# A Novel of Ideals and Fuzzy Ideals of Gamma-Semigroups

Authors: Anusorn Simuen; Aiyared Iampan; Ronnason Chinram
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we define new types of ideals, fuzzy ideals, almost ideals, and fuzzy almost ideals ofΓ-semigroups by using the elements of Γ. We investigate properties of them.


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## Body

## 1. Introduction and Preliminaries

Ideal theory in semigroups, like all other algebraic structures, plays an important role in studying them. Bi-ideals in semigroups were introduced by Good and Hughes [1] in 1952. Steinfeld [2] gave a notion and studied quasi-ideals in semigroups in 1956. In 1965, Zadeh introduced the concept of fuzzy subsets in [3]. Since then, fuzzy subsets are now applied in various fields. Kuroki [4] first studied fuzzy ideals in semigroups. Almost ideals (A-ideals) in semigroups were studied by Grošek and Satko [5–7] in 1980-1981. Next year, Bogdanović [8] studied almost bi-ideals in semigroups by using concepts of almost ideals and bi-ideals in semigroups. Sen [9] introduced Γ-semigroups where Γ is a set of their operations. This algebraic structure is generalization of semigroups. Many results in semigroup theory were generalized to the results in Γ-semigroup theory. In 2006, Chinram [10] studied quasi-Γ-ideals in Γ-semigroups, and Jirotkul joined with Chinram [11] to study bi-Γ-ideals in Γ-semigroups in 2007. Moreover, Iampan [12] gave remarkable notes on bi-Γ-ideals (or bi-ideals) in Γ-semigroups in 2009. Recently, left almost ideals (left A-ideals) in Γ-semigroups were studied by Wattanatripop and Changphas [13] in 2017. Wattanatripop et al. [14] studied fuzzy almost bi-ideals, almost quasi-ideals, and fuzzy almost ideals in semigroups in 2018. So, all of these give the inspiration to study about new types of ideals and fuzzy ideals in Γ-semigroups in this paper.First, we recall the definition ofΓ-semigroups which was defined by Sen and Saha [15].Definition 1 (see [15]).
LetS and Γ be nonempty sets. Then, S is called a Γ-semigroup if there exists a mapping S×Γ×S⟶S written as a,γ,b⟼aγb satisfying the axiom aαbβc=aαbβc for all a,b,c∈S and α,β∈Γ.Remark 1.
(1)
In caseΓ=1, the definition of Γ-semigroup is a semigroup(2)
Every semigroupS,⋅ can be considered to be a Γ-semigroup where Γ:=⋅(3)
IfS is a Γ-semigroup, then for each α∈Γ, (S,α) is a semigroup
LetS be a Γ-semigroup. For nonempty subsets A,B of S, let(1)AΓB=aαb|a∈A,b∈B,α∈Γ.
Ifx∈S and α∈Γ, we let AΓx:=AΓx, xΓA:=xΓA, and AαB:=AαB.Definition 2.
LetS be a Γ-semigroup. A nonempty subset A of S is called(1)
A sub-Γ-semigroup of S if AΓA⊆A(2)
A left ideal ofS if SΓA⊆A(3)
A right ideal ofS if AΓS⊆A(4)
An ideal ofS if it is both a left ideal and a right ideal of S(5)
A quasi-ideal ofS if SΓA∩AΓS⊆A(6)
A bi-ideal ofS if A is a sub-Γ-semigroup of S and AΓSΓA⊆A
Recently, Wattanatripop and Changphas [13] defined the concepts of left almost ideals and right almost ideals of Γ-semigroups. A Γ-semigroup containing no proper left (respectively, right) almost ideals was characterized.
Now, we recall the definitions and some notations of fuzzy subsets. A fuzzy subset of a setS is a function from S into the closed interval 0,1. For any two fuzzy subsets f and g of a set S,(1)
f∩g is a fuzzy subset of S defined by(2)f∩gx=minfx,gx,for allx∈S.(2)
f∪g is a fuzzy subset of S defined by(3)f∪gx=maxfx,gx,for allx∈S.(3)
f∘g is a fuzzy subset of S defined by(4)f∘gx=supx=abminfa,gb,ifx∈S2,0,otherwise.(4)
f⊆g if fx≤gx for all x∈S.
For a fuzzy subsetf of a set S, the support of f is defined by(5)suppf=x∈S|fx≠0.
The characteristic mapping of a subsetA of S is a fuzzy subset of S defined by(6)CAx=1,ifx∈A,0,otherwise.
The definition of fuzzy points was given by Pu and Liu [16]. For x∈S and t∈0,1, a fuzzy point xt of a set S is a fuzzy subset of S defined by(7)xty=t,ify=x,0,otherwise.
Some basic concepts of fuzzy semigroup theory can be seen in [17].
For aΓ-semigroup S, let ℱS be the set of all fuzzy subsets of S. For each α∈Γ, define a binary operation ∘α on ℱS by(8)f∘αgx=supx=aαbminfa,gb,ifx∈SαS,0,otherwise.
LetΓ⋆:=∘α|α∈Γ. Then, ℱS,Γ⋆ is a Γ-semigroup.
In 2017, Wattanatripop and Changphas [13] defined the concepts of left A-ideals and right A-ideals (almost left ideals and almost right ideals) of a Γ-semigroup as follows.
A nonempty subsetGL [GR] of a Γ-semigroup S is called a left (right) A-ideal of S if(9)sGL∩GL≠∅GRs∩GR≠∅,for alls∈S.
In 1981, Bogdanović [8] gave the definition of almost bi-ideals of semigroups as follows.
A nonempty subsetB of a semigroup S is called an almost bi-ideal of S if(10)BsB∩B≠∅,for alls∈S.
In 2018, Wattanatripop et al. [14, 18] introduced the notions of almost quasi-ideals, fuzzy almost bi-ideals, fuzzy almost left (right) ideals, and fuzzy almost quasi-ideals in semigroups as follows.
A nonempty subsetQ of a semigroup S is called an almost quasi-ideal of S if(11)sQ∩Qs∩Q≠∅,for alls∈S.
Letf be a fuzzy subset of a semigroup S such that f≠0. Then, f is called(1)
A fuzzy almost bi-ideal ofS if for all s∈S,(12)f∘Cs∘f∩f≠0.(2)
A fuzzy almost left ideal ofS if for all s∈S,(13)Cs∘f∩f≠0.(3)
A fuzzy almost right ideal ofS if for all s∈S,(14)f∘Cs∩f≠0.(4)
A fuzzy almost quasi-ideal ofS if for all s∈S,(15)Cs∘f∩f∘Cs∩f≠0.
In 1981, Kuroki [4] introduced the notion of fuzzy ideals of semigroups as follows: A fuzzy subset f of a semigroup S is called:(1)
A fuzzy left ideal ofS if fab≥fb for all a,b∈S.(2)
A fuzzy right ideal ofS if fab≥fa for all a,b∈S.(3)
A fuzzy ideal ofS if it is both a fuzzy left ideal and a fuzzy right ideal of S.
The aim of this paper is to define new types of ideals and fuzzy ideals of aΓ-semigroup S by using elements in Γ. In Section 2, we consider new types of ideals of S. In Section 3, we study new types of fuzzy ideals of S. In Section 4, we consider new types of almost ideals of S. In Section 5, we study new types of fuzzy almost ideals of S.


## 2. New Types of Ideals

In this section, we will focus onα,β-ideals, α,β-quasi-ideals, and α,β-bi-ideals of Γ-semigroups for α,β∈Γ.


### 2.1.α,β-Ideals

First, we will defineα,β-ideals of Γ-semigroups as follows.Definition 3.
LetS be a Γ-semigroup, A be a nonempty subset of S, and α,β∈Γ. Then, A is called(1)
A leftα-ideal of S if SαA⊆A.(2)
A rightβ-ideal of S if AβS⊆A.(3)
Anα,β-ideal of S if it is both a left α-ideal and a right β-ideal of S.(4)
Anα-ideal of S if it is an α,α-ideal of S.Remark 2.
(1)
Every left ideal of aΓ-semigroup S is a left α-ideal of S for all α∈Γ.(2)
Every right ideal of aΓ-semigroup S is a right β-ideal of S for all β∈Γ.(3)
Every ideal of aΓ-semigroup S is an α,β-ideal of S for all α,β∈Γ.
However, the converse of Example1 is not generally true. We can see in the following example.Example 1.
LetS=Γ=ℕ and a,γ,b⟼a+γ+b for all a,b∈S and γ∈Γ. Then, S is a Γ-semigroup. Let A=1∪6,7,8,9,…. It is easy to show that A is a left 4-ideal but not a left ideal of S.Theorem 1.
The following statements are true:(1)
IfL is a left α-ideal of a Γ-semigroup S, then L is a left ideal of a semigroup S,α.(2)
IfR is a right β-ideal of a Γ-semigroup S, then R is a right ideal of a semigroup S,β.(3)
IfI is an α-ideal of a Γ-semigroup S, then I is an ideal of a semigroup S,α.
For a nonempty subsetA of a Γ-semigroup S, let Alα,Arβ, and Aiα,β be the left α-ideal, the right β-ideal, and the α,β-ideal of S generated by A, respectively.Theorem 2.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
Alα=A∪SαA.(2)
Arβ=A∪AβS.(3)
Aiα,β=A∪SαA∪AβS∪SαAβS.Proof.
(1) LetA be a nonempty subset of a Γ-semigroup S. Let L=A∪SαA. Clearly, A⊆L. Since S is a Γ-semigroup, SαL=SαA∪SαA=SαA∪SαSαA=SαA⊆L. Therefore, L is a left α-ideal of S. Next, let C be any left α-ideal of S containing A. Since C is a left α-ideal of S and A⊆C, SαA⊆C. Therefore, L=A∪SαA⊆C. Hence, L is the smallest left α-ideal of S containing A. Therefore, Alα=L=A∪SαA, as required.
The proofs of (2) and (3) are similar to the proof of (1).Theorem 3.
LetL be a left α-ideal and R a right β-ideal of a Γ-semigroup S. Then, LγR is an α,β-ideal of S for all γ∈Γ.Proof.
LetL and R be a left α-ideal and a right β-ideal of S, respectively, and let γ∈Γ. Clear that LγR≠∅. We have SαLγR=SαLγR⊆LγR and LγRβS=LγRβS⊆LγR. Therefore, LγR is an α,β-ideal of S.Theorem 4.
LetL1 and L2 be two left α-ideals of a Γ-semigroup S. The following statements are true:(1)
L1∪L2 is a left α-ideal of S.(2)
L1∩L2 is a left α-ideal of S, where L1∩L2≠∅.Proof.
(1) LetL1 and L2 be two left α-ideals of S. Clear that L1∪L2≠∅. Then, SαL1∪L2⊆SαL1∪SαL2⊆L1∪L2. Hence, L1∪L2 is a left α-ideal of S.
(2) SinceL1∩L2≠∅, we have SαL1∩L2⊆SαL1∩SαL2⊆L1∩L2. Hence, L1∩L2 is a left α-ideals of S.Theorem 5.
LetR1 and R2 be two right β-ideals of a Γ-semigroup S. Then,(1)
R1∪R2 is a right β-ideal of S.(2)
R1∩R2 is a right β-ideal of S, where R1∩R2≠∅.Proof.
It is similar to Theorem4.Theorem 6.
LetI1 and I2 be two α,β-ideals of a Γ-semigroup S. Then,(1)
I1∪I2 is an α,β-ideal of S.(2)
I1∩I2 is an α,β-ideal of S, where I1∩I2≠∅.Proof.
It follows by Theorems4 and 5.


### 2.2.α,β-Quasi-Ideals

We defineα,β-quasi-ideals of Γ-semigroups as follows.Definition 4.
LetS be a Γ-semigroup. A nonempty subset Q of S is called(1)
Anα,β-quasi-ideal of S if SαQ∩QβS⊆Q.(2)
Anα-quasi-ideal of S if it is an α,α-quasi-ideal of S.Theorem 7.
LetS be a Γ-semigroup and Qi an α,β-quasi-ideal of S for all i∈I. If ∩i∈IQi≠∅, then ∩i∈IQi is an α,β-quasi-ideal of S.Proof.
LetS be a Γ-semigroup and Qi an α,β-quasi-ideal of S for all i∈I. Then, Sα∩i∈IQi∩∩i∈IQiβS⊆SαQi∩QiβS⊆Qi for all i∈I, so Sα∩i∈IQi∩∩i∈IQiβS⊆∩i∈IQi. Therefore, ∩i∈IQi is an α,β-quasi-ideal of S.
LetA be a nonempty subset of a Γ-semigroup S. Let Aqα,β be the α,β-quasi-ideal of S generated by A.Theorem 8.
LetA be a nonempty subset of a Γ-semigroup S. Then,(16)Aqα,β=A∪SαA∩AβS.Proof.
LetA be a nonempty subset of a Γ-semigroup S. Let Q=A∪SαA⋂AβS. Clearly, A⊆Q. We have SαQ⋂QβS=SαA∪SαA⋂AβS∩A∪SαA⋂AβSβS⊆Q. Therefore, Q is an α,β-quasi-ideal of S.LetC be any α,β-quasi-ideal of S containing A. Since C is an α,β-quasi-ideal of S and A⊆C, SαA∩AβS⊆C. Therefore, Q=A∪SαA⋂AβS⊆C.Hence,Q is the smallest α,β-quasi-ideal of S containing A. Therefore, Aqα,β=Q=A∪SαA⋂AβS, as required.Theorem 9.
LetS be a Γ-semigroup. Let L and R be a left α-ideal and right β-ideal of S, respectively. If L∩R≠∅, then L∩R is an α,β-quasi-ideal of S.Proof.
LetL and R be a left α-ideal and a right β-ideal of a Γ-semigroup S, respectively. Then, SαL∩R∩L∩RβS⊆SαL∩RβS⊆L∩R. Hence, L∩R is an α,β-quasi-ideal of S.Corollary 1.
LetS be a Γ-semigroup. Let L and R be a left α-ideal and right α-ideal of S, respectively. Then, L∩R is an α-quasi-ideal of S.Proof.
We haveRαL⊆L∩R; this implies L∩R≠∅. By Theorem 9, L∩R is an α-quasi-ideal of S.Theorem 10.
Everyα,β-quasi-ideal Q of a Γ-semigroup S is the intersection of a left α-ideal and a right β-ideal of S.Proof.
LetQ be an α,β-quasi-ideal of a Γ-semigroup S. Let L=Q∪SαQ and R=Q∪QβS. Then, SαL=SαQ∪SαQ=SαQ∪SαSαQ⊆SαQ⊆L, and also, RβS⊆R. We have that L∩R=Q∪SαQ∩Q∪QβS=Q∪SαQ∩QβS=Q. Therefore, Q=L∩R.Definition 5.
AΓ-semigroup S is called α,β-quasi-simple if S does not contain proper α,β-quasi-ideals.
AΓ-semigroup S is called α-quasi-simple if S is α,α-quasi-simple.Theorem 11.
LetS be a Γ-semigroup. Then, S is α-quasi-simple if and only if Sαs∩sαS=S for all s∈S.Proof.
Assume thatS is α-quasi-simple. Let s∈S; we claim that Sαs∩sαS is an α-quasi-ideal of S. We have sαs∈Sαs∩sαS; this implies Sαs∩sαS≠∅. Moreover, SαSαs∩sαS∩Sαs∩sαSαS⊆SαSαS∩SαsαS∩SαSαS∩sαSαS⊆SαSαs∩sαSαS = SαSαs∩sαSαS⊆Sαs∩sαS. Therefore, Sαs∩sαS is an α-quasi-ideal of S. Since S is α-quasi-simple, we have S=Sαs∩sαS.
Conversely, assume thatSαs∩sαS=S for all s∈S. Let Q be an α-quasi-ideal of S and q∈Q. By assumption, S=Sαq∩qαS. Since Q is an α-quasi-ideal of S, S=Sαq∩qαS⊆SαQ∩QαS⊆Q. Then, Q=S. Therefore, S is α-quasi-simple.Definition 6.
Anα,β-quasi-ideal Q of a Γ-semigroup S is called minimal if for all α,β-quasi-ideal C of S, if C⊆Q, then C=Q.Theorem 12.
LetS be a Γ-semigroup and Q an α,β-quasi-ideal of S. If Q is α,β-quasi-simple, then Q is a minimal α,β-quasi-ideal of S.Proof.
Assume thatS is a Γ-semigroup and Q an α,β-quasi-ideal of S. Let Q be α,β-quasi-simple. Let C be an α,β-quasi-ideal of S such that C⊆Q. Then, QαC∩CβQ⊆SαC∩CβS⊆C. Therefore, C is an α,β-quasi-ideal of Q. Since Q is α,β-quasi-simple, C=Q. Hence, Q is a minimal α,β-quasi-ideal of S.


