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# Intuitionistic $(λ,μ)$-fuzzy sets in Γ-semigroups

## Abstract

We first introduce $(λ,μ)$-fuzzy ideals and $(λ,μ)$-fuzzy interior ideals of an ordered Γ-semigroup. Then we prove that in regular and in intra-regular ordered semigroups the $(λ,μ)$-fuzzy ideals and the $(λ,μ)$-fuzzy interior ideals coincide. Lastly, we introduce $(λ,μ)$-fuzzy simple ordered Γ-semigroup and characterize the simple ordered Γ-semigroups in terms of $(λ,μ)$-fuzzy interior ideals.

## 1 Introduction and preliminaries

The formal study of semigroups began in the early twentieth century. Semigroups are important in many areas of mathematics, for example, coding and language theory, automata theory, combinatorics and mathematical analysis.

Γ-semigroups were first defined by Sen and Saha  as a generalization of semigroups and studied by many researchers .

The concept of fuzzy sets was first introduced by Zadeh  in 1965, and then the fuzzy sets have been used in the reconsideration of classical mathematics. Recently, Yuan  introduced the concept of a fuzzy subfield with thresholds. A fuzzy subfield with thresholds λ and μ is also called a $(λ,μ)$-fuzzy subfield. Yao continued to research $(λ,μ)$-fuzzy normal subfields, $(λ,μ)$-fuzzy quotient subfields, $(λ,μ)$-fuzzy subrings and $(λ,μ)$-fuzzy ideals in .

In this paper, we study $(λ,μ)$-fuzzy ideals in ordered Γ-semigroups. This can be seen as an application of  and as a generalization of [20, 21].

Let $S={x,y,z,…}$ and $Γ={α,β,γ,…}$ be two non-empty sets. An ordered Γ-semigroup $S Γ =(S,Γ,≤)$ is a poset $(S,≤)$, and there is a mapping $S×Γ×S→S$ (images to be denoted by $aαb$) such that, for all $x,y,z∈S$, $α,β,γ∈Γ$, we have

1. (1)
$(xβy)γz=xβ(yγz)$

;

2. (2)
$x≤y⇒{ x α z ≤ y α z , z α x ≤ z α y .$

If $(S,Γ,≤)$ is an ordered Γ-semigroup and A is a subset of S, we denote by $(A]$ the subset of S defined as follows:

Given an ordered Γ-semigroup S, a fuzzy subset of S (or a fuzzy set in S) is an arbitrary mapping $f:S→[0,1]$, where $[0,1]$ is the usual closed interval of real numbers. For any $t∈[0,1]$, $f t$ is defined by $f t ={x∈S|f(x)≥t}$.

For each subset A of S, the characteristic function $f A$ is a fuzzy subset of S defined by

In the following, we will use S, $S Γ$ or $(S,Γ,≤)$ to denote an ordered Γ-semigroup. In the rest of this paper, we will always assume that $0≤λ<μ≤1$.

## 2 Intuitionistic $(λ,μ)$-fuzzy Γ-ideals

In what follows, we will use S to denote a Γ-semigroup unless otherwise specified.

Definition 1 For an IFS $A=( f A , g A )$ in S, consider the following axioms:

($Γ S 1$) $f A (xγy)∨λ≥ f A (x)∧ f A (y)∧μ$,

($Γ S 2$) $g A (xγy)∧μ≤ f A (x)∨ f A (y)∨λ$

for all $x,y∈S$ and $γ∈Γ$. Then $A=( f A , g A )$ is called a first (resp. second) intuitionistic $(λ,μ)$-fuzzy Γ-subsemigroup (briefly $(λ,μ)$-$IFΓ S 1$ (resp. $(λ,μ)$-$IFΓ S 2$)) of S if it satisfies ($Γ S 1$) (resp. $Γ S 2$).

$A=( f A , g A )$

is called an intuitionistic $(λ,μ)$-fuzzy Γ-subsemigroup (briefly $(λ,μ)$-$IFΓS$) of S if it is both a first and a second intuitionistic fuzzy Γ-subsemigroup.

Theorem 1 If U is a Γ-subsemigroup of S, then $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFΓS$ of S.

