# 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.

$A\left(ab\right)\wedge \mu \le \left(A\left(a\right)\vee A\left(b\right)\right)\vee \lambda$

and

$A\left({c}^{-\mathsf{\text{1}}}\right)\wedge \mu \le A\left(c\right)\vee \lambda$

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

$\begin{array}{c}A\left({x}^{-1}y\right)\wedge \mu =A\left({x}^{-1}y\right)\wedge \mu \wedge \mu \\ \le \left(\left(A\left({x}^{-1}\right)\vee A\left(y\right)\right)\vee \lambda \right)\wedge \mu \\ =\left(A\left({x}^{-1}\right)\wedge \mu \vee A\left(y\right)\right)\vee \left(\lambda \wedge \mu \right)\\ \le \left(\left(A\left(x\right)\vee \lambda \right)\vee A\left(y\right)\right)\vee \lambda \\ =\left(A\left(x\right)\vee A\left(y\right)\right)\vee \lambda .\end{array}$

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 $A\left({x}_{0}^{-1}{y}_{0}\right)\wedge \mu =\alpha >A\left({x}_{0}\right)\vee A\left({y}_{0}\right)\vee \lambda$, then A(x0) < α, A(y0) < α and α (λ, μ]. Thus x0 A α and y0 A α . But $A\left({x}_{0}^{-1}\phantom{\rule{0.3em}{0ex}}{y}_{0}\right)\phantom{\rule{0.3em}{0ex}}\ge \phantom{\rule{0.3em}{0ex}}\alpha ,\phantom{\rule{0.3em}{0ex}}\mathsf{\text{that}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{is}}\phantom{\rule{0.3em}{0ex}}{x}_{0}^{-1}\phantom{\rule{0.3em}{0ex}}{y}_{0}\phantom{\rule{0.3em}{0ex}}\notin \phantom{\rule{0.3em}{0ex}}{A}_{\left(\alpha \right)}.$ 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

$f\left(A\right)\left(y\right)=\underset{x\in {G}_{1}}{\text{inf}}\left\{A\left(x\right)|f\left(x\right)=y\right\},\phantom{\rule{1em}{0ex}}{\forall }_{y}\in {G}_{2}.$

Proof. If f −1(y1) = or f−1(y2) = for any y1, y2 G2, then $\left(f\left(A\right)\left({y}_{1}^{-1}{y}_{2}\right)\right)\wedge \mu \le 1=\left(f\left(A\right)\left({y}_{1}\right)\vee f\left(A\right)\left({y}_{2}\right)\right)\vee \lambda .$

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

For any y1, y2 G2, we have

$\begin{array}{c}f\left(A\right)\left({y}_{1}^{-1}{y}_{2}\right)\wedge \mu =\underset{t\in {G}_{1}}{\text{inf}}\left\{A\left(t\right)|f\left(t\right)={y}_{1}^{-1}{y}_{2}\right\}\wedge \mu \\ =\underset{t\in {G}_{1}}{\text{inf}}\left\{\left(A\left(t\right)\right)\wedge \mu |f\left(t\right)={y}_{1}^{-1}{y}_{2}\right\}\\ \le \underset{{x}_{1},{x}_{2}\in {G}_{1}}{\text{inf}}\left\{\left(A\left({x}_{1}^{-1}{x}_{2}\right)\right)\wedge \mu |f\left({x}_{1}\right)={y}_{2},f\left({x}_{2}\right)={y}_{2}\right\}\\ \le \underset{{x}_{1},{x}_{2}\in {G}_{1}}{\text{inf}}\left\{\left(A\left({x}_{1}\right)\vee A\left({x}_{2}\right)\right)\vee \lambda |f\left({x}_{1}\right)={y}_{1},f\left({x}_{2}\right)={y}_{2}\right\}\\ =\left(\underset{{x}_{1}\in {s}_{1}}{\text{inf}}\left\{A\left({x}_{1}\right)|f\left({x}_{1}\right)={y}_{1}\right\}\vee \underset{{x}_{2}\in {S}_{1}}{\text{inf}}\left\{A\left({x}_{2}\right)|f\left({x}_{2}\right)={y}_{2}\right\}\right)\vee \lambda \\ =\left(f\left(A\right)\left({y}_{1}\right)\vee f\left(A\right)\left({y}_{2}\right)\right)\vee \lambda .\end{array}$