### 2.3.α,β-Bi-Ideals

We will defineα,β-bi-ideals of Γ-semigroups as follows.Definition 7.
LetS be a Γ-semigroup and α,β∈Γ. A nonempty subset B of S is called(1)
Anα,β-bi-ideal of S if BαSβB⊆B.(2)
Anα-bi-ideal of S if it is an α,α-bi-ideal of S.Theorem 13.
Everyα,β-quasi-ideal of a Γ-semigroup S is a β,α-bi-ideal of S.Proof.
LetQ be an α,β-quasi-ideal of S. Then,(17)QβSαQ⊆QβS∩SαQ⊆Q.
Hence,Q is a β,α-bi-ideal of S.Theorem 14.
LetS be a Γ-semigroup and Bi an α,β-bi-ideal of S for all i∈I. If ∩i∈IBi≠∅, then ∩i∈IBi is an α,β-bi-ideal of S.Proof.
LetS be a Γ-semigroup and Bi an α,β-bi-ideal of S for all i∈I. Then, ⋂i∈IBiαSβ⋂i∈IBi⊆BiαSβBi⊆Bi for all i∈I, so ⋂i∈IBiαSβ⋂i∈IBi⊆⋂i∈IBi. Therefore, ⋂i∈IBi is an α,β-bi-ideal of S.
LetA be a nonempty subset of a Γ-semigroup S, and let Abα,β denote the α,β-bi-ideal of S generated by A.Theorem 15.
LetA be a nonempty subset of a Γ-semigroup S and α,β∈Γ. Then,(18)Abα,β=A∪AαSβA.Proof.
LetA be a nonempty subset of a Γ-semigroup S. Let B=A∪AαSβA. Clearly, A⊆B. We have that BαSβB=A∪AαSβAαSβA∪AαSβA⊆B. Therefore, B is an α,β-bi-ideal of S.
LetC be any α,β-bi-ideal of S containing A. Since C is an α,β-bi-ideal of S and A⊆C, AαSβA⊆C. Therefore, B=A∪AαSβA⊆C.
Hence,B is the smallest α,β-bi-ideal of S containing A. Therefore, Abα,β=B=A∪AαSβA.Theorem 16.
LetS be a Γ-semigroup, A a nonempty subset of S, and B an α,β-bi-ideal of S. The following statements are true:(1)
BαA is an α,β-bi-ideal of S.(2)
AβB is an α,β-bi-ideal of S.Proof.
We have that(19)BαAαSβBαA=BαAαSβBαA⊆BαSβBαA⊆BαA.
Then,BαA is an α,β-bi-ideal of S. Similarly, AβB is an α,β-bi-ideal of S.Theorem 17.
LetS be a Γ-semigroup. Assume that B1,B2, and B3 are α,β-bi-ideals of S. Then, B1αB2βB3 is an α,β-bi-ideal of S.Proof.
SinceB1,B2, and B3 are α,β-bi-ideals of S, we have(20)B1αB2βB3αB1αB2βB3⊆B1αB2βB3,B1αB2βB3βB1αB2βB3⊆B1αB2βB3.
Then,(21)B1αB2βB3αSβB1αB2βB3=B1αB2βB3αSβB1αB2βB3⊆B1αSβB1αB2βB3⊆B1αB2βB3.
Therefore,B1αB2βB3 is an α,β-bi-ideal of S.Definition 8.
AΓ-semigroup S is called α,β-bi-simple if S does not contain proper α,β-bi-ideals.Theorem 18.
LetS be a Γ-semigroup. Then, S is α,β-bi-simple if and only if sαSβs=S for all s∈S.Proof.
Assume thatS is α,β-bi-simple. Let s∈S. We claim that sαSβs is an α,β-bi-ideal of S. We have sαsβs∈sαSβs; this implies that sαSβs≠∅. Moreover, sαSβsαSβsαSβs⊆sαSβSαSβs⊆sαSβs. Therefore, sαSβs is an α,β-bi-ideal of S. Since S is α,β-bi-simple, S=sαSβs.
Conversely, assume thatsαSβs=S for all s∈S. Let B be an α,β-bi-ideal of S and b∈B. By assumption, S=bαSβb. Since B is an α,β-bi-ideal of S, bαSβb⊆BαSβB⊆B. Then, B=S. Therefore, S is α,β-bi-simple.Definition 9.
Anα,β-bi-ideal B of a Γ-semigroup S is called minimal if for all α,β-bi-ideal C of S, if C⊆B, then C=B.Theorem 19.
LetS be a Γ-semigroup and B an α,β-bi-ideal of S. If B is α,β-bi-simple, then B is a minimal α,β-bi-ideal of S.Proof.
Assume thatS is a Γ-semigroup and B an α,β-bi-ideal of S. Let B be α,β-bi-simple. Let C be an α,β-bi-ideal of S such that C⊆B. Then, CαBβC⊆CαSβC⊆C. Therefore, C is an α,β-bi-ideal of B. Since B is α,β-bi-simple, C=B. Then, B is a minimal α,β-bi-ideal of S.


## 2.1.α,β-Ideals

First, we will defineα,β-ideals of Γ-semigroups as follows.Definition 3.
LetS be a Γ-semigroup, A be a nonempty subset of S, and α,β∈Γ. Then, A is called(1)
A leftα-ideal of S if SαA⊆A.(2)
A rightβ-ideal of S if AβS⊆A.(3)
Anα,β-ideal of S if it is both a left α-ideal and a right β-ideal of S.(4)
Anα-ideal of S if it is an α,α-ideal of S.Remark 2.
(1)
Every left ideal of aΓ-semigroup S is a left α-ideal of S for all α∈Γ.(2)
Every right ideal of aΓ-semigroup S is a right β-ideal of S for all β∈Γ.(3)
Every ideal of aΓ-semigroup S is an α,β-ideal of S for all α,β∈Γ.
However, the converse of Example1 is not generally true. We can see in the following example.Example 1.
LetS=Γ=ℕ and a,γ,b⟼a+γ+b for all a,b∈S and γ∈Γ. Then, S is a Γ-semigroup. Let A=1∪6,7,8,9,…. It is easy to show that A is a left 4-ideal but not a left ideal of S.Theorem 1.
The following statements are true:(1)
IfL is a left α-ideal of a Γ-semigroup S, then L is a left ideal of a semigroup S,α.(2)
IfR is a right β-ideal of a Γ-semigroup S, then R is a right ideal of a semigroup S,β.(3)
IfI is an α-ideal of a Γ-semigroup S, then I is an ideal of a semigroup S,α.
For a nonempty subsetA of a Γ-semigroup S, let Alα,Arβ, and Aiα,β be the left α-ideal, the right β-ideal, and the α,β-ideal of S generated by A, respectively.Theorem 2.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
Alα=A∪SαA.(2)
Arβ=A∪AβS.(3)
Aiα,β=A∪SαA∪AβS∪SαAβS.Proof.
(1) LetA be a nonempty subset of a Γ-semigroup S. Let L=A∪SαA. Clearly, A⊆L. Since S is a Γ-semigroup, SαL=SαA∪SαA=SαA∪SαSαA=SαA⊆L. Therefore, L is a left α-ideal of S. Next, let C be any left α-ideal of S containing A. Since C is a left α-ideal of S and A⊆C, SαA⊆C. Therefore, L=A∪SαA⊆C. Hence, L is the smallest left α-ideal of S containing A. Therefore, Alα=L=A∪SαA, as required.
The proofs of (2) and (3) are similar to the proof of (1).Theorem 3.
LetL be a left α-ideal and R a right β-ideal of a Γ-semigroup S. Then, LγR is an α,β-ideal of S for all γ∈Γ.Proof.
LetL and R be a left α-ideal and a right β-ideal of S, respectively, and let γ∈Γ. Clear that LγR≠∅. We have SαLγR=SαLγR⊆LγR and LγRβS=LγRβS⊆LγR. Therefore, LγR is an α,β-ideal of S.Theorem 4.
LetL1 and L2 be two left α-ideals of a Γ-semigroup S. The following statements are true:(1)
L1∪L2 is a left α-ideal of S.(2)
L1∩L2 is a left α-ideal of S, where L1∩L2≠∅.Proof.
(1) LetL1 and L2 be two left α-ideals of S. Clear that L1∪L2≠∅. Then, SαL1∪L2⊆SαL1∪SαL2⊆L1∪L2. Hence, L1∪L2 is a left α-ideal of S.
(2) SinceL1∩L2≠∅, we have SαL1∩L2⊆SαL1∩SαL2⊆L1∩L2. Hence, L1∩L2 is a left α-ideals of S.Theorem 5.
LetR1 and R2 be two right β-ideals of a Γ-semigroup S. Then,(1)
R1∪R2 is a right β-ideal of S.(2)
R1∩R2 is a right β-ideal of S, where R1∩R2≠∅.Proof.
It is similar to Theorem4.Theorem 6.
LetI1 and I2 be two α,β-ideals of a Γ-semigroup S. Then,(1)
I1∪I2 is an α,β-ideal of S.(2)
I1∩I2 is an α,β-ideal of S, where I1∩I2≠∅.Proof.
It follows by Theorems4 and 5.


## 2.2.α,β-Quasi-Ideals

We defineα,β-quasi-ideals of Γ-semigroups as follows.Definition 4.
LetS be a Γ-semigroup. A nonempty subset Q of S is called(1)
Anα,β-quasi-ideal of S if SαQ∩QβS⊆Q.(2)
Anα-quasi-ideal of S if it is an α,α-quasi-ideal of S.Theorem 7.
LetS be a Γ-semigroup and Qi an α,β-quasi-ideal of S for all i∈I. If ∩i∈IQi≠∅, then ∩i∈IQi is an α,β-quasi-ideal of S.Proof.
LetS be a Γ-semigroup and Qi an α,β-quasi-ideal of S for all i∈I. Then, Sα∩i∈IQi∩∩i∈IQiβS⊆SαQi∩QiβS⊆Qi for all i∈I, so Sα∩i∈IQi∩∩i∈IQiβS⊆∩i∈IQi. Therefore, ∩i∈IQi is an α,β-quasi-ideal of S.
LetA be a nonempty subset of a Γ-semigroup S. Let Aqα,β be the α,β-quasi-ideal of S generated by A.Theorem 8.
LetA be a nonempty subset of a Γ-semigroup S. Then,(16)Aqα,β=A∪SαA∩AβS.Proof.
LetA be a nonempty subset of a Γ-semigroup S. Let Q=A∪SαA⋂AβS. Clearly, A⊆Q. We have SαQ⋂QβS=SαA∪SαA⋂AβS∩A∪SαA⋂AβSβS⊆Q. Therefore, Q is an α,β-quasi-ideal of S.LetC be any α,β-quasi-ideal of S containing A. Since C is an α,β-quasi-ideal of S and A⊆C, SαA∩AβS⊆C. Therefore, Q=A∪SαA⋂AβS⊆C.Hence,Q is the smallest α,β-quasi-ideal of S containing A. Therefore, Aqα,β=Q=A∪SαA⋂AβS, as required.Theorem 9.
LetS be a Γ-semigroup. Let L and R be a left α-ideal and right β-ideal of S, respectively. If L∩R≠∅, then L∩R is an α,β-quasi-ideal of S.Proof.
LetL and R be a left α-ideal and a right β-ideal of a Γ-semigroup S, respectively. Then, SαL∩R∩L∩RβS⊆SαL∩RβS⊆L∩R. Hence, L∩R is an α,β-quasi-ideal of S.Corollary 1.
LetS be a Γ-semigroup. Let L and R be a left α-ideal and right α-ideal of S, respectively. Then, L∩R is an α-quasi-ideal of S.Proof.
We haveRαL⊆L∩R; this implies L∩R≠∅. By Theorem 9, L∩R is an α-quasi-ideal of S.Theorem 10.
Everyα,β-quasi-ideal Q of a Γ-semigroup S is the intersection of a left α-ideal and a right β-ideal of S.Proof.
LetQ be an α,β-quasi-ideal of a Γ-semigroup S. Let L=Q∪SαQ and R=Q∪QβS. Then, SαL=SαQ∪SαQ=SαQ∪SαSαQ⊆SαQ⊆L, and also, RβS⊆R. We have that L∩R=Q∪SαQ∩Q∪QβS=Q∪SαQ∩QβS=Q. Therefore, Q=L∩R.Definition 5.
AΓ-semigroup S is called α,β-quasi-simple if S does not contain proper α,β-quasi-ideals.
AΓ-semigroup S is called α-quasi-simple if S is α,α-quasi-simple.Theorem 11.
LetS be a Γ-semigroup. Then, S is α-quasi-simple if and only if Sαs∩sαS=S for all s∈S.Proof.
Assume thatS is α-quasi-simple. Let s∈S; we claim that Sαs∩sαS is an α-quasi-ideal of S. We have sαs∈Sαs∩sαS; this implies Sαs∩sαS≠∅. Moreover, SαSαs∩sαS∩Sαs∩sαSαS⊆SαSαS∩SαsαS∩SαSαS∩sαSαS⊆SαSαs∩sαSαS = SαSαs∩sαSαS⊆Sαs∩sαS. Therefore, Sαs∩sαS is an α-quasi-ideal of S. Since S is α-quasi-simple, we have S=Sαs∩sαS.
Conversely, assume thatSαs∩sαS=S for all s∈S. Let Q be an α-quasi-ideal of S and q∈Q. By assumption, S=Sαq∩qαS. Since Q is an α-quasi-ideal of S, S=Sαq∩qαS⊆SαQ∩QαS⊆Q. Then, Q=S. Therefore, S is α-quasi-simple.Definition 6.
Anα,β-quasi-ideal Q of a Γ-semigroup S is called minimal if for all α,β-quasi-ideal C of S, if C⊆Q, then C=Q.Theorem 12.
LetS be a Γ-semigroup and Q an α,β-quasi-ideal of S. If Q is α,β-quasi-simple, then Q is a minimal α,β-quasi-ideal of S.Proof.
Assume thatS is a Γ-semigroup and Q an α,β-quasi-ideal of S. Let Q be α,β-quasi-simple. Let C be an α,β-quasi-ideal of S such that C⊆Q. Then, QαC∩CβQ⊆SαC∩CβS⊆C. Therefore, C is an α,β-quasi-ideal of Q. Since Q is α,β-quasi-simple, C=Q. Hence, Q is a minimal α,β-quasi-ideal of S.


