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On (λ, μ)-anti-fuzzy subgroups

Abstract

We introduced the notions of (λ, μ)-anti-fuzzy subgroups, studied some properties of them and discussed the product of them.

1 Introduction and preliminaries

Fuzzy sets was first introduced by Zadeh [1] and then the fuzzy sets have been used in the reconsideration of classical mathematics. Yuan et al. [2] introduced the concept of fuzzy subgroup with thresholds. A fuzzy subgroup with thresholds λ and μ is also called a (λ, μ)-fuzzy subgroup. Yao continued to research (λ, μ)-fuzzy normal subgroups, (λ, μ)-fuzzy quotient subgroups and (λ, μ)-fuzzy subrings in [35].

Shen researched anti-fuzzy subgroups in [6] and Dong [7] studied the product of anti-fuzzy subgroups.

By a fuzzy subset of a nonempty set X we mean a mapping from X to the unit interval 0[1]. If A is a fuzzy subset of X, then we denote A(α)= {x X|A(x) < α} for all α 0[1].

Throughout this article, we will always assume that 0 ≤ λ < μ ≤ 1.

Let G, G1, and G2 always denote groups in the following. 1, 11, and 12 are identities of G, G1, and G2, respectively.

2 (λ, μ)-anti-fuzzy subgroups

Definition 1. A fuzzy set A of a group G is called a (λ, μ)-anti-fuzzy subgroup of G if a, b, c G.

A a b μ ( A a A b ) λ

and

A c - 1 μ A c λ

where c-1 is the inverse element of c.

Proposition 1. If A is a (λ, μ)-anti-fuzzy subgroup of a group G, then A(1) μ ≤ A(x) λ for all × G, where 1 is the identity of G.

Proof. x G and let x−1 be the inverse element of x. Then A(1) μ = A(xx−1) μ = (A(xx−1) μ) μ ≤ ((A(x) A(x−1)) λ) μ = (A(x) μ) (A(x−1) μ) (λ μ) ≤ A(x) (A(x) λ) λ = A(x) λ.

Theorem 1. Let A be a fuzzy subset of a group G. Then A is a (λ, μ)-anti-fuzzy subgroup of G if and only if A(x-1y) μ ≤ (A(x) A(y)) λ, x, y G.

Proof. Let A is a (λ, μ)-anti-fuzzy subgroup of G, then

A ( x - 1 y ) μ = A ( x - 1 y ) μ μ ( ( A ( x - 1 ) A ( y ) ) λ ) μ = ( A ( x - 1 ) μ A ( y ) ) ( λ μ ) ( ( A ( x ) λ ) A ( y ) ) λ = ( A ( x ) A ( y ) ) λ .

Conversely, suppose A(x-1y) μ ≤ (A(x) A(y)) λ, x, y G, then

A(1) μ = A(x-1x) μA(x) A(x) λ = A(x) λ.

So

A(x−1) μ = A(x−11) μ = A(x−11) μ μ ≤ (A(x) A(1) λ) μ = (A(1) μ) ((A(x) λ) μ) ≤ (A(x) λ) ((A(x) λ) μ) = A(x) λ.

A(xy) μ = A((x-1)-1 y) μ = A((x-1)-1y) μ μ ≤ (A(x−1) A(y) λ) μ = (A(x−1) μ) ((A(y) λ) μ) ≤ (A(x) λ) (A(y) λ) = (A(x) A(y)) λ.

So A is a (λ, μ)-anti-fuzzy subgroup of G.

Theorem 2. Let A be a fuzzy subset of a group G. Then the following are equivalent:

(1) A is a (λ, μ)-anti-fuzzy subgroup of G;

(2) A(α)is a subgroup of G, for any α (λ, μ], where A(α).

Proof. "(1) (2)"

Let A be a (λ, μ)-anti-fuzzy subgroup of G. For any α (λ, μ], such that A α , we need to show that x−1 y A(α), for all x,y A(α).