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

${f}^{-1}\phantom{\rule{0.3em}{0ex}}\left(B\right)\phantom{\rule{0.3em}{0ex}}\left(x\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}B\left(f\left(x\right)\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\forall }_{x}\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{G}_{1}.$

Proof. For any x1, x2 G1,

$\begin{array}{c}{f}^{-1}\left(B\right)\left({x}_{1}^{-1}{x}_{2}\right)\wedge \mu =B\left(f\left({x}_{1}^{-1}{x}_{2}\right)\right)\wedge \mu \\ =B\left({\left(f\left({x}_{1}\right)\right)}^{-1}f\left({x}_{2}\right)\right)\wedge \mu \\ \le \left(B\left(f\left({x}_{1}\right)\right)\vee B\left(f\left({x}_{2}\right)\right)\right)\vee \lambda \\ =\left({f}^{-1}\left(B\right)\left({x}_{1}\right)\vee {f}^{-1}\left(B\right)\left({x}_{2}\right)\right)\vee \lambda .\end{array}$

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

$\left(A\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}B\right)\phantom{\rule{0.3em}{0ex}}\left(x,\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}A\left(x\right)\phantom{\rule{0.3em}{0ex}}\vee \phantom{\rule{0.3em}{0ex}}B\left(y\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\forall \left(x,\phantom{\rule{0.3em}{0ex}}y\right)\phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{G}_{1}\phantom{\rule{0.3em}{0ex}}×\phantom{\rule{0.3em}{0ex}}{G}_{2}.$

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

$\begin{array}{c}\left(\left(A×B\right){\left(x,a\right)}^{-1}\left(y,b\right)\right)\wedge \mu =\left(\left(A×B\right)\left({x}^{-1},{a}^{-1}\right)\left(y,b\right)\right)\wedge \mu \\ =\left(A\left({x}^{-1}y\right)\vee B\left({a}^{-1}b\right)\right)\wedge \mu \\ =\left(A\left({x}^{-1}y\right)\wedge \mu \right)\vee \left(B\left({a}^{-1}b\right)\wedge \mu \right)\\ \le \left(A\left(x\right)\vee A\left(y\right)\vee \lambda \right)\vee \left(B\left(a\right)\vee B\left(b\right)\vee \lambda \right)\\ =\left(A\left(x\right)\vee B\left(a\right)\right)\vee \left(A\left(y\right)\vee B\left(b\right)\right)\vee \lambda \\ =\left(\left(A×B\right)\left(x,a\right)\right)\vee \left(\left(A×B\right)\left(y,b\right)\right)\vee \lambda .\end{array}$

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.

$A\left({1}_{1}\right)\wedge \mu \le B\left(a\right)\vee \lambda ,\phantom{\rule{1em}{0ex}}{\forall }_{a}\in {G}_{2}$

and

$B\left({1}_{2}\right)\wedge \mu \le A\left(x\right)\vee \lambda ,\phantom{\rule{1em}{0ex}}\forall x\in {G}_{1}.$

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

$\begin{array}{c}A\left(xy\right)\wedge \mu \le \left(A\left(xy\right)\vee B\left({1}_{2}{1}_{2}\right)\right)\wedge \mu \\ =\left(\left(A×B\right)\left(xy,{1}_{2}{1}_{2}\right)\right)\wedge \mu \\ =\left(\left(A×B\right)\left(\left(x,{1}_{2}\right)\left(y,{1}_{2}\right)\right)\right)\wedge \mu \\ \le \left(\left(A×B\right)\left(x,{1}_{2}\right)\vee \left(A×B\right)\left(y,{1}_{2}\right)\right)\vee \lambda \\ =A\left(x\right)\vee B\left({1}_{2}\right)\vee A\left(y\right)\vee B\left({1}_{2}\right)\vee \lambda \\ =\left(A\left(x\right)\vee A\left(y\right)\right)\vee \lambda .\end{array}$

and

$\begin{array}{c}A\left({1}_{1}\right)\wedge \mu \le \left(A\left({1}_{1}\right)\vee B\left({1}_{2}\right)\right)\wedge \mu \\ =\left(\left(A×B\right)\left({1}_{1},{1}_{2}\right)\right)\wedge \mu \\ \le \left(A×B\right)\left(x,{1}_{2}\right)\vee \lambda \\ =A\left(x\right)\vee B\left({1}_{2}\right)\vee \lambda \\ =A\left(x\right)\vee \lambda .\end{array}$

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 .

## References

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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.

### Competing interests

The authors declare that they have no competing interests.

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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