Proof Let $x,y∈S$ and $γ∈Γ$.

1. (1)

If $x,y∈U$, then $xγy∈U$ from the hypothesis. Thus

$χ U (xγy)∨λ=1∨λ=1≥ χ U (x)∧ χ U (y)∧μ$

and

$χ U ˜ (xγy)∧μ= ( 1 − χ U ( x γ y ) ) ∧μ=(1−1)∧μ=0≤ χ U ˜ (x)∨ χ U ˜ (y)∨λ.$
1. (2)

If $x∉U$ or $y∉U$, then $χ U (x)=0$ or $χ U (y)=0$. Thus

$χ U (xγy)∨λ≥0= χ U (x)∧ χ U (y)∧μ$

and

$χ U ˜ (xγy)∧μ≤1= χ U ˜ (x)∨ χ U ˜ (y)∨λ.$

And we complete the proof. □

Theorem 2 Let U be a non-empty subset of S. If $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFΓ S 1$ or $(λ,μ)$-$IFΓ S 2$ of S, then U is a Γ-subsemigroup of S.

Proof (1) Suppose that $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFΓ S 1$ of S. For any $u,v∈U$ and $γ∈Γ$, we need to show that $uγv∈U$. From ($Γ S 1$), we know that

$χ U (uγv)∨λ≥ χ U (u)∧ χ U (v)∧μ=1∧1∧μ=μ.$

Notice that $λ<μ$, thus $χ U (uγv)≥μ>0$.

And also because U is a crisp set of S, then we conclude that $χ U (uγv)=1$; that is, $uγv∈U$. Thus U is a Γ-subsemigroup of S.

1. (2)

Now assume that $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFΓ S 2$ of S. For any $u,v∈U$ and $γ∈Γ$, we also need to show that $uγv∈U$. It follows from ($Γ S 2$) that

$χ U ˜ (xγy)∧μ≤ χ U ˜ (x)∨ χ U ˜ (y)∨λ=0∨0∨λ=λ.$

Notice that $λ<μ$, thus $χ U ˜ (xγy)≤λ$.

And also because U is a crisp set of S, then we conclude that $χ U ˜ (xγy)=0$, i.e., $χ U (uγv)=1$. That is, $uγv∈U$. Thus U is a Γ-subsemigroup of S. □

Definition 2 For an IFS $A=( f A , g A )$ in S, consider the following axioms:

($LΓ I 1$) $f A (xγy)∨λ≥ f A (y)∧μ$,

($LΓ I 2$) $g A (xγy)∧μ≤ g A (y)∨λ$

for all $x,y∈S$ and $γ∈Γ$. Then $A=( f A , g A )$ is called a first (resp. second) intuitionistic $(λ,μ)$-fuzzy left Γ-ideal (briefly $(λ,μ)$-$IFLΓ I 1$ (resp. $(λ,μ)$-$IFLΓ I 2$)) of S if it satisfies ($LΓ I 1$) (resp. ($LΓ I 2$)).

$A=( f A , g A )$

is called an intuitionistic $(λ,μ)$-fuzzy left Γ-ideal (briefly $(λ,μ)$-$IFLΓI$) of S if it is both a first and a second intuitionistic $(λ,μ)$-fuzzy left Γ-ideal.

Definition 3 For an IFS $A=( f A , g A )$ in S, consider the following axioms:

($RΓ I 1$) $f A (xγy)∨λ≥ f A (x)∧μ$,

($RΓ I 2$) $g A (xγy)∧μ≤ g A (x)∨λ$

for all $x,y∈S$ and $γ∈Γ$. Then $A=( f A , g A )$ is called a first (resp. second) intuitionistic $(λ,μ)$-fuzzy right Γ-ideal (briefly $(λ,μ)$-$IFRΓ I 1$ (resp. $(λ,μ)$-$IFRΓ I 2$)) of S if it satisfies ($RΓ I 1$) (resp. ($RΓ I 2$)).

$A=( f A , g A )$

is called an intuitionistic $(λ,μ)$-fuzzy right Γ-ideal (briefly $(λ,μ)$-$IFRΓI$) of S if it is both a first and a second intuitionistic $(λ,μ)$-fuzzy right Γ-ideal.

