A Novel of Ideals and Fuzzy Ideals of Gamma-Semigroups

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. %is 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 quasiideals, 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 of Γ-semigroups which was defined by Sen and Saha [15]. Definition 1 (see [15]). Let S and Γ be nonempty sets.%en, S is called a Γ-semigroup if there exists a mapping S × Γ × S⟶ S written as (a, c, b)⟼ acb satisfying the axiom (aαb)βc � aα(bβc) for all a, b, c ∈ S and α, β ∈ Γ.


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][6][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. is 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 quasiideals, 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 of Γ-semigroups which was defined by Sen and Saha [15].

Remark 1
(1) In case |Γ| � 1, the definition of Γ-semigroup is a semigroup (2) Every semigroup (S, ·) can be considered to be a Γsemigroup where Γ : � · { } (3) If S is a Γ-semigroup, then for each α ∈ Γ, (S, α) is a semigroup Let S be a Γ-semigroup. For nonempty subsets A, B of S, let If x ∈ S and α ∈ Γ, we let AΓx : � AΓ x { }, xΓA : � x { }ΓA, and AαB : � A α { }B. 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 set S 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 , for all x ∈ S. (2) (2) f ∪ g is a fuzzy subset of S defined by , for all x ∈ S. (3) (3) f ∘ g is a fuzzy subset of S defined by For a fuzzy subset f of a set S, the support of f is defined by e characteristic mapping of a subset A of S is a fuzzy subset of S defined by e definition of fuzzy points was given by Pu and Liu [16]. For x ∈ S and t ∈ (0, 1], a fuzzy point x t of a set S is a fuzzy subset of S defined by Some basic concepts of fuzzy semigroup theory can be seen in [17]. For a Γ-semigroup S, let F(S) be the set of all fuzzy subsets of S. For each α ∈ Γ, define a binary operation ∘ α on F(S) by Let Γ ⋆ : � ∘ α |α ∈ Γ . en, (F(S), 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.
In 1981, Bogdanović [8] gave the definition of almost biideals of semigroups as follows.
A nonempty subset B of a semigroup S is called an almost bi-ideal of S if 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 subset Q of a semigroup S is called an almost quasi-ideal of S if Let f be a fuzzy subset of a semigroup S such that f ≠ 0. en, f is called (4) A fuzzy almost quasi-ideal of S if for all s ∈ S, In 1981, Kuroki [4] introduced the notion of fuzzy ideals of semigroups as follows: A fuzzy subset f of a semigroup S is called: (3) A fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S. e aim of this paper is to define new types of ideals and fuzzy ideals of a Γ-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.
Definition 3. Let S be a Γ-semigroup, A be a nonempty subset of S, and α, β ∈ Γ. en, A is called However, the converse of Example 1 is not generally true. We can see in the following example.
It is easy to show that A is a left 4ideal but not a left ideal of S.

Theorem 1.
e following statements are true: For a nonempty subset A of a Γ-semigroup S, let (A) l(α) , (A) r(β) , and (A) i(α,β) be the left α-ideal, the right β-ideal, and the (α, β)-ideal of S generated by A, respectively.

Theorem 2. Let A be a nonempty subset of a Γ-semigroup S.
en,

Proof
(1) Let A be a nonempty subset of a Γ-semigroup S. Let erefore, 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.
as required. e proofs of (2) and (3) are similar to the proof of (1).

Theorem 3. Let L be a left α-ideal and R a right β-ideal of a Γ-semigroup S. en, LcR is an
Proof. Let L and R be a left α-ideal and a right β-ideal of S, respectively, and let c ∈ Γ. Clear that LcR ≠ ∅. We have Sα(LcR) � (SαL)cR⊆LcR and (LcR)βS � Lc(RβS)⊆LcR. erefore, LcR is an (α, β)-ideal of S.

Theorem 4.
Let L 1 and L 2 be two left α-ideals of a Γsemigroup S. e following statements are true:

Proof
(1) Let L 1 and L 2 be two left α-ideals of S. Clear that Theorem 5. Let R 1 and R 2 be two right β-ideals of a Γsemigroup S. en, Proof. It is similar to eorem 4. Theorem 6. Let I 1 and I 2 be two (α, β)-ideals of a Γsemigroup S. en, Proof. It follows by eorems 4 and 5.
Let A be a nonempty subset of a Γ-semigroup S. Let (A) q(α,β) be the (α, β)-quasi-ideal of S generated by A.

