EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1592-1607 ISSN 1307-5543 – ejpam.com Published by New York Business Global Application of Bipolar Fuzzy Set to a Novel of Fuzzy Ideal in Γ-Semigroups P. Khamrot1, T. Gaketem2,∗ 1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna of Phitsanulok, Phitsanulok, Thailand 2 Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. In this paper we define new types of bipolar fuzzy ideals, bipolar fuzzy almost ideals in Γ-semigroups. We discussed properties of bipolar fuzzy ideals, bipolar fuzzy almost ideals, minimal bipolar fuzzy ideal, minimal bipolar almost ideal in Γ-semigroups. Moreover, we prove connection between almost ideals and bipolar fuzzy almost ideals of Γ-semigroups. 2020 Mathematics Subject Classifications: 20M12, 06F05 Key Words and Phrases: Bipolar fuzzy ideals, bipolar fuzzy almost ideals, minimal bipolar fuzzy ideals, minimal biploar fuzzy almost ideals 1. Introduction As a theory that deals with uncertainty, the fuzzy set theory was studied by Zadeh in 1965 [12]. It has been applied to many areas, such as medical science, robotics, com- puter science, information science, control engineering, measure theory, logic, set theory, topology and others. In 1994, Zhang [13] introduced the concept of bipolar fuzzy sets. In 2000, Lee [8] extended a fuzzy set to theory of a bipolar fuzzy set whose function ranges from the interval [−1, 0] ∪ [0, 1]. As application of the bipolr set theory affects and effec- tiveness and efficiency of decision making. Therefore, it is used to solve problem related to decision-making, organization problems, economic problems, and evaluation, risk man- agement, environmental and social impact assessments. Later in 2012, S.K. Majumder [9] studied the bipolar fuzzy set in Γ-semigroups and integration properties of bipolar fuzzy ideals in Γ-semigroups. The ideal theory is an essential structures in semigroups and many researchers have applied the knowledge of ideals in Γ-semigroups studies in a fuzzy semi- group. For example, Chinram et al. [1] studied almost quasi-Γ-ideal and fuzzy almost quasi-Γ-ideals in Γ-semigroup, M. K. R. Marapureddy and PRV S. R. Doradla [7] investi- gated weak interior ideals of Γ-semigroups, S.K. Majumder and M. Mandal [4] examimed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4788 Email addresses: thiti.ga@up.ac.th (T. Gaketem), pk g@rmutl.ac.th (P. Khamrot) https://www.ejpam.com 1592 © 2023 EJPAM All rights reserved. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1593 a fuzzy generalized bi-ideal in Γ-semigroups. In 2021, T. Gaketem and P. Khamrot [2] discussed bipolar fuzzy weakly interior ideals in semigroups. In the same year, [11] A. Simuen et al. studied a novel of ideals and fuzzy ideals of Γ-semigroups. Recently, in 2022-2023, T. Gaketem and P. Khamrot [3, 5, 6] investigated propertes of a novel of ideals on intuitionistic fuzzy ideals, cubic ideals and interval valued fuzzy ideals of Γ-semigroups. In this paper, we extend the new fuzzy ideal to the bipolar fuzzy ideal of Γ-semigroups and investigate their properties. We prove connection between almost ideals and bipolar fuzzy almost ideals of Γ-semigroups. 2. Preliminaries In this section, we review some basic concepts that are necessary to understand our section. A sub-Γ-semigroup of a Γ-semigroup S is a non-empty set K of S such that KΓK ⊆ K. A left (right) ideal of a Γ-semigroup S is a non-empty set K of S such that SΓK ⊆ K (KΓS ⊆ K). By an ideal of a Γ-semigroup S, we mean a non-empty set of S which is both a left and a right ideal of S. A quasi-ideal of a Γ-semigroup S is a non-empty set K of S such that KΓS ∩ SΓK ⊆ K. A sub-Γ-semigroup K of a Γ-semigroup S is called a bi-ideal of S if KΓSΓK ⊆ K. Definition 1. [11] Let S be a Γ-semigroup, K be a non-empty subset of S and α, β ∈ Γ. Then K is said to be [(i)] (i) A left (right) almost ideal of Γ-semigroup S which is a non-empty set K such that (sΓK) ∩K ̸= ∅ ((KΓs) ∩K ̸= ∅) for all s ∈ S. (ii) An almost bi-ideal of Γ-semigroup S which is a non-empty set K such that (KΓsΓK)∩ K ̸= ∅ for all s ∈ S. (iii) An almost quasi-ideal of Γ-semigroup S which is a non-empty set K such that (sΓK∩ KΓs) ∩K ̸= ∅ for all s ∈ S. (iv) A left α-ideal of a Γ-semigroup S which is a non-empty set K such that SαK ⊆ K. A right α-ideal of a Γ-semigroup S is a non-empty set K such that KβS ⊆ K. (v) An (α, β)-ideal of a Γ-semigroup S which is a non-empty set K such that it is both a left α-ideal and a right β-ideal of S. For any mi ∈ [0, 1], i ∈ A, define ∨ i∈A mi := sup i∈A {mi} and ∧ i∈A mi := inf i∈A {mi}. We see that for any m,n ∈ [0, 1], we have m ∨ n = max{m,n} and m ∧ n = min{m,n}. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1594 Definition 2. [12] A fuzzy set ξ of a non-empty set T is a function ξ : T → [0, 1]. For any two fuzzy sets ξ and ς of a non-empty set T , define ≥,=,∧, and ∨ as follows: [(i)]ξ ≥ ς ⇔ ξ(k) ≥ ς(k) for all k ∈ T, ξ = ς ⇔ ξ ≥ ς and ς ≥ ξ, (ξ ∧ ς)(k) = min{ξ(k), ς(k)} = ξ(k)∧ς(k) for all k ∈ T, (ξ∨ς)(k) = max{ξ(k), ς(k)} = ξ(k)∨ς(k) for all k ∈ T. For the symbol ξ ≤ ς, we mean ς ≥ ξ. For any two fuzzy sets of ξ and ς of a non-empty of T , we define the support of ξ instead of supp(ξ) = {k ∈ T | ξ(k) ̸= 0}, ξ ⊆ ς if ξ(k) ≤ ς(k), (ξ ∪ ς)(k) = max{ξ(k), ς(k)} and (ξ ∩ ς)(k) = min{ξ(k), ς(k)} for all k ∈ T. For any element k in a semigroup S, define the set Fk by Fk := {(y, z) ∈ S×S | k = yz}. For two fuzzy sets ξ and ς on a semigroup S, define the product ξ ◦ ς as follows: for all k ∈ S, (ξ ◦ ς)(k) =  ∨ (y,z)∈Fk {ξ(y) ∧ ς(z)} if Fk ̸= ∅, 0 if Fk = ∅. Definition 3. Let I be a non-empty set of a semigroup S. A positive characteristic function and a negative characteristic function are respectively defined by λI : S → [0, 1], k 7→ λI(u) := { 1 k ∈ I, 0 k /∈ I, The following definitions are types of fuzzy almost ideal on semigroups. Definition 4. [11] A fuzzy set ξ of a semigroup S and s ∈ S, is said to be [(i)] (i) a fuzzy almost left (right) ideal of S if λs ◦ ξ ∩ ξ ̸= 0 (ξ ◦ λs ∩ ξ ̸= 0) for all s ∈ S, (ii) a fuzzy almost ideal of S if it is both a fuzzy almost left ideal and a fuzzy almost right ideal of S for all s ∈ S, (iii) a fuzzy almost bi-ideal of S if ξ ◦ λs ◦ ξ ∩ ξ ̸= 0 for all s ∈ S, (iv) a fuzzy almost quasi-ideal of S if λs ◦ ξ ∩ ξ ◦ λs ∩ ξ ̸= 0. Theorem 1. [10] Let ξ be a nonzero fuzzy set of a semigroup S. Then ξ is a fuzzy subsemigroup (ideal) of S if and only if supp(ξ) is a subsemigroup (ideal) of S. The following definitions are types of fuzzy subsemigroups on Γ-semigroups. Definition 5. [10] A fuzzy set ξ of a Γ-semigroup S is said to be [(i)] (i) A fuzzy subsemigroup of S if ξ(uγv) ≥ ξ(u) ∧ ξ(v) for all u, v ∈ S and γ ∈ Γ. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1595 (ii) A fuzzy left (right) ideal of S if ξ(uγv) ≥ ξ(v) (ξ(uγv) ≥ ξ(u)) for all u, v ∈ S and γ ∈ Γ. (iii) A fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S. (iv) A fuzzy bi-ideal of S if ξ is a fuzzy subsemigroup of S and ξ(uγvβw) ≥ ξ(u) ∧ ξ(w) for all u, v, w ∈ S and γ, β ∈ Γ. Now, we review the definition of a bipolar valued fuzzy set and basic properties used in the next section. Definition 6. [8] Let S be a non-empty set. A bipolar fuzzy set (BF set) ξ on S is an object having the form ξ := {(k, ξp(k), ξn(k)) | k ∈ S}, where ξp : S → [0, 1] and ξn : S → [−1, 0]. Remark 1. For the sake of simplicity, we shall use the symbol ξ = (S; ξp, ξn) for the BF set ξ = {(k, ξp(k), ξn(k)) | k ∈ S}. The following presents an example of a BF set. Example 1. Let S = {21, 22, 23...}. Define ξp : S → [0, 1] as a function ξp(u) = { 0 if u is an old number 1 if u is an even number and ξn : S → [−1, 0] as a function ξn(u) = { −1 if u is an old number 0 if u is an even number. Then ξ = (S; ξp, ξn) is a BF set. For bipolar fuzzy sets ξ = (S; ξp, ξn) and ς = (S; ςp, ςn), define the products ξp ◦ ςp and ξn ◦ ςn as follows: For u ∈ S (ξp ◦ ςp)(k) =  ∨ (y,z)∈Fk {ξp(y) ∧ ςp(z)} if k = yz 0 if otherwise. and (ξn ◦ ςn)(k) =  ∧ (y,z)∈Fk {ξn(y) ∨ ςn(z)} if k = yz 0 if otherwise. Definition 7. Let I be a non-empty set of a semigroup S. A positive characteristic function and a negative characteristic function are respectively defined by P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1596 λp I : S → [0, 1], k 7→ λp I(u) := { 1 k ∈ I, 0 k /∈ I, and λn I : S → [−1, 0], k 7→ λn I (k) := { −1 k ∈ I, 0 k /∈ I. Remark 2. For the sake of simplicity, we shall use the symbol λI = (S;λp I , λ n I ) for the BF set λI := {(k, λp I(k), λ n I (k)) | k ∈ I}. For u ∈ S and (t, s) ∈ [0, 1]× [−1, 0], a bipolar fuzzy point x(t,s) = (S;xpt , x n s ) of a set S is a bipolar set of S defined by xpt (u) = { t if u = x 0 if u ̸= x and xns (u) = { s if u = x 0 if u ̸= x. Next, we study the intersection of the BF set as defined. Let ξ = (S; ξp, ξn) and ς = (S; ςp, ςn) be BF sets of a semigroup S. Define ξ ∩ ς = (ξp ∩ ςp, ξn ∩ ςn) where (ξp ∩ ςp)(k) = ξp(k) ∧ ςp(k) and (ξn ∩ ςn)(k) = ξn(k) ∨ ςn(k) for all k ∈ S. Definition 8. [9] A BF set ξ = (S; ξp, ξn) on a Γ-semigroup S is called a BF sub- semigroup on S if it satisfies