Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups

Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Γ-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Γ-semigroups. Many results in ideal theory of Γ-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Γ-semigroups. In Section 2, we define almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups by using the concept ideas of interior Γ-ideals and almost Γ-ideals of Γ-semigroups. Every almost interior Γ-ideal of a Γ-semigroup S is clearly a weakly almost interior Γ-ideal of S but the converse is not true in general. The notions of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups are generalizations of the notion of interior Γ-ideal of a Γ-semigroup S. We investigate basic properties of both almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Γ-semigroups. In Section 3, we introduce fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γ-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and fuzzy almost interior Γ-ideals [fuzzy weakly almost interior Γ-ideals] of Γ-semigroups.


Introduction and preliminaries
In 1965, Zadeh [25] first introduced the notion of fuzzy subsets. Applications of fuzzy subsets have been developed in many fields. Rosenfeld applied the fuzzy subsets to define fuzzy subgroups of groups in [14]. Applications of fuzzy subsets in semigroups were first considered by Kuroki [11,12]. Now, fuzzy subsets were studied in many algebraic structures (for example, in ternary semigroups ( [21]), in non-associative ordered semigroups ( [1]), etc). The definition of almost ideals of semigroups (or A-ideals) was first studied by Grosek and Satko [5,6,7] in 1980. Recently, Wattanatripop, Chinram and Changphas [23,24] introduced the notion of quasi almost ideals (or quasi-A-ideals) of semigroups and gave their basic properties. Moreover, they applied fuzzy subsets to define fuzzy almost ideals, fuzzy almost bi-ideals and fuzzy quasi almost ideals of semigroups and showed relationship between almost ideals [almost bi-ideals, quasi almost ideals] and their fuzzifications of semigroups. Furthermore, Kaopusek, Kaewnoi and Chinram [9] using the concepts of interior ideals and almost ideals of semigroups, defined the notions of almost interior ideals and weakly almost interior ideals of semigroups. Moreover, they investigated their basic properties. Now, the notion of almost ideals in semigroups were extend to some generalizations of semigroups, for example, almost ideals in ternary semigroup [21], almost hyperideals in semihypergroups [20], etc.
Firstly, we recall the definitions and some notations of fuzzy sets. A fuzzy subset of a set S is a membership function from S into the closed unit interval [0, 1]. Let S be any set and f and g be any two fuzzy subsets of S.
(1) The intersection of f and g (f ∩ g) is a fuzzy subset of S defined as follows: (2) The union of f and g (f ∪ g) is a fuzzy subset of S defined as follows: for all x ∈ S, and we say that f is a fuzzy subset of g.
For any fuzzy subset f of any set S, the support of f (supp(f )) is a subset of S defined by For any subset A of any set S, the characteristic mapping C A of A is a fuzzy subset of S defined by For any element x of any set S and a real number t ∈ (0, 1], a fuzzy point x t ( [13]) of S is a fuzzy subset of S defined by Now, we recall the definition of Γ-semigroups defined by Sen and Saha in [17]. (1) for all a, b ∈ S and for all α ∈ Γ, aαb ∈ S and (2) for all a, b, c ∈ S and for all α, β ∈ Γ, (aαb)βc = aα(bβc).

For nonempty subsets
If x ∈ S, we let AΓx := AΓ{x} and xΓB := {x}ΓB. For α ∈ Γ, we let Let S be a Γ-semigroup and F(S) be the set of all fuzzy subsets of S. For each α in Γ, let • α be a binary operation on F(S) defined by Definition 1.2. Let S be a Γ-semigroup and T be a nonempty subset of S.
Next, we recall the definitions of interior Γ-ideals of Γsemigroups that we will use later.
In 2010, Sardar, Davvaz and Majumder studied fuzzy interior Γ-ideals (or interior ideals) of Γ-semigroups and investigated some of their basic properties in [15]. They obtained the characterization of simple Γ-semigroups in terms of fuzzy interior Γ-ideals. Definition 1.4. [15] A fuzzy subsemigroup f of a Γ-semigroup S is called a fuzzy interior Γ-ideal of S if it satisfies, for all a, x, y ∈ S and α, β ∈ Γ.
Wattanatripop and Changphas first defined and studied almost Γ-ideals in Γ-semigroups [22] by using the concept of almost ideal in semigroups defined by Grosek and Satko. They defined left (right) almost Γ-ideals in Γ-semigroups as the following.
