On Intuitionistic Fuzzy R-Ideals of Semi ring

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1 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril On Intuitionistic Fuzzy R-Ideals of Sei ring Ragavan.C 1, Solairaju. 2 and Kaviyarasu.M 3 1, 3 sst Prof. Departent of Matheatics, Sri Vidya Mandir rts & Science College, Uthangarai, T.N. India 2 ssociate Professor of Matheatics Jaal Mohaed College, Trichy, T.N, India. 1 e-ail:ragavanshana@gail.co, 2 solairaa@yahoo.co.in, 3 anjjukavi@gail.co bstract In this paper, we introduce the notion of intuitionistic fuzzy ideal and intuitionistic fuzzy R-ideal in sei ring and investigate soe properties of intuitionistic fuzzy R-ideals of seiring. 1. Introduction: The concept of fuzzy set μ of a set X was introduced by L.. Zadch [9] as a function fro X in [0, 1]. The concept of fuzzy ideals in a ring was introduced by W. L. Liu [8]. T. K. Dutta and B. K. Biwa s [3, 4] studied fuzzy ideals, fuzzy prie ideals of sei rings and they defined fuzzy R-ideals and fuzzy prie R-ideals of sei rings. Y. B. Jun, J. Naggers and M. S. Ki [5] extended the concept of an L-fuzzy left (resp. right) ideals of a ring to a sei ring. The concept of the idea of intuitionistic fuzzy set was first published by K. T. tanassov [1, 2], as a generalization of the notion of fuzzy set. K.H. Ki and J. G. Lee [6] studied the intuitionistic Fuzzifications of the concept of several ideals in a sei groups and investigate soe properties of such ideals. K. H. Ki [7] introduced the notion of intuitionistic Q-fuzzy seipriality in a sei group and investigates soe properties of intuitionistic Q-Fuzzifications of the concept of several ideals. In this paper we introduce the notion of intuitionistic fuzzy R-ideal of sei ring and investigate soe properties of intuitionistic fuzzy R-ideal of sei rings. Throughout this paper R is a seiring. 2. Preliinaries: Let (R, +, ) be a searing. By a left (right) ideal of R we ean a non-epty subset of R such that + and R (R ). By ideal, we ean a non-epty subset of R which both left and right ideal of R. left ideal of R is said to be a left R-ideal if t, x R and if t + x or x + t then x. Right R-ideal is defined dually, and two sided R-ideal or siply a R-ideal is both a left and a right R-ideal. By a fuzzy set μ of a non-epty set R we ean a function μ: R [0, 1], and the copleent of μ, denoted by μ, is the fuzzy set in R given by μ (x)=1- μ (x), for all x R. fuzzy set μ in R is called fuzzy left (resp. right) ideal of R if for any x, y R, μ(x+y) in{μ(x),μ(y) and μ (xy) μ (y) ( μ (xy) μ (x)) and μ is called fuzzy ideal if μ both fuzzy left and right ideal of R. fuzzy ideal μ of R is called R-fuzzy ideal of R if for any x,y R, μ(x) in{ax{μ((x+z)+(z+y)),μ((z+x)+(y+z)), μ (y). intuitionistic fuzzy set (IFS for short) in a nonepty set R is an object have the for: = {(x: (x), (x)) / x R Where the function μ : R [0, 1] and : R [0, 1] denoted the degree of ebership and the degree of non-ebership, respectively, and 0 μ (x) + (x) 1 n intuitionistic fuzzy set = {(x : μ (x), (x)) / x R in R can be identified to ordered pair (μ, ) in I R x I R. We shall use the sybol = (, ) for the IFS: = {(x: μ (x), (x)) / x R 3. Intuitionistic fuzzy R-ideal: Definition 3.1: n IFS = (μ, ) in R is called an intuitionistic fuzzy left (resp. right) ideal of R 1- μ (x+y) in{μ (x), μ (y) and μ (xy) μ (y) (resp. μ (xy) μ (x)). 