## 2.3.α,β-Bi-Ideals

We will defineα,β-bi-ideals of Γ-semigroups as follows.Definition 7.
LetS be a Γ-semigroup and α,β∈Γ. A nonempty subset B of S is called(1)
Anα,β-bi-ideal of S if BαSβB⊆B.(2)
Anα-bi-ideal of S if it is an α,α-bi-ideal of S.Theorem 13.
Everyα,β-quasi-ideal of a Γ-semigroup S is a β,α-bi-ideal of S.Proof.
LetQ be an α,β-quasi-ideal of S. Then,(17)QβSαQ⊆QβS∩SαQ⊆Q.
Hence,Q is a β,α-bi-ideal of S.Theorem 14.
LetS be a Γ-semigroup and Bi an α,β-bi-ideal of S for all i∈I. If ∩i∈IBi≠∅, then ∩i∈IBi is an α,β-bi-ideal of S.Proof.
LetS be a Γ-semigroup and Bi an α,β-bi-ideal of S for all i∈I. Then, ⋂i∈IBiαSβ⋂i∈IBi⊆BiαSβBi⊆Bi for all i∈I, so ⋂i∈IBiαSβ⋂i∈IBi⊆⋂i∈IBi. Therefore, ⋂i∈IBi is an α,β-bi-ideal of S.
LetA be a nonempty subset of a Γ-semigroup S, and let Abα,β denote the α,β-bi-ideal of S generated by A.Theorem 15.
LetA be a nonempty subset of a Γ-semigroup S and α,β∈Γ. Then,(18)Abα,β=A∪AαSβA.Proof.
LetA be a nonempty subset of a Γ-semigroup S. Let B=A∪AαSβA. Clearly, A⊆B. We have that BαSβB=A∪AαSβAαSβA∪AαSβA⊆B. Therefore, B is an α,β-bi-ideal of S.
LetC be any α,β-bi-ideal of S containing A. Since C is an α,β-bi-ideal of S and A⊆C, AαSβA⊆C. Therefore, B=A∪AαSβA⊆C.
Hence,B is the smallest α,β-bi-ideal of S containing A. Therefore, Abα,β=B=A∪AαSβA.Theorem 16.
LetS be a Γ-semigroup, A a nonempty subset of S, and B an α,β-bi-ideal of S. The following statements are true:(1)
BαA is an α,β-bi-ideal of S.(2)
AβB is an α,β-bi-ideal of S.Proof.
We have that(19)BαAαSβBαA=BαAαSβBαA⊆BαSβBαA⊆BαA.
Then,BαA is an α,β-bi-ideal of S. Similarly, AβB is an α,β-bi-ideal of S.Theorem 17.
LetS be a Γ-semigroup. Assume that B1,B2, and B3 are α,β-bi-ideals of S. Then, B1αB2βB3 is an α,β-bi-ideal of S.Proof.
SinceB1,B2, and B3 are α,β-bi-ideals of S, we have(20)B1αB2βB3αB1αB2βB3⊆B1αB2βB3,B1αB2βB3βB1αB2βB3⊆B1αB2βB3.
Then,(21)B1αB2βB3αSβB1αB2βB3=B1αB2βB3αSβB1αB2βB3⊆B1αSβB1αB2βB3⊆B1αB2βB3.
Therefore,B1αB2βB3 is an α,β-bi-ideal of S.Definition 8.
AΓ-semigroup S is called α,β-bi-simple if S does not contain proper α,β-bi-ideals.Theorem 18.
LetS be a Γ-semigroup. Then, S is α,β-bi-simple if and only if sαSβs=S for all s∈S.Proof.
Assume thatS is α,β-bi-simple. Let s∈S. We claim that sαSβs is an α,β-bi-ideal of S. We have sαsβs∈sαSβs; this implies that sαSβs≠∅. Moreover, sαSβsαSβsαSβs⊆sαSβSαSβs⊆sαSβs. Therefore, sαSβs is an α,β-bi-ideal of S. Since S is α,β-bi-simple, S=sαSβs.
Conversely, assume thatsαSβs=S for all s∈S. Let B be an α,β-bi-ideal of S and b∈B. By assumption, S=bαSβb. Since B is an α,β-bi-ideal of S, bαSβb⊆BαSβB⊆B. Then, B=S. Therefore, S is α,β-bi-simple.Definition 9.
Anα,β-bi-ideal B of a Γ-semigroup S is called minimal if for all α,β-bi-ideal C of S, if C⊆B, then C=B.Theorem 19.
LetS be a Γ-semigroup and B an α,β-bi-ideal of S. If B is α,β-bi-simple, then B is a minimal α,β-bi-ideal of S.Proof.
Assume thatS is a Γ-semigroup and B an α,β-bi-ideal of S. Let B be α,β-bi-simple. Let C be an α,β-bi-ideal of S such that C⊆B. Then, CαBβC⊆CαSβC⊆C. Therefore, C is an α,β-bi-ideal of B. Since B is α,β-bi-simple, C=B. Then, B is a minimal α,β-bi-ideal of S.


## 3. New Types of Fuzzy Ideals

### 3.1. Fuzzyα,β-Ideals

We will define fuzzyα,β-ideals of Γ-semigroups as follows.Definition 10.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called(1)
A fuzzy leftα-ideal of S if fxαy≥fy for all x,y∈S.(2)
A fuzzy rightβ-ideal of S if fxβy≥fx for all x,y∈S.(3)
A fuzzyα,β-ideal of S if it is both a fuzzy left α-ideal and a fuzzy right β-ideal of S.
The following theorems show the relationship betweenα,β-ideals and fuzzy α,β-ideals.Theorem 20.
LetA be a nonempty subset of a Γ-semigroup S. Then, the following statements are true:(1)
A is a left α-ideal of S if and only if CA is a fuzzy left α-ideal of S.(2)
A is a right β-ideal of S if and only if CA is a fuzzy right β-ideal of S.(3)
A is an α,β-ideal of S if and only if CA is a fuzzy α,β-ideal of S.Proof.
(1) Suppose thatA is a left α-ideal of S. Let x,y∈S.
Ify∈A, then xαy∈A. Thus, CAxαy=1 so that CAxαy≥CAy.
Ify∉A, then CAy=0≤CAxαy.
Therefore,CA is a fuzzy left α-ideal of S.
Conversely, assume thatCA is a fuzzy left α-ideal of S. Let x∈S and y∈A. Then, CAy=1. Since CAxαy≥CAy, we have xαy∈A. Hence, A is a left α-ideal of S.
(2) is similar to (1).
(3) follows by (1) and (2).Theorem 21.
Letf be a fuzzy subset of a Γ-semigroup S. Then, the following properties hold:(1)
f is a fuzzy left α-ideal of S if and only if S∘αf⊆f.(2)
f is a fuzzy right β-ideal of S if and only if f∘βS⊆f.(3)
f is a fuzzy (α,β)-ideal of S if and only if S∘αf⊆f and f∘βS⊆f.Proof.
(1) Assume thatf is a fuzzy left α-ideal of a Γ-semigroup S. Let x∈S and α∈Γ.
Ifx∉SαS, then S∘αfx=0. So, S∘αfx≤fx.
Ifx∈SαS, then there exist y,z∈S such that x=yαz. Since f is a fuzzy left α-ideal of S, we have(22)S∘αfx=supx=yαzminSy,fz≤minSy,fz=fz≤fyαz=fx.
We conclude thatS∘αf⊆f.
Conversely, assume thatS∘αf⊆f. Let x,y,z∈S be such that x=yαz. Then,(23)fyαz=fx≥S∘αfx=supx=yαzminSy,fz≥minSy,fz=min1,fz=fz.
Hence,f is a fuzzy left α-ideal of S.
(2) and (3) can be seen in a similar fashion.Theorem 22.
Letf and g be fuzzy left α-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy left α-ideal of S.(2)
f∪g is a fuzzy left α-ideal of S.Proof.
Letx,y∈S. Then,(24)f∩gxαy=minfxαy,gxαy≥minfy,gy=f∩gy.
Hence,f∩g is a fuzzy left α-ideal of S. Next,(25)f∪gxαy=maxfxαy,gxαy≥maxfy,gy=f∪gy.
Hence,f∪g is a fuzzy left α-ideal of S.Theorem 23.
Letf and g be fuzzy right β-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy right β-ideal of S.(2)
f∪g is a fuzzy right β-ideal of S.Proof.
It is similar to Theorem22.Theorem 24.
Letf and g be fuzzy α,β-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy α,β-ideal of S.(2)
f∪g is a fuzzy α,β-ideal of S.Proof.
It follows by Theorems22 and 23.Theorem 25.
Letf be a nonzero fuzzy subset of a Γ-semigroup S and ft=x∈S|fx≥t. The following statements are true:(1)
f is a fuzzy left α-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is a left α-ideal of S.(2)
f is a fuzzy right β-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is a right β-ideal of S.(3)
f is a fuzzy α,β-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is an α,β-ideal of S.Proof.
(1) Suppose thatf is a fuzzy left α-ideal of S. Then, fxαy≥fy for all x,y∈S. Let t∈0,1 be such that ft≠∅. Let x∈ft and s∈S. Since fx≥t, fsαx≥fx≥t. Thus, sαx∈ft. Hence, ft is a left α-ideal of S.
Conversely, assume thatft is a left α-ideal of S if t∈0,1 and ft≠∅. Let x,y∈S and t=fy. Since fy≥t, we have y∈ft so that ft≠∅. Thus, ft is a left α-ideal of S. Since y∈ft and x∈S, we have xαy∈ft. Then, fxαy≥t=fy. Hence, f is a fuzzy left α-ideal of S.
(2) is similar to (1).
(3) follows by (1) and (2).


### 3.2. Fuzzyα,β-Quasi-Ideals

We will define fuzzyα,β-quasi-ideals of Γ-semigroups as follows.Definition 11.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy α,β-quasi-ideal of S if S∘αf∩f∘βS⊆f.
A fuzzy subsetf of S is called a fuzzy α-quasi-ideal of S if f is a fuzzy α,β-quasi-ideal of S.Theorem 26.
LetS be a Γ-semigroup. Let f and g be a fuzzy left α-ideal and a fuzzy right α-ideal of S, respectively. Then, f∩g is a fuzzy α-quasi-ideal of S.Proof.
Letf and g be a fuzzy left α-ideal and a fuzzy right α-ideal of a Γ-semigroup S, respectively. We have g∘αf⊆f∩g; this implies f∩g≠∅. Then, S∘αf∩g∩f∩g∘αS⊆S∘αf∩f∘αS⊆f∩g. Hence, f∩g is a fuzzy α-quasi-ideal of S.Theorem 27.
Every fuzzy (α,β)-quasi-ideal of a Γ-semigroup S is the intersection of a fuzzy left α-ideal and a fuzzy right β-ideal of S.Proof.
Letf be a fuzzy (α,β)-quasi-ideal of a Γ-semigroup S. Let g=f∪S∘αf and h=f∪f∘βS. Then, S∘αg=S∘αf∪S∘αf=S∘αf∪S∘αS∘αf⊆S∘αf⊆g, and also, h∘βS⊆h. Thus, g and h are a fuzzy left α-ideal and a fuzzy right β-ideal of S, respectively. We claim that f=g∩h. That is, f⊆f∪S∘αf∩f∪f∘βS⊆g∩h, and conversely, g∩h=f∪S∘αf∩f∪f∘βS⊆f∪S∘αf∩f∘βS⊆f. Therefore, f=g∩h.Theorem 28.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is an α,β-quasi-ideal of S if and only if CQ is a fuzzy α,β-quasi-ideal of S.Proof.
Assume thatQ is an α,β-quasi-ideal of a Γ-semigroup S. If y∉SαQ∩QβS, S∘αCQ∩CQ∘βSy=0≤CQy. Let y∈SαQ∩QβS. Then, y∈Q. So CQy=1. Hence, S∘αCQ∩CQ∘βS⊆CQ. Therefore, CQ is a fuzzy α,β-quasi-ideal of S.
Conversely, assume thatCQ is a fuzzy α,β-quasi-ideal of S. Then, S∘αCQ∩CQ∘βS⊆CQ. Let x∈SαQ∩QβS. Then, S∘αCQ∩CQ∘βSx=1, and this implies that CQx=1. So, sαQ∩Qβs⊆Q. Consequently, Q is an α,β-quasi-ideal of S.