Simuen, Wattanatripop and Chinram [19] using the concepts of quasi Γ-ideals in Γ-semigroups and almost Γ-ideals in Γ-semigroups, studied the basic properties of both almost quasi Γ-ideals and fuzzy almost quasi Γ-ideals of Γsemigroups. Moreover, they gave the remarkable relationship between almost quasi-Γ-ideals and their fuzzification. In addition, Simuen, Abdullah, Yonthanthum and Chinram [18] introduced the concepts of almost bi Γ-ideals and fuzzy almost bi-Γ-ideals of Γ-semigroups. Also, they gave some properties 304 Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in Gamma-semigroups and investigated relationship between almost bi Γ-ideals of Γsemigroups and their fuzzification.
Similar to quasi-Γ-ideals and bi-Γ-ideals, interior Γ-ideals are elementary ideals in Γ-semigroups. This is motivation to study almost interior Γ-ideals and their fuzzification. The purpose of this paper is to introduce and study almost interior Γideals and weakly almost interior Γ-ideals of Γ-semigroups. Moreover, we investigate their properties. Furthermore, we define the fuzzifications of almost interior Γ-ideals [weakly almost interior Γ-ideals] of Γ-semigroups and give some relationship between almost interior Γ-ideals [weakly almost interior Γ-ideals] and their fuzzification.

Almost interior Γ-ideals in Γsemigroups
In this section, we give the definitions of almost interior Γideals and weakly almost interior Γ-ideals of Γ-semigroups. Moreover, we investigate their properties.
Proof. Assume that I is any interior Γ-ideal of a Γ-semigroup S and let a, b ∈ S. Then aΓIΓb ⊆ I. This implies that aΓIΓb ∩ I = ∅. It is conclude that I is an almost interior Γ-ideal of S. Example 2.1 shows that the converse of Theorem 2.1 is not generally true.
Theorem 2.2. Let I and H be any two nonempty subsets of a Γ-semigroup S such that I ⊆ H. If I is an almost interior Γ-ideal of S, then H is also an almost interior Γ-ideal of S.
Proof. Let H be a subset of S containing I and let a, b ∈ S.
From Theorem 2.2, we have the following corollary. Corollary 2.3. Let S be a Γ-semigroup and I 1 , I 2 be any two almost interior Γ-ideals of a Γ-semigroup S. Thus I 1 ∪ I 2 is also an almost interior Γ-ideal of S.
Proof. Since I 1 ⊆ I 1 ∪I 2 , by Theorem 2.2, I 1 ∪I 2 is an almost interior Γ-ideal of S.
It is easy to show that I 1 and I 2 are almost interior Γ-ideals of

Remark 2.1 follows from Example 2.2.
Remark 2.1. If I 1 and I 2 are almost interior Γ-ideals of Γsemigroups, then I 1 ∩ I 2 need not be an almost interior Γ-ideal of Γ-semigroups.
From Example 2.3, we have the following remark.
However, the converse of Remark 2.2 is not true by the following example.
Theorem 2.4. Let I and H be any two nonempty subsets of a Γ-semigroup S such that I ⊆ H. If I is a weakly almost interior Γ-ideal of S, then H is also a weakly almost interior Γ-ideal of S.
Proof. Let a be any element of S. Since I ⊆ H, then aΓIΓa ∩ I ⊆ aΓHΓa∩H. Thus aΓHΓa∩H = ∅ because aΓIΓa∩I = ∅. Hence, H is a weakly almost interior Γ-ideal of S.
From Theorem 2.4, The following corollary holds.
Corollary 2.5. Let I 1 and I 2 be weakly almost interior Γideals of a Γ-semigroup S. Then I 1 ∪ I 2 is a weakly interior Γ-ideal of S.
Proof. Since I 1 ⊆ I 1 ∪ I 2 , it follows from Theorem 2.4 that I 1 ∪ I 2 is a weakly almost interior Γ-ideal of S. We see that I 1 and I 2 are weakly almost interior Γ-ideals of Z 4 but I 1 ∩ I 2 = ∅, so it is not a weakly almost interior Γ-ideal of Z 4 .
The following remark holds by Example 2.5.  3 Fuzzy almost interior Γ-ideals in Γsemigroups In this section, we give the definitions of fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γsemigroups. Moreover, we investigate their properties.
