Almost Bi-Γ-Ideals and Fuzzy Almost Bi-Γ-Ideals of Γ-Semigroups

In this paper, we introduce the notions of almost bi-Γ-ideals and fuzzy almost bi-Γideals of Γ-semigroups and give properties of them. Moreover, we investigate relationships between almost bi-Γ-ideals and fuzzy almost bi-Γ-ideals. 2020 Mathematics Subject Classifications: 20M99


Introduction and Preliminaries
Ideal theory in semigroups, like all other algebraic structures, plays an important role in studying them. Good and Hughes [8] introduced the notion of bi-ideals of semigroups in 1952. An introductory definition of left, right, two-sided almost ideals of semigroups was launched by Grosek and Satko [9] in 1980. They gave the characterization of these ideals when a semigroup S contains no proper left, right, two-sided almost ideals in [9], and afterwards, they discovered the minimal almost ideals and the smallest almost ideals of semigroups in [10] and [11], respectively. In 1981, Bogdanovic [3] introduced the definition of almost bi-ideals in semigroups by using the definitions of almost ideals and bi-ideals in semigroups. In [5], Wattanatripop, Chinram and Changphas gave the properties of quasialmost-ideals and first defined the concept of fuzzy almost ideals in semigroups. Moreover, they provided the relationships between almost ideals and their fuzzification. Furthermore, they investigated fuzzification of almost bi-ideals in semigroups in [4]. Almost (m, n)ideals and their fuzzification in semigroups were studied by Suebsung, Wattanatripop and Chinram in [23]. Moreover, the idea of almost ideals and their fuzzification were extended to n-ary semigroups in [21].
A bi-Γ-ideal in Γ-semigroups was sometimes called a bi-ideal (see [14]). Some generalizations of this ideal were studied in [2] and [16]. Recently, Wattanatripop and Changphas first studied the concept of almost ideals in Γ-semigroups. In [22], they defined the definitions of left [right] almost ideals in Γ-semigroups. Moreover, a Γ-semigroup containing no proper left [right] almost ideals was characterized.
In 1965, Zadeh [24] introduced the concept of fundamental fuzzy sets. Since then, fuzzy sets have been studied in various fields. A function from a set M into the closed unit interval [0, 1] is called a fuzzy subset of M . Let f and g be any two fuzzy subsets of a set M .
(1) A fuzzy subset f ∩ g of M is defined by (2) A fuzzy subset f ∪ g of M is defined by (3) If f (m) ≤ g(m) for all m ∈ M , we say that f is a subset of g, and use the notation f ⊆ g and sometimes we will say that f is contained in g.
For a fuzzy subset f of any set M , the support of f is the set of points in M defined by For a subset A of any set M , the characteristic function χ A of A is a fuzzy subset of M defined by For any element m of any set M and t ∈ (0, 1], a fuzzy point m t of M is a fuzzy subset of M defined by (see [15]).

Almost bi-Γ-ideals
First, we define almost bi-Γ-ideals of Γ-semigroups as follows: From Example 2, we see that an almost bi-Γ-ideal of Γ-semigroup S need not be a bi-Γ-ideal of S.
Therefore, B is an almost bi-Γ-ideal of M .

Fuzzy almost bi-Γ-ideals
We define fuzzification of almost bi-Γ-ideals in Γ-semigroups as follows: Theorem 4. Assume that f and g are fuzzy subsets of a Γ-semigroup M such that f ⊆ g.
If f is a fuzzy almost bi-Γ-ideal of M , then g is also a fuzzy almost bi-Γ-ideal of M .
Proof. Since f is a fuzzy almost bi-Γ-ideal of M , for each fuzzy point Hence, g is also a fuzzy almost bi-Γ-ideal of M . We can easily to check that [((f ∩g)• α 0 t • β (f ∩g))∩(f ∩g)](a) = 0 for all α, β ∈ Γ, t ∈ (0, 1] and a ∈ Z 5 , so f ∩ g is not a fuzzy almost bi-Γ-ideal of Z 5 . The following remark follows from Example 5.  Proof. Assume that B is an almost bi-Γ-ideal of a Γ-semigroup M and let m t be any fuzzy point of M . Then BΓmΓB ∩ B = ∅. Thus there exists b ∈ B such that b ∈ BαmβB for some α, β ∈ Γ. This implies that (χ B • α m t • β χ B )(b) = 0 and χ B (b) = 0. Hence,