### 3.3. Fuzzyα,β-Bi-Ideals

Definition 12.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy α,β-bi-ideal of S if f∘αS∘βf⊆f.Theorem 29.
LetS be a Γ-semigroup, g a fuzzy subset of S, and f a fuzzy α,β-bi-ideal of S. Then, the following statements are true:(1)
f∘αg is a fuzzy α,β-bi-ideal of S.(2)
g∘βf is a fuzzy α,β-bi-ideal of S.Proof.
(1) Letxt be a fuzzy point of S. Then,(26)f∘αg∘αS∘βf∘αg=f∘αg∘αxt∘βf∘αg⊆f∘αxt∘βf∘αg⊆f∘αg.
Hence,f∘αg is a fuzzy α,β-bi-ideal of S.
(2) is similar to (1).Theorem 30.
LetS be a Γ-semigroup. If f1,f2 and f3 are fuzzy α,β-bi-ideals of S, then f1∘αf2∘βf3 is a fuzzy bi-(α,β)-ideal of S.Proof.
Letxt be a fuzzy point of S. Then,(27)f1∘αf2∘βf3∘αxt∘βf1∘αf2∘βf3⊆f1∘αf2∘βS∘αf2∘βf3⊆f1∘αf2∘βf3.
Hence,f∘αf2∘βf3 is a fuzzy α,β-bi-ideal of S.Theorem 31.
LetS be a Γ-semigroup and B a nonempty subset of S. Then, B is an α,β-bi-ideal of S if and only if CB is a fuzzy α,β-bi-ideal of S.Proof.
Assume thatB is an α,β-bi-ideal of S. Then, BαSβB⊆B. If z∉BαSβB, we have CB∘αS∘βCBz=0≤CBz. Let z∈BαSβB. Then, z∈B. Thus, CB∘αS∘βCBz=1; then, CBz=1. This implies that CB∘αS∘βCB⊆CB. Hence, CB is a fuzzy α,β-bi-ideal of S.
Conversely, assume thatCB is a fuzzy α,β-bi-ideal of S. Then, CB∘αS∘βCB⊆CB. Let z∈BαSβB. Then, CB∘αS∘βCBz=1; thus, CBz=1. Hence, z∈B so that BαSβB⊆B. Thus, B is an α,β-bi-ideal of S.Theorem 32.
Letf and g be two fuzzy (α,β)-bi-ideals of a Γ-semigroup S. If f∩g≠∅, then f∩g is a fuzzy (α,β)-bi-ideal of S.Proof.
Letf and g be fuzzy (α,β)-bi-ideals of a Γ-semigroup S. Then, f∩g∘αS∘βf∩g⊆f∘αS∘βf∩f∘αS∘βf∩g⊆f∩g. Hence, f∩g is a fuzzy α,β-bi-ideal of S.


## 3.1. Fuzzyα,β-Ideals

We will define fuzzyα,β-ideals of Γ-semigroups as follows.Definition 10.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called(1)
A fuzzy leftα-ideal of S if fxαy≥fy for all x,y∈S.(2)
A fuzzy rightβ-ideal of S if fxβy≥fx for all x,y∈S.(3)
A fuzzyα,β-ideal of S if it is both a fuzzy left α-ideal and a fuzzy right β-ideal of S.
The following theorems show the relationship betweenα,β-ideals and fuzzy α,β-ideals.Theorem 20.
LetA be a nonempty subset of a Γ-semigroup S. Then, the following statements are true:(1)
A is a left α-ideal of S if and only if CA is a fuzzy left α-ideal of S.(2)
A is a right β-ideal of S if and only if CA is a fuzzy right β-ideal of S.(3)
A is an α,β-ideal of S if and only if CA is a fuzzy α,β-ideal of S.Proof.
(1) Suppose thatA is a left α-ideal of S. Let x,y∈S.
Ify∈A, then xαy∈A. Thus, CAxαy=1 so that CAxαy≥CAy.
Ify∉A, then CAy=0≤CAxαy.
Therefore,CA is a fuzzy left α-ideal of S.
Conversely, assume thatCA is a fuzzy left α-ideal of S. Let x∈S and y∈A. Then, CAy=1. Since CAxαy≥CAy, we have xαy∈A. Hence, A is a left α-ideal of S.
(2) is similar to (1).
(3) follows by (1) and (2).Theorem 21.
Letf be a fuzzy subset of a Γ-semigroup S. Then, the following properties hold:(1)
f is a fuzzy left α-ideal of S if and only if S∘αf⊆f.(2)
f is a fuzzy right β-ideal of S if and only if f∘βS⊆f.(3)
f is a fuzzy (α,β)-ideal of S if and only if S∘αf⊆f and f∘βS⊆f.Proof.
(1) Assume thatf is a fuzzy left α-ideal of a Γ-semigroup S. Let x∈S and α∈Γ.
Ifx∉SαS, then S∘αfx=0. So, S∘αfx≤fx.
Ifx∈SαS, then there exist y,z∈S such that x=yαz. Since f is a fuzzy left α-ideal of S, we have(22)S∘αfx=supx=yαzminSy,fz≤minSy,fz=fz≤fyαz=fx.
We conclude thatS∘αf⊆f.
Conversely, assume thatS∘αf⊆f. Let x,y,z∈S be such that x=yαz. Then,(23)fyαz=fx≥S∘αfx=supx=yαzminSy,fz≥minSy,fz=min1,fz=fz.
Hence,f is a fuzzy left α-ideal of S.
(2) and (3) can be seen in a similar fashion.Theorem 22.
Letf and g be fuzzy left α-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy left α-ideal of S.(2)
f∪g is a fuzzy left α-ideal of S.Proof.
Letx,y∈S. Then,(24)f∩gxαy=minfxαy,gxαy≥minfy,gy=f∩gy.
Hence,f∩g is a fuzzy left α-ideal of S. Next,(25)f∪gxαy=maxfxαy,gxαy≥maxfy,gy=f∪gy.
Hence,f∪g is a fuzzy left α-ideal of S.Theorem 23.
Letf and g be fuzzy right β-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy right β-ideal of S.(2)
f∪g is a fuzzy right β-ideal of S.Proof.
It is similar to Theorem22.Theorem 24.
Letf and g be fuzzy α,β-ideals of a Γ-semigroup S. Then,(1)
f∩g is a fuzzy α,β-ideal of S.(2)
f∪g is a fuzzy α,β-ideal of S.Proof.
It follows by Theorems22 and 23.Theorem 25.
Letf be a nonzero fuzzy subset of a Γ-semigroup S and ft=x∈S|fx≥t. The following statements are true:(1)
f is a fuzzy left α-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is a left α-ideal of S.(2)
f is a fuzzy right β-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is a right β-ideal of S.(3)
f is a fuzzy α,β-ideal of S if and only if for all t∈0,1, if ft is a nonempty set, then ft is an α,β-ideal of S.Proof.
(1) Suppose thatf is a fuzzy left α-ideal of S. Then, fxαy≥fy for all x,y∈S. Let t∈0,1 be such that ft≠∅. Let x∈ft and s∈S. Since fx≥t, fsαx≥fx≥t. Thus, sαx∈ft. Hence, ft is a left α-ideal of S.
Conversely, assume thatft is a left α-ideal of S if t∈0,1 and ft≠∅. Let x,y∈S and t=fy. Since fy≥t, we have y∈ft so that ft≠∅. Thus, ft is a left α-ideal of S. Since y∈ft and x∈S, we have xαy∈ft. Then, fxαy≥t=fy. Hence, f is a fuzzy left α-ideal of S.
(2) is similar to (1).
(3) follows by (1) and (2).


## 3.2. Fuzzyα,β-Quasi-Ideals

We will define fuzzyα,β-quasi-ideals of Γ-semigroups as follows.Definition 11.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy α,β-quasi-ideal of S if S∘αf∩f∘βS⊆f.
A fuzzy subsetf of S is called a fuzzy α-quasi-ideal of S if f is a fuzzy α,β-quasi-ideal of S.Theorem 26.
LetS be a Γ-semigroup. Let f and g be a fuzzy left α-ideal and a fuzzy right α-ideal of S, respectively. Then, f∩g is a fuzzy α-quasi-ideal of S.Proof.
Letf and g be a fuzzy left α-ideal and a fuzzy right α-ideal of a Γ-semigroup S, respectively. We have g∘αf⊆f∩g; this implies f∩g≠∅. Then, S∘αf∩g∩f∩g∘αS⊆S∘αf∩f∘αS⊆f∩g. Hence, f∩g is a fuzzy α-quasi-ideal of S.Theorem 27.
Every fuzzy (α,β)-quasi-ideal of a Γ-semigroup S is the intersection of a fuzzy left α-ideal and a fuzzy right β-ideal of S.Proof.
Letf be a fuzzy (α,β)-quasi-ideal of a Γ-semigroup S. Let g=f∪S∘αf and h=f∪f∘βS. Then, S∘αg=S∘αf∪S∘αf=S∘αf∪S∘αS∘αf⊆S∘αf⊆g, and also, h∘βS⊆h. Thus, g and h are a fuzzy left α-ideal and a fuzzy right β-ideal of S, respectively. We claim that f=g∩h. That is, f⊆f∪S∘αf∩f∪f∘βS⊆g∩h, and conversely, g∩h=f∪S∘αf∩f∪f∘βS⊆f∪S∘αf∩f∘βS⊆f. Therefore, f=g∩h.Theorem 28.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is an α,β-quasi-ideal of S if and only if CQ is a fuzzy α,β-quasi-ideal of S.Proof.
Assume thatQ is an α,β-quasi-ideal of a Γ-semigroup S. If y∉SαQ∩QβS, S∘αCQ∩CQ∘βSy=0≤CQy. Let y∈SαQ∩QβS. Then, y∈Q. So CQy=1. Hence, S∘αCQ∩CQ∘βS⊆CQ. Therefore, CQ is a fuzzy α,β-quasi-ideal of S.
Conversely, assume thatCQ is a fuzzy α,β-quasi-ideal of S. Then, S∘αCQ∩CQ∘βS⊆CQ. Let x∈SαQ∩QβS. Then, S∘αCQ∩CQ∘βSx=1, and this implies that CQx=1. So, sαQ∩Qβs⊆Q. Consequently, Q is an α,β-quasi-ideal of S.


## 3.3. Fuzzyα,β-Bi-Ideals

Definition 12.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy α,β-bi-ideal of S if f∘αS∘βf⊆f.Theorem 29.
LetS be a Γ-semigroup, g a fuzzy subset of S, and f a fuzzy α,β-bi-ideal of S. Then, the following statements are true:(1)
f∘αg is a fuzzy α,β-bi-ideal of S.(2)
g∘βf is a fuzzy α,β-bi-ideal of S.Proof.
(1) Letxt be a fuzzy point of S. Then,(26)f∘αg∘αS∘βf∘αg=f∘αg∘αxt∘βf∘αg⊆f∘αxt∘βf∘αg⊆f∘αg.
Hence,f∘αg is a fuzzy α,β-bi-ideal of S.
(2) is similar to (1).Theorem 30.
LetS be a Γ-semigroup. If f1,f2 and f3 are fuzzy α,β-bi-ideals of S, then f1∘αf2∘βf3 is a fuzzy bi-(α,β)-ideal of S.Proof.
Letxt be a fuzzy point of S. Then,(27)f1∘αf2∘βf3∘αxt∘βf1∘αf2∘βf3⊆f1∘αf2∘βS∘αf2∘βf3⊆f1∘αf2∘βf3.
Hence,f∘αf2∘βf3 is a fuzzy α,β-bi-ideal of S.Theorem 31.
LetS be a Γ-semigroup and B a nonempty subset of S. Then, B is an α,β-bi-ideal of S if and only if CB is a fuzzy α,β-bi-ideal of S.Proof.
Assume thatB is an α,β-bi-ideal of S. Then, BαSβB⊆B. If z∉BαSβB, we have CB∘αS∘βCBz=0≤CBz. Let z∈BαSβB. Then, z∈B. Thus, CB∘αS∘βCBz=1; then, CBz=1. This implies that CB∘αS∘βCB⊆CB. Hence, CB is a fuzzy α,β-bi-ideal of S.
Conversely, assume thatCB is a fuzzy α,β-bi-ideal of S. Then, CB∘αS∘βCB⊆CB. Let z∈BαSβB. Then, CB∘αS∘βCBz=1; thus, CBz=1. Hence, z∈B so that BαSβB⊆B. Thus, B is an α,β-bi-ideal of S.Theorem 32.
Letf and g be two fuzzy (α,β)-bi-ideals of a Γ-semigroup S. If f∩g≠∅, then f∩g is a fuzzy (α,β)-bi-ideal of S.Proof.
Letf and g be fuzzy (α,β)-bi-ideals of a Γ-semigroup S. Then, f∩g∘αS∘βf∩g⊆f∘αS∘βf∩f∘αS∘βf∩g⊆f∩g. Hence, f∩g is a fuzzy α,β-bi-ideal of S.