Definition 4 For an IFS $A=( f A , g A )$ in S, it is called an intuitionistic $(λ,μ)$-fuzzy Γ-ideal (briefly $(λ,μ)$-$IFΓI$) of S if it is both an intuitionistic fuzzy left and an intuitionistic fuzzy right Γ-ideal.

Theorem 3 If $A=( f A , g A )$ is a $(λ,μ)$-$IFLΓ I 1$ of S. U is a left-zero Γ-subsemigroup of S. For any $x,y∈U$, one of the following must hold:

1. (1)
$f A (x)= f A (y)$

;

2. (2)
$f A (x)≠ f A (y)⇒$

($f A (x)∨ f A (y)≤λ$, or $f A (x)∧ f A (y)≥μ$).

Proof Let $x,y∈U$. Since U is left-zero, we have that $xγy=x$ and $yγx=y$ for all $γ∈Γ$.

From the hypothesis, we have that

$f A (x)∨λ= f A (xγy)∨λ≥ f A (y)∧μ$

and

$f A (y)∨λ= f A (yγx)∨λ≥ f A (x)∧μ.$

Obviously, if $f A (x)= f A (y)$, then the previous two inequalities hold.

Suppose $f A (x)< f A (y)$. If $f A (x)∨ f A (y)>λ$ and $f A (x)∧ f A (y)<μ$, four cases are possible:

1. (1)

If $f A (x)>λ$ and $f A (x)<μ$, then $f A (x)∨λ= f A (x)< f A (y)$. Note that $f A (x)<μ$, we obtain that $f A (x)< f A (y)∧μ$; that is, $f A (x)∨λ< f A (y)∧μ$. This is a contradiction to the previous proposition.

2. (2)

If $f A (x)>λ$ and $f A (y)<μ$, then $f A (x)∨λ= f A (x)< f A (y)= f A (y)∧μ$. This is a contradiction to the previous proposition.

3. (3)

If $f A (y)>λ$ and $f A (x)<μ$, then from $f A (x)<μ$ and $f A (x)< f A (y)$, we obtain that $f A (x)< f A (y)∧μ$. From $λ< f A (y)$ and $λ<μ$, we conclude that $λ< f A (y)∧μ$. So, $f A (x)∨λ< f A (y)∧μ$. This is a contradiction to the previous proposition.

4. (4)

If $f A (y)>λ$ and $f A (y)<μ$, then from $F A (X)< f A (y)$ and $λ< f A (y)$, we obtain $F A (X)∨λ< f A (y)= f A (y)∧μ$. This is a contradiction to the previous proposition.

If $f A (y)< f A (x)$, we can prove the results dually.

Thus if $f A (y)≠ f A (x)$, then $f A (x)∨ f A (y)≤λ$ or $f A (x)∧ f A (y)≥μ$. □

Similarly, we can prove the following three theorems.

Theorem 4 If $A=( f A , g A )$ is a $(λ,μ)$-$IFRΓ I 1$ of S. U is a right-zero Γ-subsemigroup of S. For any $x,y∈U$, one of the following must hold:

1. (1)
$f A (x)= f A (y)$

;

2. (2)
$f A (x)≠ f A (y)⇒( f A (x)∨ f A (y)≤λ, or f A (x)∧ f A (y)≥μ)$

.

Theorem 5 If $A=( f A , g A )$ is a $(λ,μ)$-$IFLΓ I 2$ of S. U is a left-zero Γ-subsemigroup of S. For any $x,y∈U$, one of the following must hold:

1. (1)
$g A (x)= g A (y)$

;

2. (2)
$g A (x)≠ g A (y)⇒( g A (x)∨ g A (y)≤λ, or g A (x)∧ g A (y)≥μ)$

.

Theorem 6 If $A=( f A , g A )$ is a $(λ,μ)$-$IFRΓ I 2$ of S. U is a right-zero Γ-subsemigroup of S. For any $x,y∈U$, one of the following must hold:

1. (1)
$g A (x)= g A (y)$

;

2. (2)
$g A (x)≠ g A (y)⇒( g A (x)∨ g A (y)≤λ, or g A (x)∧ g A (y)≥μ)$

.

