Intuitionistic Fuzzy Ideals of Near Rings

International Mathematical Forum, Vol. 7, 202, no. 6, 769-776 Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India pksharma@davjalandhar.com Abstract In this paper, an attempt has been made to study some algebraic nature of intuitionistic fuzzy ideals of near ring N and their properties with the help of their (α, β) cut sets. Mathematics Subject Classification: 03F55, 03E72, 6Y30 Keywords: Intuitionistic fuzzy set (IFS), Intuitionistic fuzzy subgroup (IFSG), Near ring, Intuitionistic fuzzy ideal (IFI), (α,β) cut set, (Near ring)- Homomorphism. Introduction The idea of Intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. Biswas applied the concept of Intuitionistic fuzzy sets to the theory of groups and studied intuitionistic fuzzy subgroups of a group. The notion of an intuitionistic fuzzy R-subgroups of a near ring is given by Jun, Yon and Kim. Zhan Jianming and Ma Xueling also discussed the various properties on intuitionistic fuzzy ideals of near rings. Sharma studied intuitionistic fuzzy subgroups of a group with the help of their.(α,β)-cut sets. Here in this paper we study the properties of Intuitionistic fuzzy ideals of near ring with the help of their.(α,β)-cut sets. By a near ring, we mean a non-empty set N with two binary operations + and. satisfying the following axioms (i) ( N, + ) is a group (ii) ( N,. ) is a semigroup (iii) ( x + y ).z = x.z + y.z for all x, y, z N

770 P. K. Sharma Precisely speaking it is a right near ring because it satisfies the right distributive law. If the condition (iii) is replace by z.( x + y ). = z.x + z.y for all x, y, z N, then it is called left near ring. In this paper we use the word near ring instead of right near ring.we denote xy instead of x.y. A near ring N is called zero symmetric if x.0 = 0 for all x N. An ideal of a near ring N is a subset I of N such that (i) ( I, + ) is normal subgroup of ( N, + ) (ii) I N I (iii) y ( x + i ) yx I ; for all x, y N and i I Note that I is right ideal of N if I satisfies (i) and (ii), and I is left ideal of N if I satisfies (i) and (iii).we note that intersection of a family of right (resp. left) ideals in N is a right (resp. left) ideal in N. If N and N be two near rings. A map f : N N is called a (near ring) homomorphism if f (x + y) = f (x) + f (y) and f (xy) = f(x)f(y) for any x, y N.The homomorphic image of right ( resp. left ) ideal in N is right ( resp. left ) ideal. The detail of results about near rings can be found in the book Near rings by G. Pilz. An Intuitionistic fuzzy set A = {< x, μ A (x), ν A (x) > : x X }in X can be identified with an ordered pair ( μ A, ν A ) in I X I X. For sake of simplicity, we shall use the symbol A = ( μ A, ν A ) for the IFS A = {< x, μ A (x), ν A (x) > : x X }. Clearly, every fuzzy set μ A is an IFS of the form (μ A, μ A c ), where μ A c = - μ A. 2. Preliminaries Definition (2.)[4] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy subnear-ring of N if for all x, y N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (xy) min{μ A (x), μ A (y)} and ν A (xy) max{ν A (x), ν A (y)} Definition (2.2)[4] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy N-subgroup of N if for all x, y, n N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (nx) μ A (x) and ν A (nx) ν A (x) (iii) μ A (xn) μ A (x) and ν A (xn) ν A (x) If the condition (i) and (ii) holds, then A is called Intuitionistic fuzzy left N-subgroup of N and if the condition (i) and (iii) holds, then A is called Intuitionistic fuzzy right N-subgroup of N.