## 4. New Types of Almost Ideals

### 4.1. Almostα,β-Ideals

Definition 13.
LetS be a Γ-semigroup and L,R, and I be nonempty subsets of S. Let α,β∈Γ. Then,(1)
L is called an almost left α-ideal of S if SαL∩L≠∅.(2)
R is called an almost right β-ideal of S if RβS∩R≠∅.(3)
I is called an almost α,β-ideal of S if it is both an almost left α-ideal and an almost right β-ideal of S.Theorem 33.
LetS be a Γ-semigroup. If L is a left α-ideal of S, then L is an almost left α-ideal of S.Proof.
LetL be a left α-ideal of S. Then, SαL⊆L so that SαL∩L≠∅. Thus, L is an almost left α-ideal of S.Theorem 34.
LetS be a Γ-semigroup. If R is a right β-ideal of S, then R is an almost right β-ideal of S.Proof.
It is similar to Theorem33.Theorem 35.
LetS be a Γ-semigroup. If I is an (α,β)-ideal of S, then I is an almost (α,β)-ideal of S.Proof.
It follows by Theorems33 and 34.Theorem 36.
LetS be a Γ-semigroup. If L is an almost left α-ideal of S and L⊆H⊆S, then H is an almost left α-ideal of S.Proof.
LetL be an almost left α-ideal of S and L⊆H⊆S. Since SαL∩L≠∅ and SαL∩L⊆SαH∩H, we have SαH∩H≠∅. Therefore, H is an almost left α-ideal of S.Theorem 37.
LetS be a Γ-semigroup. If R is an almost right β-ideal of S and R⊆H⊆S, then H is an almost right β-ideal of S.Proof.
It is similar to Theorem36.Theorem 38.
LetS be a Γ-semigroup. If I is an almost (α,β)-ideal of S and I⊆H⊆S, then H is an almost (α,β)-ideal of S.Proof.
It follows by Theorems36 and 37.Corollary 2.
LetS be a Γ-semigroup. If L1 and L2 are almost left α-ideals of S, then L1∪L2 is an almost left α-ideal of S.Corollary 3.
LetS be a Γ-semigroup. If R1 and R2 are almost right β-ideals of S, then R1∪R2 is an almost right β-ideal of S.Corollary 4.
LetS be a Γ-semigroup. If I1 and I2 are almost (α,β)-ideals of S, then I1∪I2 is an almost (α,β)-ideal of S.Example 2.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We havea,b,d and a,c,d are almost left α-ideals of S. However, a,b,d∩a,c,d=a,d is not an almost left α-ideal of S.Remark 3.
The intersection of two almost leftα-ideals of a Γ-semigroup S need not be an almost left α-ideal of S.Remark 4.
The intersection of two almost rightβ-ideals of a Γ-semigroup S need not be an almost right β-ideal of S.Remark 5.
The intersection of two almost (α,β)-ideals of a Γ-semigroup S need not be an almost (α,β)-ideal of S.Definition 14.
AΓ-semigroup S is called(1)
Almost leftα-simple if S does not contain proper almost left α-ideals.(2)
Almost rightβ-simple if S does not contain proper almost right β-ideals.(3)
Almost (α,β)-simple if S does not contain proper almost (α,β)-ideals.Theorem 39.
LetS be a Γ-semigroup. Then, S is almost left α-simple if and only if for each a∈S there exists s∈S such that sαS∖a=a.Proof.
Assume thatS is almost left α-simple. Then, S has no proper almost left α-ideals. Let a∈S. Then, S∖a is not an almost left α-ideal of S. Thus, there exists s∈S such that sαS∖a∩S∖a=∅; this implies that sαS∖a=a.
Conversely, for eacha∈S, there exists s∈S such that sαS∖a=a. Assume that L is a proper almost left α-ideal of S, and let a∈S∖L. Since L⊂S∖a, we have S∖a is an almost left α-ideal of S by Theorem 36. Thus, sαS∖a∩S∖a≠∅; we get a contradiction. Hence, S has no proper left almost α-ideals. Therefore, S is almost left α-simple.Theorem 40.
LetS be a Γ-semigroup. Then, S is almost right β-simple if and only if for each a∈S, there exists s∈S such that S∖aβs=a.Proof.
It is similar to Theorem39.


### 4.2. Almostα,β-Quasi-Ideals

Definition 15.
A nonempty subsetQ of a Γ-semigroup S is called an almost α,β-quasi-ideal of S if sαQ∩Qβs∩Q≠∅ for all s∈S.Proposition 1.
Everyα,β-quasi-ideal of a Γ-semigroup S is either sαQ∩Qβs=∅ for some s∈S or an almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an α,β-quasi-ideal of a Γ-semigroup S. Assume that sαQ∩Qβs≠∅ for all s∈S. Let s∈S. Then, sαQ∩Qβs⊆SαQ∩QβS⊆Q. That is, sαQ∩Qβs∩Q≠∅. Hence, Q is an almost α,β-quasi-ideal of S.Theorem 41.
Every almostα,β-quasi-ideal of a Γ-semigroup S is an almost left α-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S. Let s∈S. Then, ∅≠sαQ∩Qβs∩Q⊆sαQ∩Q. Hence, Q is an almost left α-ideal of S.
Similarly, every almostα,β-quasi-ideal of a Γ-semigroup S is an almost right β-ideal of S, but the converse is not true.Theorem 42.
IfQ is an almost α,β-quasi-ideal of a Γ-semigroup S and Q⊆H⊆S, then H is an almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S and Q⊆H⊆S. Let s∈S. Then, ∅≠sαQ∩Qβs∩Q⊆sαH∩Hβs∩H. Therefore, H is an almost α,β-quasi-ideal of S.Corollary 5.
The union of two almostα,β-quasi-ideals of a Γ-semigroup S is an almost α,β-quasi-ideal of S.Example 3.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We haveb,c,e and b,d,e are almost α,β-quasi-ideals of S.Remark 6.
The intersection of two almostα,β-quasi-ideals of a Γ-semigroup S need not be an almost α,β-quasi-ideal of S.Definition 16.
AΓ-semigroup S is called almost α,β-quasi-simple if S does not contain proper almost α,β-quasi-ideals.Theorem 43.
AΓ-semigroup S is almost α,β-quasi-simple if and only if for any a∈S, there exists sa such that saαS\a∩S\aβsa⊆a.Proof.
Assume that aΓ-semigroup S is almost α,β-quasi-simple and S\a is not an almost α,β-quasi-ideal of S. Then, there exists sa∈S such that saαS\a∩S\aβsa∩S\a=∅. Therefore, saαS\a∩S\aβsa⊆a.
Conversely, assume that, for anya∈S, there exists sa such that saβS\a∩S\aβsa⊆a. Then, saαS\a∩S\aβsa∩S\a=∅. Hence, S\a is not an almost α,β-quasi-ideal of S. Let A be a proper almost α,β-quasi-ideal of S. Then, A⊆S\a⊆S for some a∈S; this is a contradiction. Therefore, S has no proper almost α,β-quasi-ideals. Hence, S is almost α,β-quasi-simple.


### 4.3. Almostα,β-Bi-Ideals

Definition 17.
A nonempty subsetB of a Γ-semigroup S is called an almost α,β-bi-ideal of S if BαsβB∩B≠∅ for all s∈S.Theorem 44.
IfB is an almost α,β-bi-ideal of a Γ-semigroup S and B⊆C⊆S, then C is an almost α,β-bi-ideal of S.Proof.
LetB be an almost α,β-bi-ideal of a Γ-semigroup S and B⊆C⊆S. Since BαsβB∩B≠∅ for all s∈S and BαsβB∩B⊆CαsβC∩C, we have CαsβC∩C≠∅. Therefore, C is an almost α,β-bi-ideal of S.Corollary 6.
The union of two almostα,β-bi-ideals of a Γ-semigroup S is also an almost α,β-bi-ideal of S.Example 4.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We haveb,c,e and b,d,e are almost α,β-bi-ideals of S.Remark 7.
The intersection of two almostα,β-bi-ideals of a Γ-semigroup S need not be an almost α,β-bi-ideal of S.Theorem 45.
AΓ-semigroup S has a proper almost α,β-bi-ideal if and only if there exists an element a∈S such that S\aαsβS\a∩S\a≠∅ for all a∈S.Proof.
Assume that aΓ-semigroup S contains a proper almost α,β-bi-ideal B and a∉B. Then, B⊂S\a⊂S. By Theorem 44, S\a is a proper almost α,β-bi-ideal of S, that is, S\aαsβS\a∩S\a≠∅ for all s∈S.
Conversely, leta∈S such that S\aαsβS\a∩S\a≠∅ for all s∈S. Since S\a⊂S, we have S\a is a proper almost α,β-bi-ideal of S.Definition 18.
AΓ-semigroup S is called almost α,β-bi-simple if S does not contain proper almost α,β-bi-ideals.Theorem 46.
AΓ-semigroup S is almost α,β-bi-simple if and only if for all a∈S, there exists s∈S such that S∖aαsβS∖a=a.Proof.
Assume that aΓ-semigroup S is almost α,β-bi-simple. Then, S has no proper almost α,β-bi-ideals, and let a∈S. Then, S∖a is not an almost α,β-bi-ideal of S. Thus, there exists s∈S such that S∖aαsβS∖a∩S∖a≠∅. This implies that S∖aαsβS∖a=a.
Conversely, suppose thatS has a proper almost α,β-bi-ideal B, that is, B⊂S. Since B⊆S∖a⊂S, we have S∖a is an almost α,β-bi-ideal of S by Theorem 44; this is a contradiction. Hence, S has no proper almost α,β-bi-ideals. Therefore, S is almost α,β-bi-simple.


## 4.1. Almostα,β-Ideals

Definition 13.
LetS be a Γ-semigroup and L,R, and I be nonempty subsets of S. Let α,β∈Γ. Then,(1)
L is called an almost left α-ideal of S if SαL∩L≠∅.(2)
R is called an almost right β-ideal of S if RβS∩R≠∅.(3)
I is called an almost α,β-ideal of S if it is both an almost left α-ideal and an almost right β-ideal of S.Theorem 33.
LetS be a Γ-semigroup. If L is a left α-ideal of S, then L is an almost left α-ideal of S.Proof.
LetL be a left α-ideal of S. Then, SαL⊆L so that SαL∩L≠∅. Thus, L is an almost left α-ideal of S.Theorem 34.
LetS be a Γ-semigroup. If R is a right β-ideal of S, then R is an almost right β-ideal of S.Proof.
It is similar to Theorem33.Theorem 35.
LetS be a Γ-semigroup. If I is an (α,β)-ideal of S, then I is an almost (α,β)-ideal of S.Proof.
It follows by Theorems33 and 34.Theorem 36.
LetS be a Γ-semigroup. If L is an almost left α-ideal of S and L⊆H⊆S, then H is an almost left α-ideal of S.Proof.
LetL be an almost left α-ideal of S and L⊆H⊆S. Since SαL∩L≠∅ and SαL∩L⊆SαH∩H, we have SαH∩H≠∅. Therefore, H is an almost left α-ideal of S.Theorem 37.
LetS be a Γ-semigroup. If R is an almost right β-ideal of S and R⊆H⊆S, then H is an almost right β-ideal of S.Proof.
It is similar to Theorem36.Theorem 38.
LetS be a Γ-semigroup. If I is an almost (α,β)-ideal of S and I⊆H⊆S, then H is an almost (α,β)-ideal of S.Proof.
It follows by Theorems36 and 37.Corollary 2.
LetS be a Γ-semigroup. If L1 and L2 are almost left α-ideals of S, then L1∪L2 is an almost left α-ideal of S.Corollary 3.
LetS be a Γ-semigroup. If R1 and R2 are almost right β-ideals of S, then R1∪R2 is an almost right β-ideal of S.Corollary 4.
LetS be a Γ-semigroup. If I1 and I2 are almost (α,β)-ideals of S, then I1∪I2 is an almost (α,β)-ideal of S.Example 2.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We havea,b,d and a,c,d are almost left α-ideals of S. However, a,b,d∩a,c,d=a,d is not an almost left α-ideal of S.Remark 3.
The intersection of two almost leftα-ideals of a Γ-semigroup S need not be an almost left α-ideal of S.Remark 4.
The intersection of two almost rightβ-ideals of a Γ-semigroup S need not be an almost right β-ideal of S.Remark 5.
The intersection of two almost (α,β)-ideals of a Γ-semigroup S need not be an almost (α,β)-ideal of S.Definition 14.
AΓ-semigroup S is called(1)
Almost leftα-simple if S does not contain proper almost left α-ideals.(2)
Almost rightβ-simple if S does not contain proper almost right β-ideals.(3)
Almost (α,β)-simple if S does not contain proper almost (α,β)-ideals.Theorem 39.
LetS be a Γ-semigroup. Then, S is almost left α-simple if and only if for each a∈S there exists s∈S such that sαS∖a=a.Proof.
Assume thatS is almost left α-simple. Then, S has no proper almost left α-ideals. Let a∈S. Then, S∖a is not an almost left α-ideal of S. Thus, there exists s∈S such that sαS∖a∩S∖a=∅; this implies that sαS∖a=a.
Conversely, for eacha∈S, there exists s∈S such that sαS∖a=a. Assume that L is a proper almost left α-ideal of S, and let a∈S∖L. Since L⊂S∖a, we have S∖a is an almost left α-ideal of S by Theorem 36. Thus, sαS∖a∩S∖a≠∅; we get a contradiction. Hence, S has no proper left almost α-ideals. Therefore, S is almost left α-simple.Theorem 40.
LetS be a Γ-semigroup. Then, S is almost right β-simple if and only if for each a∈S, there exists s∈S such that S∖aβs=a.Proof.
It is similar to Theorem39.


## 4.2. Almostα,β-Quasi-Ideals

Definition 15.
A nonempty subsetQ of a Γ-semigroup S is called an almost α,β-quasi-ideal of S if sαQ∩Qβs∩Q≠∅ for all s∈S.Proposition 1.
Everyα,β-quasi-ideal of a Γ-semigroup S is either sαQ∩Qβs=∅ for some s∈S or an almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an α,β-quasi-ideal of a Γ-semigroup S. Assume that sαQ∩Qβs≠∅ for all s∈S. Let s∈S. Then, sαQ∩Qβs⊆SαQ∩QβS⊆Q. That is, sαQ∩Qβs∩Q≠∅. Hence, Q is an almost α,β-quasi-ideal of S.Theorem 41.
Every almostα,β-quasi-ideal of a Γ-semigroup S is an almost left α-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S. Let s∈S. Then, ∅≠sαQ∩Qβs∩Q⊆sαQ∩Q. Hence, Q is an almost left α-ideal of S.
Similarly, every almostα,β-quasi-ideal of a Γ-semigroup S is an almost right β-ideal of S, but the converse is not true.Theorem 42.
IfQ is an almost α,β-quasi-ideal of a Γ-semigroup S and Q⊆H⊆S, then H is an almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S and Q⊆H⊆S. Let s∈S. Then, ∅≠sαQ∩Qβs∩Q⊆sαH∩Hβs∩H. Therefore, H is an almost α,β-quasi-ideal of S.Corollary 5.
The union of two almostα,β-quasi-ideals of a Γ-semigroup S is an almost α,β-quasi-ideal of S.Example 3.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We haveb,c,e and b,d,e are almost α,β-quasi-ideals of S.Remark 6.
The intersection of two almostα,β-quasi-ideals of a Γ-semigroup S need not be an almost α,β-quasi-ideal of S.Definition 16.
AΓ-semigroup S is called almost α,β-quasi-simple if S does not contain proper almost α,β-quasi-ideals.Theorem 43.
AΓ-semigroup S is almost α,β-quasi-simple if and only if for any a∈S, there exists sa such that saαS\a∩S\aβsa⊆a.Proof.
Assume that aΓ-semigroup S is almost α,β-quasi-simple and S\a is not an almost α,β-quasi-ideal of S. Then, there exists sa∈S such that saαS\a∩S\aβsa∩S\a=∅. Therefore, saαS\a∩S\aβsa⊆a.
Conversely, assume that, for anya∈S, there exists sa such that saβS\a∩S\aβsa⊆a. Then, saαS\a∩S\aβsa∩S\a=∅. Hence, S\a is not an almost α,β-quasi-ideal of S. Let A be a proper almost α,β-quasi-ideal of S. Then, A⊆S\a⊆S for some a∈S; this is a contradiction. Therefore, S has no proper almost α,β-quasi-ideals. Hence, S is almost α,β-quasi-simple.


