EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 620-630 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Almost Bi-Γ-Ideals and Fuzzy Almost Bi-Γ-Ideals of Γ-Semigroups Anusorn Simuen1, Saleem Abdullah2, Winita Yonthanthum1, Ronnason Chinram 1,3,∗ 1 Algebra and Applications Research Unit, Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand 2 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan 3 Centre of Excellence in Mathematics, Si Ayuthaya Road, Bangkok 10400, Thailand Abstract. In this paper, we introduce the notions of almost bi-Γ-ideals and fuzzy almost bi-Γ- ideals 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 Key Words and Phrases: bi-Γ-ideals, almost bi-Γ-ideals, fuzzy almost bi-Γ-ideals 1. 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 quasi- almost-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]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3759 Email addresses: asimuen96@gmail.com (A. Simuen), saleemabdullah@awkum.edu.pk (S. Abdullah), winita.m@psu.ac.th (W. Yonthanthum), ronnason.c@psu.ac.th (R. Chinram) https://www.ejpam.com 620 c© 2020 EJPAM All rights reserved. R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 621 The notion of Γ-semigroups has been first studied by Sen [18] in 1981. In 1986, Sen and Saha [19] improved more general definition as follows: Definition 1. ([19]) Let M and Γ be non-empty sets. (M,Γ) is called a Γ-semigroup if it satisfies the following laws. (1) aαb ∈M for all a, b ∈M and α ∈ Γ. (2) M is associative under Γ, that is (aαb)βc = aα(bβc) for all a, b, c ∈M and all α, β ∈ Γ. Every semigroup (S, ·) can be considered as a Γ-semigroup S by choosing Γ = {·}. Then a Γ-semigroup is one of the generalizations of semigroups. The investigation on Γ-semigroups was done by certain mathematicians which are parallel to some results of semigroups, for example, one may see [6, 7, 17–19]. Similar to semigroups, ideal theory in Γ-semigroups plays an important role (for example, we can see in [1, 6, 7, 12–14, 20]). Let M be a Γ-semigroup. For nonempty subsets A and B of M , let AΓB = {aαb | a ∈ A, b ∈ B,α ∈ Γ}. If m ∈M , we let AΓm = AΓ{m} and mΓA = {m}ΓA. If α ∈ Γ, we let AαB = {aαb | a ∈ A, b ∈ B}. Definition 2. (see [7]) Let M be a Γ-semigroup. (1) A nonempty subset T of M is called a sub Γ-semigroup of M if TΓT ⊆ T . (2) A sub Γ-semigroup B of M is called a bi-Γ-ideal of M if BΓMΓB ⊆ B. A bi-Γ-ideal in Γ-semigroups was sometimes called a bi-ideal (see [14]). Some general- izations 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 defi- nitions 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 (f ∩ g)(m) = min{f(m), g(m)} for all m ∈M . R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 622 (2) A fuzzy subset f ∪ g of M is defined by (f ∪ g)(m) = max{f(m), g(m)} for all m ∈M . (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 supp(f) = {m ∈M | f(m) 6= 0}. For a subset A of any set M , the characteristic function χA of A is a fuzzy subset of M defined by χA(m) = { 1 m ∈ A, 0 m /∈ A. For any element m of any set M and t ∈ (0, 1], a fuzzy point mt of M is a fuzzy subset of M defined by mt(x) = { t x = m, 0 x 6= m (see [15]). 