Intuitionistic fuzzy ideals of near rings 77 Definition (2.3)[3] An IFS A = ( μ A, ν A ) of a near ring N is called an Intuitionistic fuzzy Ideal of N if for all x, y, n N (i) μ A (x-y) min{μ A (x), μ A (y)} and ν A (x-y) max{ν A (x), ν A (y)} (ii) μ A (xn) μ A (x) and ν A (xn) ν A (x) (iii) μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) (iv) μ A (n( x + y)- nx) μ A (y) and ν A (n( x + y)- nx) ν A (y) Theorem (2.4) If A = (μ A, ν A ) be Intuitionistic fuzzy ideal of a near ring N, then (i) μ A (x) μ A (0) and ν A (x) ν A (0) for all x N (ii) μ A (- x ) = μ A (x) and ν A (- x ) = ν A (x) for all x N (iii) μ A (x + y) = μ A (y + x) and ν A (x + y) = ν A (y + x) for all x, y N Proof. Trivial Proof Proposition (2.5) If A = (μ A,ν A ) be Intuitionistic fuzzy ideal of a near ring N,then (i) μ A (x - y ) μ A ( 0 ) μ A ( x ) = μ A ( y ) (ii) ν A ( x- y ) ν A ( 0 ) ν A ( x ) = ν A ( y ) Proof. Trivial Proof Definition (2.6)[7] Let N and N be two near rings. Then the mapping f : N N is called (near ring) homomorphism if for all x, y N, the following holds (i) f ( x + y ) = f ( x ) + f ( y ) and (ii) f ( x y ) = f ( x ) f ( y ) Lemma ( 2.7) Let N and N be two near rings and let f : N N a near ring epimorphism Let 0 and 0 be additive identity element in N and N respectively such that f (0) = 0. If A = (μ A, ν A ) and B = (μ B, ν B ) are IFI s in N and N respectively. Then (i) μ f(a) ( 0 ) = μ A (0) and ν f(a) ( 0 ) = ν A (0) (ii) - μ f (B) ( 0 ) = μ B (0 ) and - ν f (B) ( 0 ) = ν B (0 ) Proof. Trivial Proof 3. ( α,β )-Cut of Intuitionistic fuzzy set (IFS) and their properties Definition (3.): ( α, β ) Cut of Intuitionistic fuzzy set Let A be Intuitionistic fuzzy set of a universe set X. Then ( α, β )-cut of A is a crisp subset C α, β (A) of the IFS A is given by C α, β (A) ={ x: x X s.t. μ A (x) α, ν A (x) β}, where α, β [0,] with α+β. Proposition (3.2)[ 9 ] If A and B be two IFS s of a universe set X, then

772 P. K. Sharma following holds (i) C α, β (A) C δ, θ(a) if α δ and β θ (ii) C -β, β (A) C α, β(a) C α, -α (A) (iii) A B implies C α, β (A) C α, β (B) (iv) C α, β (A B) = C α, β (A) C α, β (B) (v) C α, β (A B) C α, β (A) C α, β (B) equality hold if α + β = (vi) C α, β ( A i ) = C α, β (A i ) (vii) C 0, (A) = X. Proposition (3.3)[0] : Let f ; X Y be a mapping. Then the following holds (i) ( ) f C C ( f ( A )), A IFS( X ) α, β α, β (ii) ( ) f C ( B) = C ( f ( B )), B IFS( Y ) α, β α, β Proposition (3.4) If A = ( μ A, ν A ) be IFI in near ring N, then C α, β(a) is ideal of N if μ A (0) α, ν A (0) β Proof. Let μ A (0) α, ν A (0) β.. Clearly C α, β(a). Let x, y C α, β(a). Then μ A (x) α, ν A (x) β and μ A (y) α, ν A (y) β min{μ A (x), μ A (y) } α and max{ν A (x), ν A (y) } β Now μ A (x-y) min{μ A (x), μ A (y)} α and ν A (x-y) max{ ν A (x), ν A (y) } β μ A (x-y) α and ν A (x-y) β and so x y C α, β(a) Thus (C α, β(a), + ) is a subgroup of ( N, + ) To show that (C α, β(a), + ) is normal subgroup of ( N, + ) Let x C α, β(a) be any element and y N. we have μ A (x) α, ν A (x) β Since A is IFI of near ring N. μ A (y + x - y) μ A (x) α and ν A (y + x - y) ν A (x) β implies that y + x y C α, β(a). Thus (C α, β(a), + ) is normal subgroup of ( N, + ) Next to show that C α, β(a) N C α, β(a). Let x C α, β(a) and n N. Therefore we have μ A (x) α, ν A (x) β. As A is IFI of near ring N. μ A (xn) μ A (x) α and ν A (xn) ν A (x) β μ A (xn) α and ν A (xn) β And so xn C α, β(a). Thus C α, β(a) N C α, β(a) Next to show that n ( x + i ) nx C α, β(a) ; for all x, n N and i C α, β(a) Let i C α, β(a) be any element. Therefore we have μ A ( i ) α, ν A ( i ) β As A is IFI of near ring N, therefore we have μ A (n( x + i)- nx) μ A ( i ) α and ν A (n( x + i )- nx) ν A ( i ) β

