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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 [1] as a generalization of semigroups and studied by many researchers [213].

The concept of fuzzy sets was first introduced by Zadeh [14] in 1965, and then the fuzzy sets have been used in the reconsideration of classical mathematics. Recently, Yuan [15] 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 [1619].

In this paper, we study (λ,μ)-fuzzy ideals in ordered Γ-semigroups. This can be seen as an application of [19] 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×SS (images to be denoted by aαb) such that, for all x,y,zS, α,β,γΓ, we have

  1. (1)
    (xβy)γz=xβ(yγz)

    ;

  2. (2)
    xy{ 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:

(A]={tS|ta for some aA}.

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 ={xS|f(x)t}.

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

f A (x)={ 1 if  x A , 0 if  x A .

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 0λ<μ1.

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,yS and γΓ. 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,yS and γΓ.

  1. (1)

    If x,yU, then xγyU from the hypothesis. Thus

    χ U (xγy)λ=1λ=1 χ U (x) χ U (y)μ

and

χ U ˜ (xγy)μ= ( 1 χ U ( x γ y ) ) μ=(11)μ=0 χ U ˜ (x) χ U ˜ (y)λ.
  1. (2)

    If xU or yU, then χ U (x)=0 or χ U (y)=0. Thus

    χ U (xγy)λ0= χ U (x) χ U (y)μ

and

χ U ˜ (xγy)μ1= 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,vU and γΓ, we need to show that uγvU. From (Γ S 1 ), we know that

χ U (uγv)λ χ U (u) χ U (v)μ=11μ=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, uγvU. 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,vU and γΓ, we also need to show that uγvU. It follows from (Γ S 2 ) that

    χ U ˜ (xγy)μ χ U ˜ (x) χ U ˜ (y)λ=00λ=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, uγvU. 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,yS and γΓ. 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,yS and γΓ. 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,yU, 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,yU. Since U is left-zero, we have that xγy=x and yγx=y for all γΓ.

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,yU, 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,yU, 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,yU, 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,yS and γΓ.

  1. (1)

    If yU, then xγyU since U is a left Γ-ideal of S. It follows that

    χ U (xγy)λ=1λ=1 χ U (y)μ

and

χ U ˜ (xγy)μ=0μ=0 χ U ˜ (y)λ.
  1. (2)

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

    χ U (xγy)λ0= χ U (y)μ

and

χ U ˜ (xγy)μ1= 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 xS and βΓ.

  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,yS and β,γΓ. 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,yS and β,γΓ, 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,yS. 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 αΓ. 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,yS and β,γΓ.

  1. (1)

    If sU, then xβsγyU since U is an interior Γ-ideal of S. So,

    χ U (xβsγy)λ=1λ=1 χ U (s)μ

and

χ U ˜ (xβsγy)μ=0μ=0 χ U ˜ (s)λ.
  1. (2)

    If sU, then χ U (s)=0. Thus

    χ U (xβsγy)λ0= χ U (s)μ

and

χ U ˜ (xβsγy)μ1= 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 xSΓUΓS. Thus x=sβuγt for some s,tS, uU and β,γΓ. It follows from (IΓ I 1 ) that

χ U (x)λ= χ U (sβuγt)λ χ U (u)μ=1μ=μ.

Notice that λ<μ, we obtain that χ U (x)μ, that is, χ U (x)=1. So, xU. 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 xSΓUΓS. Then x=sβuγt for some s,tS, uU and β,γΓ. Using (IΓ I 2 ), we conclude that

χ U ˜ (x)μ= χ U ˜ (sβuγt)μ χ U ˜ (u)λ=0λ=λ.

Notice that λ<μ, we obtain that χ U ˜ (x)λ, that is, χ U ˜ (x)=0. So, xU. 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,bS.

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 [19], 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.

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Correspondence to Yuming Feng.

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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

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Keywords

  • Γ-semigroup
  • (λ,μ)-fuzzy interior ideal
  • (λ,μ)-fuzzy simple
  • regular ordered Γ-semigroup
  • intra-regular ordered Γ-semigroup