Since A(x) < α and A(y) < α, Then A(x−1 y) μA(x) A(y) λ < α α λ = α λ = α. Note that αμ, we obtain A(x−1 y) < α. So x−1 y A(α).

"(2) (1)"

Conversely, let A(α)is a subgroup of G. We need to prove that A(x−1 y) μA(x) A(y) λ, x G. If there exist x0, y0 G such that A x 0 - 1 y 0 μ=α>A x 0 A y 0 λ, then A(x0) < α, A(y0) < α and α (λ, μ]. Thus x0 A α and y0 A α . But A x 0 - 1 y 0 α, that is x 0 - 1 y 0 A α . This is a contradiction with that A(α)is a subgroup of G. Hence A(x-1 y) μA(x) A(y) λ holds for any x, y G.

Therefore, A is a (λ, μ)-anti-fuzzy subgroup of G.

We set inf = 1, where is the empty set.

Theorem 3. Let f: G1G2 be a homomorphism and let A be a (λ, μ)-anti-fuzzy subgroup of G1. Then f(A) is a (λ, μ)-anti-fuzzy subgroup of G2, where

f A y = inf x G 1 A x | f x = y , y G 2 .

Proof. If f −1(y1) = or f−1(y2) = for any y1, y2 G2, then f A y 1 - 1 y 2 μ1= f A y 1 f A y 2 λ.

Suppose that f−1(y1) ≠ , f−1(y2) = for any y1, y2 G2. Then

For any y1, y2 G2, we have

f A y 1 - 1 y 2 μ = inf t G 1 A t | f t = y 1 - 1 y 2 μ = inf t G 1 A t μ | f t = y 1 - 1 y 2 inf x 1 , x 2 G 1 A x 1 - 1 x 2 μ | f x 1 = y 2 , f x 2 = y 2 inf x 1 , x 2 G 1 A x 1 A x 2 λ | f x 1 = y 1 , f x 2 = y 2 = ( inf x 1 s 1 A x 1 | f x 1 = y 1 inf x 2 S 1 A x 2 | f x 2 = y 2 ) λ = f A y 1 f A y 2 λ .

So, f(A) is a (λ, μ)-anti-fuzzy subgroup of G2.

Theorem 4. Let f : G1G2 be a homomorphism and let B be a (λ, μ)-anti-fuzzy subgroup of G2. Then f−1(B) is a (λ, μ)-anti-fuzzy subgroup of G1, where

f - 1 B x = B f x , x G 1 .

Proof. For any x1, x2 G1,

f - 1 B x 1 - 1 x 2 μ = B f x 1 - 1 x 2 μ = B f x 1 - 1 f x 2 μ B f x 1 B f x 2 λ = f - 1 B x 1 f - 1 B x 2 λ .

So, f−1(B) is a (λ, μ)-anti-fuzzy subgroup of G1.

Let G1 be a group with the identity 11 and G2 be a group with the identity 12, then G1 × G2 is a group with the identity (11, 12) if we define (x1, y1) (x2, y2) = (x1x2, y1y2) for all (x1, y1), (x2, y2) G1 × G2. Moreover, the inverse element of any (x, a) G1 × G2 is (y, b) G1 × G2 if and only if y is the inverse element of x in G1 and b is the inverse element of a in G2.

Theorem 5. Let A, B be two (λ, μ)-anti-fuzzy subgroups of groups G1 and G2, respectively. The product of A and B, denoted by A × B, is a (λ, μ)-anti-fuzzy subgroup of the group G1 × G2, where

A × B x , y = A x B y , x , y G 1 × G 2 .

Proof. Let (x−1, a−1) be the inverse element of (x, a) in G1 × G2. Then x−1 is the inverse element of x in G1 and a−1 is the inverse element of a in G2. Hence A(x−1) μA(x) λ and B(a−1) μB(a) λ. For all (y, b) G1 × G2. We have

A × B x , a - 1 y , b μ = A × B x - 1 , a - 1 y , b μ = A x - 1 y B a - 1 b μ = A x - 1 y μ B a - 1 b μ A x A y λ B a B b λ = A x B a A y B b λ = A × B x , a A × B y , b λ .