2- (x+y) ax{ (x), (y) and (xy) (y) (resp. (xy) (x)). For all x, y R: Definition 3.2: n intuitionistic fuzzy ideal = (μ, ) of R is called an intuitionistic fuzzy R- ideal of R if 1. μ (xy) μ (y), μ (xy) μ (x). 2 μ (x) in {ax {μ ((x+z) + (z+y)), μ ((z+x) +(y+z)), μ (y) and 3. (xy) (y), (xy) (x)). 4 (x) ax{in{ ((x+z)+(z+y)), ((z+x)+(y+z)), (y). x, y R. Theore 3.3: Let = (μ, ) an intuitionistic fuzzy set in R such that μ is fuzzy R-ideal of R then d = (μ, μ R) is an intuitionistic R-ideal of R. Proof: Let x, y R, since μ is a fuzzy R-ideal of R μ is a fuzzy ideal. So μ (x+y) in {μ (x), μ (y) and μ (xy) μ (y), μ (xy) μ (x), μ (x+y) =1-μ (x+y) 1-in {μ (x), μ (y) =ax {1-μ (x), 1-μ (y) = ax { μ (x), μ (y) μ (x + y) ax { μ (x), μ (y), μ (xy) = 1 μ (xy) 1 - μ (y) = μ (y), also μ (xy) μ (x) Therefore d = (μ, μ ) is an intuitionistic fuzzy ideal of R. Let μ (x) in {ax {μ ((x+z) + (z+y)), μ ((z+x) +(y+z)), μ (y) μ (x)=1 - μ (x) in {ax {μ ((x+z) + (z+y)), μ ((z+x) +(y+z)), μ (y),

2 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril μ (x) =ax {1- ax {μ ((x+z) + (z+y)), μ ((z+x) +(y+z)), μ (y), 1-μ (y) = ax {in {1- μ ((x+z)+(z+y)),1-μ ((z+x)+(y+z)),1- μ (y) =ax {in { μ ((x+z) + (z+y)), μ ((z+x)+(y+z)), μ (y) Ragavan.C 1, Solairaju. 2 and Kaviyarasu.M 3 so μ (x) ax {in { μ ((x+z) + (z+y)), μ ((z+x)+(y+z)), μ (y) d = (μ, μ ) is an intuitionistic R-ideal Theore 3.4: n IFS = (μ, ) is an intuitionistic fuzzy R-ideal of R if and only if the fuzzy sets μ and are fuzzy R-ideals of R. Proof: Suppose that an IFS = (μ, ) is an intuitionistic fuzzy R-ideal of R. Clearly μ is a fuzzy R-ideal. Let x, y R. Since = (μ, ) is an is an intuitionistic fuzzy R-ideal. (x + y) ax { (x), (y) and (xy) (y), (xy) (x) (x) ax {in { ((x+z) + (z+y)), ((z+x) +(y+z)), (y) R (x + y) = 1 - (x + y) 1 ax { (x), (y) = in { R (x), R (y) So R (x + y) in { R (x), R (y) R (xy) = 1- (xy) 1 - (y) = R (y), R (xy) (x) 1 ax R (y). lso we can get that R (xy) R (x). R (x) = 1 (x) 1 ax {in { ((x+z) + (z+y)), ((z+x) +(y+z)), (y) = ax {1 in { ((x+z) + (z+y)), ((z+x) +(y+z)), 1 - (y) = in {ax {1- ((x+z) + (z+y)), 1- ((z+x) +(y+z)), R (y) = in {ax { R ((x+z) + (z+y)), R ((z+x) +(y+z)), R (y) So R (x) in {ax { R ((x+z) + (z+y)), R ((z+x) +(y+z)), R (y) Therefore R fuzzy R-ideals of R. suppose that μ and are fuzzy R-ideals of R Let x, y R. Since μ is a fuzzy R-ideals of R. μ (x +y) in {μ (x), μ (y) μ (x y) in {μ (x), μ (y) and μ (x y) μ (y), μ (xy) μ (x) μ (x) in {ax {μ ((x+z)+(z+y)), μ ((z+x)+(y+z)),μ (y) (x + y) = 1 R (x + y). Since R is a fuzzy ideal of R we get that (x + y) = 1 in { (x), R (y) = ax {1- R (x), 1- R (y) = ax {R (x), 1- R (y) So R (x + y) ax { R (x), R (y) lso we get that (xy) (y), (xy) (x) (x) =1 R(x) Since is a fuzzy R-ideals of R we get that (x) 1 in {ax { R ((x+z) + (z+y)), R ((z+x) + (y+z)), R (y) = ax {1 - ax { R ((x+z) + (z+y)), R ((z+x) + (y+z)), 1- R (y) = ax {in { 1 R ((x+z) + (z+y)), 1- R ((z+x) +(y+z)), R (y) = ax {in { R ((x+z) + (z+y)), R ((z+x)+(y+z)), R (y) So R (x) ax {in { R ((x+z) + (z+y)), R ((z+x) +(y+z)), R (y) Therefore= (μ, ) is an is an intuitionistic fuzzy R-ideal Theore 3.7: n IFS = (μ, ) is an intuitionistic fuzzy R-ideal of R iff for any t [a, b] such that (μ ) t Φ and ( )t Φ. (μ ) t and ( ) t are R-ideal of R, where (μ )t ={x R / μ (x) t. Proof: Suppose that IFS = (μ, ) is an