Relationships between almost bi-Γ-ideals and their fuzzification
To prove the converse, we assume that χ B is a fuzzy almost bi-Γ-ideal of M and let m ∈ M . Then there exist α, β ∈ Γ such that ( On the other hand, we assume that supp(f ) is an almost bi-Γ-ideal of M. It follows from Theorem 5 that χ supp(f ) is a fuzzy almost bi-Γ-ideal of M. Let m t be any fuzzy point We conclude that f is a fuzzy almost bi-Γ-ideal of M.
Next, we will study the minimality of fuzzy almost bi-Γ-ideals. Proof. Let A be a minimal almost bi-Γ-ideal of a Γ-semigroup M . By Theorem 5, we have that χ A is a fuzzy almost bi-Γ-ideal of M. Assume that g is a fuzzy almost bi- for all fuzzy points m t of M . Thus χ supp(g) is a fuzzy almost bi-Γ-ideal of M. By Theorem 5, supp(g) is an almost bi-Γ-ideal of M. Because of A is a minimal, then supp(g) = A = supp(χ A ). Therefore, χ A is minimal.
To prove the converse, assume that χ A is a minimal fuzzy almost bi-Γ-ideal of M and B is an almost bi-Γ-ideal of M contained in A. Then χ B is a fuzzy almost bi-Γ-ideal of M and χ B ⊆ χ A . Thus, B = supp(χ B ) = supp(χ A ) = A. We conclude that A is minimal. To prove the converse, we let B be any almost bi-Γ-ideal of M . Follow by Theorem 5, we have that χ B is a fuzzy almost bi-Γ-ideal of M . By assumption, we get B = supp(χ B ) = M. This implies that M has no proper almost bi-Γ-ideals. Definition 6. Let M be a Γ-semigroup and α ∈ Γ.
for any x, y ∈ M .
for any x, y ∈ M .
Next, we investigate relationship between α-prime almost bi-Γ-ideals and their fuzzificaion. Proof. Let A be any α-prime almost bi-Γ-ideal of M . Then χ A is a fuzzy almost bi-Γ-ideal of M by Theorem 5. Let x and y be elements in M . If xαy ∈ A, then x ∈ A or y ∈ A. This implies that If xαy ∈ A, then χ A (xαy) = 0 ≤ max{χ A (x), χ A (y)}.
To prove the converse, suppose that χ A is an α-prime fuzzy almost bi-Γ-ideal of M . By Theorem 5, we have that A is an almost bi-Γ-ideal of M . Let x and y be elements in M such that xαy ∈ A. Thus, χ A (xαy) = 1. By assumption, we have that χ A (xαy) ≤ max{χ A (x), χ A (y)}. Therefore, max{χ A (x), χ A (y)} = 1. We can conclude that x ∈ A or y ∈ A. Hence, A is an α-prime almost bi-Γ-ideal of M . (1) An almost bi- Finally, we give relationship between α-semiprime almost bi-Γ-ideals and their fuzzification. Proof. Let A be an α-semiprime almost bi-Γ-ideal of M . By Theorem 5, χ A is a fuzzy almost bi-Γ-ideal of M . Let m ∈ M . If mαm ∈ A, then m ∈ A. So, χ A (m) = 1. Hence, χ A (mαm) ≤ χ A (m). If mαm ∈ A, then χ A (mαm) = 0 ≤ χ A (m). By both cases, we conclude that χ A (mαm) ≤ χ A (m) for all m ∈ M . Thus, χ A is an α-semiprime fuzzy almost bi-Γ-ideal of M .
Conversely, assume that χ A is an α-semiprime fuzzy almost bi-Γ-ideal of M . By Theorem 5, we have that A is an almost bi-Γ-ideal of M . Let m ∈ M be such that mαm ∈ A. Thus χ A (mαm) = 1. By assumption, we have that χ A (mαm) ≤ χ A (m). Since χ A (mαm) = 1, it follows that χ A (m) = 1. Therefore, m ∈ A. Consequently, A is an α-semiprime almost bi-Γ-ideal of M .

Conclusion
In this paper, we define almost bi-Γ-ideals and their fuzzification of Γ-semigroups. Every bi-Γ-ideal is an almost bi-Γ-ideal but the converse is not true in general. We show that the union of two almost bi-Γ-ideals is also an almost bi-Γ-ideal. However, it is not generally true in case the intersection. Similarly, we have that the union of two fuzzy almost bi-Γ-ideals is also a fuzzy almost bi-Γ-ideal but it is not generally true in case the intersection. Moreover, the relationships between almost bi-Γ-ideals and their fuzzification were shown in Section 4.