Lemma 1 If U is a left Γ-ideal of S, then $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFLΓI$ of S.

Proof Let $x,y∈S$ and $γ∈Γ$.

1. (1)

If $y∈U$, then $xγy∈U$ since U is a left Γ-ideal of S. It follows that

$χ U (xγy)∨λ=1∨λ=1≥ χ U (y)∧μ$

and

$χ U ˜ (xγy)∧μ=0∧μ=0≤ χ U ˜ (y)∨λ.$
1. (2)

If $y∉U$, then $χ U (y)=0$. It follows that

$χ U (xγy)∨λ≥0= χ U (y)∧μ$

and

$χ U ˜ (xγy)∧μ≤1= χ U ˜ (y)∨λ.$

Consequently, $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFLΓI$ of S. □

Theorem 7 Let S be regular. If $E S$ is a left-zero Γ-subsemigroup of S, then, for all $e, e ′ ∈ E S$, we have

$χ L [ e ] ( e ′ ) = χ L [ e ] (e)$

or

$χ L [ e ] ( e ′ ) ∨ χ L [ e ] (e)≤λ,$

or

$χ L [ e ] ( e ′ ) ∧ χ L [ e ] (e)≥μ.$

Proof Since S is regular, $E S$ is non-empty. Let $e=eγe, e ′ = e ′ γ ′ e ′ ∈ E S$, where $γ, γ ′ ∈Γ$. Because S is regular, $L[e]=SΓe$. From the fact that $L[e]$ is a left Γ-ideal of S, we obtain that $L [ e ] ˜ =( χ L [ e ] , χ L [ e ] ˜ )$ is a $(λ,μ)$-$IFLΓ I 1$ of S by the previous lemma.

Applying Theorem 3, we obtain the results. □

The following theorem can be proved in a similar way.

Theorem 8 Let S be regular. If $E S$ is a left-zero Γ-subsemigroup of S, then, for all $e, e ′ ∈ E S$, we have

$χ L [ e ] ˜ ( e ′ ) = χ L [ e ] ˜ (e)$

or

$χ L [ e ] ˜ ( e ′ ) ∨ χ L [ e ] ˜ (e)≤λ,$

or

$χ L [ e ] ˜ ( e ′ ) ∧ χ L [ e ] ˜ (e)≥μ.$

Theorem 9 Let S be regular. If for all $e, e ′ ∈ E S$ we have

$χ L [ e ] ( e ′ ) = χ L [ e ] (e)$

or

$χ L [ e ] ( e ′ ) ∨ χ L [ e ] (e)≤λ,$

or

$χ L [ e ] ( e ′ ) ∧ χ L [ e ] (e)≥μ,$

then $E S$ is a left-zero Γ-subsemigroup of S.

Proof Since S is regular, $E S$ is non-empty. Let $e=eγe, e ′ = e ′ γ ′ e ′ ∈ E S$, where $γ, γ ′ ∈Γ$. Because S is regular, $L[e]=SΓe$. From the fact that $L[e]$ is a left Γ-ideal of S, we obtain that $L [ e ] ˜ =( χ L [ e ] , χ L [ e ] ˜ )$ is a $(λ,μ)$-$IFLΓ I 1$ of S by the previous lemma.

1. (1)

If $χ L [ e ] ( e ′ )= χ L [ e ] (e)=1$, then $e ′ ∈L[e]=SΓe$. Thus

$e ′ =xβe=xβ(eγe)=(xβe)γe= e ′ γe$

for some $x∈S$ and $β∈Γ$.

1. (2)
$χ L [ e ] ( e ′ )∨ χ L [ e ] (e)≤λ$

will never happen since $χ L [ e ] ( e ′ )∨ χ L [ e ] (e)=1$.

2. (3)

If $χ L [ e ] ( e ′ )∧ χ L [ e ] (e)≥μ$, that is, $χ L [ e ] ( e ′ )≥μ$, then $χ L [ e ] ( e ′ )=1$. And so $e ′ ∈L[e]=SΓe$. The following proof will be the same as in case (1).

