a result Mukherjee and Bhattacharya [ 5 ] showed that almost all the global notions of fuzzy subgroups can be characterized through its level subgroups. Anthony and Sherwood [ 2 ] redefined fuzzy subgroups in terms of a t-norm which replaced the minimum operation and they characterized basic properties of t-fuzzy subgroups in [2, 3]. Chakrabatty and Khare [ 6 ] introduced the notion of fuzzy homomorphism between two groups and studied its effect to the fuzzy subgroups. In [ 1 ] Ajmal defined the notion of containment of an

2 48 P.K. Sharma ordinary kernel of a group homomorphism in fuzzy subgroups and gave the idea of quotient group in a natural way. Consequently, the fundamental theorem of homomorphism was established in fuzzy subgroups. Many new notions in fuzzy subgroups has been defined, for eample Solairaju and Nagarajan in [15] defined the notion of Q-fuzzy subgroups and Massa deh [12] studied the concept of fuzzy subgroups with operators and defined normal fuzzy subgroups with operator. In this paper, we introduce the notion of -fuzzy set of a set with regard to fuzzy set and then define the concept of -fuzzy subgroup, -fuzzy normal subgroup and -fuzzy quotient group. Many related results have been derived. Preliminaries Definition(2.1) A fuzzy set A of a set X is a function A : X [0,1]. Fuzzy sets taking the values 0 and 1 are called Crisp sets. Let A and B be two fuzzy subsets of a set X. Then the following epressions are defined in [ 9 ], [ 10 ] and [ 18 ] A B if and only if A() B(), for all X A = B if and only if A B and B A, The complement of the fuzzy set A is A c and is defined as A c () = 1 A() (A B) () = min{a(), B()}, X (A B)() = ma{a(), B()}, X Definition(2.2)[ 9 ] A function A: [0,1] is a called fuzzy subgroup ( in short FS ) of if A(y) min {A(), A(y)} A( -1 ) A(),, y It is easy to show that a fuzzy subgroup of a group satisfies A() A(e) and A( -1 ) = A(),for all, where e is the identity element of. Proposition(2.3)[ 9 ] A function A : [0,1] is a FS of a group if and only if A(y -1 ) min {A( ), A(y )},, y Proposition(2.4)[ 9 ] If A : [0,1] is a FS of a group, then A() A(e),, where e is the identity element of A(y -1 ) = A(e) A() = A(y),, y Proposition(2.5)[ 9 ] Let A be a fuzzy subgroup of a group, then it is called fuzzy normal subgroup ( FNS ) of if A(y) = A(y),, y Definition(2.6)[ 9 ] Let A : [0,1] is a FNS of group. For any

3 - Fuzzy Subgroups 49, the fuzzy set A : [0,1] defined by (A)(y) = A( -1 y), y is called a left fuzzy coset of A. Similarly, the fuzzy set A : [0,1] defined by (A)(y) = A(y -1 ), y is called a right fuzzy coset of A. Definition (2.7)[ 7 ] Let A be a fuzzy set of a group. For t [0,1], the upper level subset of A is the set U(A, t) = { : A() t }. Clearly, U(A, 0) = and if t 1 > t 2, then U(A, t 1 ) U(A, t 2 ) Proposition (2.8)[ 7 ] Let be a group and A be a fuzzy subset of such that U(A, t) is a subgroup of, t [0,1] with t A(e ). Then A is a FS of. Definition (2.9)[ 7 ] Let A be a FS of a group. The subgroups U(A, t), t [0,1] with t A(e ) are called upper level subgroups of A. Definition (2.10)[ 1, 6 ] Let f : 1 2 be a homomorphism of group 1 into a group 2. Let A and B be fuzzy subsets of 1 and 2 respectively, then f (A) and f -1 (B) are respectively the image of fuzzy set A and the inverse image of fuzzy set B, defined as 1 1 Sup{ A( ) : f ( y)} ; if f ( y) f ( A)( y), for every y ; if f ( y) 1 and f B B f ( )( ) ( ( )), for every 1 Remark(2.11) (i) Clearly, f (A)(f()) A(), for every element 1 (ii) When f is a bijective map, then f (A)(f()) = A(), Fuzzy subsets and their properties Definition(3.1) Let A be a fuzzy subset of a group. Let [0,1]. Then the fuzzy set A of is called the - fuzzy subset of (w.r.t. fuzzy set A) and is defined as A () = min{ A( ), }, for all Remark (3.2) Clearly, 1 A = A and 0 A 0 Some Results (3.3)(i) Let A and B be two fuzzy subsets of X. Then (A B) = A B (ii) Let f : X Y be a mapping and A and B be two fuzzy subsets of X and Y respectively, then f -1 ( B ) = ( f -1 ( B )) (b) f (A ) = (f (A) ) Proof. (i) Now, (A B) () = min { (A B)(), }