2. Almost bi-Γ-ideals First, we define almost bi-Γ-ideals of Γ-semigroups as follows: Definition 3. A non-empty subset B of a Γ-semigroup M is called an almost bi-Γ-ideal of S if BΓmΓB ∩B 6= ∅ for all m ∈M . Example 1. Let B be any bi-Γ-ideal of a Γ-semigroup M . Then BΓMΓB ⊆ B. This implies that for any m ∈ M,BΓmΓB ⊆ BΓMΓB ⊆ B. So BΓmΓB ∩ B = BΓmΓB 6= ∅ for all m ∈M . Then B is an almost bi-Γ-ideal of M . By Example 1, we conclude that every bi-Γ-ideal of a Γ-semigroup M is an almost bi-Γ-ideal of M . Example 2. Consider the Γ-semigroup Z8 with Γ = { 0, 1, 2 } under the usual addition. Let B = {4, 6}. We see that R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 623 (B + Γ + 0 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 1 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 2 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 3 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 4 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 5 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 6 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅, (B + Γ + 7 + Γ +B) ∩B = Z8 ∩ {4, 6} 6= ∅. Therefore, B is an almost bi-Γ-ideal of Z8. However, B is not a bi-Γ-ideal of Z8 because B + Γ + Z8 + Γ +B = Z8 6⊆ B. From Example 2, we see that an almost bi-Γ-ideal of Γ-semigroup S need not be a bi-Γ-ideal of S. Example 3. Consider the Γ-semigroup M = {a, b, c, d} with Γ = {α, β} and the multi- plication table: α a b c d a a c c a b c a a c c c a a c d a c c a β a b c d a c a a c b a c c a c a c c a d c a a c Let B = {a, c}. Then BΓaΓB ∩B = {a, c} ∩ {a, c} = {a, c} 6= ∅, BΓbΓB ∩B = {a, c} ∩ {a, c} = {a, c} 6= ∅, BΓcΓB ∩B = {a, c} ∩ {a, c} = {a, c} 6= ∅, BΓdΓB ∩B = {a, c} ∩ {a, c} = {a, c} 6= ∅. Therefore, B is an almost bi-Γ-ideal of M . Theorem 1. Assume that B is an almost bi-Γ-ideal of a Γ-semigroup M . If A is any subset of M containing B, then A is also an almost bi-Γ-ideal of M . Proof. Since B is an almost bi-Γ-ideal of M and B ⊆ A, we have BΓmΓB∩B 6= ∅ and BΓmΓB∩B ⊆ AΓmΓA∩A for all m ∈M , respectively. This implies that AΓmΓA∩A 6= ∅ for all m ∈M . Therefore, A is an almost bi-Γ-ideal of M . Corollary 1. The union of any two almost bi-Γ-ideals of a Γ-semigroup M is also an almost bi-Γ-ideal of M . Proof. Let A and B be any two almost bi-Γ-ideals of M . Since A ⊆ A ∪ B ⊆ M , it follows from Theorem 1 that A ∪B is an almost bi-Γ-ideal of M . R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 624 Example 4. Consider the Γ-semigroup Z8 with Γ = { 0, 1, 2 } under the usual addition. Let A = {2, 3} and B = {4, 6}. Clearly, A and B are almost bi-Γ-ideals of Z8 but A ∩B = ∅, so it is not an almost bi-Γ-ideal of Z8. By Example 4, we have the following remark. Remark 1. The intersection of any two almost bi-Γ-ideals of a Γ-semigroup M need not be an almost bi-Γ-ideal of M . Theorem 2. A Γ-semigroup M contains a proper almost bi-Γ-ideal if and only if there exists an element m of M such that M r {m} is an almost bi-Γ-ideal of M . Proof. Assume that a Γ-semigroup M contains a proper almost bi-Γ-ideal B and let m ∈M rB. Then B ⊆M r {m} ⊂M . By Theorem 1, M r {m} is an almost