Theorem 8. Let A be a nonempty subset of a Γ-semigroup S.
en, Proof. Let A be a nonempty subset of a Γ-semigroup S. Let Theorem 9. Let S be a Γ-semigroup. Let L and R be a left α-ideal and right β-ideal of S, respectively.
Proof. Let L and R be a left α-ideal and a right β-ideal of a Γsemigroup S, respectively. en,  Proof. Assume that S 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, Conversely, assume that Sαs ∩ sαS � S for all s ∈ S. Let Q be an α-quasi-ideal of S and q ∈ Q. By assumption, Hence, Q is a (β, α)-bi-ideal of S.
Let A be a nonempty subset of a Γ-semigroup S, and let Theorem 15. Let A be a nonempty subset of a Γ-semigroup S and α, β ∈ Γ. en,
Definition 10. Let α, β ∈ Γ and f be a fuzzy subset of a Γsemigroup S. en, f is called α-ideal and a fuzzy right β-ideal of S.
Theorem 20. Let A be a nonempty subset of a Γ-semigroup S. en, the following statements are true: (2) is similar to (1).
Theorem 21. Let f be a fuzzy subset of a Γ-semigroup S. en, the following properties hold:

Journal of Mathematics
Hence, f is a fuzzy left α-ideal of S. (2) and (3) can be seen in a similar fashion.
Theorem 22. Let f and g be fuzzy left α-ideals of a Γsemigroup S. en, (25) Theorem 23. Let f and g be fuzzy right β-ideals of a Γsemigroup S. en, Proof. It is similar to eorem 22.
Theorem 25. Let f be a nonzero fuzzy subset of a Γsemigroup S and f t � x ∈ S|f(x) ≥ t . e following statements are true: Conversely, assume that f t is a left α-ideal of S if t ∈ (0, 1] and f t ≠ ∅. Let x, y ∈ S and t � f(y). Since f(y) ≥ t, we have y ∈ f t so that f t ≠ ∅. us, f t is a left α-ideal of S. Since y ∈ f t and x ∈ S, we have xαy ∈ f t . en, f(xαy) ≥ t � f(y). Hence, f is a fuzzy left α-ideal of S.

Fuzzy
Definition 11. Let α, β ∈ Γ and f be a fuzzy subset of a Γ- Theorem 26. Let S be a Γ-semigroup. Let f and g be a fuzzy left α-ideal and a fuzzy right α-ideal of S, respectively. en, f ∩ g is a fuzzy α-quasi-ideal of S.
Proof. Let f and g be a fuzzy left α-ideal and a fuzzy right α-ideal of a Γ-semigroup S, respectively. We have g ∘ α f⊆ f ∩ g; this implies f ∩ g ≠ ∅. en, Journal of Mathematics

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. Let f be a fuzzy (α, β)-quasi-ideal of a Γ-semigroup S.
and also, h ∘ β S⊆h. us, g and h are a fuzzy left α-ideal and a fuzzy right β-ideal of S, respectively. We claim that Theorem 28. Let Q be a nonempty subset of a Γ-semigroup S.

Fuzzy (α, β)-Bi-Ideals
Definition 12. Let α, β ∈ Γ and f be a fuzzy subset of a Γsemigroup S. en, f is called a fuzzy (α, Theorem 29. Let S be a Γ-semigroup, g a fuzzy subset of S, and f a fuzzy (α, β)-bi-ideal of S. en, the following statements are true:

Proof
(1) Let x t be a fuzzy point of S. en, Proof. Let x t be a fuzzy point of S. en, Theorem 31. Let S be a Γ-semigroup and B a nonempty Conversely, assume that C B is a fuzzy (α, β)-bi-ideal of S.

Theorem 33. Let S be a Γ-semigroup. If L is a left α-ideal of S, then L is an almost left α-ideal of S.
Proof. Let L be a left α-ideal of S. en, SαL⊆L so that SαL ∩ L ≠ ∅. us, L is an almost left α-ideal of S.

Theorem 34. Let S 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 eorem 33.

Theorem 35. Let S 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 eorems 33 and 34.

Theorem 36. Let S 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. Let L be an almost left α-ideal of S and L⊆H⊆S. Since SαL ∩ L ≠ ∅ and SαL ∩ L⊆SαH ∩ H, we have SαH ∩ H ≠ ∅.
erefore, H is an almost left α-ideal of S.

Theorem 37. Let S 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 eorem 36.

Theorem 38. Let S 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 eorems 36 and 37.
Similarly, every almost (α, β)-quasi-ideal of a Γ-semigroup S is an almost right β-ideal of S, but the converse is not true. H is an almost (α, β)-quasiideal of S.

Fuzzy Almost (α, β)-Ideals
Definition 19. Let α, β ∈ Γ. Let f be a fuzzy subset of a Γsemigroup S. en, f is called for all fuzzy point x t of S. Proof. Assume that f is a fuzzy almost left α-ideal of a Γsemigroup S and g is a fuzzy subset of S such that f⊆g. en, for each fuzzy point erefore, g is a fuzzy almost left α-ideal of S.
Theorem 48. Let f be a fuzzy almost right β-ideal of a Γsemigroup S and g be a fuzzy subset of S such that f⊆g. en, g is a fuzzy almost right β-ideal of S.
Proof. It is similar to eorem 47.
Theorem 49. Let f be a fuzzy almost (α, β)-ideal of a Γsemigroup S and g be a fuzzy subset of S such that f⊆g. en, g is a fuzzy almost (α, β)-ideal of S.