4 50 P.K. Sharma = min{ min{a(), B()}, } = min { min{a(), }, min{b(), }} = min { A (), B () } = A B (), for all X Hence (A B) = A B (ii) (a) f -1 ( B )() = B (f ()) = min { B(f ()), } = min { f -1 ( B )(), } = ( f -1 ( B )) (), for all X Hence f -1 ( B ) = ( f -1 ( B )) (ii)(b) f (A )(y) = Sup {( A )() : f () = y } = Sup { min{ A(), } : f () = y } = min { Sup {A() : f () = y }, } = min { f (A)(y), } = (f (A) ) (y), for all y Y Hence f (A ) = (f (A) ) - Fuzzy Subgroups In this section, we introduce the notion of -fuzzy subgroups and - fuzzy normal subgroups. Here we prove that every fuzzy subgroup (normal subgroup) is also -fuzzy subgroup (normal subgroup), but converse need not be true. We also obtain the conditions when the converse is also true. The notion of - fuzzy coset has also been defined and discussed deeply and the notion of quotient group with regard to an -fuzzy normal subgroup which results into a natural homomorphism from into quotient group has been obtained. Finally, a one-one correspondence between the quotient group of the -fuzzy normal subgroup and the quotient group of with regard to the normal subgroup in is obtained. The homomorphic image and pre-image of -fuzzy subgroup (normal subgroup) are also obtained. Theorem(4.1) Let A be a fuzzy subset of a group. Let [0,1]. Then A is called -fuzzy subgroup ( in short -FS) of if A is FS of i.e. if the following conditions hold A (y) min { A (), A (y) } A ( -1 ) = A (), for all, y. Proposition(4.2) If A : [0,1] is a -FS of a group, then

5 - Fuzzy Subgroups 51 A () A (e),, where e is the identity element of A (y -1 ) = A (e) A () = A (y),, y Proof. (i) A (e) = A ( -1 ) min{a (), A ( -1 ) }= min{a (), A ()}= A () (ii) A () = A (y -1 y) min{a (y -1 ), A (y)}= min{a (e), A (y)}= A (y) = A (y -1 ) min{a (y -1 ), A ()} min{a (y -1 ), A ()}= A () Thus A () = A (y),, y Proposition(4.3) If A be a FS of the group, then A is also -FS of Proof. Let, y be any elements of the group A (y) = min { A(y), } min{ min { A(), A(y) }, } = min { min { A(), }, min { A(y), }} = min { A (), A (y) } Thus A (y) min { A (), A (y) } Also, A ( -1 ) = min { A( -1 ), } = min { A(), } = A () Hence A is -FS of. Remark(4.4) The converse of above proposition need not be true Eample (4.5) Let = { e, a, b, ab }, where a 2 = b 2 = e and ab = ba be the Klein four group. Let the fuzzy set A of be defined as A = {< e, 0.1 >, < a, 0.3 >, < b, 0.3 >, < ab, 0.2 > }. Clearly, A is not a FS of. Take = Then A() > for all. So that A ( ) min{ A( ), }, for all Therefore, A ( y) min{ A( ), A( y)} hold Further, as a a, b b, ( ab) ( ab). So A ( ) A ( ) hold Hence A is a - FS of. Proposition(4.6) Let A be a fuzzy subset of a group such that A( -1 ) = A() hold for all. Let p, where p = Inf { A() : }. Then A is -FS of Proof. Since p p i.e. Inf { A() : } A() and so min{ A(), }=, for all i.e. A () =, for all Thus, A (y) min { A (), A (y) } hold for all, y