Consequently, $E S$ is a left-zero Γ-subsemigroup of S. □

The following theorem can be proved similarly.

Theorem 10 Let S be regular. If for all $e, e ′ ∈ E S$ we have

$χ L [ e ] ˜ ( e ′ ) = χ L [ e ] ˜ (e)$

or

$χ L [ e ] ˜ ( e ′ ) ∨ χ L [ e ] ˜ (e)≤λ,$

or

$χ L [ e ] ˜ ( e ′ ) ∧ χ L [ e ] ˜ (e)≥μ,$

then $E S$ is a left-zero Γ-subsemigroup of S.

## 3 Intuitionistic $(λ,μ)$-fuzzy interior Γ-ideals

Definition 5 For an IFS $A=( f A , g A )$ in S, consider the following axioms:

($IΓ I 1$) $f A (xβsγy)∨λ≥ f A (s)∧μ$,

($IΓ I 2$) $g A (xβsγy)∧μ≤ g A (s)∨λ$

for all $s,x,y∈S$ and $β,γ∈Γ$. Then $A=( f A , g A )$ is called a first (resp. second) intuitionistic $(λ,μ)$-fuzzy interior Γ-ideal (briefly $(λ,μ)$-$IFIΓ I 1$ (resp. $(λ,μ)$-$IFIΓ I 2$)) of S if it satisfies ($IΓ I 1$) (resp. ($IΓ I 2$)).

$A=( f A , g A )$

is called an intuitionistic $(λ,μ)$-fuzzy interior Γ-ideal (briefly $(λ,μ)$-$IFIΓI$) of S if it is both a first and a second intuitionistic $(λ,μ)$-fuzzy interior Γ-ideal.

Theorem 11 Every $(λ,μ)$-$IFΓI$ of S is a $(λ,μ)$-$IFIΓI$ of S.

Proof Let $A=( f A , g A )$ be a $(λ,μ)$-$IFΓI$ of S. For all $s,x,y∈S$ and $β,γ∈Γ$, we have

$f A ( x β s γ y ) ∨ λ = f A ( x β s γ y ) ∨ λ ∨ λ ≥ ( f A ( x β s ) ∧ μ ) ∨ λ = ( f A ( x β s ) ∨ λ ) ∧ ( μ ∨ λ ) ≥ f ( s ) ∧ μ .$

Similarly, we have $g A (xβsγy)∧μ≤ g A (s)∨λ$.

So, $A=( f A , g A )$ is a $(λ,μ)$-$IFIΓI$ of S. □

Theorem 12 If S is regular, then every $(λ,μ)$-$IFIΓI$ of S is a $(λ,μ)$-$IFΓI$ of S.

Proof Let $A=( f A , g A )$ be a $(λ,μ)$-$IFIΓI$ of S and $x,y∈S$. Since S is regular, there exist $s, s ′ ∈S$ and $β, β ′ ,γ, γ ′ ∈Γ$ such that $x=xβsγx$ and $y=y β ′ s ′ γ ′ y$. Thus

$f A (xαy)∨λ= f A ( x α ( y β ′ s ′ γ ′ y ) ) ∨λ= f A ( x α y β ′ ( s ′ γ ′ y ) ) ∨λ≥ f A (y)∧μ$

and

$g A (xαy)∧μ= g A ( x α ( y β ′ s ′ γ ′ y ) ) ∧μ= g A ( x α y β ′ ( s ′ γ ′ y ) ) ∧μ≤ g A (y)∨λ$

for all $α∈Γ$. It follows that $A=( f A , g A )$ is a $(λ,μ)$-$IFLΓI$ of S. Similarly, we can prove that A is a $(λ,μ)$-$IFRΓI$ of S. This completes the proof. □

Theorem 13 If U is an interior Γ-ideal of S, then $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFIΓI$ of S.

Proof Let $s,x,y∈S$ and $β,γ∈Γ$.