intuitionistic fuzzy R-ideal of R So by theore (2.3) μ and R are fuzzy R-ideal of R μ and are fuzzy ideal of R By [4] for any t [0, 1] such that (μ )t Φ and ( )t Φ (μ )t and ( )t are ideal of R. Let x (μ ) t and y R and ((x +z)+(z+ y)) (μ )t or ((z + x)+(y+z)) (μ )t. μ (x) t and μ ((x +z)+(z+ y)) t or μ ((z + x)+(y+z)) t. ax {μ ((x +z)+(z+ y)), μ ((z + x)+(y+z)) t. Since μ is fuzzy R-ideal of Rμ (y) ini {ax {μ ((x +z)+(z+ y)), μ ((z + x)+(y+z)), μ (x) t μ (y) t so y (μ )t. Therefore (μ )t is R-ideal of R.Siilarly we can prove that ( )t is R-ideal of R.Suppose that for any t [0, 1] such (μ )t Φ and ( )t Φ, (μ )t and ( )t are R-ideal of R.So (μ )t and ( )t are ideal of R.By [4] μ and are fuzzy ideal of R.Let x, y R and μ (y) = r 1, μ ((x +z)+(z+ y))=r 2, μ ((z + x)+(y+z))=r 3, (ri [0, 1])Let t= in{ax{r 1, r 3, r 1 y (μ )t and ((x +z)+(z+ y)) (μ )t or ((z + x)+(y+z)) (μ )t Since (μ )t is R-ideal of R.So x (μ )t μ (x) t, μ (x) in{ax{μ ((x +z)+(z+ y)),μ ((z + x)+(y+z)),μ (y)therefore μ is fuzzy R-ideal of R.Siilarly we can prove that is fuzzy R-ideal of R.By theore (2.5) we get that =(μ, ) is an intuitionistic fuzzy R-ideal of R.Recall a function f fro a sei-ring R into sei-ring T hooorphis if f (x + y) = f (x) + f (y) and f (xy) = f(x) f(y) for any x,y R Let f be a function

3 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril fro a set X into a set Y respectively, then the iage of under f, denoted by f() and the preiage of B under f, denoted by f-1(b), are IFSs in X and Y respectively and defined by f () = (f (μ ), f( )), f -1 (B)=(f -1 (μ B ), f -1 ( B )) Theore 3.9: Let f: R T be onto hooorphis of sei rings. If B= (μ B, B ) is an intuitionistic fuzzy R-ideal of T then the preiage f -1 (B) = (f -1 (μ B ), f -1 ( B )) of B under f is an intuitionistic fuzzy R-ideal of R. Proof:1- let x, y R. f -1 (μ B ) (x + y) = μ B (f (x + y) = μ B (f (x) + f(y)) in {μ B (f(x)), μ B (f(y)) =in {f-1(μ B ) (x), f-1(μ B ) (y). So f -1 (μ B ) (x + y) in {f -1 (μ B ) (x), f -1 (μ B ) (y) On Intuitionistic Fuzzy R-Ideals f -1 (μ B ) (xy) = μ B (f (xy)) = μ B (f(x) f(y)) μ B (f(y)) = f -1 (μ B ) (y). So f -1 (μ B ) (xy) f -1 (μ B ) (y).lso f -1 (μ B ) (xy) f -1 (μ B ) (x), f -1 ( B ) (x+y) = B (f(x+y)) = B (f(x) +f(y)) ax { B (f(x)), B (f(y)) = ax { f -1 ( B ) (x), f -1 ( B )(y) So f -1 (B) (x+y) ax { f -1 ( B )(x), f -1 ( B )(y). f -1 ( B ) (xy) = B (f(xy)) = B (f(x) f(y)) B (f(y)) = f -1 ( B ) (y). So f -1 ( B ) (xy) f -1 ( B ) (y).lso f -1 ( B ) (xy) f -1 ( B ) (x) 2- Let x, y R f(x), f(y) T. f -1 (μ B )(x) μ B (f(x)) in {ax {μ B (f (x+z) +f (z+y)), μb (f (z+x) + f (y+z)), μ B (f(y)) = in{ax{ f -1 (μ B )( (x+z)+(z+y)), f -1 (μ B )((z+x)+(y+z)), f -1 (μ B )(y) f -1 ( B )(x)= B (f(x)) ax {in { B (f(x+z) +f (z+y)), B (f (z+x) +f(y+z)), B (f(y)) = ax{in{ f -1 ( B )((x+z)+(z+y)), f -1 ( B )((z+x)+(y+z)), f -1 ( B )(y) Therefore f -1 (B) = (f -1 (μ B ), f -1 ( B )) is an intuitionistic fuzzy R-ideal of R. Theore 5. If μ is a intuitionistic fuzzy R- ideal in sei ring R then is also intuitionistic fuzzy R- ideal in sei ring R. Proof: For all x, y R. { μ( x) { in{ ax{ μ( ( x + z) + ( z + y) ), μ( ( z + x) + ( y + z) ), μ( y) μ( x) in{ ax{ μ( ( x + z) + ( z + y) ), μ( ( z + x) + ( y + z) ), μ( y) μ ( x) in ax{ μ( ( x + z) + ( z + y) ), μ( ( z + x) + ( y + z) ), μ( y) { { { μ( ( ) ( )), μ( ( ) ( )), μ( ) in ax x + z + z + y z + x + y + z y μ( x) in{ ax{ μ( ( x + z) + ( z + y) ), μ( ( z + x) + ( y + z) ), μ ( y) { ( x) ax in { ( ( x+ z) + ( z+ y) ), ( ( z+ x) + ( y+ z) ), ( y) { { ax{ in { ( ), (( + ) + ( + )), ( ) ax in { ( ), ( ( ) ( )), ( ) ( ) ( ) ( ) { { { ( ( x z) ( z y) ) ( ( z x) ( y z) ) ( y) ( ) ( + ) + ( + ) x x z z y z x y z y x x z z y z x y z y ax in + + +, + + +, { { ( ) ( ) ax in (( + ) + ( + )), (( + ) + ( + )), x x z z y z x y z y Therefore is a fuzzy ideal in sei ring R Theore 7: If μ is a fuzzy ideal in sei ring R then B is also fuzzy ideal in sei ring R. Proof: For all x, y R. μ (x+y) in {μ (x), μ (y) and μ (xy) μ (y), μ (xy) μ (x) μ B (x+y) in{μ B (x), μ B (y) and μ B (xy) μ B (y), μ B (xy) μ B (x)

4 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril in { ( x+ y), B( x+ y) in{ in { ( x) + ( y), in { B( x) + B( y) in { ( xy), B( xy) in { (y), B( y), in { ( xy), B( xy) in { (x), B(x) B(x+ y) in{ in { ( x), B( x),{in{ B(x), B( y) = in { B(x), B(y) B(xy) B(y), B( xy) B(x). Theore 8.If μ is a fuzzy R- ideal in sei ring R then B is also fuzzy R- ideal in sei ring R. Proof: For all x, y R. x, y R. Now, (x) in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y) in{ (x), B(x) in{in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y) If oneis contained in another. Ragavan.C 1, Solairaju. 2 and Kaviyarasu.M 3 B(x) in{ax{in{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), { ((z + x) + (y + z)), B((z + x) + (y + z)), ax{in{ (y), B(y) B(x) in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y) Theore 9: If μ is a fuzzy ideal in sei ring R then B is also fuzzy ideal in sei ring R. Proof: For all x, y R. μ (x+y) in {μ (x), μ (y) and μ (xy) μ (y), μ (xy) μ (x) μ B (x+y) in{μ B (x), μ B (y) and μ B (xy) μ B (y), μ B (xy) μ B (x) { ax ( x+ y), B( x+ y) ax{ in { ( x) + ( y), in { B( x) + B( y) If oneis contained in another. ax { ( xy), B( xy) ax { (y), B( y), ax { ( xy), B( xy) ax { (x), B(x) B(x+ y) in{ ax { ( x), B( x),{ax{ B(x), B( y) = in { B(x), B (y) B(xy) B(y), B(xy) B(x). Theore10.If μ is a fuzzy R- ideal in sei ring R then B is also fuzzy R- ideal in sei ring R. Proof: For all x, y R. x, y R. Now, (x) in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y) ax{ (x), B(x) ax{in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y) If one is contained in another. B(x) in{ax{ax{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), { ((z + x) + (y + z)), B((z + x) + (y + z)), ax{ax{ (y), B(y) B(x) in{ax{ B(( x+ z) + (z+ y)), { B((z+ x) + (y + z )), B(y) Theore 11.If μ is an intuitionistic fuzzy ideal of sei ring R then Bis an intuitionistic fuzzy ideal of sei ring R of one is contains another. Proof: If μ is a fuzzy ideal in sei ring R then B is also fuzzy ideal in sei ring R. Proof: For all x, y R. μ (x+y) in {μ (x), μ (y) and μ (xy) μ (y), μ (xy) μ (x) μ B (x+y) in{μ B (x), μ B (y) and μ B (xy) μ B (y), μ B (xy) μ B (x) in { ( x+ y), B( x+ y) in{ in { ( x), ( y), in { B( x), B( y) B(x+ y) in{ in { ( x), B( x),{in{ B(x), B( y) B(x+ y) in { B(x), B(y) in { ( xy), B( xy) in { (y), B( y), and in { ( xy), B( xy) in { (x), B(x) B(xy) B(y), B(xy) B(x). Now x, y R, now (x+y) in { (x), (y), (xy) (y), (xy) (x)