Proof
(1) Assume that A is an almost left α-ideal of a Γ-semigroup S. en, xαA ∩ A ≠ ∅ for all x ∈ S. us, there exists y ∈ xαA, and y ∈ A. So, (x t ∘ α C A )(y) � 1 and Conversely, assume that C A is a fuzzy almost left α -ideal of S. Let x ∈ S. en, e proofs of (2) and (3) are similar to the proof of (1).
Theorem 51. Let f be a fuzzy subset of a α-semigroup S. en, (

1) f is a fuzzy almost left α -ideal of S if and only if supp(f) is an almost left α -ideal of S. (2) f is a fuzzy almost right β -ideal of S if and only if supp(f) is an almost right β-ideal of S. (3) f is a fuzzy almost (α, β)-ideal of S if and only if supp(f) is an almost (α, β)-ideal of S.
Proof Conversely, assume that supp(f) is an almost left α -ideal of S. By eorem 50, C supp(f) is a fuzzy almost left α -ideal of S. en, en, there exists y ∈ S such that a � xαy, f(a) ≠ 0, e proofs of (2) and (3) are similar to the proof of (1).
Definition 20. A fuzzy almost left α -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 supp(g) � supp(f).

Theorem 52. Let A be a nonempty subset of a Γ-semigroup S. en, (1) A is a minimal almost left α-ideal of S if and only if C A 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 C A 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
C A is a minimal fuzzy almost (α, β)-ideal of S.

Proof
(1) Assume that A is a minimal almost left α-ideal of a Γsemigroup S. By eorem 50 (1), C A is a fuzzy almost left α-ideal of S. Let g be a fuzzy almost left α-ideal of S such that g⊆C A . By eorem 51 (1), supp(g) is an almost left α-ideal of S.
en, supp(g)⊆supp(C A ) � A. Since A is minimal, supp(g) � A � supp(C A ). erefore, C A is minimal.
Conversely, assume that C A is a minimal fuzzy almost left α-ideal of S. By eorem 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 eorem 50 (1), C L is a fuzzy almost left α-ideal of S such that (2) and (3) can be proved similarly.

Fuzzy Almost (α, β)-Quasi-Ideals
Definition 21. Let α, β ∈ Γ and f be a fuzzy subset of a Γsemigroup S. en, f is called a fuzzy almost Theorem 53. Let f be a fuzzy almost (α, β)-quasi-ideal of a Γ-semigroup S and g a fuzzy subset of S such that f⊆g. en, g is a fuzzy almost (α, β)-quasi-ideal of S.
Proof. Assume that f 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. en, for all fuzzy point erefore, g is a fuzzy almost (α, β)-quasi-ideal of S.
and g:
Theorem 54. Let Q be a nonempty subset of a Γ-semigroup S. en, Q is an almost (α, β)-quasi-ideal of S if and only if C Q is a fuzzy almost (α, β)-quasi-ideal of S.
Definition 22. A fuzzy almost (α, β)-quasi-ideal f of a Γsemigroup is called minimal if for each fuzzy almost (α, β)-quasi-ideal g of S such that g⊆f, we have supp(g) � supp(f).
Theorem 56. Let Q be a nonempty subset of a Γ-semigroup S. en, Q is a minimal almost (α, β)-quasi-ideal of S if and only if C Q is a minimal fuzzy almost (α, β)-quasi-ideal of S.
Proof. Assume that Q is a minimal almost (α, β)-quasi-ideal of a Γ-semigroup S. By eorem 54, C Q is a fuzzy almost (α, β)-quasi-ideal of S. Let g be a fuzzy almost (α, β)-quasiideal of S such that g⊆C Q . en, supp(g)⊆supp(C Q ) � Q.
erefore, C Q is minimal.
Conversely, assume that C Q 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 eorem 54, C Q′ is a fuzzy almost (α, β)-quasi-ideal of S such that C Q′ ⊆C Q . Since C Q is minimal, Q ′ � supp(C Q′ ) � supp(C Q ) � Q. erefore, Q is minimal.

Fuzzy Almost (α, β)-Bi-Ideals
Definition 23. Let α, β ∈ Γ and f be a fuzzy subset of a Γsemigroup S. en, f is called a fuzzy almost (α, β)-bi-ideal of S if (f ∘ α x t ∘ β f) ∩ f ≠ 0 for all fuzzy point x t of S.
Theorem 57. Let f be a fuzzy almost (α, β)-bi-ideal of a Γsemigroup S and g be a fuzzy subset of S such that f⊆g. en, g is a fuzzy almost (α, β)-bi-ideal of S.
Theorem 59. Let f be a fuzzy subset of a Γ-semigroup S. en, f is a fuzzy almost (α, β)-bi-ideal of S if and only if supp(f) is an almost (α, β)-bi-ideal of S.
en, there exists x ∈ S such that

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

Data Availability
No data were used to support this research.

Conflicts of Interest
e authors declare that they have no conflicts of interest.