Hence A × B is a (λ, μ)-anti-fuzzy subgroup of G1 × G2.

Theorem 6. Let A and B be two fuzzy subsets of groups G1 and G2, respectively. If A × B is a (λ, μ)-anti-fuzzy subgroup of G1 × G2, then at least one of the following statements must hold.

A 1 1 μ B a λ , a G 2

and

B 1 2 μ A x λ , x G 1 .

Proof. Let A × B be a (λ, μ)-anti-fuzzy subgroup of the group G1 × G2.

By contraposition, suppose that none of the statements hold. Then we can find x G1 and a G2 such that A(x) λ < B(12) μ and B(a) λ < A(11) μ. Now

(A×B) (x, a) λ = (A(x)B(a))λ = (A(x)λ)(B(a)λ) < (A(11)μ) (B(12)μ) = (A×B) (11,12) μ.

Thus A × B is a (λ, μ)-anti-fuzzy subgroup of the group G1 × G2 satisfying (A × B)(x, a) λ < (A × B) (11, 12) μ. This is a contradict with that (11, 12) iss the identity of G1 × G2 .

Theorem 7. Let A and B be fuzzy subsets of groups G1 and G2, respectively, such that B(12) μA(x) λ for all × G1. If A × B is a (λ, μ)-anti-fuzzy subgroup of G1 × G2, then A is a (λ, μ)-anti-fuzzy subgroup of G1 .

Proof. From B(12) μA(x) λ we obtain that μA(x) λ or B(12) ≤ A(x) λ, for all x G1.

Let x, y G1, then (x, 12), (y, 12) G1 × G2.

Two cases are possible:

  1. (1)

    If μA(x) λ for all x G1. Then

A(xy) μμA(x) λ ≤ (A(x) A(y)) λ

and A(11) μμA(x) λ.

  1. (2)

    If B(12) ≤ A(x) λ for all x G1. Then

    A x y μ A x y B 1 2 1 2 μ = A × B x y , 1 2 1 2 μ = A × B x , 1 2 y , 1 2 μ A × B x , 1 2 A × B y , 1 2 λ = A x B 1 2 A y B 1 2 λ = A x A y λ .

and

A 1 1 μ A 1 1 B 1 2 μ = A × B 1 1 , 1 2 μ A × B x , 1 2 λ = A x B 1 2 λ = A x λ .

Hence A is a (λ, μ)-anti-fuzzy subgroup of G1.

Analogously, we have

Theorem 8. Let A and B be fuzzy subsets of groups G1 and G2, respectively, such that A(11) μ ≤ B(a) λ for all a G2. If A × B is a (λ, μ)-anti-fuzzy subgroup of G1 × G2, then B is a (λ, μ)-anti-fuzzy subgroup of G2 .

From the previous theorems, we have the following corollary

Corollary 1. Let A and B be fuzzy subsets of groups G1 and G2, respectively. If A × B is a (λ, μ)-anti-fuzzy subgroup of G1 × G2, then either A is a (λ, μ)-anti-fuzzy subgroup of G1 or B is a (λ, μ)-anti-fuzzy subgroup of G2 .

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    Yao B: ( λ, μ )-fuzzy subrings and ( λ, μ )-fuzzy ideals. J Fuzzy Math 2007, 15(4):981–987.

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Acknowledgements

YF wished to thank Prof. Michela for her help with the language.

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Correspondence to Yuming Feng.

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The authors declare that they have no competing interests.

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YF had posed ideals and typed this article with a computer. BY had given some good advice. All authors read and approved the final manuscript.

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Feng, Y., Yao, B. On (λ, μ)-anti-fuzzy subgroups. J Inequal Appl 2012, 78 (2012). https://doi.org/10.1186/1029-242X-2012-78

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Keywords

  • product
  • (λ, μ)-fuzzy
  • subgroup
  • ideal