## 4.3. Almostα,β-Bi-Ideals

Definition 17.
A nonempty subsetB of a Γ-semigroup S is called an almost α,β-bi-ideal of S if BαsβB∩B≠∅ for all s∈S.Theorem 44.
IfB is an almost α,β-bi-ideal of a Γ-semigroup S and B⊆C⊆S, then C is an almost α,β-bi-ideal of S.Proof.
LetB be an almost α,β-bi-ideal of a Γ-semigroup S and B⊆C⊆S. Since BαsβB∩B≠∅ for all s∈S and BαsβB∩B⊆CαsβC∩C, we have CαsβC∩C≠∅. Therefore, C is an almost α,β-bi-ideal of S.Corollary 6.
The union of two almostα,β-bi-ideals of a Γ-semigroup S is also an almost α,β-bi-ideal of S.Example 4.
Consider aΓ-semigroup S=a,b,c,d,e with Γ=α,β and
We haveb,c,e and b,d,e are almost α,β-bi-ideals of S.Remark 7.
The intersection of two almostα,β-bi-ideals of a Γ-semigroup S need not be an almost α,β-bi-ideal of S.Theorem 45.
AΓ-semigroup S has a proper almost α,β-bi-ideal if and only if there exists an element a∈S such that S\aαsβS\a∩S\a≠∅ for all a∈S.Proof.
Assume that aΓ-semigroup S contains a proper almost α,β-bi-ideal B and a∉B. Then, B⊂S\a⊂S. By Theorem 44, S\a is a proper almost α,β-bi-ideal of S, that is, S\aαsβS\a∩S\a≠∅ for all s∈S.
Conversely, leta∈S such that S\aαsβS\a∩S\a≠∅ for all s∈S. Since S\a⊂S, we have S\a is a proper almost α,β-bi-ideal of S.Definition 18.
AΓ-semigroup S is called almost α,β-bi-simple if S does not contain proper almost α,β-bi-ideals.Theorem 46.
AΓ-semigroup S is almost α,β-bi-simple if and only if for all a∈S, there exists s∈S such that S∖aαsβS∖a=a.Proof.
Assume that aΓ-semigroup S is almost α,β-bi-simple. Then, S has no proper almost α,β-bi-ideals, and let a∈S. Then, S∖a is not an almost α,β-bi-ideal of S. Thus, there exists s∈S such that S∖aαsβS∖a∩S∖a≠∅. This implies that S∖aαsβS∖a=a.
Conversely, suppose thatS has a proper almost α,β-bi-ideal B, that is, B⊂S. Since B⊆S∖a⊂S, we have S∖a is an almost α,β-bi-ideal of S by Theorem 44; this is a contradiction. Hence, S has no proper almost α,β-bi-ideals. Therefore, S is almost α,β-bi-simple.


## 5. New Types of Fuzzy Almost Ideals

### 5.1. Fuzzy Almostα,β-Ideals

Definition 19.
Letα,β∈Γ. Let f be a fuzzy subset of a Γ-semigroup S. Then, f is called(1)
A fuzzy almost leftα-ideal of S if xt∘αf∩f≠0 for all fuzzy point xt of S.(2)
A fuzzy almost rightβ-ideal of S if f∘βxt∩f≠0 for all fuzzy point xt of S.(3)
A fuzzy almostα,β-ideal of S if it is both a fuzzy almost left α-ideal and a fuzzy almost right β-ideal of S.Theorem 47.
Letf be a fuzzy almost left α-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost left α-ideal of S.Proof.
Assume thatf is a fuzzy almost left α-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for each fuzzy point xt, xt∘αf∩f≠0. We have xt∘αf∩f⊆xt∘αg∩g; this implies xt∘αg∩g≠0. Therefore, g is a fuzzy almost left α-ideal of S.Theorem 48.
Letf be a fuzzy almost right β-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost right β-ideal of S.Proof.
It is similar to Theorem47.Theorem 49.
Letf be a fuzzy almost (α,β)-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost (α,β)-ideal of S.Proof.
It follows by Theorem47 and Theorem 48.Corollary 7.
The union of two fuzzy almost leftα-ideals of a Γ-semigroup S is a fuzzy almost left α-ideal of S.Corollary 8.
The union of two fuzzy almost rightβ-ideals of a Γ-semigroup S is a fuzzy almost right β-ideal of S.Corollary 9.
The union of two fuzzy almost (α,β)-ideals of a Γ-semigroup S is a fuzzy almost (α,β)-ideal of S.Theorem 50.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
A is an almost left α-ideal of S if and only if CA is a fuzzy almost left α-ideal of S.(2)
A is an almost right β-ideal of S if and only if CA is a fuzzy almost right β-ideal of S.(3)
A is an almost (α,β)-ideal of S if and only if CA is a fuzzy almost (α,β)-ideal of S.Proof.
(1) Assume thatA is an almost left α-ideal of a Γ-semigroup S. Then, xαA∩A≠∅ for all x∈S. Thus, there exists y∈xαA, and y∈A. So, xt∘αCAy=1 and CAy=1. Hence, xt∘αCA∩CA≠0. Therefore, CA is a fuzzy almost left α -ideal of S.
Conversely, assume thatCA is a fuzzy almost left α -ideal of S. Let x∈S. Then, xt∘αCA∩CA≠0. Then, there exists a∈S such that xt∘αCA∩CAa≠0. Hence, a∈xαA∩A. So, xαA∩A≠∅. Consequently, A is an almost left α -ideal of S.
The proofs of (2) and (3) are similar to the proof of (1).Theorem 51.
Letf be a fuzzy subset of a α-semigroup S. Then,(1)
f is a fuzzy almost left α -ideal of S if and only if suppf is an almost left α -ideal of S.(2)
f is a fuzzy almost right β -ideal of S if and only if suppf is an almost right β-ideal of S.(3)
f is a fuzzy almost (α,β)-ideal of S if and only if suppf is an almost (α,β)-ideal of S.Proof.
(1) Assume thatf is a fuzzy almost left α -ideal of a Γ -semigroup S. Let x∈S. Then, xt∘αf∩f≠0. Hence, there exists a∈S such that xt∘αf∩fa≠0 . So, there exists b∈S such that a=xαb,fa≠0,fb≠0. That is, a,b∈suppf. Thus, xt∘αCsuppfa≠0 and Csuppfa≠0. Therefore, Cx∘αCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost left α -ideal of S. By Theorem 50, suppf is an almost left α-ideal of S.
Conversely, assume thatsuppf is an almost left α -ideal of S. By Theorem 50, Csuppf is a fuzzy almost left α -ideal of S. Then, xt∘αCsuppf∩Csuppf≠0 for all x∈S. Then, there exists a∈S such that xt∘αCsuppf∩Csuppfa≠0. Hence, xt∘αCsuppfa≠0 and Csuppfa≠0. Then, there exists y∈S such that a=xαy,fa≠0,fy≠0. This means xt∘αf∩f≠0. Therefore, f is a fuzzy almost left α ideal of S.
The proofs of (2) and (3) are similar to the proof of (1).Definition 20.
A fuzzy almost leftα -ideal f of a Γ -semigroup S is minimal if for all fuzzy almost left α -ideal g of S such that g⊆f, we obtain suppg=suppf.Theorem 52.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
A is a minimal almost left α-ideal of S if and only if CA is a minimal fuzzy almost left α-ideal of S.(2)
A is a minimal almost right β-ideal of S if and only if CA is a minimal fuzzy almost right β-ideal of S.(3)
A is a minimal almost (α,β)-ideal of S if and only if CA is a minimal fuzzy almost (α,β)-ideal of S.Proof.
(1) Assume thatA is a minimal almost left α-ideal of a Γ-semigroup S. By Theorem 50 (1), CA is a fuzzy almost left α-ideal of S. Let g be a fuzzy almost left α-ideal of S such that g⊆CA. By Theorem 51 (1), suppg is an almost left α-ideal of S. Then, suppg⊆suppCA=A. Since A is minimal, suppg=A=suppCA. Therefore, CA is minimal.
Conversely, assume thatCA is a minimal fuzzy almost left α-ideal of S. By Theorem 50 (1), A is an almost left α-ideal of S. Let L be an almost left α-ideal of S such that L⊆A. By Theorem 50 (1), CL is a fuzzy almost left α-ideal of S such that CL⊆CA. Hence, L=suppCL=suppCA=A. Therefore, A is minimal.
(2) and (3) can be proved similarly.Corollary 10.
LetA be a sub-Γ-semigroup of a Γ-semigroup S. Then,(1)
A is almost left α-simple if and only if for each fuzzy almost left α-ideal f of S, suppf=A.(2)
A is almost right β-simple if and only if for each fuzzy almost right β-ideal f of S, suppf=A.(3)
A is almost (α,β)-simple if and only if for each fuzzy almost (α,β)-ideal of S, suppf=A.


### 5.2. Fuzzy Almostα,β-Quasi-Ideals

Definition 21.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy almost α,β-quasi-ideal of S if f∘αxt∩xt∘βf∩f≠0 for all fuzzy points xt of S.Theorem 53.
Letf be a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S and g a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for all fuzzy point xt of S, f∘αxt∩xt∘βf∩f≠0. We have that f∘αxt∩xt∘βf∩f⊆g∘αxt∩xt∘βg∩g. This implies that g∘αxt∩xt∘βg∩g≠0. Therefore, g is a fuzzy almost α,β-quasi-ideal of S.Corollary 11.
Letf and g be fuzzy almost α,β-quasi-ideals of a Γ-semigroup S. Then, f∪g is a fuzzy almost α,β-quasi-ideal of S.Proof.
Sincef⊆f∪g, by Theorem 53, f∪g is a fuzzy almost α,β-quasi-ideal of S.Example 5.
Consider theΓ-semigroup ℤ5 where Γ=0¯,2¯,4¯ and a¯γb¯=a¯+γ+b¯, where a¯,b¯∈ℤ5 and γ∈Γ. Let f:ℤ5⟶0,1 defined by(28)f0¯=0,f1¯=0.6,f2¯=0,f3¯=0.4,f4¯=0.4,and g:ℤ5⟶0,1 defined by(29)g0¯=0,g1¯=0.3,g2¯=0.6,g3¯=0,g4¯=0.8.
We havef and g are fuzzy almost 0¯,0¯-quasi-ideals of ℤ5, but f∩g is not a fuzzy almost 0¯,0¯-quasi-ideal of ℤ5.Theorem 54.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is an almost α,β-quasi-ideal of S if and only if CQ is a fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S, and let xt be a fuzzy point of S. Then, Qαx∩xβQ∩Q≠∅. Thus, there exists y∈Qαx∩xβQ and y∈Q. So, CQ∘αxt∩xt∘βCQy≠0 and CQy=1. Hence, CQ∘αxt∩xt∘βCQ∩CQ≠0. Therefore, CQ is a fuzzy almost α,β-quasi-ideal of S.
Conversely, assume thatCQ is a fuzzy almost α,β-quasi-ideal of S. Let s∈S. Then, CQ∘αst∩st∘βCQ∩CQ≠0. Then, there exists x∈S such that CQ∘αst∩st∘βCQ∩CQx≠0. Hence, x∈Qαs∩sβQ∩Q. So, Qαs∩sβQ∩Q≠∅. Consequently, Q is an almost α,β-quasi-ideal of S.Theorem 55.
Letf be a fuzzy subset of a Γ-semigroup S. Then, f is a fuzzy almost α,β-quasi-ideal of S if and only if suppf is an almost α,β-quasi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S. Let s∈S and t∈0,1. Then, f∘αst∩st∘βf∩f≠0. Hence, there exists x∈S such that(30)f∘αst∩st∘βf∩fx≠0.
So, there existy1,y2∈S such that x=y1αs=sβy2,fx≠0,fy1≠0, and fy2≠0. That is, x,y1,y2∈suppf. Thus, Csuppf∘αst∩st∘βCsuppfx≠0 and Csuppfx≠0. Therefore, Csuppf∘αst∩st∘βCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost α,β-quasi-ideal of S. By Theorem 54, suppf is an almost α,β-quasi-ideal of S.
Conversely, assume thatsuppf is an almost α,β-quasi-ideal of S. By Theorem 54, Csuppf is a fuzzy almost α,β-quasi-ideal of S. Then, for each fuzzy point st of S, we have Csuppf∘αst∩st∘βCsuppf∩Csuppf≠0. Then, there exists x∈S such that(31)Csuppf∘αst∩st∘βCsuppf∩Csuppfx≠0.
Hence,Csuppf∘αst∩st∘βCsuppfx≠0 and Csuppfx≠0. Then, there exist y1,y2∈S such that x=y1αs=sβy2,fx≠0,fy1≠0, and fy2≠0. This means that f∘αst∩st∘βf∩f≠0. Therefore, f is a fuzzy almost α,β-quasi-ideal of S.
Next, we define minimal fuzzy almostα,β-quasi-ideals in Γ-semigroups and give some relationship between minimal almost α,β-quasi-ideals and minimal fuzzy almost α,β-quasi-ideals of Γ-semigroups.Definition 22.
A fuzzy almostα,β-quasi-ideal f of a Γ-semigroup is called minimal if for each fuzzy almost α,β-quasi-ideal g of S such that g⊆f, we have suppg=suppf.Theorem 56.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is a minimal almost α,β-quasi-ideal of S if and only if CQ is a minimal fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatQ is a minimal almost α,β-quasi-ideal of a Γ-semigroup S. By Theorem 54, CQ is a fuzzy almost α,β-quasi-ideal of S. Let g be a fuzzy almost α,β-quasi-ideal of S such that g⊆CQ. Then, suppg⊆suppCQ=Q. Since g⊆Csuppg and by Theorem 53, we have Csuppg is a fuzzy almost α,β-quasi-ideal of S. By Theorem 54, suppg is an almost α,β-quasi-ideal of S. Since Q is minimal, suppg=Q=suppCQ. Therefore, CQ is minimal.
Conversely, assume thatCQ is a minimal fuzzy almost α,β-quasi-ideal of S. Let Q′ be an almost α,β-quasi-ideal of S such that Q′⊆Q. By Theorem 54, CQ′ is a fuzzy almost α,β-quasi-ideal of S such that CQ′⊆CQ. Since CQ is minimal, Q′=suppCQ′=suppCQ=Q. Therefore, Q is minimal.Corollary 12.
LetQ be a sub Γ-semigroup of a Γ-semigroup S. Then, Q is almost α,β-quasi-simple if and only if for all fuzzy almost α,β-quasi-ideal f of S, suppf=Q.