Intuitionistic fuzzy ideals of near rings 773 i.e. μ A (n( x + i)- nx) α and ν A (n( x + i)- nx) β and so n( x + i )- nx C α, β(a) Hence C α, β(a) is Ideal in near ring N. Theorem (3.5) If A = ( μ A, ν A ) is an IFS of a near ring N, then A is IFI if and only if C α, β(a) is an ideal of N, for all α, β [0, ] with α + β and μ A (0) α, ν A (0) β. Proof If A be IFI of a near ring N, then C α, β(a) is an ideal of N, for all α, β [0,] with α + β and μ A (0) α, ν A (0) β follows from Proposition (3.4) Conversely, let A = ( μ A, ν A ) be IFS of a near ring N such that C α, β(a) is an ideal of N, for all α, β [0, ] with α + β and μ A (0) α, ν A (0) β. To show that A is IFI of near ring N. Let x, y N and α = min{ μ A (x), μ A (y)} and β = max{ ν A (x), ν A (y) } μ A (x) α, μ A (y) α and ν A (x) β, ν A (y) β μ A (x) α, ν A (x) β and μ A (y) α, ν A (y) β Therefore x, y C α, β(a). As C α, β(a) is ideal in near ring N x y C α, β(a) μ A (x - y) α = min{ μ A (x), μ A (y)} and ν A (x y) β = max{ ν A (x), ν A (y) } Thus μ A (x-y) min{μ A (x), μ A (y)} and ν A (x y) max{ν A (x), ν A (y) }.() As C α, β(a) N C α, β(a) holds for all α, β [0,] with α + β and μ A (0) α, ν A (0) β. Let x C α, β(a) be s.t μ A (x) = α and ν A (x) = β and n N be any element. Then xn C α, β(a) and so μ A (xn) α = μ A (x) and ν A (xn) β = ν A (x) i.e. μ A (xn) μ A (x) and ν A (xn) ν A (x) And if x C α, β(a) be such that μ A (x) = α and ν A (x) = β -α, where α α. Then x C A α, β ( ). As C α, β is Ideal in near ring N xn Cα, β μ A (xn) α = μ A (x) and ν A (xn) β = ν A (x) i.e. μ A (xn) μ A (x) and ν A (xn) ν A (x) Thus μ A (xn) μ A (x) and ν A (xn) ν A (x) holds for all x, n N (2) Next to show that μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) holds for all x, y N. As ( C α, β(a), + ) be normal subgroup of ( N, + ) Let x C α, β(a) be such that μ A (x) = α and ν A (x) = β and y N be any element. Then ( y + x y ) C α, β(a) μ A (y + x - y) α = μ A (x) and ν A (y + x - y) β = ν A (x) Now, if x C α, β(a) be s.t. μ A (x) = α and ν A (x) = β -α, where α α Then x Cα, β As C α, β is normal subgroup of N. So ( y+ x- y) C α, β μ A (y + x - y) α = μ A (x) and ν A (y + x - y) β = ν A (x)

774 P. K. Sharma i.e. μ A (y + x - y) μ A (x) and ν A (y + x - y) ν A (x) holds for all x, y N (3) Next to show that μ A (n( x + i)- nx) μ A (i) and ν A (n( x + i)- nx) ν A (i) holds for all x, n N, i A. Take i C α, β(a) be an element such that μ A ( i ) = α and ν A ( i ) = β. Since C α, β(a) is ideal of the near ring N Therefore, for x, n N, we have n( x + i)- nx C α, β(a) μ A (n( x + i)- nx) α = μ A (i) and ν A (n( x + i)- nx) β = ν A (i) and if i C α, β(a) be such that μ A (i) = α and ν A (i) = β -α, where α α Then i Cα, β As C α, β is ideal of N. n( x + i)- nx C α, β μ A (n( x + i)- nx) α = μ A (i) and ν A (n( x + i)- nx) β = ν A (i) i.e. μ A ( n (x + i)- nx) μ A (i) and ν A (n( x + i)- nx) ν A (i) (4) From (), (2), (3) and (4) we find that A is IFI of near ring N. Example (3.6) Let A = ( μ A, ν A ) be IFI of near ring N, then the set M = { x N : μ A (x) = μ A (0) and ν A (x) = ν A (0) } is ideal in near ring N. Proof. Easy to verify Corollary (3.7) Let N be near ring. Then the IFS A ={< x, μ A (x), ν A (x)>: x N : μ A (x) = μ A (0) and ν A (x) = ν A (0) } of N is IFI of the near ring N. Proof. Taking α = μ A (0) and β = ν A (0), then C α, β(a) = M Therefore by Theorem (3.5), A is IFI of near ring N. Theorem (3.8) If A and B be two IFI s of a near ring N, then A B is also IFI of N. Proof. Since A and B be two IFI s of near ring N. By Proposition (3.4), we have C α, β(a) and C α, β(b) are ideals in near ring N. Since intersection of two ideals in near ring is ideal in N. Therefore C α, β(a) C α, β(b) is ideal in N C α, β(a B) is ideal in N (by using Proposition 3.2(iv)) A B is IFI in near ring N ( by Theorem (3.5)converse part ) Corollary (3.9) Intersection of a family of IFI s of near ring N is IFI in N Theorem (3.0) Let N and N be two near rings and let f : N N be near ring homomorphism. If B = ( μ B, ν B ) is an IFI in N, then the pre-image f - ( B) of B under f is an IFI of N. Proof. Since B is IFI in near ring N C α, β(b) is ideal in N, for all α, β [0,] with α + β and μ B (0) α, ν B (0) β [ By Proposition (3.4) ]. f - (C α, β(b)) is ideal in N. But f - (C α, β(b)) = C α, β( f - (B))[By Theorem(3.3)] - C α, β( f ( B)) is ideal in N and by using Theorem (3.5), we get f - ( B) is IFI in near ring N.