the following conditions: ξp(uαv) ≥ ξp(u) ∧ ξp(v) and ξn(uαv) ≤ ξn(u) ∨ ξn(v) for all u, v ∈ S and α ∈ Γ. The following presents an example of a BF subsemigroup. Example 2. Let S be the set of all negative integers and Γ be the set of all non positive even intergers. Then S is a Γ-semigroup by usual multiplication of integers. Let ξ = (S; ξp, ξn) be a BF set of S, as defined follows ξp(u) =  1 if u = 0 − 1 u if u = −1,−2 − 1 u−2 if u < −2 and ξn(u) =  −1 if u = 0 1 u if u = −1,−2 1 u−2 if u < −2 Then ξ = (S; ξp, ξn) is a BF subsemigroup of S. Definition 9. [9] A BF set ξ = (S; ξp, ξn) on a Γ-semigroup S is called a BF left (right) ideal on S if it satisfies the following conditions: ξp(uαv) ≥ ξp(v) (ξp(uαv) ≥ ξp(u)) and ξn(uαv) ≤ ξn(v) (ξn(uαv) ≤ ξn(u)) for all u, v ∈ S and α ∈ Γ. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1597 3. New Types of Bipolar Valued Fuzzy Ideals In this section, we define the bipolar fuzzy (α, β)-ideal and study its basic properties. Definition 10. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S and α, β ∈ Γ. Then ξ = (S; ξp, ξn) is called [(i)]A BF left α-ideal of S if ξp(uαv) ≥ ξp(v) and ξn(uαv) ≤ ξn(v) for all u, v ∈ S. A BF right β-ideal of S if ξp(uβv) ≥ ξ(u) and ξn(uβv) ≤ ξn(u) for all u, v ∈ S. A BF (α, β)-ideal of S if it is both a BF left α-ideal and a BF right β-ideal of S. A BF α-ideal of S if it is a BF (α, α)-ideal of S. Theorem 2. Let K be a non-empty subset of Γ-semigroup S. Then K is a left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if λK = (S;λp K , λn K) is a BF left α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Suppose that K is a left α-ideal of S and u, v ∈ S. If v ∈ K, then uαv ∈ K. Thus, λp K(v) = λp K(uαv) = 1 and λn K(v) = λn K(uαv) = −1. Hence, λp K(uαv) ≥ λp K(v) and λn K(uαv) ≤ λn K(v). If v /∈ K, then uαv ∈ K. Thus, λp K(v) = λn K(v) = 0 and λp K(uαv) = 1, λn K(uαv) = −1. Hence, λp K(uαv) ≥ λp K(v) and λn K(uαv) ≤ λn K(v). Therefore, λK = (S;λp K , λn K) is a BF left α-ideal of S. Conversely, assume that λK = (S;λp K , λn K) is a BF left α-ideal of S and u, v ∈ S with v ∈ K. Then λp K(v) = 1 and λp I(v) = −1. By assumption, λp K(uαv) ≥ λp K(v) and λn K(uαv) ≤ λn K(v) Thus, uαv ∈ K. Hence, K is a left α-ideal of S. Theorem 3. The positive and negative of the intersection and union of any two BF left α- ideals (right β-ideals, (α, β)-ideals) of a Γ-semigroup S is a BF left α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Let ξ = (S; ξp, ξn) and ς = (S; ςp, ςn) be BF left α-ideals of S and let u, v ∈ S. Then (ξp ∩ ςp)(uαv) = ξp(uαv) ∧ ςp(uαv) ≥ ξp(v) ∧ ςp(v) = (ξp ∩ ςp)(v) and (ξn ∩ ςn)(uαv) = ξn(uαv) ∨ ςn(uαv) ≤ ξn(v) ∨ ςn(v) = (ξn ∩ ςn)(v). Similarly, (ξp ∪ ςp)(uαv) = ξp(uαv) ∨ ςp(uαv) ≥ ξp(v) ∨ ςp(v) = (ξp ∪ ςp)(v) and (ξn ∪ ςn)(uαv) = ξn(uαv) ∧ ςn(uαv) ≤ ξn(v) ∧ ςn(v) = (ξn ∩ ςn)(v). Thus, ξ ∩ ς and ξ ∪ ς are BF left α-ideals of S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1598 Theorem 4. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S and ξ(l,m) = (S; ξpl , ξ n m) be BF point with ξpl = {x ∈ S | ξpl (x) ≥ l} and ξnm = {x ∈ S | ξnm(x) ≤ m}. Then ξ = (S; ξp, ξn) is a BF left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if ξ(l,m) = (S; ξpl , ξ n m) is a non-empty set and ξ(l,m) is a left α-ideal (right β-ideal, (α, β)-ideal) of S for all (l,m) ∈ (0, 1]× [−1, 0). Proof. Suppose that ξ = (S; ξp, ξn) is a BF left α-ideal of S. Then ξp(uαr) ≥ ξp(r) and ξn(uαr) ≤ ξn(r) for all u, r ∈ S. Let (l,m) ∈ (0, 1] × [−1, 0) be such that ξ(l,m) ̸= ∅. Let r ∈ ξ(l,m) and u ∈ S. Then ξp(r) ≥ l and ξn(r) ≤ m. Thus, ξp(uαr) ≥ ξp(r) ≥ l and ξp(uαr) ≤ ξn(r) ≤ m. So, uαr ∈ ξ(l,m). Hence, ξ(l,m) = (S; ξpl , ξ n m) is a left α-ideal of S. Conversely, assume that ξ(l,m) = (S; ξpl , ξ n m) is a left α-ideal of S if (l,m) ∈ (0, 1] × [−1, 0) and ξ(l,m) ̸= ∅. Let u, v ∈ S and l = ξp(v),m = ξn(v). By assumption, ξp(v) ≥ l and ξn(v) ≤ m. Then v ∈ ξ(l,m). Thus, ξl ̸= ∅. Hence, ξl is a left α-ideal of S. Since v ∈ ξ(l,m) and u ∈ S, we have xαv ∈ ξ(l,m). Thus, ξp(uαv) ≥ l = ξp(v) and ξn(uαv) ≤ m = ξn(v). Hence, ξ = (S; ξp, ξn) is a BF left α-ideal of S. Next, we will define the (α, β)-product. For BF sets ξ = (S; ξp, ξn) and ς = (S; ςp, ςn), define the product ξp ◦α ςp and ξn ◦α ςn as follows: For u ∈ S (ξp ◦α ςp)(u) =  ∨ (y,α,z)∈Fuα {ξp(y) ∧ ςp(z)} if u = yαz 0 if otherwise. and (ξn ◦α ςn)(u) =  ∧ (y,α,z)∈Fuα {ξn(y) ∨ ςn(z)} if u = yαz 0 if otherwise, where Fuα = {(y, z) ∈ S × Γ× S | u = yαz}, for u ∈ S and α ∈ Γ. Next, we define BF (α, β)-bi-ideal and study its basic properties. Definition 11. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S and α, β ∈ Γ. Then ξ = (S; ξp, ξn) is called a BF (α, β)-bi-ideal of S if ξp◦αλp S◦βξp ≥ ξp and ξn◦αλn S◦βξn ≤ ξn where λS = (S;λp S , λ n S) is a BF set mapping every element of S to [−1, 1]. Theorem 5. Let K be a non-empty subset of Γ-semigroup S. Then K is an (α, β)-bi-ideal of S if and only if the characteristic function λK = (S;λp K , λn K) is a BF (α, β)-bi-ideal of S. Proof. Suppose that K is an (α, β)-bi-ideal of S and KαSβK ⊆ K. If u ∈ KαSβK, then λp K(u) = (ξp ◦α λp S ◦β ξp)(u) = 1 and λn K(u) = (ξn ◦α λn S ◦β ξn)(u) = −1. Hence, (ξp ◦α λp S ◦β ξp)(u) ≥ ξp(u) and (ξn ◦α λn S ◦β ξn)(u) ≤ ξn(u) If u /∈ KαSβK, then λp K(u) = (ξp ◦α λp S ◦β ξp)(u) = 0 and λn K(u) = (ξn ◦α λn S ◦β ξn)(u) = 0. Hence, (ξp ◦α λp S ◦β ξp)(u) ≥ ξp(u) and (ξn ◦α λn S ◦β ξn)(u) ≤ ξn(u) P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1599 Therefore, λK = (S;λp K , λn K) is a BF (α, β)-bi-ideal of S. Conversely, assume that λK = (S;λp K , λn K) is a BF (α, β)-bi-ideal of S and u ∈ KαSβK. Then (ξp ◦α λp S ◦β ξp)(u) = 1 and (ξn ◦α λn S ◦β ξn)(u) = −1. By assumption, (ξp ◦α λp S ◦β ξp)(u) ≥ ξp(u) and (ξn ◦α λn S ◦β ξn)(u) ≤ ξn(u). Thus, u ∈ K. Hence, K is an (α, β)-bi-ideal of S. Theorem 6. The positive and negative of intersection of any two BF (α, β)-bi-ideals of a Γ-semigroup S are a BF (α, β)-bi-ideal of S. Proof. Let ξ = (S; ξp, ξn) and ς = (S; ςp, ςn) be BF (α, β)-bi-ideals of S and u ∈ S. Then ((ξp ∩ ςp) ◦α λp S ◦β (ξp ∩ ςp))(u) ≥ (ξp ◦α λp S ◦β ξp)(u) ∧ (ςp ◦α λp S ◦β ςp)(u) ≥ (ξp ∩ ςp)(u) and ((ξn ∩ ςn) ◦α λn S ◦β (ξn ∩ ςn))(u) ≤ (ξn ◦α λn S ◦β ξn)(u)∨ (ςn ◦α λn S ◦β ςn)(u) ≤ (ξn ∩ ςn)(u). Thus, ξ ∩ ς is a BF (α, β)-bi-ideal of S. Theorem 7. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S and ξ(l,m) = (S; ξpl , ξ n m) be BF point with ξpl = {x ∈ S | ξpl (x) ≥ l} and ξnm = {x ∈ S | ξnm(x) ≤ m}. Then ξ = (S; ξp, ξn) is a BF (α, β)-bi-ideal of S if and only if ξ(l,m) = (S; ξpl , ξ n m) is a non-empty set and ξ(l,m) is an (α, β)-bi-ideal of S for all (l,m) ∈ (0, 1]× [−1, 0). Proof. Suppose that ξ = (S; ξp, ξn) is a BF (α, β)-bi-ideal of S. Then ξp(uαvβw) ≥ ξp(u)∧ξp(w) and ξn(uαvβw) ≤ ξn(u)∨ξn(w) for all u, v, w ∈ S. Let (l,m) ∈ (0, 1]×[−1, 0) be such that ξ(l,m) ̸= ∅. Let u,w ∈ ξ(l,m) and v ∈ S. Then ξp(u) ≥ l, ξp(w) and ξn(u) ≤ m, ξn(w) ≤ m. Thus, ξp(uαvβw) ≥ ξp(u) ∧ ξp(w) ≥ l and ξp(uαvβw) ≤ ξn(u) ∨ ξn(w) ≤ m. So, uαvβw ∈ ξ(l,m). Hence, ξ(l,m) = (S; ξpl , ξ n m) is an (α, β)-bi-ideal of S. Conversely, assume that ξ(l,m) = (S; ξpl , ξ n m) is an (α, β)-bi-ideal of S if (l,m) ∈ (0, 1]× [−1, 0) and ξ(l,m) ̸= ∅. Let u, v, w ∈ S and l = ξp(u), l = ξp(u), m = ξn(u), m = ξn(w). By assumption, ξp(u)∧ξp(w) ≥ l and ξn(u)∨ξn(w) ≤ m. Then u,w ∈ ξ(l,m). Thus, ξl ̸= ∅. Hence, ξl is an (α, β)-bi-ideal of S. Since u,w ∈ ξ(l,m) and v ∈ S, we have uαvβw ∈ ξ(l,m). Thus, ξp(uαvβw) ≥ l = ξp(u) ∧ ξ(p)(w) and ξn(uαvβw) ≤ m = ξn(u) ∨ ξn(w). Hence, ξ = (S; ξp, ξn) is a BF (α, β)-bi-ideal of S. Next, we define a BF (α, β)-quasi-ideal and study its basic properties. Definition 12. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S and α, β ∈ Γ. Then ξ = (S; ξp, ξn) is called a BF (α, β)-quasi-ideal of S if λp S ◦α ξp ∩ ξp ◦β λp S ⊆ ξp and λn S ◦α ξn ∪ ξn ◦β λn S ⊇ ξn. Theorem 8. If ξ = (S; ξp, ξn) and ς = (S; ςp, ςn) is a BF left α-ideal and a BF right α-ideal of a Γ-semigroup S, respectively, then ξ ∩ ς is a BF α-quasi-ideal of S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1600 Proof. Let ξ = (S; ξp, ξn) and ς = (S; ςp, ςn) is a BF left α-ideal and a BF right α-ideal of S, respectively. Then ςp ◦α ξp ⊆ λp S ◦α ξp ⊆ ξp and ςp ◦α ξp ⊆ ςp ◦α λp S ⊆ ςp. Thus, ςp ◦α ξp ⊆ ξp ∩ ςp. So, λp S ◦α (ξp ∩ ςp) ∩ (ξp ∩ ςp) ◦α λp S ⊆ λp S ◦α (ξp ∩ ςp) ◦α λp S ⊆ ξp ∩ ςp. Thus, ξp ∩ ςp is a BF α-quasi-ideal of S. Similarly, we can show that ξn ∩ ςn is a BF α-quasi-ideal of S. Hence, ξ ∩ ς is a BF α-quasi-ideal of S. Theorem 9. Every BF (α, β)-quasi-ideal of Γ-semigroup S is the intersection of a BF left α-ideal and a BF right β-ideal of S Proof. Let ξ = (S; ξp, ξn) be a BF (α, β)-quasi-ideal of S. Consider ςp = ξp ∪ (λp S ◦α ξp) and ςn = ξn ∪ (λn S ◦α ξn) where ς = (S; ςp, ςn), κp = ξp ∪ (ξp ◦β λp S) and κn = ξn ∪ (ξn ◦β λn S) where κ = (S;κp, κn). Then λp S ◦α ςp = λp S ◦α (ξp ∪ (λp S ◦α ξp)) = (λp S ◦α ξp) ∪ (λp S ◦α (λp S ◦α ξp)) = (λp S ◦α ξp) ∪ ((λp S ◦α λp S) ◦α ξp) = (λp S ◦α ξp) ∪ (λp S ◦α ξp) ⊆ ξp ∪ (λp S ◦α ξp) = ςp. And κp ◦β λp S = (ξp ∪ (ξp ◦β λp S)) ◦α λp S = (ξp ◦α λp S) ∪ (ξp ◦β λp S ◦α λp S) = (ξp ◦α λp S) ∪ ξp ◦β (λp S ◦α λp S) = (ξp ◦α λp S) ∪ (ξp ◦β λp S) ⊆ ξp ∪ (ξp ◦β λp S) = κp. Similarly, we can show that λn S ◦α ςn ⊇ ςn and κn ◦β λn S ⊇ κn. Thus, ς = (S; ςp, ςn) and κ = (S;κp, κn) is a BF left α-ideal and a BF right β-ideal of S. Now, ξp ⊆ (ξp ∪ (λp S ◦α ξp)) ∩ (ξp ∪ (ξp ◦β λp S)) = ςp ∩ κp and ςp ∩κp = (ξp ∪ (λp S ◦α ξp))∩ (ξp ∪ (ξp ◦β λp S)) = ξp ∩ ((λp S ◦α ξ p)∪ (ξp ◦β λp S)) ⊆ ξp ∩ ξp = ξp. Hence, ξp = ςp ∩ κp. Simlarly, we can show that ξn = ςn ∩ κn. Theorem 10. Let K be a non-empty subset of Γ-semigroup S. Then K is a (α, β)-quasi- ideal of S if and only if the characteristic function λK = (S;λp K , λn K) is a BF (α, β)-quasi- ideal of S. Proof. Suppose that K is a (α, β)-quasi-ideal of S and u ∈ S. If u ∈ (SαK) ∩ (KβS), then u ∈ K. Thus, λp K(u) = 1 and λn K(u) = −1. Hence, ((λp K ◦α λp S) ∩ (λp S ◦β λp K))(u) ≤ λp K(u) and ((λn K ◦α λn S) ∪ (λn S ◦β λn K))(u) ≥ λn K(u) If u /∈ (SαK) ∩ (KβS), then λp K(u) = 0 and λp K(u) = 0. Hence, ((λp K ◦α λp S) ∩ (λp S ◦β λp K))(u) ≤ λp K(u) and ((λn K ◦α λn S) ∪ (λn S ◦β λn K))(u) ≥ λn K(u). Therefore, λK = (S;λp K , λn K) is a BF (α, β)-quasi-ideal of S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1601 Conversely, assume that λK = (S;λp K , λn K) is a BF (α, β)-quasi-ideal of S and u ∈ (SαK) ∩ (KβS). Then ((λp K ◦α λp S) ∩ (λp S ◦β λp K))(u) = 1 and ((λn K ◦α λn S)∩ (λn S ◦β λn K))(u) = −1. By assumption, ((λp K ◦α λp S)∩ (λp S ◦β λp K))(u) ≤ λp K(u) and ((λn K ◦αλn S)∪(λn S ◦β λn K))(u) ≥ λn K(u). Thus, u ∈ K. Hence, K is an (α, β)-quasi-ideal of S. 4. New Types of Bipolar Fuzzy Almost Ideals Definition 13. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S, and α, β ∈ Γ is said to be [(i)] (i) A BF almost left α-ideal of S if (xpt ◦α ξp) ∧ ξp ̸= 0 and (xns ◦α ξn) ∨ ξn ̸= 0. (ii) A BF almost right β-ideal of S if ξp ◦β (xpt ) ∧ ξp ̸= 0 and (ξn ◦β xns ) ∨ ξn ̸= 0. (iii) A BF almost (α, β)-ideal of S if it is both a BF almost left α-ideal and a BF almost right β-ideal of S. Theorem 11. If ξ = (S; ξp, ξn) is a BF almost left α-ideal (right β-ideal, (α, β)-ideal) of a Γ-semigroup S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς, then ς = (S; ςp, ςn) is a BF left almost α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Suppose that ξ = (S; ξp, ξn) is a BF almost left α-ideal of S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς. Then (xpt ◦α ξp) ∧ ξp ̸= 0 and (xns ◦α ξn) ∨ ξn ̸= 0. Thus, (xpt ◦α ξp) ∧ ξp ⊆ (xpt ◦α ςp) ∧ ςp ̸= 0 and (xns ◦α ξn) ∨ ξn ⊆ (xns ◦α ςn) ∨ ςn ̸= 0. Hence, ς = (S; ςp, ςn) is a BF left almost α-ideal of S. Theorem 12. Let K be a non-empty subset of Γ-semigroup S. Then K is an almost left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if the characteristic function λK = (S;λp K , λn K) is a BF almost left α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Suppose that K is an almost left α-ideal of S. Then uαK ∩K ̸= ∅ for all u ∈ S. Thus, there exists v ∈ uαK and v ∈ K. So, (xpt ◦α λp K)(v) = λp K(v) = 1 and (xns ◦α λn K)(v) = λn K(v) = −1. Hence, (xpt ◦α λp K) ∧ λp K ̸= 0 and (xns ◦α λn K) ∨ λn K ̸= 0. Therefore, λK = (S;λp K , λn K) is a BF almost left α-ideal of S. Conversely, assume that λK = (S;λp K , λn K) is a BF almost left α-ideal of S and u ∈ S. Then (xpt ◦α λp K) ∧ λp K ̸= 0 and (xns ◦α λn K) ∨ λn K ̸= 0. Thus, there exists r ∈ S such that ((xpt ◦α λp K) ∧ λp K)(r) ̸= 0 and ((xns ◦α λn K) ∨ λn K)(r) ̸= 0. Hence, r ∈ uαK ∩ K implies uαK ∩K ̸= ∅. Therefore, K is an almost left α-ideal of S. Next, we review the definition of supp(ξ), and we study the properties between supp(ξ) and BF almost left α-ideal (right β-ideal, (α, β)-ideal) of Γ-semigroups. Let ξ = (S; ξp, ξn) be a BF set of a non-empty of S. Then the support of ξ instead of supp(ξ) = {u ∈ S | ξ(u) ̸= 0} where ξp(u) ̸= 0 and ξn(u) ̸= 0 for all u ∈ S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1602 Theorem 13. Let ξ = (S; ξp, ξn) be a BF set of a non-empty of a Γ-semigroup S. Then ξ = (S; ξp, ξn) is a BF almost left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if supp(ξ) is an almost left α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Let ξ = (S; ξp, ξn) be a BF almost left α-ideal of S and u ∈ S. Then (xpt ◦α ξp) ∧ ξp ̸= 0 and (xns ◦α ξn) ∨ ξn ̸= 0. Thus, there exists r ∈ S such that ((xpt ◦α ξp) ∧ ξp)(r) ̸= 0 and ((xns ◦α ξn) ∨ ξn)(r) ̸= 0. So, there exists k ∈ S such that r = uαk, xpt (r) ̸= 0, xns (r) ̸= 0 and xpt (k) ̸= 0, xns (k) ̸= 0. It implies that r, k ∈ supp(ξ). Thus, (xpt ◦α λ p supp(ξ))(r) ̸= 0, (xns ◦α λn supp(ξ))(r) ̸= 0, and λp supp(ξ) ̸= 0, λn supp(ξ) ̸= 