The definition of fuzzy almost interior Γ-ideals of Γsemigroups is defined as follows: A fuzzy subset f of a Γ-semigroup S is said to be a fuzzy almost interior Γ-ideal of S if for all fuzzy points x t1 , y t2 of S, there exist α, β ∈ Γ such that x t1 • α f • β y t2 ∩f = 0. It is easy to see that f is a fuzzy almost interior Γ-ideal of Z 5 .
Moreover, fuzzy weakly almost interior Γ-ideals of Γsemigroups is defined as follows: A fuzzy subset f of a Γ-semigroup S is said to be a fuzzy weakly almost interior Γ-ideal of S if for all fuzzy points x t1 , x t2 of S, there exist α, β ∈ Γ such that x t1 • α f • β x t2 ∩ f = 0. We have that f is a fuzzy weakly almost interior Γ-ideal of Z 6 . Furthermore, we have the following remark.  We have that f is a fuzzy weakly almost interior Γ-ideal but not a fuzzy almost interior Γ-ideal of Z 6 . Theorem 3.1. Let f and g be any two fuzzy subsets of a Γsemigroup S such that f ⊆ g. Suppose that f is a fuzzy almost interior Γ-ideal of S. Then g is also a fuzzy almost interior Γ-ideal of S.
Proof. Let x t1 and x t2 be fuzzy points of S. Since f is a fuzzy almost interior Γ-ideal of S, there exist α and β of Γ such that From Theorem 3.1, we have the following corollary.
Corollary 3.2. Let f and g be any two fuzzy almost interior Γ-ideals of a Γ-semigroup S. Thus f ∪ g is also a fuzzy almost interior Γ-ideal of S.
Proof. Since f ⊆ f ∪ g, by Theorem 3.1, f ∪ g is a fuzzy almost interior Γ-ideal of S. Theorem 3.3. Let f and g be any two fuzzy subsets of a Γsemigroup S such that f ⊆ g. Assume that f is a fuzzy weakly almost interior Γ-ideal of S. Then g is also a fuzzy weakly almost interior Γ-ideal of S.
Proof. The proof of this theorem is similar to the proof of Theorem 3.1.
From Theorem 3.3, we have the following corollary.    We have that f and g are fuzzy weakly almost interior Γ-ideals of Z 6 but f ∩ g is not a fuzzy weakly almost interior Γ-ideal of Z 6 .
Likewise, we have the following remark. Remark 3.3. If f and g are any two fuzzy weakly almost interior Γ-ideals of Γ-semigroups, then f ∩ g need not be a fuzzy weakly almost interior Γ-ideal of Γ-semigroups.
Theorem 3.5. Let I be a nonempty subset of a Γ-semigroup S. We have that I is an almost interior Γ-ideal of S if and only if C I is a fuzzy almost interior Γ-ideal of S.
Proof. Suppose that I is an almost interior Γ-ideal of a Γsemigroup S and let x t1 and y t2 be any two fuzzy points of S. Then xΓIΓy ∩ I = ∅. Thus there exists an element z ∈ S such that z ∈ xΓIΓy and z ∈ I. Thus z ∈ xαIβy for some α, β ∈ Γ. So (x t1 • α C I • β y t2 )(z) = 0 and C I (z) = 1. Hence, x t1 • α C I • β y t2 ∩ C I = 0. Therefore, C I is a fuzzy almost interior Γ-ideal of S.
Conversely, suppose that C I is a fuzzy almost interior Γideal of S. Let x, y ∈ S. Then x t1 • α C I • β y t2 ∩ C I = 0 for some α and β in Γ. Then there exists z ∈ S such that [(x t1 • α C I • β y t2 ) ∩ C I ](z) = 0. Hence z ∈ xΓIΓy ∩ I. So xΓIΓy ∩ I = ∅. Consequently, I is an almost interior Γ-ideal of S. Theorem 3.6. Let I be a nonempty subset of a Γ-semigroup S. We have that I is a weakly almost interior Γ-ideal of S if and only if C I is a fuzzy weakly almost interior Γ-ideal of S.