6 52 P.K. Sharma Further, A( -1 ) = A() hold for all (given) implies that A ( -1 ) = A () Hence A is -FS of. Proposition(4.7) Intersection of two -FS s of a group is also -FS of Proof. Let A and B be two -FS s of a group. Let, y be any element, then (A B) (y) = (A B )(y) [ By using Result (3.3)(i)] = min { A (y), B (y) } min { min {A (), A (y) }, min {B (), B (y) }} = min { min {A (), B ()}, min {A (y), B (y) }} = min { (A B) (), (A B) (y) } Thus (A B) (y) min { (A B) (), (A B) (y) } Also, (A B) ( -1 ) = (A = min { A ( -1 ), B ( -1 ) } = min { A (), B () } = (A B) () Hence (A B) is -FS of. B )( -1 ) [ By using Result (3.3)(i)] Corollary (4.8) Intersection of a family of - FS s of a group is again a - FS of. Remark(4.9) Union of two - FS s of a group need not be - FS of. Eample(4.10) Let = Z, the group of integers under ordinary addition of integers. Define the two fuzzy sets A and B by 0.3, if = 3Z 0.15, if = 2Z A( ) and B( ) 0, otherwise 0.05, otherwise It can be easily verified that A and B are 1-FS of Z. Now, (A B )( ) ma{ A( ), B( )} 0.3 ; if 3Z Therefore, (A B)( ) 0.15 ; if 2Z-3Z 0.05 ; if 2Z or 3Z

7 - Fuzzy Subgroups 53 Take = 9 and y = 4 then (AB)( ) 0.3, (A B)( y) 0.15 Now, (A B)( y) (A B)(9 4) (A B)(5) 0.05 and min{(a B)( ), (A B)( y)} min{0.3, 0.15} 0.15 Clearly, (A B)( y) min{(a B)( ), (A B)( y)} Thus A B is not 1- FS of. Hence, we see that, the union of two - FS s of need not be a - FS of. Eample(4.11) Let = Z, the group of integers under ordinary addition of integers. Define the two fuzzy sets A and B by 0.15, if = 2Z 1, if = 2Z A( ) and B( ) 0.05, otherwise 0, otherwise 1 ; if 2Z ( A B)( ) 0.05 ; otherwise It is easy to verified that A, B and A B are 1-FS s of. Definition (4.12) Let A be - FS of a group, where [0,1]. For any, define a fuzzy set A of, called fuzzy right coset of A in as follows 1 ( ) min{ ( ), } A g A g, for all, g. Similarly, we define the fuzzy left coset A of A in as follows 1 A ( g) min{ A( g), }, for all, g. Definition (4.13) Let A be -FS of a group, where [0,1]. Then A is called fuzzy normal subgroup (- FNS) of if and only if A = A, for all. Note (i) Clearly, 1- FNS is ordinary FNS of 1 1 (ii) A ( g) A ( g ) and A ( g) A ( g), for all g Remark (4.14) If A is a FNS of a group, then A is also a - FNS of. Proof. Let A be a FNS of. Then for any, we have A = A Therefore, for any g, we have (A)(g) = (A)(g) i.e. A( -1 g) = A(g -1 ) So min { A( -1 g), } = min { A(g -1 ), } i.e. A (g) = A (g). So we have A = A, for all Hence A is a - FNS of