1. (1)

If $s∈U$, then $xβsγy∈U$ since U is an interior Γ-ideal of S. So,

$χ U (xβsγy)∨λ=1∨λ=1≥ χ U (s)∧μ$

and

$χ U ˜ (xβsγy)∧μ=0∧μ=0≤ χ U ˜ (s)∨λ.$
1. (2)

If $s∉U$, then $χ U (s)=0$. Thus

$χ U (xβsγy)∨λ≥0= χ U (s)∧μ$

and

$χ U ˜ (xβsγy)∧μ≤1= χ U ˜ (s)∨λ.$

Consequently, we obtain that $U ˜$ is a $(λ,μ)$-$IFIΓI$ of S. □

Theorem 14 Let S be regular and U be a non-empty subset of S. If $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFIΓ I 1$ or $(λ,μ)$-$IFIΓ I 2$ of S, then U is an interior Γ-ideal of S.

Proof It is obvious that U is a Γ-subsemigroup of S by Theorem 2.

Case 1. Suppose that $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFIΓ I 1$ of S and $x∈SΓUΓS$. Thus $x=sβuγt$ for some $s,t∈S$, $u∈U$ and $β,γ∈Γ$. It follows from ($IΓ I 1$) that

$χ U (x)∨λ= χ U (sβuγt)∨λ≥ χ U (u)∧μ=1∧μ=μ.$

Notice that $λ<μ$, we obtain that $χ U (x)≥μ$, that is, $χ U (x)=1$. So, $x∈U$. Thus U is an interior Γ-ideal of S.

Case 2. Suppose that $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFIΓ I 2$ of S and $x∈SΓUΓS$. Then $x=sβuγt$ for some $s,t∈S$, $u∈U$ and $β,γ∈Γ$. Using ($IΓ I 2$), we conclude that

$χ U ˜ (x)∧μ= χ U ˜ (sβuγt)∧μ≤ χ U ˜ (u)∨λ=0∨λ=λ.$

Notice that $λ<μ$, we obtain that $χ U ˜ (x)≤λ$, that is, $χ U ˜ (x)=0$. So, $x∈U$. Thus U is an interior Γ-ideal of S. □

## 4 Intuitionistic $(λ,μ)$-fuzzy simple Γ-semigroups

Definition 6 S is called first (resp. second) intuitionistic $(λ,μ)$-fuzzy left simple if for any $(λ,μ)$-$IFLΓ I 1$ (resp. $(λ,μ)$-$IFLΓ I 2$) $A=( f A , g A )$ of S, we have $f A (a)∨λ≥ f A (b)∧μ$ (resp. $g A (a)∨λ≥ g A (b)∧μ$) for all $a,b∈S$.

S is said to be intuitionistic $(λ,μ)$-fuzzy left simple if it is both first and second intuitionistic $(λ,μ)$-fuzzy left simple.

Theorem 15 If S is left simple, then S is intuitionistic $(λ,μ)$-fuzzy left simple.

Proof Let $A=( f A , g A )$ be a $(λ,μ)$-$IFLΓI$ of S and $x, x ′ ∈S$. Because S is left simple, there exist $s, s ′ ∈S$ and $γ, γ ′ ∈Γ$ such that $x=sγ x ′$ and $x ′ = s ′ γ ′ x$. Thus, since A is a $(λ,μ)$-$IFLΓI$ of S, we have that and Consequently, S is intuitionistic $(λ,μ)$-fuzzy left simple. □

Theorem 16 If S is first or second intuitionistic $(λ,μ)$-fuzzy left simple, then S is left simple.

Proof Let U be a left Γ-ideal of S. Suppose that S is first (or second) intuitionistic $(λ,μ)$-fuzzy left simple. Because $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFLΓI$ of S, $U ˜ =( χ U , χ U ˜ )$ is a $(λ,μ)$-$IFLΓ I 1$ (and $(λ,μ)$-$IFLΓ I 2$) of S. □

## 5 Conclusion and further research

In this paper, we generalized the results of [20, 21]. We introduced $(λ,μ)$-fuzzy ideals and $(λ,μ)$-fuzzy interior ideals of an ordered Γ-semigroup and we got some interesting results. When $λ=0$ and $μ=1$, we meet ordinary fuzzy ideals and fuzzy interior ideals. From this point of view, $(λ,μ)$-fuzzy ideals and $(λ,μ)$-fuzzy interior ideals are more general concepts than fuzzy ones.