### 5.3. Fuzzy Almostα,β-Bi-Ideals

Definition 23.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy almost α,β-bi-ideal of S if f∘αxt∘βf∩f≠0 for all fuzzy point xt of S.Theorem 57.
Letf be a fuzzy almost α,β-bi-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-bi-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for each fuzzy point xt, f∘αxt∘βf∩f≠0. We have f°αxt°βf∩f⊆g°αxt°βg∩g; this implies g∘αxt∘βg∩g≠0. Therefore, g is a fuzzy almost α,β-bi-ideal of S.Corollary 13.
Letf and g be fuzzy almost α,β-bi-ideals of a Γ-semigroup S. Then, f∪g is a fuzzy almost α,β-bi-ideal of S.Example 6.
Consider theΓ-semigroup ℤ5 where Γ=0¯,4¯ and a¯γb¯=a¯+γ+b¯, where a¯,b¯∈ℤ5 and γ∈Γ. Let f:ℤ5⟶0,1 defined by(32)f0¯=0,f1¯=0.7,f2¯=0,f3¯=0.6,f4¯=0.4,and g:ℤ5⟶0,1 defined by(33)g0¯=0,g1¯=0.1,g2¯=0.6,g3¯=0,g4¯=0.8.
We havef and g are fuzzy almost 0¯,4¯-bi-ideals of ℤ5.Remark 8.
The intersection of two fuzzy almost (α,β)-bi-ideals of a Γ-semigroup S need not be a fuzzy almost (α,β)-bi-ideal of S.Theorem 58.
LetB be a nonempty subset of a Γ-semigroup S. Then, B is an almost α,β-bi-ideal of S if and only if CB is a fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatB is an almost α,β-bi-ideal of a Γ-semigroup S. Then, BαxβB∩B≠∅ for all x∈S. Thus, there exists y∈BαxβB and y∈B. So, CB∘αxt∘βCBy=1 and CBy=1. Hence, CB∘αxt∘βCB∩CB≠0. Therefore, CB is a fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCB is a fuzzy almost α,β-bi-ideal of S. Let s∈S. Then, CB∘αs1∘βCB∩CB≠0. Thus, there exists x∈S such that CB∘αs1∘βCB∩CBx≠0. Hence, x∈BαsβB∩B. So, BαsβB∩B≠∅. Consequently, B is an almost α,β-bi-ideal of S.Theorem 59.
Letf be a fuzzy subset of a Γ-semigroup S. Then, f is a fuzzy almost α,β-bi-ideal of S if and only if suppf is an almost α,β-bi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-bi-ideal of a Γ-semigroup S. Let s∈S. Then, f∘αst∘βf∩f≠0. Hence, there exists x∈S such that fα∘st∘βf∩fx≠0. So, there exist y1,y2∈S such that x=y1αsβy2,fx≠0,fy1≠0, and fy2≠0. That is, x,y1,y2∈suppf. Thus, Csuppf∘αst∘βCsuppfx≠0 and Csuppfx≠0. Therefore, Csuppf∘αst∘βCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost α,β-bi-ideal of S. By Theorem 58, suppf is an almost α,β-bi-ideal of S.
Conversely, assume thatsuppf is an almost α,β-bi-ideal of S. By Theorem 58, Csuppf is a fuzzy almost α,β-bi-ideal of S. Then, Csuppf∘αst∘βCsuppf∩Csuppf≠0 for all s∈S. Then, there exists x∈S such that Csuppf∘αst∘βCsuppf∩Csuppfx≠0. Hence, Csuppf∘αst∘βCsuppfx≠0 and Csuppfx≠0. Then, there exist y1,y2∈S such that x=y1αsβy2,fx≠0,fy1≠0, and fy2≠0. This means f∘αst∘βf∩f≠0. Therefore, f is a fuzzy almost α,β-bi-ideal of S.
We define minimal fuzzy almostα,β-bi-ideals in Γ-semigroups and give some relationship between minimal almost α,β-bi-ideals and minimal fuzzy almost α,β-bi-ideals of Γ-semigroups.Definition 24.
A fuzzy almostα,β-bi-ideal f of a Γ-semigroup S is called minimal if for all fuzzy almost α,β-bi-ideal g of S such that g⊆f, we have suppg=suppf.Theorem 60.
LetB be a nonempty subset of a Γ-semigroup S. Then, B is a minimal almost α,β-bi-ideal of S if and only if CB is a minimal fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatB is a minimal almost α,β-bi-ideal of a Γ-semigroup S. By Theorem 58, CB is a fuzzy almost α,β-bi-ideal of S. Let g be a fuzzy almost α,β-bi-ideal of S such that g⊆CB. Then, suppg⊆suppCB=B. Since g⊆Csuppg and by Theorem 57, we have Csuppg is a fuzzy almost bi-ideal of S. By Theorem 58, suppg is an almost α,β-bi-ideal of S. Since B is minimal, suppg=B=suppCB. Therefore, CB is minimal.
Conversely, assume thatCB is a minimal fuzzy almost α,β-bi-ideal of S. Let B′ be an almost α,β-bi-ideal of S such that B′⊆B. Then, CB′ is a fuzzy almost α,β-bi-ideal of S such that CB′⊆CB. Hence, B′=suppCB′=suppCB=B. Therefore, B is minimal.Corollary 14.
LetB be a sub-Γ-semigroup of a Γ-semigroup S. Then, B is almost α,β-bi-simple if and only if for all fuzzy almost α,β-bi-ideal f of S, suppf=B.
Next, we give the relationship betweenα-prime almost α,β-bi-ideals and α-prime fuzzy almost α,β-bi-ideals.Definition 25.
LetS be a Γ-semigroup and γ∈Γ.(1)
An almostα,β-bi-ideal A of S is called γ-prime if for all x,y∈S,xγy∈A implies x∈A or y∈A.(2)
A fuzzy almostα,β-bi-ideal f of S is called γ-prime if for all x,y∈S,fxγy≤maxfx,fy.Theorem 61.
LetA be a nonempty subset of a Γ-semigroup S. Then, A is a γ-prime almost α,β-bi-ideal of S if and only if CA is a γ-prime fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatA is a γ-prime almost α,β-bi-ideal of S. By Theorem 58, CA is a fuzzy almost α,β-bi-ideal of S. Let x,y∈S. We consider two cases:Case 1:xγy∈A. So, x∈A or y∈A. Then, maxCAx,CAy=1≥CAxγy.Case 2:xγy∉A. Then, CAxγy=0≤maxCAx,CAy.
Thus,CA is a γ-prime fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCA is a γ-prime fuzzy almost α,β-bi-ideal of S. By Theorem 58, A is an almost α,β-bi-ideal of S. Let x,y∈S be such that xγy∈A. Then, CAxγy=1. By assumption, CAxγy≤maxCAx,CAy. Therefore, maxCAx,CAy=1. Hence, x∈A or y∈A. Thus, A is a γ-prime almost α,β-bi-ideal of S.
In this section, we give the relationship betweenγ-semiprime almost α,β-bi-ideals and γ-semiprime fuzzy almost α,β-bi-ideals.Definition 26.
LetS be a Γ-semigroup and α∈Γ.(1)
An almostα,β-bi-ideal A of S is called a γ-semiprime if for all x∈S,xγx∈A implies x∈A.(2)
A fuzzy almostα,β-bi-ideal f of S is called a γ-semiprime if for all x∈S,fxγx≤fx.Theorem 62.
LetA be a nonempty subset of S. Then, A is a γ-semiprime almost α,β-bi-ideal of S if and only if CA is a γ-semiprime fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatA is a γ-semiprime almost α,β-bi-ideal of S. By Theorem 58, CA is a fuzzy almost α,β-bi-ideal of S. Let x∈S. We consider two cases:Case 1:xγx∈A. Then, x∈A. So, CAx=1. Hence, CAx≥CAxγx.Case 2:xγx∉A. Then, CAxγx=0≤CAx.
Thus,CA is a γ-semiprime fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCA is a γ-semiprime fuzzy almost α,β-bi-ideal of S. By Theorem 58, A is an almost α,β-bi-ideal of S. Let x∈S be such that xγx∈A. Then, CAxγx=1. By assumption, CAxγx≤CAx. Since CAxγx=1, CAx=1. Hence, x∈A. Thus, A is a γ-semiprime almost α,β-bi-ideal of S.


## 5.1. Fuzzy Almostα,β-Ideals

Definition 19.
Letα,β∈Γ. Let f be a fuzzy subset of a Γ-semigroup S. Then, f is called(1)
A fuzzy almost leftα-ideal of S if xt∘αf∩f≠0 for all fuzzy point xt of S.(2)
A fuzzy almost rightβ-ideal of S if f∘βxt∩f≠0 for all fuzzy point xt of S.(3)
A fuzzy almostα,β-ideal of S if it is both a fuzzy almost left α-ideal and a fuzzy almost right β-ideal of S.Theorem 47.
Letf be a fuzzy almost left α-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost left α-ideal of S.Proof.
Assume thatf is a fuzzy almost left α-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for each fuzzy point xt, xt∘αf∩f≠0. We have xt∘αf∩f⊆xt∘αg∩g; this implies xt∘αg∩g≠0. Therefore, g is a fuzzy almost left α-ideal of S.Theorem 48.
Letf be a fuzzy almost right β-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost right β-ideal of S.Proof.
It is similar to Theorem47.Theorem 49.
Letf be a fuzzy almost (α,β)-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost (α,β)-ideal of S.Proof.
It follows by Theorem47 and Theorem 48.Corollary 7.
The union of two fuzzy almost leftα-ideals of a Γ-semigroup S is a fuzzy almost left α-ideal of S.Corollary 8.
The union of two fuzzy almost rightβ-ideals of a Γ-semigroup S is a fuzzy almost right β-ideal of S.Corollary 9.
The union of two fuzzy almost (α,β)-ideals of a Γ-semigroup S is a fuzzy almost (α,β)-ideal of S.Theorem 50.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
A is an almost left α-ideal of S if and only if CA is a fuzzy almost left α-ideal of S.(2)
A is an almost right β-ideal of S if and only if CA is a fuzzy almost right β-ideal of S.(3)
A is an almost (α,β)-ideal of S if and only if CA is a fuzzy almost (α,β)-ideal of S.Proof.
(1) Assume thatA is an almost left α-ideal of a Γ-semigroup S. Then, xαA∩A≠∅ for all x∈S. Thus, there exists y∈xαA, and y∈A. So, xt∘αCAy=1 and CAy=1. Hence, xt∘αCA∩CA≠0. Therefore, CA is a fuzzy almost left α -ideal of S.
Conversely, assume thatCA is a fuzzy almost left α -ideal of S. Let x∈S. Then, xt∘αCA∩CA≠0. Then, there exists a∈S such that xt∘αCA∩CAa≠0. Hence, a∈xαA∩A. So, xαA∩A≠∅. Consequently, A is an almost left α -ideal of S.
The proofs of (2) and (3) are similar to the proof of (1).Theorem 51.
Letf be a fuzzy subset of a α-semigroup S. Then,(1)
f is a fuzzy almost left α -ideal of S if and only if suppf is an almost left α -ideal of S.(2)
f is a fuzzy almost right β -ideal of S if and only if suppf is an almost right β-ideal of S.(3)
f is a fuzzy almost (α,β)-ideal of S if and only if suppf is an almost (α,β)-ideal of S.Proof.
(1) Assume thatf is a fuzzy almost left α -ideal of a Γ -semigroup S. Let x∈S. Then, xt∘αf∩f≠0. Hence, there exists a∈S such that xt∘αf∩fa≠0 . So, there exists b∈S such that a=xαb,fa≠0,fb≠0. That is, a,b∈suppf. Thus, xt∘αCsuppfa≠0 and Csuppfa≠0. Therefore, Cx∘αCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost left α -ideal of S. By Theorem 50, suppf is an almost left α-ideal of S.
Conversely, assume thatsuppf is an almost left α -ideal of S. By Theorem 50, Csuppf is a fuzzy almost left α -ideal of S. Then, xt∘αCsuppf∩Csuppf≠0 for all x∈S. Then, there exists a∈S such that xt∘αCsuppf∩Csuppfa≠0. Hence, xt∘αCsuppfa≠0 and Csuppfa≠0. Then, there exists y∈S such that a=xαy,fa≠0,fy≠0. This means xt∘αf∩f≠0. Therefore, f is a fuzzy almost left α ideal of S.
The proofs of (2) and (3) are similar to the proof of (1).Definition 20.
A fuzzy almost leftα -ideal f of a Γ -semigroup S is minimal if for all fuzzy almost left α -ideal g of S such that g⊆f, we obtain suppg=suppf.Theorem 52.
LetA be a nonempty subset of a Γ-semigroup S. Then,(1)
A is a minimal almost left α-ideal of S if and only if CA is a minimal fuzzy almost left α-ideal of S.(2)
A is a minimal almost right β-ideal of S if and only if CA is a minimal fuzzy almost right β-ideal of S.(3)
A is a minimal almost (α,β)-ideal of S if and only if CA is a minimal fuzzy almost (α,β)-ideal of S.Proof.
(1) Assume thatA is a minimal almost left α-ideal of a Γ-semigroup S. By Theorem 50 (1), CA is a fuzzy almost left α-ideal of S. Let g be a fuzzy almost left α-ideal of S such that g⊆CA. By Theorem 51 (1), suppg is an almost left α-ideal of S. Then, suppg⊆suppCA=A. Since A is minimal, suppg=A=suppCA. Therefore, CA is minimal.
Conversely, assume thatCA is a minimal fuzzy almost left α-ideal of S. By Theorem 50 (1), A is an almost left α-ideal of S. Let L be an almost left α-ideal of S such that L⊆A. By Theorem 50 (1), CL is a fuzzy almost left α-ideal of S such that CL⊆CA. Hence, L=suppCL=suppCA=A. Therefore, A is minimal.
(2) and (3) can be proved similarly.Corollary 10.
LetA be a sub-Γ-semigroup of a Γ-semigroup S. Then,(1)
A is almost left α-simple if and only if for each fuzzy almost left α-ideal f of S, suppf=A.(2)
A is almost right β-simple if and only if for each fuzzy almost right β-ideal f of S, suppf=A.(3)
A is almost (α,β)-simple if and only if for each fuzzy almost (α,β)-ideal of S, suppf=A.