8 54 P.K. Sharma The converse of the above result need not be true Eample (4.15) Let = D 3 = < a, b : a 3 = b 2 = e, ba = a 2 b > be the dihedral group with si elements. Define the FS A of D 3 by 0.8 if < b > A() = 0.7 if otherwise 2 2 Note that A is not a FNS of, for A( a ( ab)) = = A( ab( a )). Now take = 0.6, we get A 1 1 (g) = min{ A( g), } min{ A( g ), } = A (g),, g. Hence A is a - FNS of. Proposition(4.16). Let A be a - FNS of a group. Then 1 A ( y y) A ( ) or equivalently, A ( y) A ( y), holds for all, y Proof. Since A be - FNS of a group. A = A (y -1 ) = A (y -1 ) hold for y min{ A( y ), } = min{ A( y ), } A ( y ) A ( y ) A y A y A 1 1 (( ) ) (( ) ) A ( y) A ( y) [ as A is - FS of so A g hold for all 1 ( ) A ( g), for all g ] Net, we show that for some specific values of, every -FS A of will always be -FNS of. In this direction, we have the following: Proposition(4.17). Let A be an -FS of a group such that p, where p = Inf{ A() : for all }. Then A is also a -FNS of. Proof.. Since p implies that p Inf{ A() : for all } A() for all and so min{a(), } = 1 Thus A ( g) min{ A( g ), }. Similarly g A g g g 1, A ( ) min{ ( ), } i.e. A ( ) A ( ), for all g Therefore A A, for all. Hence A is a -FNS of. Proposition(4.18) Let A be a -FNS of a group, then the set { : A ( ) A ( e)} is a normal subgroup of. A Proof. Clearly, Let, y A A, for e. A be any element. Then we have A (y -1 ) min {A (), A (y)} = min { A (e), A (e)} = A (e)

9 - Fuzzy Subgroups 55 A (y -1 ) A (e), but A (y -1 ) A (e) always Therefore A (y -1 ) = A (e) y -1 Thus A A is a subgroup of. Further, let and y, we have A (y -1 y) = A () = A (e) y -1 y A. So A is a normal subgroup of. A Proposition(4.19) Let A be a -FNS of a group, then A = ya if and only if -1 y A A = A y if and only if y -1 A Proof. (i) Firstly, let A = ya A ( -1 y) = min {A( -1 y), } = (A )(y) = (ya )(y) = min {A(y -1 y), }= min{ A(e), }= A (e) Thus A ( -1 y) = A (e) -1 y Conversely, let -1 y A A ( -1 y) = A (e). Let z be any A element Now, (A )(z) = min { A( -1 z), } = A ( -1 z) = A ( ( -1 y)(y -1 z)) min {A ( -1 y), A (y -1 z)} = min { A (e), A (y -1 z)} = A (y - 1 z) = (ya )(z) Interchanging the role of and y, we get (A )(z) = (ya )(z), for all z Hence A = ya (ii) This follows similarly as part (i) Proposition (4.20) Let A be a -FNS of a group and, y, u, v be any element in. If A = ua and ya = va, then ya = uva Proof. Since A = ua and ya = va -1 u, y -1 v A Now, (y) -1 (uv) = y -1 ( -1 u)v = y -1 ( -1 u) (yy -1 )v =[ y -1 ( -1 u) y] (y -1 v) uva [ As A is a normal subgroup of ]. So (y) -1 (uv) A ya = Proposition(4.21) Let / A denote the collection of all fuzzy cosets of a -FNS A of. i.e. / A = { A : }. Then the binary operations defined on the set / A as follows: A A t A, for all, y y y A