bi-Γ-ideal of M . Conversely, let m ∈ M be such that M r {m} is an almost bi-Γ-ideal of M . Since M r {m} (M , we get M r {m} is a proper almost bi-Γ-ideal of M . Theorem 3. Let M be a Γ-semigroup such that |M | > 1. Then M has no proper almost bi-Γ-ideals if and only if for all m ∈M there exists a ∈M such that (M r {m})ΓaΓ(M r {m}) = {m}. Proof. Assume that M has no proper almost bi-Γ-ideals and let m ∈M . By Theorem 2, M r {m} is not an almost bi-Γ-ideal of M . Thus there exists an element a of M such that (Mr{m})ΓaΓ(Mr{m})∩(Mr{m}) = ∅. Hence, (Mr{m})ΓaΓ(Mr{m}) = {m}. Conversely, suppose M contains a proper almost bi-Γ-ideal B. Let m ∈ M r B. By assumption, we have (M r {m})ΓaΓ(M r {m}) = {m} for some element a in M . Since B ⊆ M r {m} ⊂ M , we get M r {m} is an almost bi-Γ-ideal of M by Theorem 1. This implies that ∅ = {m} ∩ (M r {m}) = (M r {m})ΓaΓ(M r {m})∩ (M r {m}) 6= ∅, which is a contradiction. Therefore, M has no proper almost bi-Γ-ideals. 3. Fuzzy almost bi-Γ-ideals For a Γ-semigroup M , let F(M) be the set of all fuzzy subsets of M . For each α ∈ Γ, define a binary operation ◦α on F(M) by (f ◦α g)(m) =  sup m=aαb {min{f(a), g(b)}} if m ∈MαM, 0 otherwise. Let Γ? := {◦α | α ∈ Γ}. Then (F(M),Γ?) is a Γ-semigroup. Proposition 1. For fuzzy subsets f and g of a Γ-semigroup M such that f ⊆ g and α ∈ Γ, if h is any fuzzy subset of M , then h ◦α f ⊆ h ◦α g and f ◦α h ⊆ g ◦α h. R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 625 We define fuzzification of almost bi-Γ-ideals in Γ-semigroups as follows: Definition 4. A fuzzy subset f of a Γ-semigroup M is called a fuzzy almost bi-Γ-ideal of M if for all fuzzy points mt of M , there exist α, β ∈ Γ such that (f ◦α mt ◦β f) ∩ f 6= 0. 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 mt of M , there exist α, β ∈ Γ such that (f ◦α mt ◦β f) ∩ f 6= 0. We have that (f ◦α mt ◦β f) ∩ f ⊆ (g ◦αmt ◦β g) ∩ g, this implies that (g ◦αmt ◦β g) ∩ g 6= 0. Hence, g is also a fuzzy almost bi-Γ-ideal of M . Corollary 2. If f and g are fuzzy almost bi-Γ-ideals of a Γ-semigroup M , then f ∪ g is also a fuzzy almost bi-Γ-ideal of M . Proof. It follows by Theorem 4 because of f ⊆ f ∪ g. Example 5. Consider the Γ-semigroup Z5 where Γ = {0} and aγb := a + γ + b. Let f and g be fuzzy subsets of Z5 defined by f(0) = 0, f(1) = 0.5, f(2) = 0, f(3) = 0.1, f(4) = 0.4 and g(0) = 0, g(1) = 0.3, g(2) = 0.7, g(3) = 0, g(4) = 0.2. It is easy to check that [(f ◦α mt ◦β f) ∩ f ](4) 6= 0 and [(g ◦α mt ◦β g) ∩ g](4) 6= 0 for all α, β ∈ Γ,m ∈ Z5 and t ∈ (0, 1]. So f and g are fuzzy almost bi-Γ-ideals of Z5. From the definition of the intersection of two fuzzy subsets, we have (f ∩ g)(0) = 0, (f ∩ g)(1) = 0.3, (f ∩ g)(2) = 0, (f ∩ g)(3) = 0, (f ∩ g)(4) = 0.2. We can easily to check that [((f∩g)◦α0t◦β (f∩g))∩(f∩g)](a) = 0 for all α, β ∈ Γ, t ∈ (0, 1] and a ∈ Z5, so f ∩ g is not a fuzzy almost bi-Γ-ideal of Z5. The following remark follows from Example 5. Remark 2. The intersection of two fuzzy almost bi-Γ-ideals of a Γ-semigroup M need not be a fuzzy almost bi-Γ-ideal of M . 