5 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril B (x+y) in { B (x), B (y) and B (xy) B (y), B (xy) B (x) in { ( x+ y), B( x+ y) in{ ax { ( x), ( y), ax { B( x) + B( y) B(x+ y) ax{ in { ( x), B( x),{in{ B(x), B( y) B(x+ y) ax{ in { ( x), B( x),{in{ (y), B( y) (x+ y) ax { (x), (y) B B B Theore 12if μ is an intuitionistic fuzzy- R ideal of sei ring R then Bis an intuitionistic fuzzy R ideal of sei ring R if one is contains another. proof: for all x, y, R (x) in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and (xy) (y), B(xy) B(y) in{ (x), B(x) in{in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y)and, B B B On Intuitionistic Fuzzy R-Ideals in{ (xy), (xy) in{ (y), (y) B B(x) in{ax{in{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), ((z + x) + (y + z)), B((z + x) + (y + z)), in{ (y), B(y)and B(xy) B(y) B(x) in{ax{ B(( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), B(y), and B(xy) B(y) (x) ax{in{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) ax{in{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and (xy) (y), B(xy) B(y) ax{ (x), B(x) ax{ax{in{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), ax{in{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and, ax{ (xy), B(xy) ax{ (y), B(y) B(x) ax{in{ax{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), ((z + x) + (y + z)), B((z + x) + (y + z)), ax{ (y), B(y)and B(xy) B(y) B(x) ax{in{ B(( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), B(y), and B(xy) B(y) 13 if μ is an intuitionistic fuzzy- R ideal of sei ring R then Bis an intuitionistic fuzzy R ideal of sei ring R if one is contains another.

6 International Journal of Scientific & Engineering Research, Volue 5, Issue 4, pril (x) in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and (xy) (y), B(xy) B(y) ax{ (x), B(x) ax{in{ax{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), in{ax{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and, ax{ (xy), (xy) ax{ (y), B(y) B(x) ax{in{ax{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), ((z + x) + (y + z)), B((z + x) + (y + z)), ax{ (y), B(y)and B(xy) B(y) B(x) in{ax{ B(( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), B(y), and B(xy) B(y) (x) ax{in{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y) B(x) ax{in{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and (xy) (y), B(xy) B(y) ax{ (x), B(x) ax{ax{in{ (( x+ z) + (z+ y)), ((z + x) + (y + z)), (y), ax{in{ B(( x+ z) + (z+ y)), B((z + x) + (y + z)), B(y)and, ax{ (xy), B(xy) ax{ (y), B(y) B(x) ax{in{ax{ (( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), ((z + x) + (y + z)), B((z + x) + (y + z)), ax{ (y), B(y)and B(xy) B(y) B(x) ax{in{ B(( x+ z) + (z+ y)), B(( x+ z) + (z+ y)), B(y), and B(xy) B(y) References 1. K. T. tanassov, Intuitionistic fuzzy sets, Fuzzy sets and systes, 20(1986), K. T. tanassov, New operations defined over the Intuitionistic fuzzy sets, Fuzzy sets and systes 61(1994), T. K. Dutta and B. K. Biwa s, Fuzzy prie ideals of sei ring, Bull Malaysian Math. Soc.17 (1994), T. K. Dutta and B. K. Biwa s, Fuzzy k-ideals of sei rings, Bull Malaysian Math. Soc.87 (1995), Y. B. Jun., J. Naggers and M.s.ki, On L-fuzzy ideals in sei rings I, Czech. Math.J.48 (1998) K. H. Ki and J. G.Lee, On Intuitionistic fuzzy Bi-ideals of sei group, Turk J. Math. 20(2005), K. H. Ki and J. G. Lee, On Intuitionistic Q-fuzzy sei ring ideals in sei groups, dvances in fuzzy Matheatics, Vol.1 No.1 (2006), W. J. Liu, Fuzzy invariants subgroups and fuzzy ideals, Fuzzy sets and systes 8(1987),