Intuitionistic fuzzy ideals of near rings 775 Theorem (3.) Let N and N be two near rings and let f : N N be epimorphism If A = ( μ A, ν A ) is IFI of N, then f is IFI of N. Proof. In view of Theorem (3.5), it is enough to show that C α, β( f ) is ideal in N, for all α, β [0,] with α + β and μ f(a) (0 ) α, ν f(a) (0 ) β. Let y, y 2 C α, β( f (A)) be any two elements, then μ ( )( y) α, ν A ( y) β and μ ( y2) α, ν ( y2) β f f f f By Proposition (3.3)(i), we have f C C ( f ( A )), A IFS( N) ( ) α, β α, β Therefore s x and x 2 in N such that f (x ) = y, f ( x 2 ) = y 2 and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β A f A f f 2 f 2 A 2 f 2 μ( x) α, ν ( x) β and μ( x) α, ν ( x) β A A A 2 A 2 A x A x2 A x A x2 min{ μ ( ), μ ( )} α and max { ν ( ), ν ( )} β As A is IFI of near ring N. Therefore μa( x x2) min{ μa( x), μa( x2)} α and νa( x x2) max{ νa( x), νa( x2)} β μ ( x x ) α and ν ( x x ) β A 2 A 2 ( ) ( ) x x C f x x f C C ( f ) 2 α, β 2 α, β α, β f ( x ) f ( x2 ) Cα, β( f ) y- y 2 Cα, β( f ) Hence ( C ( f ), + ) is a subgroup of ( N, +) α, β Next to show that ( C, ( f ), + ) is normal subgroup of ( N, +) Let y C α, β( f (A)) and n be any elements, then as above s x and n in N such that f (x ) = y, f ( n ) = n and μ ( x ) μ ( y ) α, ν ( x ) ν ( y ) β μ ( x ) α, ν ( x ) β α β A f A f A A x C α, β(a). As A is IFI of N, therefore C α, β(a) is ideal in N ( n + x n ) C α, β(a) f (( n + x n )) f (C α, β(a)) C α, β ( f(a)) f ( n ) + ( x ) ( n ) C α, β ( f(a)) i.e. ( n + y - n ) C α, β ( f(a)) Thus ( C ( f ), + ) is normal subgroup of ( N, +) α, β Now to show that C α, β(f (A))N N. As above, we get x n N f (x n ) = f (x ) f (n) = y n f ( N ) = N Further, to show that n ( y + y 2 ) - n y C α, β( f (A)) for all y, y 2, n N. As above s x and x 2 in N such that f (x ) = y, f (x 2 ) = y 2 and f(n) = n such that x, x 2 C α, β(a). As C α, β(a) is ideal in N. Therefore ( x + x 2 ) - n x C α, β(a) f ( n ( x + x 2 ) - n x ) f ( C α, β(a)) C α, β ( f(a)) f ( n ) ( f ( x ) + f ( x 2 )) f ( n) f ( x ) = n ( y + y 2 ) - n y C α, β( f (A))

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