10 56 P.K. Sharma is a well defined operation Proof. Let A = A and A y A y, for some, y,, y Let g be any element, then [ A A ]( g ) ( A )( g ) y y Now A g A g y A gy A gy ( y )( ) min{ ( ( ) ), } min{ (( ) ), } ( ) A gy A gy A g y ( ) min{ (( ) ), } min{ (( ) ), } A g A g A 1 1 y ( ) y ( ) min{ (( g) y ), } min{ A( y ( g)), } min{ A(( y ) g), } 1 1 = min{ A(( y ) g), } min{ A( g( y ) ), } = A ( g) y Therefore is well defined operation on / A. Proposition(4.22) The set / A of all - fuzzy cosets of -FNS A of a group, form a group under the well-defined operations. Proof. It is easy to check that the identity element of / A is is the identity element of the group, and the inverse of an element A. 1 A e, where e A is Definition (4.23) The group / A of - fuzzy coset of the -FNS A of is called the factor group or the quotient group of by A. Theorem(4.24) A natural mapping f : / A, where is a group and / A is the set of all -fuzzy cosets of the -FNS A of defined by f () = A, is an onto homomorphism with ker f = Proof. Let, y be any element, then f (y) = A A A f ( ) f ( y). Therefore f is a homomorphism. y y A Moreover, f is surjective ( obvious ) Now, Ker f = { : f ( ) A } = { : A A } e e 1 = { : e } { : } A A A Using above Theorem (4.24), we can easily verify the following theorem Theorem (4.25) The group / A of - fuzzy cosets of the -FNS A of

11 - Fuzzy Subgroups 57 is isomorphic to the quotient group / A of. The isomorphic correspondence is given by A ( ). A 5. Homomorphism of - fuzzy groups Theorem (5.1) Let f : 1 2 be a homomorphism of group 1 into a group 2. Let B be -FS of group 2. Then f -1 (B) is -FS of group 1. Proof. Let B be a -FS of group 2. Let 1, 2 1 be any element. Then 1 1 f ( B) ( ) f ( B )( ) B (( f ( )) B ( f ( ) f ( )) min{ B ( f ( )), B ( f ( )} = min { f ( B )( 1 ), f ( B )( 2)} 1 1 f B = min f ( B) ( ), ( ) ( ) [ using Result (3.3)(ii)(a)] f Thus f ( B) ( ) min f ( B) ( ), ( B) ( ) f , ( ) ( ) ( )( ) (( ( 1 f B )) (( ( ) 1 ) (( ( )) f 1 ( B )( ) Also f B B f B f B f Thus ( B) ( ) f ( B )( ) 1 Hence f B ( ) is -FS of. 1 Theorem (5.2) Let f : 1 2 be a homomorphism of a group 1 into a group 2. Let B be a -FNS of group 2, then f -1 (B) is a -FNS of group 1. Proof. Let B be a -FNS of group 2. Let 1, 2 1 be any element. Then 1 1 f ( B) ( ) f ( B )( ) B (( f ( )) B ( f ( ) f ( )) B ( f ( ) f ( )) f 2 1 = B ( f ( )) ( B) ( ) 1 1 f B f B Thus ( ) ( ) ( ) ( ) Hence f ( B) is -FNS of 1. Theorem (5.3) Let f : 1 2 be a bijective homomorphism of group 1 onto a group 2. Let A be a -FS of group 1. Then f (A) is a -FS of group 2.