Proof. The proof of this theorem is similar to the proof of Theorem 3.5. Proof. Let f be a fuzzy almost interior Γ-ideal of a Γsemigroup S. Let x t1 and y t2 be an two fuzzy points of S. Then Hence C supp(f ) is a fuzzy almost interior Γideal of S. By Theorem 3.5, supp(f ) is an almost interior Γideal of S.
To prove the converse, suppose that supp(f ) is an almost interior Γ-ideal of S. By Theorem 3.5, we have that C supp(f ) is a fuzzy almost interior Γ-ideal of S. Let x t1 and y t2 be any two fuzzy points of S.  Proof. The proof of this theorem is similar to the proof of Theorem 3.7.
Next, we define minimal fuzzy almost interior Γ-ideals in Γ-semigroups and give some relationship between minimal almost interior Γ-ideals and minimal fuzzy almost interior Γideals of Γ-semigroups. Proof. Let I be a minimal almost interior Γ-ideal of S. By Theorem 3.5, we have that C I is a fuzzy almost interior Γideal of S. Let g be any fuzzy almost interior Γ-ideal of S such that g ⊆ C I . By Theorem 3.7, we have that supp(g) is an almost interior Γ-ideal of S. Moreover, we have that supp(g) ⊆ supp(C I ) = I. Since I is minimal, supp(g) = I = supp(C I ).
This implies that C I is minimal.
To prove the converse, assume that C I is a minimal fuzzy almost interior Γ-ideal of S. Let I be any almost interior Γideal of S such that I ⊆ I. Thus C I is a fuzzy almost interior Γ-ideal of S such that C I ⊆ C I . Hence, I = supp(C I ) = supp(C I ) = I. It is conclude that I is minimal.
From Theorem 3.9, we have the following corollary. Proof. Let f be a fuzzy almost interior Γ-ideal of S. By Theorem 3.7, we have that supp(f ) is an almost interior Γideal of S. Since S has no proper almost interior Γ-ideal, supp(f ) = S.
Conversely, suppose that I is an proper almost interior Γideal of S. By Theorem 3.9, we have that C I is a fuzzy almost interior Γ-ideal of S. Thus supp(C I ) = I = S, a contradiction. Hence, S has no proper almost interior Γ-ideal.
Next, we will study the minimality of fuzzy weakly almost interior Γ-ideals of Γ-semigroups.
Definition 3.4. A fuzzy weakly almost interior Γ-ideal f is called minimal if for each fuzzy weakly almost interior Γ-ideal g of S such that g ⊆ f , we have supp(g) = supp(f ).
Theorem 3.11. A nonempty subset I of a Γ-semigroup A is a minimal weakly almost interior Γ-ideal of S if and only if C I is a minimal fuzzy weakly almost interior Γ-ideal of S.
Proof. The proof of this theorem is similar to the proof of Theorem 3.9.
From Theorem 3.11, we have the following corollary. Proof. The proof of this corollary is similar to the proof of Corollary 3.10.

Conclusions
In this paper, we define some novel of ideals and fuzzy ideals of Γ-semigroups and investigate their remarkable properties. In Section 2, we define almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups by using ideas of almost interior Γ-ideals and weakly almost interior Γ-ideals of Γ-semigroups in [9]. Moreover, The union of two (weakly) almost interior Γ-ideals is also a (weakly) almost interior Γideals but the intersection of them need not be a (weakly) almost interior Γ-ideals in Γ-semigroups. Moreover, we investigate the necessary and sufficient condition of Γ-semigroups having no proper weakly almost interior Γ-ideals in Theorem 2.6. In Section 3, we introduce fuzzy almost interior Γ-ideals and fuzzy weakly almost interior Γ-ideals of Γ-semigroups. Moreover, The union of two fuzzy (weakly) almost interior Γ-ideals is also a fuzzy (weakly) almost interior Γ-ideals but the intersection of them need not be a fuzzy (weakly) almost interior Γ-ideals in Γ-semigroups. We give some relationship between almost interior Γ-ideals and their fuzzification were shown in Theorem 3.5-3.9 and Theorem 3.11. We give the necessary and sufficient condition of Γ-semigroups having no proper (weakly) almost interior Γ-ideals by checking all (weakly) almost interior Γ-ideals in Corollary 3.10 and 3.12. Moreover, the results in this paper generalized the results in [9] and [10].
In the future work, we can study other kinds of almost ideals and fuzzifications in Γ-semigroups or almost ideals and fuzzifications in other algebraic structures.