## 5.2. Fuzzy Almostα,β-Quasi-Ideals

Definition 21.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy almost α,β-quasi-ideal of S if f∘αxt∩xt∘βf∩f≠0 for all fuzzy points xt of S.Theorem 53.
Letf be a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S and g a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for all fuzzy point xt of S, f∘αxt∩xt∘βf∩f≠0. We have that f∘αxt∩xt∘βf∩f⊆g∘αxt∩xt∘βg∩g. This implies that g∘αxt∩xt∘βg∩g≠0. Therefore, g is a fuzzy almost α,β-quasi-ideal of S.Corollary 11.
Letf and g be fuzzy almost α,β-quasi-ideals of a Γ-semigroup S. Then, f∪g is a fuzzy almost α,β-quasi-ideal of S.Proof.
Sincef⊆f∪g, by Theorem 53, f∪g is a fuzzy almost α,β-quasi-ideal of S.Example 5.
Consider theΓ-semigroup ℤ5 where Γ=0¯,2¯,4¯ and a¯γb¯=a¯+γ+b¯, where a¯,b¯∈ℤ5 and γ∈Γ. Let f:ℤ5⟶0,1 defined by(28)f0¯=0,f1¯=0.6,f2¯=0,f3¯=0.4,f4¯=0.4,and g:ℤ5⟶0,1 defined by(29)g0¯=0,g1¯=0.3,g2¯=0.6,g3¯=0,g4¯=0.8.
We havef and g are fuzzy almost 0¯,0¯-quasi-ideals of ℤ5, but f∩g is not a fuzzy almost 0¯,0¯-quasi-ideal of ℤ5.Theorem 54.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is an almost α,β-quasi-ideal of S if and only if CQ is a fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatQ is an almost α,β-quasi-ideal of a Γ-semigroup S, and let xt be a fuzzy point of S. Then, Qαx∩xβQ∩Q≠∅. Thus, there exists y∈Qαx∩xβQ and y∈Q. So, CQ∘αxt∩xt∘βCQy≠0 and CQy=1. Hence, CQ∘αxt∩xt∘βCQ∩CQ≠0. Therefore, CQ is a fuzzy almost α,β-quasi-ideal of S.
Conversely, assume thatCQ is a fuzzy almost α,β-quasi-ideal of S. Let s∈S. Then, CQ∘αst∩st∘βCQ∩CQ≠0. Then, there exists x∈S such that CQ∘αst∩st∘βCQ∩CQx≠0. Hence, x∈Qαs∩sβQ∩Q. So, Qαs∩sβQ∩Q≠∅. Consequently, Q is an almost α,β-quasi-ideal of S.Theorem 55.
Letf be a fuzzy subset of a Γ-semigroup S. Then, f is a fuzzy almost α,β-quasi-ideal of S if and only if suppf is an almost α,β-quasi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-quasi-ideal of a Γ-semigroup S. Let s∈S and t∈0,1. Then, f∘αst∩st∘βf∩f≠0. Hence, there exists x∈S such that(30)f∘αst∩st∘βf∩fx≠0.
So, there existy1,y2∈S such that x=y1αs=sβy2,fx≠0,fy1≠0, and fy2≠0. That is, x,y1,y2∈suppf. Thus, Csuppf∘αst∩st∘βCsuppfx≠0 and Csuppfx≠0. Therefore, Csuppf∘αst∩st∘βCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost α,β-quasi-ideal of S. By Theorem 54, suppf is an almost α,β-quasi-ideal of S.
Conversely, assume thatsuppf is an almost α,β-quasi-ideal of S. By Theorem 54, Csuppf is a fuzzy almost α,β-quasi-ideal of S. Then, for each fuzzy point st of S, we have Csuppf∘αst∩st∘βCsuppf∩Csuppf≠0. Then, there exists x∈S such that(31)Csuppf∘αst∩st∘βCsuppf∩Csuppfx≠0.
Hence,Csuppf∘αst∩st∘βCsuppfx≠0 and Csuppfx≠0. Then, there exist y1,y2∈S such that x=y1αs=sβy2,fx≠0,fy1≠0, and fy2≠0. This means that f∘αst∩st∘βf∩f≠0. Therefore, f is a fuzzy almost α,β-quasi-ideal of S.
Next, we define minimal fuzzy almostα,β-quasi-ideals in Γ-semigroups and give some relationship between minimal almost α,β-quasi-ideals and minimal fuzzy almost α,β-quasi-ideals of Γ-semigroups.Definition 22.
A fuzzy almostα,β-quasi-ideal f of a Γ-semigroup is called minimal if for each fuzzy almost α,β-quasi-ideal g of S such that g⊆f, we have suppg=suppf.Theorem 56.
LetQ be a nonempty subset of a Γ-semigroup S. Then, Q is a minimal almost α,β-quasi-ideal of S if and only if CQ is a minimal fuzzy almost α,β-quasi-ideal of S.Proof.
Assume thatQ is a minimal almost α,β-quasi-ideal of a Γ-semigroup S. By Theorem 54, CQ is a fuzzy almost α,β-quasi-ideal of S. Let g be a fuzzy almost α,β-quasi-ideal of S such that g⊆CQ. Then, suppg⊆suppCQ=Q. Since g⊆Csuppg and by Theorem 53, we have Csuppg is a fuzzy almost α,β-quasi-ideal of S. By Theorem 54, suppg is an almost α,β-quasi-ideal of S. Since Q is minimal, suppg=Q=suppCQ. Therefore, CQ is minimal.
Conversely, assume thatCQ is a minimal fuzzy almost α,β-quasi-ideal of S. Let Q′ be an almost α,β-quasi-ideal of S such that Q′⊆Q. By Theorem 54, CQ′ is a fuzzy almost α,β-quasi-ideal of S such that CQ′⊆CQ. Since CQ is minimal, Q′=suppCQ′=suppCQ=Q. Therefore, Q is minimal.Corollary 12.
LetQ be a sub Γ-semigroup of a Γ-semigroup S. Then, Q is almost α,β-quasi-simple if and only if for all fuzzy almost α,β-quasi-ideal f of S, suppf=Q.


## 5.3. Fuzzy Almostα,β-Bi-Ideals

Definition 23.
Letα,β∈Γ and f be a fuzzy subset of a Γ-semigroup S. Then, f is called a fuzzy almost α,β-bi-ideal of S if f∘αxt∘βf∩f≠0 for all fuzzy point xt of S.Theorem 57.
Letf be a fuzzy almost α,β-bi-ideal of a Γ-semigroup S and g be a fuzzy subset of S such that f⊆g. Then, g is a fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-bi-ideal of a Γ-semigroup S and g is a fuzzy subset of S such that f⊆g. Then, for each fuzzy point xt, f∘αxt∘βf∩f≠0. We have f°αxt°βf∩f⊆g°αxt°βg∩g; this implies g∘αxt∘βg∩g≠0. Therefore, g is a fuzzy almost α,β-bi-ideal of S.Corollary 13.
Letf and g be fuzzy almost α,β-bi-ideals of a Γ-semigroup S. Then, f∪g is a fuzzy almost α,β-bi-ideal of S.Example 6.
Consider theΓ-semigroup ℤ5 where Γ=0¯,4¯ and a¯γb¯=a¯+γ+b¯, where a¯,b¯∈ℤ5 and γ∈Γ. Let f:ℤ5⟶0,1 defined by(32)f0¯=0,f1¯=0.7,f2¯=0,f3¯=0.6,f4¯=0.4,and g:ℤ5⟶0,1 defined by(33)g0¯=0,g1¯=0.1,g2¯=0.6,g3¯=0,g4¯=0.8.
We havef and g are fuzzy almost 0¯,4¯-bi-ideals of ℤ5.Remark 8.
The intersection of two fuzzy almost (α,β)-bi-ideals of a Γ-semigroup S need not be a fuzzy almost (α,β)-bi-ideal of S.Theorem 58.
LetB be a nonempty subset of a Γ-semigroup S. Then, B is an almost α,β-bi-ideal of S if and only if CB is a fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatB is an almost α,β-bi-ideal of a Γ-semigroup S. Then, BαxβB∩B≠∅ for all x∈S. Thus, there exists y∈BαxβB and y∈B. So, CB∘αxt∘βCBy=1 and CBy=1. Hence, CB∘αxt∘βCB∩CB≠0. Therefore, CB is a fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCB is a fuzzy almost α,β-bi-ideal of S. Let s∈S. Then, CB∘αs1∘βCB∩CB≠0. Thus, there exists x∈S such that CB∘αs1∘βCB∩CBx≠0. Hence, x∈BαsβB∩B. So, BαsβB∩B≠∅. Consequently, B is an almost α,β-bi-ideal of S.Theorem 59.
Letf be a fuzzy subset of a Γ-semigroup S. Then, f is a fuzzy almost α,β-bi-ideal of S if and only if suppf is an almost α,β-bi-ideal of S.Proof.
Assume thatf is a fuzzy almost α,β-bi-ideal of a Γ-semigroup S. Let s∈S. Then, f∘αst∘βf∩f≠0. Hence, there exists x∈S such that fα∘st∘βf∩fx≠0. So, there exist y1,y2∈S such that x=y1αsβy2,fx≠0,fy1≠0, and fy2≠0. That is, x,y1,y2∈suppf. Thus, Csuppf∘αst∘βCsuppfx≠0 and Csuppfx≠0. Therefore, Csuppf∘αst∘βCsuppf∩Csuppf≠0. Hence, Csuppf is a fuzzy almost α,β-bi-ideal of S. By Theorem 58, suppf is an almost α,β-bi-ideal of S.
Conversely, assume thatsuppf is an almost α,β-bi-ideal of S. By Theorem 58, Csuppf is a fuzzy almost α,β-bi-ideal of S. Then, Csuppf∘αst∘βCsuppf∩Csuppf≠0 for all s∈S. Then, there exists x∈S such that Csuppf∘αst∘βCsuppf∩Csuppfx≠0. Hence, Csuppf∘αst∘βCsuppfx≠0 and Csuppfx≠0. Then, there exist y1,y2∈S such that x=y1αsβy2,fx≠0,fy1≠0, and fy2≠0. This means f∘αst∘βf∩f≠0. Therefore, f is a fuzzy almost α,β-bi-ideal of S.
We define minimal fuzzy almostα,β-bi-ideals in Γ-semigroups and give some relationship between minimal almost α,β-bi-ideals and minimal fuzzy almost α,β-bi-ideals of Γ-semigroups.Definition 24.
A fuzzy almostα,β-bi-ideal f of a Γ-semigroup S is called minimal if for all fuzzy almost α,β-bi-ideal g of S such that g⊆f, we have suppg=suppf.Theorem 60.
LetB be a nonempty subset of a Γ-semigroup S. Then, B is a minimal almost α,β-bi-ideal of S if and only if CB is a minimal fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatB is a minimal almost α,β-bi-ideal of a Γ-semigroup S. By Theorem 58, CB is a fuzzy almost α,β-bi-ideal of S. Let g be a fuzzy almost α,β-bi-ideal of S such that g⊆CB. Then, suppg⊆suppCB=B. Since g⊆Csuppg and by Theorem 57, we have Csuppg is a fuzzy almost bi-ideal of S. By Theorem 58, suppg is an almost α,β-bi-ideal of S. Since B is minimal, suppg=B=suppCB. Therefore, CB is minimal.
Conversely, assume thatCB is a minimal fuzzy almost α,β-bi-ideal of S. Let B′ be an almost α,β-bi-ideal of S such that B′⊆B. Then, CB′ is a fuzzy almost α,β-bi-ideal of S such that CB′⊆CB. Hence, B′=suppCB′=suppCB=B. Therefore, B is minimal.Corollary 14.
LetB be a sub-Γ-semigroup of a Γ-semigroup S. Then, B is almost α,β-bi-simple if and only if for all fuzzy almost α,β-bi-ideal f of S, suppf=B.
Next, we give the relationship betweenα-prime almost α,β-bi-ideals and α-prime fuzzy almost α,β-bi-ideals.Definition 25.
LetS be a Γ-semigroup and γ∈Γ.(1)
An almostα,β-bi-ideal A of S is called γ-prime if for all x,y∈S,xγy∈A implies x∈A or y∈A.(2)
A fuzzy almostα,β-bi-ideal f of S is called γ-prime if for all x,y∈S,fxγy≤maxfx,fy.Theorem 61.
LetA be a nonempty subset of a Γ-semigroup S. Then, A is a γ-prime almost α,β-bi-ideal of S if and only if CA is a γ-prime fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatA is a γ-prime almost α,β-bi-ideal of S. By Theorem 58, CA is a fuzzy almost α,β-bi-ideal of S. Let x,y∈S. We consider two cases:Case 1:xγy∈A. So, x∈A or y∈A. Then, maxCAx,CAy=1≥CAxγy.Case 2:xγy∉A. Then, CAxγy=0≤maxCAx,CAy.
Thus,CA is a γ-prime fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCA is a γ-prime fuzzy almost α,β-bi-ideal of S. By Theorem 58, A is an almost α,β-bi-ideal of S. Let x,y∈S be such that xγy∈A. Then, CAxγy=1. By assumption, CAxγy≤maxCAx,CAy. Therefore, maxCAx,CAy=1. Hence, x∈A or y∈A. Thus, A is a γ-prime almost α,β-bi-ideal of S.
In this section, we give the relationship betweenγ-semiprime almost α,β-bi-ideals and γ-semiprime fuzzy almost α,β-bi-ideals.Definition 26.
LetS be a Γ-semigroup and α∈Γ.(1)
An almostα,β-bi-ideal A of S is called a γ-semiprime if for all x∈S,xγx∈A implies x∈A.(2)
A fuzzy almostα,β-bi-ideal f of S is called a γ-semiprime if for all x∈S,fxγx≤fx.Theorem 62.
LetA be a nonempty subset of S. Then, A is a γ-semiprime almost α,β-bi-ideal of S if and only if CA is a γ-semiprime fuzzy almost α,β-bi-ideal of S.Proof.
Assume thatA is a γ-semiprime almost α,β-bi-ideal of S. By Theorem 58, CA is a fuzzy almost α,β-bi-ideal of S. Let x∈S. We consider two cases:Case 1:xγx∈A. Then, x∈A. So, CAx=1. Hence, CAx≥CAxγx.Case 2:xγx∉A. Then, CAxγx=0≤CAx.
Thus,CA is a γ-semiprime fuzzy almost α,β-bi-ideal of S.
Conversely, assume thatCA is a γ-semiprime fuzzy almost α,β-bi-ideal of S. By Theorem 58, A is an almost α,β-bi-ideal of S. Let x∈S be such that xγx∈A. Then, CAxγx=1. By assumption, CAxγx≤CAx. Since CAxγx=1, CAx=1. Hence, x∈A. Thus, A is a γ-semiprime almost α,β-bi-ideal of S.


## 6. Discussion and Conclusion

In this paper, we define new types of ideals and fuzzy ideals by using elements inΓ. We show interesting properties of these ideals and fuzzy ideals. Moreover, we show the relationships between these ideals and their fuzzifications.



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