4. Relationships between almost bi-Γ-ideals and their fuzzification Theorem 5. A non-empty subset B of a Γ-semigroup M is an almost bi-Γ-ideal of M if and only if χB is a fuzzy almost bi-Γ-ideal of M . R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 626 Proof. Assume that B is an almost bi-Γ-ideal of a Γ-semigroup M and let mt be any fuzzy point of M . Then BΓmΓB ∩B 6= ∅. Thus there exists b ∈ B such that b ∈ BαmβB for some α, β ∈ Γ. This implies that (χB ◦α mt ◦β χB)(b) 6= 0 and χB(b) 6= 0. Hence, (χB ◦α mt ◦β χB) ∩ χB 6= 0. Therefore, χB is a fuzzy almost bi-Γ-ideal of M . To prove the converse, we assume that χB is a fuzzy almost bi-Γ-ideal of M and let m ∈M . Then there exist α, β ∈ Γ such that (χB ◦αmt ◦β χB) ∩ χB 6= 0, so [(χB ◦αmt ◦β χB) ∩ χB](y) 6= 0 for some y ∈ M. Hence, y ∈ B and y = aαmβb for some a, b ∈ B and α, β ∈ Γ. Therefore, y ∈ BΓmΓB∩B. So BΓmΓB∩B 6= ∅. Consequently, B is an almost bi-Γ-ideal of M Theorem 6. A fuzzy subset f of a Γ-semigroup M is a fuzzy almost bi-Γ-ideal of M if and only if supp(f) is an almost bi-Γ-ideal of M . Proof. Assume that f is a fuzzy almost bi-Γ-ideal of a Γ-semigroup M and let m ∈M and t ∈ (0, 1]. Then there exist α, β ∈ Γ such that (f ◦αmt◦β f)∩f 6= 0. Hence, [(f ◦αmt◦β f) ∩ f ](x) 6= 0 for some x ∈ M . So there exist y1, y2 ∈ S such that x = y1αmβy2, f(x) 6= 0, f(y1) 6= 0 and f(y2) 6= 0. That is x, y1, y2 ∈ supp(f). Thus [χsupp(f)◦αst◦βχsupp(f)](x) 6= 0 and χsupp(f)(x) 6= 0. Therefore, (χsupp(f) ◦αmt ◦β χsupp(f))∩χsupp(f) 6= 0. Hence, χsupp(f) is a fuzzy almost bi-Γ-ideal of M. By Theorem 5, supp(f) is an almost bi-Γ-ideal of M. On the other hand, we assume that supp(f) is an almost bi-Γ-ideal of M. It follows from Theorem 5 that χsupp(f) is a fuzzy almost bi-Γ-ideal of M. Let mt be any fuzzy point of M. Thus, (χsupp(f) ◦αmt ◦β χsupp(f))∩χsupp(f) 6= 0 for some α, β ∈ Γ. Then there exists an element x in M such that [(χsupp(f) ◦α mt ◦β χsupp(f)) ∩ χsupp(f)](x) 6= 0. Therefore, (χsupp(f)◦αmt◦βχsupp(f))(x) 6= 0 and χsupp(f)(x) 6= 0. Then there exist y1, y2 ∈M such that x = y1αmβy2, f(x) 6= 0, f(y1) 6= 0 and f(y2) 6= 0. This means that (f ◦αmt ◦β f) ∩ f 6= 0. 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. Definition 5. A fuzzy almost bi-Γ-ideal f of a Γ-semigroup M is called minimal if for all fuzzy almost bi-Γ-ideal g of M contained in f , we must have supp(g) = supp(f). Now, we provide the relationship between minimal almost bi-Γ-ideals and their fuzzi- fication. Theorem 7. A non-empty subset A of a Γ-semigroup M is a minimal almost bi-Γ-ideal of M if and only if χA is a minimal fuzzy almost bi-Γ-ideal of M . Proof. Let A be a minimal almost bi-Γ-ideal of a Γ-semigroup M . By Theorem 5, we have that χA is a fuzzy almost bi-Γ-ideal of M. Assume that g is a fuzzy almost bi-Γ-ideal of M contained in χA. Thus, supp(g) ⊆ supp(χA) = A. Because of g ⊆ χsupp(g), we have (g ◦αmt ◦β g)∩g ⊆ (χsupp(g) ◦αmt ◦β χsupp(g))∩χsupp(g) for all fuzzy points mt of M . Thus χsupp(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(χA). Therefore, χA is minimal. To prove the converse, assume that χA is a minimal fuzzy almost bi-Γ-ideal of M and B is an almost bi-Γ-ideal of M contained in A. Then χB is a fuzzy almost bi-Γ-ideal of M and χB ⊆ χA. Thus, B = supp(χB) = supp(χA) = A. We conclude that A is minimal. R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 627 Corollary 3. A Γ-semigroup M has no proper almost bi-Γ-ideals if and only if for all fuzzy almost bi-Γ-ideal f of M , supp(f) = M. Proof. Assume that M has no proper almost bi-Γ-ideals and let f be a fuzzy almost bi- Γ-ideal of M . By Theorem 6, we have supp(f) is almost bi-Γ-ideal of M . Thus supp(f) = M . To prove the converse, we letB be any almost bi-Γ-ideal ofM . Follow by Theorem 5, we have that χB is a fuzzy almost bi-Γ-ideal ofM . By assumption, we getB = supp(χB) = M. This implies that M has no proper almost bi-Γ-ideals. Definition 6. Let M be a Γ-semigroup and α ∈ Γ. (1) An almost bi-Γ-ideal B of M is called α-prime if xαy ∈ B ⇒ x ∈ B or y ∈ B for any x, y ∈M . (2) A fuzzy almost bi-Γ-ideal f of M is called α-prime if f(xαy) ≤ max{f(x), f(y)} for any x, y ∈M . Next, we investigate relationship between α-prime almost bi-Γ-ideals and their fuzzi- ficaion. Theorem 8. A nonempty subset A of a Γ-semigroup M is an α-prime almost bi-Γ-ideal of M if and only if χA is an α-prime fuzzy almost bi-Γ-ideal of M . Proof. Let A be any α-prime almost bi-Γ-ideal of M . Then χA 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 χA(xαy) = 1 ≤ max{χA(x), χA(y)}. If xαy 6∈ A, then χA(xαy) = 0 ≤ max{χA(x), χA(y)}. We conclude that χA(xαy) ≤ max{χA(x), χA(y)} for all x, y ∈ M . Therefore, χA is an α-prime fuzzy almost bi-Γ-ideal of M . To prove the converse, suppose that χA 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, χA(xαy) = 1. By assumption, we have that χA(xαy) ≤ max{χA(x), χA(y)}. Therefore, max{χA(x), χA(y)} = 1. We can conclude that x ∈ A or y ∈ A. Hence, A is an α-prime almost bi-Γ-ideal of M . R. Chinram et al. / Eur. J. Pure Appl. Math, 13 (3) (2020), 620-630 628 Definition 7. Let M be a Γ-semigroup and α ∈ Γ. (1) An almost bi-Γ-ideal A of M is called α-semiprime if mαm ∈ A⇒ m ∈ A for all m ∈M. (2) A fuzzy almost bi-Γ-ideal f of M is called α-semiprime if f(mαm) ≤ f(m) for all m ∈M. Finally, we give relationship between α-semiprime almost bi-Γ-ideals and their fuzzifi- cation. Theorem 9. A nonempty subset A of a Γ-semigroup M is an α-semiprime almost bi-Γ- ideal of M if and only if χA is an α-semiprime fuzzy almost bi-Γ-ideal of M . Proof. Let A be an α-semiprime almost bi-Γ-ideal of M . By Theorem 5, χA is a fuzzy almost bi-Γ-ideal of M . Let m ∈ M . If mαm ∈ A, then m ∈ A. So, χA(m) = 1. Hence, χA(mαm) ≤ χA(m). If mαm 6∈ A, then χA(mαm) = 0 ≤ χA(m). By both cases, we conclude that χA(mαm) ≤ χA(m) for all m ∈ M . Thus, χA is an α-semiprime fuzzy almost bi-Γ-ideal of M . Conversely, assume that χA 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 χA(mαm) = 1. By assumption, we have that χA(mαm) ≤ χA(m). Since χA(mαm) = 1, it follows that χA(m) = 1. Therefore, m ∈ A. Consequently, A is an α-semiprime almost bi-Γ-ideal of M . 5. 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. Acknowledgements This work was supported by the Faculty of Sciences Research Fund, Prince of Songkla University, Contract no. 1-2562-02-013. 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