0. Hence, (xpt ◦α λp supp(ξ)) ∧ λp supp(ξ) ̸= 0 and (xns ◦α λn supp(ξ)) ∨ λn supp(ξ) ̸= 0. Therefore, λsupp(ξ) is a BF almost left α-ideal of S. This shows that supp(ξ) is an almost left α-ideal of S. Conversely, let supp(ξ) be an almost left α-ideal of S. Then, by Theorem 12, λsupp(ξ) is a BF almost left α-ideal of S. Thus, (xpt ◦α λp supp(ξ))∧λp supp(ξ) ̸= 0 and (xns ◦α λn supp(ξ))∨ λn supp(ξ) ̸= 0. So, there exists r ∈ S such that ((xpt ◦α λp supp(ξ)) ∧ λp supp(ξ))(r) ̸= 0 and ((xns ◦αλn supp(ξ))∨λ n supp(ξ))(r) ̸= 0. It implies that (xpt ◦αλ p supp(ξ))(r) ̸= 0, (xnt ◦αλn supp(ξ))(r) ̸= 0 and λp K(r) ̸= 0, λn K(r) ̸= 0. Thus, there exists k ∈ S such that r = uαk, xpt (r) ̸= 0, xns (r) ̸= 0 and xpt (k) ̸= 0, xns (k) ̸= 0. Hence, (xpt ◦α ξp) ∧ ξp ̸= 0 and (xns ◦α ξn) ∨ ξn ̸= 0. Therefore, ξ = (S; ξp, ξn) is a BF almost left α-ideal of S. Definition 14. An almost ideal I of a Γ-semigroup S is called minimal if for every almost ideal of J of S such that J ⊆ I, we have J = I. Definition 15. A BF almost left α-ideal (right β-ideal, (α, β)-ideal) ξ = (S; ξp, ξn) of a Γ-semigroup S is minimal if for all BF almost left α-ideal (right β-ideal, (α, β)-ideal) ς = (S; ςp, ςn) of S such that ς ⊆ ξ, then supp(ς) = supp(ξ). Theorem 14. Let K be a non-empty subset of a Γ-semigroup S. Then K is a minimal almost left α-ideal (right β-ideal, (α, β)-ideal) if and only if λK = (S;λp K , λn K) is a minimal BF almost left α-ideal (right β-ideal, (α, β)-ideal) of S. Proof. Suppose that K is a minimal almost left α-ideal of S. Then K is an almost left α-ideal of S. Thus, by Theorem 12, λK = (S;λp K , λn K) is a BF left α-ideal of S. Let ς = (S; ςp, ςn) be a BF left α-ideal of S such that ς ⊆ ξ. Then, by Theorem 13, supp(ς) is an almost left α-ideal of S. Thus, supp(ς) ⊆ supp(λK) = K. By assumption, supp(ς) = K = supp(λK). Thus, λK = (S;λp K , λn K) is a minimal BF almost left α-ideal of S. Conversely, suppose that λK = (S;λp K , λn K) is a minimal BF almost left α-ideal of S. Then, by Theorem 12, K is an almost left α-ideal of S. Let J be an almost left α-ideal of S such that J ⊆ K. Then by Theorem 12, λJ = (S;λp J , λ n J) is a BF left α-ideal of S such that λJ ⊆ λK . Thus, J = supp(λJ) = supp(λK) = K. Hence, K is a minimal almost left α-ideal of S. Corollary 1. Let S be a Γ-semigroup S. Then S has no proper almost left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if for any BF almost left α-ideal (right β-ideal, (α, β)-ideal) ξ = (S; ξp, ξn) of S, supp(ξ) = S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1603 Next, we define the BF almost (α, β)-quasi-ideals, and we study their properties. Definition 16. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S, and α, β ∈ Γ is said to be a BF almost (α, β)-quasi-ideal of S if (ξ ◦α x(t,s)) ∩ (x(t,s) ◦β ξ) ̸= 0. Theorem 15. If ξ = (S; ξp, ξn) is a BF almost (α, β)-quasi-ideal of a Γ-semigroup S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς, then ς = (S; ςp, ςn) is a BF (α, β)-quasi- ideal of S. Proof. Suppose that ξ = (S; ξp, ξn) is a BF almost (α, β)-quasi-ideal of S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς. Then (xpt ◦α ξp) ∧ (ξp ◦β xpt ) ̸= 0 and (xns ◦α ξn) ∨ (ξn ◦β xns ) ̸= 0. Thus, (xpt ◦α ξp) ∧ (ξp ◦β xpt ) ⊆ (xpt ◦α ςp) ∧ (ςp ◦β xpt ) ̸= 0, and (xns ◦α ξn) ∨ (ξn ◦β xns ) ⊆ (xns ◦α ςn) ∧ (ςn ◦β xns ) ̸= 0. Hence, ς = (S; ςp, ςn) is a BF (α, β)-quasi-ideal of S. Theorem 16. Let K be a non-empty subset of Γ-semigroup S. Then K is an almost (α, β)-quasi-ideal of S if and only if the characteristic function λK = (S;λp K , λn K) is a BF almost (α, β)-quasi-ideal of S. Proof. Suppose that K is an almost (α, β)-quasi-ideal of S. Then (Kαu) ∩ (uβK) ∩ K ̸= ∅ for all u ∈ S. Thus, there exists v ∈ (Kαu) ∩ (uβK) and v ∈ K. So, ((xpt ◦α λp K) ∧ (λp K ◦β xpt ))(v) ̸= 0 and ((xns ◦α λn K) ∨ (λn K ◦β xns ))(v) ̸= 0. Hence, (λK ◦α x(t,s)) ∩ (x(t,s) ◦β λK) ̸= 0. Therefore, λK = (S;λp K , λn K) is a BF almost (α, β)-quasi-ideal of S. Conversely, assume that λK = (S;λp K , λn K) is a BF almost (α, β)-quasi-ideal of S and u ∈ S. Then (λK ◦α x(t,s)) ∩ (x(t,s) ◦β λK) ̸= 0. Thus, there exists r ∈ S such that ((xpt ◦α λp K) ∧ (λp K ◦β xpt ))(r) ̸= 0 and ((xns ◦α λn K) ∨ (λn K ◦β xns ))(r) ̸= 0. Hence, r ∈ (Kαu) ∩ (uβK) ∩ K implies (Kαu) ∩ (uβK) ∩ K ̸= ∅. Therefore, K is an almost (α, β)-quasi-ideal of S. Next, we study the properties between supp(ξ) and a BF almost (α, β)-quasi-ideal of Γ-semigroups. Theorem 17. Let ξ = (S; ξp, ξn) be a BF set of a non-empty of a Γ-semigroup S. Then ξ = (S; ξp, ξn) is a BF almost (α, β)-quasi-ideal of S if and only if supp(ξ) is an almost (α, β)-quasi-ideal of S. Proof. Let ξ = (S; ξp, ξn) be a BF almost (α, β)-quasi-ideal of S and u ∈ S. Then (λK ◦αx(t,s))∩(x(t,s)◦βλK) ̸= 0. Thus, there exists r ∈ S such that (ξp◦αxt)∩(xt◦β ξp) ̸= 0 and (ξn ◦α xs) ∪ (xs ◦β ξn) ̸= 0 So, there exists k1, k2 ∈ S such that r = k1αu = uβk2, xpt (r) ̸= 0, xns (r) ̸= 0 and xpt (k1) ̸= 0, xns (k1) ̸= 0. It implies that r, k1, k2 ∈ supp(ξ). Thus ((ξp ◦α xt) ∩ (xt ◦β ξp))(r) ̸= 0 and λp supp(ξ) ̸= 0. Similalry, (ξn ◦α xs) ∪ (xs ◦β ξn)(r) ̸= 0 and λn supp(ξ) ̸= 0. Hence, (λp supp(ξ) ◦α xpt ) ∩ (xpt ◦β λp supp(ξ)) ∩ λp supp(ξ) ̸= 0 and (λn supp(ξ) ◦α xns ) ∪ (xns ◦β λn supp(ξ)) ∪ λn supp(ξ) ̸= 0. Therefore, λsupp(ξ) is a BF almost (α, β)-quasi-ideal of S. This shows that supp(ξ) is an almost (α, β)-quasi-ideal of S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1604 Conversely, let supp(ξ) be an almost (α, β)-quasi-ideal of S. Then, by Theorem 16, χsupp(ξ) is a BF (α, β)-quasi-ideal of S. Thus, [(λp supp(ξ)◦αx p t )∩(x p t ◦βλ p supp(ξ))]∩λ p supp(ξ) ̸= 0 and [(λn supp(ξ) ◦α xns ) ∩ (xns ◦β λn supp(ξ))] ∩ λn supp(ξ) ̸= 0. So, there exists r ∈ S such that [(λp supp(ξ) ◦α x p t )∩ (xpt ◦β λ p supp(ξ))]∩λp supp(ξ)(r) ̸= 0 and ([(λn supp(ξ) ◦α x n s )∩ (xns ◦β λn supp(ξ))]∩ λn supp(ξ))(r) ̸= 0. It implies that ((λsupp(ξ) ◦α xt) ∩ (xt ◦β λsupp(ξ)))(r) ̸= 0 and, λK(r) ̸= 0. Thus, there exist k1, k2 ∈ S such that r = k1αu = uβk2, ξp(r) ̸= 0, ξn(r) ̸= 0, and ξp(k) ̸= 0, ξn(k) ̸= 0. Hence, (ξp ◦α xt) ∩ (xt ◦β ξp) ̸= 0, and (ξn ◦α xs) ∪ (xs ◦β ξn) ̸= 0. So, (ξ ◦α x(t,s)) ∩ (x(t,s) ◦β ξ) ̸= 0. Therefore, ξ is a BF almost (α, β)-quasi-ideal of S. Definition 17. A BF almost (α, β)-quasi-ideal ξ = (S; ξp, ξn) of a Γ-semigroup S is minimal if for all BF almost (α, β)-quasi-ideal ς = (S; ςp, ςn) of S such that ς ⊆ ξ, then supp(ς) = supp(ξ). Theorem 18. Let K be a non-empty subset of a Γ-semigroup S. Then K is a minimal almost (α, β)-quasi-ideal if and only if λK = (S;λp K , λn K) is a minimal BF almost (α, β)- quasi-ideal of S. Proof. Suppose thatK is a minimal almost (α, β)-quasi-ideal of S. ThenK is an almost (α, β)-quasi-ideal of S. Thus, by Theorem 16, λK = (S;λp K , λn K) is a BF (α, β)-quasi-ideal of S. Let ς = (S; ςp, ςn) be a BF (α, β)-quasi-ideal of S such that ς ⊆ λK . Then, by Theorem 15, supp(ς) is an almost (α, β)-quasi-ideal of S. Thus, supp(ς) ⊆ supp(λK) = K. By assumption, supp(ς) = K = supp(λK). Thus, λK = (S;λp K , λn K) is a minimal BF almost (α, β)-quasi-ideal of S. Conversely, suppose that λK = (S;λp K , λn K) is a minimal BF almost (α, β)-quasi-ideal of S. Then, by Theorem 17, K is an almost (α, β)-quasi-ideal of S. Let J be an almost (α, β)-quasi-ideal of S such that J ⊆ K. Then, by Theorem 17, λJ = (S;λp J , λ n J) is a BF (α, β)-quasi-ideal of S such that λJ ⊆ λK . Thus, J = supp(λJ) = supp(λK) = K. Hence, K is a minimal almost (α, β)-quasi-ideal of S. Corollary 2. Let S be a Γ-semigroup. Then S has no proper almost (α, β)-quasi-ideal of S if and only if for any BF almost (α, β)-quasi-ideal ξ = (S; ξp, ξn) of S, supp(ξ) = K. Next, we define the BF almost (α, β)-bi-ideals and we study their properties. Definition 18. Let ξ = (S; ξp, ξn) be a BF set of a Γ-semigroup S, and α, β ∈ Γ is said to be BF almost (α, β)-bi-ideal of S if (ξp ◦αxpt ◦β ξp)∧ξp ̸= 0 and (ξn ◦αxns ◦β ξn)∨ξn ̸= 0. Theorem 19. If ξ = (S; ξp, ξn) is a BF almost (α, β)-bi-ideal of a Γ-semigroup S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς, then ς = (S; ςp, ςn) is a BF (α, β)-bi-ideal of S. Proof. Suppose that ξ = (S; ξp, ξn) is a BF almost (α, β)-bi-ideal of S, and ς = (S; ςp, ςn) is a BF set of S such that ξ ⊆ ς. Then (ξp ◦α xpt ◦β ξp)∧ ξp ̸= 0, and (ξn ◦α xns ◦β ξn)∨ξn ̸= 0. Thus, (ξp◦αxpt ◦β ξp)∧ξp ⊆ (ςp◦αxpt ◦β ςp)∧ςp ̸= 0, and (ξn◦αxns ◦β ξn)∨ξn ⊆ (ςn ◦α xns ◦β ςn) ∨ ςn ̸= 0. Hence, ς = (S; ςp, ςn) is a BF (α, β)-bi-ideal of S. P. Khamrot, T. Gaketem / Eur. J. Pure Appl. Math, 16 (3) (2023), 1592-1607 1605 Theorem 20. Let K be a non-empty subset of Γ-semigroup S. Then K is an almost (α, β)-bi-ideal of S if and only if the characteristic function λK = (S;λp K , λn K) is a BF almost (α, β)-bi-ideal of S. Proof. Suppose that K is an almost (α, β)-bi-ideal of S. Then KαuβK ∩K ̸= ∅ for all u ∈ S. Thus, there exists v ∈ KαuβK and v ∈ K. So, ((λp K ◦α xpt ◦β λ p K))(v) = λp K(v) = 1, and ((λn K ◦α xns ◦β λn K))(v) = λn K(v) = −1. Hence, (λp K ◦α xpt ◦β λp K) ∧ λp K ̸= 0 and (λn K ◦α xns ◦β λn K) ∨ λn K ̸= 0. Therefore, λK = (S;λp K , λn K) is a BF almost (α, β)-bi-ideal of