In , Yao gave the definition of $(λ,μ)$-fuzzy bi-ideals in semigroups. One can study $(λ,μ)$-fuzzy bi-ideals in ordered Γ-semigroups. We would like to explore this in next papers.

## References

1. 1.

Sen MK, Saha NK: On Γ -semigroup I. Bull. Calcutta Math. Soc. 1986, 78: 180–186.

2. 2.

Chattopadyay S: Right inverse Γ -semigroup. Bull. Calcutta Math. Soc. 1986, 78: 180–186.

3. 3.

Chattopadyay S: Right orthodox Γ -semigroup. Southeast Asian Bull. Math. 2005, 29: 23–30.

4. 4.

Dutta TK, Adhikari NC: On Γ -semigroup with right and left unities. Soochow J. Math. 1993, 19(4):461–474.

5. 5.

Dutta TK, Adhikari NC: On prime radical of Γ -semigroup. Bull. Calcutta Math. Soc. 1994, 86(5):437–444.

6. 6.

Hila K: On regular, semiprime and quasi-reflexive Γ -semigroup and minimal quasi-ideals. Lobachevskii J. Math. 2008, 29: 141–152. 10.1134/S1995080208030050

7. 7.

Hila K: On some classes of le- Γ -semigroup and minimal quasi-ideals. Algebras Groups Geom. 2007, 24: 485–495.

8. 8.

Chinram R: On quasi- Γ -ideals in Γ -semigroups. ScienceAsia 2006, 32: 351–353. 10.2306/scienceasia1513-1874.2006.32.351

9. 9.

Saha NK: On Γ -semigroup II. Bull. Calcutta Math. Soc. 1987, 79: 331–335.

10. 10.

Sen MK, Chattopadhyay S: Semidirect product of a monoid and a Γ -semigroup. East-West J. Math. 2004, 6: 131–138.

11. 11.

Sen MK, Seth A: On po- Γ -semigroups. Bull. Calcutta Math. Soc. 1993, 85: 445–450.

12. 12.

Sen MK, Saha NK: Orthodox Γ -semigroups. Int. J. Math. Math. Sci. 1990, 13: 527–534. 10.1155/S016117129000076X

13. 13.

Uckun M, Ozturk MA, Jun YB: Intuitionistic fuzzy sets in Γ -semigroups. Bull. Korean Math. Soc. 2007, 44(2):359–367. 10.4134/BKMS.2007.44.2.359

14. 14.

Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-X

15. 15.

Yuan X, Zhang C, Ren Y: Generalized fuzzy groups and many-valued implications. Fuzzy Sets Syst. 2003, 138: 205–211. 10.1016/S0165-0114(02)00443-8

16. 16.

Yao B:$(λ,μ)$-fuzzy normal subfields and $(λ,μ)$-fuzzy quotient subfields. J. Fuzzy Math. 2005, 13(3):695–705.

17. 17.

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

18. 18.

Yao B: Fuzzy Theory on Group and Ring. Science and Technology Press, Beijing; 2008. (in Chinese)

19. 19.

Yao B:$μλ(,)$-fuzzy ideal in semigroups. Fuzzy Syst. Math. 2009, 23(1):123–127.

20. 20.

Kehayopulu N, Tsingelis M: Fuzzy interior ideals in ordered semigroups. Lobachevskii J. Math. 2006, 21: 65–71.

21. 21.

Sardar S, Davvaz B, Majumder S: A study on fuzzy interior ideals of Γ -semigroups. Comput. Math. Appl. 2010, 60: 90–94. 10.1016/j.camwa.2010.04.033

## Author information

Authors

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Correspondence to Yuming Feng.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

After a long time discuss between YF and BL, the paper is finally finished. During YF’s writing of this paper, BL gave some very good advice and she helped to draft the manuscript. All authors read and approved the final manuscript.

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Li, B., Feng, Y. Intuitionistic $(λ,μ)$-fuzzy sets in Γ-semigroups. J Inequal Appl 2013, 107 (2013). https://doi.org/10.1186/1029-242X-2013-107

• $(λ,μ)$-fuzzy interior ideal
• $(λ,μ)$-fuzzy simple 