12 58 P.K. Sharma Proof. Let A be a -FS of group 1. Let y 1, y 2 2 be any element. Then there eists unique element 1, 2 1 such that f ( 1 ) = y 1 and f ( 2 ) = y 2. ( f ( A)) ( y y ) = min{ f ( A)( y y ), } = min { f ( A)( f ( ) f ( )), }} = min{ f ( A)( f ( )), }} 1 2 = min{ A( ), }= A ( ) = min{ A ( ), A ( )}, for all, such that f ( ) y and f ( ) y min{ {A ( ) : f ( ) y }, {A ( ) : f ( ) y }} = min{ f ( A )( y ), f ( A )( y )} = min{( f ( A)) ( y ), ( f ( A)) ( y )} [ usin g Result (3.3)(ii)(b)] 1 2 Thus ( f ( A)) ( y y ) min{( f ( A)) ( y ), ( f ( A)) ( y )} , ( ( )) ( ) ( )( ) { ( ) : ( ) } Further f A y f A y A f y = { A ( ) : f ( ) y} = f ( A )( y) Hence f ( A) is - FS of. 2 Theorem(5.4): Let f: 1 2 be bijective homomorphism and A be a - FNS of group 1. Then f (A) is a -FNS of group 2. Proof. Since A is a -FNS of group 1. Let y 1, y 2 2 be any element. Then, there eists unique element 1, 2 1 such that f ( 1 ) = y 1 and f ( 2 ) = y 2. ( f ( A)) ( y y ) = min{ f ( A)( y y ), } = min { f ( A)( f ( ) f ( )), }} = min{ f ( A)( f ( )), }} 1 2 = min{ A( ), }= A ( ) A ( ) min{ A( ), } = min{ f ( A)( f ( )), }= min { f ( A)( f ( ) f ( )), }} = min{ f ( A)( y y ), }= ( f ( A)) ( y y ) Thus ( f ( A)) ( y y ) ( f ( A)) ( y y ) Hence f ( A) is - FNS of Conclusion In this paper, we have introduced the concept of -fuzzy subgroup and - fuzzy cosets in a group and used it to introduce the concept of -fuzzy normal subgroup and discussed various related properties. We have also studied effect on the image and inverse image of -fuzzy subgroup (normal subgroup) under group homomorphism. In the net studies, we will formulate the concept of (, )-fuzzy subgroups and applied it to study some properties such as (, )-fuzzy cosets and consequently introduce the notion of (, )-fuzzy normal subgroups.

13 - Fuzzy Subgroups 59 References [1] N. Ajmal, Homomorphism of groups, correspondence theorem and fuzzy quotient groups, Fuzzy Sets and Systems, 61(1994), [2] J.M. Anthony and H. Sherwood, Fuzzy group redefined, J. Math. Anal. Appl.69(1979), [3] J.M. Anthony and H. Sherwood, A Characterization of fuzzy subgroups, Fuzzy Sets and Systems, 7 (1987), [4] S.K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems, 51(1992), [5] P. Bhattacharya and N.P Mukherjee, Fuzzy groups: Some group theoretical and analogues, Inform Sci. 39 (1986), [6] A.B. Chakrabatty and S.S. Khare, Fuzzy homomorphism and algebraic structures, Fuzzy Sets and Systems 51 (1993), [7] P. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84(1981), [8] Y. L. Liu, Quotient groups induced by fuzzy subgroups, Quasigroups and Related Systems 11 (2004), [9] R. Kumar, Fuzzy Algebra, University of Delhi Publication Division, 1993 [10] D.S. Malik, J.N Mordeson and P.S. Nair, Fuzzy Normal Subgroups in Fuzzy groups, J. Korean Math. Soc. 29 (1992), No. 1, pp. 1 8 [11] D.S. Malik and J.N Mordeson, Fuzzy commutative algebra, World Scientific Publishing Pvt. Ltd [12] M.O. Massa deh, On fuzzy subgroups with operators, Asian Journal of Mathematics and Statistics 5(4), , (2012) [13] P.K. Sharma, -Anti Fuzzy Subgroups, International Review of Fuzzy Mathematics (July-Dec 2012) ( Accepted ) [14] P.K. Sharma, (, )- Fuzzy Subgroups, Fuzzy Sets Rough Sets and Multivalued Operations and Applications (July-Dec 2012) Accepted) [15] A. Solairaju and R. Nagarajan, A New structure and construction of Q- fuzzy groups, Advances in Fuzzy Mathematics, 4, No.1(2009), [16] A. Solairaju and R. Nagarajan, Some structure properties of Q- cyclic fuzzy group family, Antarctica Journal of mathematics,vol.7, No, 1, [17] A. Rosenfeld, Fuzzy roups, Journal of mathematical analysis and application, 35(1971), [18] L.A Zadeh, Fuzzy sets, Information and Control 8, (1965),

14 60 P.K. Sharma