S. Conversely, assume that λK = (S;λp K , λn K) is a BF almost (α, β)-bi-ideal of S, and u ∈ S. Then (λp K ◦α xpt ◦β λ p K)∧λp K ̸= 0, and (λn K ◦α xns ◦β λn K)∨λn K ̸= 0. Thus, there exists r ∈ S such that ((λp K ◦αxpt ◦β λ p K)∧λp K)(r) ̸= 0, and ((λn K ◦αxns ◦β λn K)∨λn K)(r) ̸= 0. Hence, r ∈ KαuβK ∩K implies that KαuβK ∩K ̸= ∅. Therefore, K is an almost (α, β)-bi-ideal of S. Next, we study the properties between supp(ξ) and a BF almost (α, β)-bi-ideal of Γ-semigroups. Theorem 21. Let ξ = (S; ξp, ξn) be a fuzzy set of a non-empty of a Γ-semigroup S. Then ξ = (S; ξp, ξn) is a BF almost (α, β)-bi-ideal of S if and only if supp(ξ) is an almost (α, β)-bi-ideal of S. Proof. Let ξ = (S; ξp, ξn) be a BF almost (α, β)-bi-ideal of S, and u ∈ S. Then (ξp ◦α xpt ◦β ξp) ∧ ξp ̸= 0 and (ξn ◦α xns ◦β ξn) ∨ ξn ̸= 0. Thus, there exists r ∈ S such that ((ξp ◦α xpt ◦β ξp)∧ ξp)(r) ̸= 0, and ((ξn ◦α xns ◦β ξn)∨ ξn)(r) ̸= 0. So, there exists k1, k2 ∈ S such that r = k1αβk2, ξ p(r) ̸= 0, ξn(r) ̸= 0, and ξp(k) ̸= 0, ξn(k) ̸= 0. It implies that r, k1, k2 ∈ supp(ξ). Thus, (λp supp(ξ) ◦α xpt ◦β λp supp(ξ))(r) ̸= 0 and λp supp(ξ) ̸= 0. Similarly, (λn supp(ξ) ◦α xns ◦β λn supp(ξ))(r) ̸= 0, and λn supp(ξ) ̸= 0. Hence, [(λp supp(ξ) ◦α xpt ◦β λp supp(ξ))] ∧ λp supp(ξ) ̸= 0, and [(λn supp(ξ) ◦α xns ◦β λn supp(ξ))] ∨ λn supp(ξ) ̸= 0. Therefore, λsupp(ξ) is a BF almost (α, β)-bi-ideal of S. This shows that supp(ξ) is an almost (α, β)-bi-ideal of S. Conversely, let supp(ξ) be an almost (α, β)-bi-ideal of S. Then, by Theorem 20, λsupp(ξ) is a BF almost (α, β)-bi-ideal of S. Thus, [(λp supp(ξ) ◦α xpt ◦β λp supp(ξ))] ∧ λp supp(ξ) ̸= 0, and [(λn supp(ξ) ◦α xns ◦β λn supp(ξ))] ∨ λn supp(ξ) ̸= 0. So, there exists r ∈ S such that ((λp supp(ξ)◦αx p t ◦βλ p supp(ξ))∧λ p supp(ξ))(r) ̸= 0, and ((λn supp(ξ)◦αx n s ◦βλn supp(ξ))∨λ n supp(ξ))(r) ̸= 0. It implies that (λp supp(ξ) ◦α xpt ◦β λp supp(ξ))(r) ̸= 0, and λp supp(ξ)(r) ̸= 0. Similarly, (λn supp(ξ) ◦α xns ◦β λn supp(ξ))(r) ̸= 0, and λn supp(ξ)(r) ̸= 0. Thus, there exist k1, k2 ∈ S such that r = k1αuβk2, ξ p(r) ̸= 0, ξn(r) ̸= 0, and ξp(k) ̸= 0, ξn(k) ̸= 0. Hence, (ξp ◦α xpt ◦β ξp)∧ ξp ̸= 0, and (ξn ◦α xns ◦β ξn)∨ ξn ̸= 0. Therefore, ξ is a BF almost (α, β)-bi-ideal of S. Definition 19. A BF almost (α, β)-bi-ideal ξ = (S; ξp, ξn) of a Γ-semigroup S is minimal if for all BF almost (α, β)-bi-ideal ς = (S; ςp, ςn) of S such that ς ⊆ ξ, then supp(ς) = supp(ξ). REFERENCES 1606 Theorem 22. Let K be a non-empty subset of a Γ-semigroup S. Then K is a minimal almost (α, β)-bi-ideal if and only if λK = (S;λp K , λn K) is a minimal BF almost (α, β)-bi- ideal of S. Proof. Suppose that K is a minimal almost (α, β)-bi-ideal of S. Then K is an almost (α, β)-quasi-ideal of S. Thus, by Theorem 20, λK = (S;λp K , λn K) is a BF (α, β)-quasi- ideal of S. Let ς = (S; ςp, ςn) be a BF (α, β)-bi-ideal of S such that ς ⊆ λK . Then, by Theorem 21, supp(ς) is an almost (α, β)-bi-ideal of S. Thus, supp(ς) ⊆ supp(λK) = K. By assumption, supp(ς) = K = supp(λK). Thus, λK = (S;λp K , λn K) is a minimal BF almost (α, β)-bi-ideal of S. Conversely, let λK = (S;λp K , λn K) be a minimal BF almost (α, β)-bi-ideal of S. Then, by Theorem 20, K is an almost (α, β)-bi-ideal of S. Let J be an almost (α, β)-bi-ideal of S such that J ⊆ K. Then by Theorem 20, λJ is a BF (α, β)-bi-ideal of S such that λJ ⊆ λK . Thus, J = supp(λJ) = supp(λK) = K. Hence, K is a minimal almost (α, β)-bi-ideal of S. Corollary 3. Let S be a Γ-semigroup S. Then S has no proper almost (α, β)-bi-ideal of S if and only if for any BF almost (α, β)-bi-ideal ξ = (S; ξp, ξn) of S, supp(ξ) = S. 5. Conclusion In this article, we introduce the concept of a new bipolar fuzzy ideal and bipolar fuzzy almost ideals in Γ-semigroups. We also study properties of new bipolar fuzzy ideals and bipolar fuzzy almost ideals. We hope that the present study will be useful mathematical tools. In further, we extend to hesitant fuzzy almost ideals and algebraic systems. Acknowledgements This research project was supported by the Thailand Science Research and Innova- tion Fund and the University of Phayao (Grant No. FF66-UoE017) Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand. References [1] R. Chinram. On quasi-gamma-ideals in γ-semigroups. Science Asia, 32:351–353, 2016. [2] T. Gaketem and P. Khamrot. On some semigroups characterized in terms of biplar fuzzy weakly interior ideals. IAENG International Journal of Computer Science, 48(2):250–256, 2021. [3] T. Gaketem and P. Khamrot. 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