Int. J. Anal. Appl. (2022), 20:69 A Novel of Cubic Ideals in Γ-Semigroups Pannawit Khamrot1, Thiti Gaketem2,∗ 1Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna of Phitsanulok, Phitsanulok, Thailand 2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics School of Science, University of Phayao, Phayao 56000, Thailand ∗Corresponding author: thiti.ga@up.ac.th Abstract. In this paper, we give the concepts of new types of cubic ideals in Γ-semigroups. We study properties and relationships between cubic (α,β)-ideals and ideals in semigroups. Furthermore, we proved some basic properties of cubic almost ideals in semigroups. 1. Introduction The theory for dealing with uncertainty, fuzzy set theory, was discovered by Zadeh in 1965 [18], mathematical tool for describing the behavior of the systems that are too complex or illdefined to admit precise mathematical analysis by classical methods and tools. The studies of cubic sets and cubic subgroups were presented by Jun et al in 2012 [10,11]. Later V. Chinnadurai and K. Bharathivelan [2] studies cubic ideals in Γ-semigroups and proved basic properties of cubic ideals in Γ-semigroups. The theory of ideal is structured important in semigroups and many researchers used knowledge of ideals in Γ-semigroups discussed in fuzzy semigroup such as Chinram et al. [3] discussed almost quasi-Γ-ideal and fuzzy almost quasi-Γ-ideals in Γ-semigroup, M. K. R. Marapureddy and PRV S. R. Doradla [14] discussed weak interior ideals of Γ-semigroups, S.K. Majumder and M. Mandal [9] discussed fuzzy generalized bi-ideal in Γ-semigroups. In the study of the concept of cubic ideals, many researchers expanded on this idea [4,6,7,7,8,13,15]. Recently, in 2021 [17] A. Simuen et al. discussed a novel of ideals and fuzzy ideals of Γ-semigroups. Received: Nov. 14, 2022. 2010 Mathematics Subject Classification. 20M12, 06F05. Key words and phrases. (α,β)-cubci ideal; cubic almost ideal; Γ-semigroups. https://doi.org/10.28924/2291-8639-20-2022-69 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-69 2 Int. J. Anal. Appl. (2022), 20:69 In this paper we extend new fuzzy ideals to cubic ideals of Γ-semigroups and we investigate the properties of new types cubic ideals. of Γ-semigroups. 2. Preliminaries In this section, we review concepts basic definitions and the theorem used to prove all result in the next 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 it 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 2.1. [17] Let S be a Γ-semigroup, K be a non-empty subset of S, for all e ∈ S and α,β ∈ Γ. Then K is said to be (1) a left (right) almost ideal of Γ-semigroup S is a non-empty set K such that (eΓK) ∩K 6= ∅ ((KΓe) ∩K 6= ∅) (2) an almost bi-ideal of Γ-semigroup S is a non-empty set K such that (KΓeΓK) ∩K 6= ∅. (3) an almost quasi-ideal of Γ-semigroup S is a non-empty set K such that (eΓK∩KΓe)∩K 6= ∅. (4) a left α-ideal of a Γ-semigroup S is a non-empty set K such that SαK ⊆K. A right β-ideal of a Γ-semigroup S is a non-empty set K such that KβS ⊆K. (5) an (α,β)-ideal of a Γ-semigroup S is a non-empty set K such that it is both a left α-ideal and a right β-ideal of S. We see that for any ζ1,ζ2 ∈ [0, 1], we have ζ1 ∨ζ2 = max{ζ1,ζ2} and ζ1 ∧ζ2 = min{ζ1,ζ2}. A fuzzy set υ of a non-empty set T is function from T into unit closed interval [0, 1] of real numbers, i.e., υ : T → [0, 1]. For any two fuzzy sets υ and ν of a non-empty set T , define ≥, =,∧, and ∨ as follows: (1) υ ≥ ν ⇔ υ(e) ≥ ν(e) for all e ∈T , (2) υ = ν ⇔ υ ≥ ν and ν ≥ υ, (3) (υ ∧ν)(e) = min{υ(e),ν(e)} = υ(e) ∧ν(e) for all e ∈T , (4) (υ ∨ν)(e) = max{υ(e),ν(e)} = υ(e) ∨ν(e) for all e ∈T . For the symbol υ ≤ ν, we mean ν ≥ υ. The following definitions are types of fuzzy subsemigroups on Γ-semigroups. Definition 2.2. [17] A fuzzy set υ of a Γ-semigroup S is said to be (1) a fuzzy subsemigroup of S if υ(eγf ) ≥ υ(e) ∧υ(f ) for all e,f ∈S and γ ∈ Γ, (2) a fuzzy left (right) ideal of S if υ(eγf ) ≥ υ(f ) (υ(eγf ) ≥ υ(e)) for all e,f ∈S and γ ∈ Γ, Int. J. Anal. Appl. (2022), 20:69 3 (3) a fuzzy ideal of S if it is both a fuzzy left ideal and a fuzzy right ideal of S, (4) a fuzzy bi-ideal of S if υ is a fuzzy subsemigroup of S and υ(eγf βh) ≥ υ(e) ∧υ(h) for all e,f ,h ∈S and γ,β ∈ Γ. Now, we review the concept of interval valued fuzzy sets. Let CS[0, 1] be the set of all closed subintervals of [0, 1], i.e., CS[0, 1] = {ω̌ = [ωl,ωu] | 0 ≤ ωl ≤ ωu ≤ 1}, where ωl is a lower interval value of ω̌ and ωu is an upper interval value of ω̌. We note that [ω,ω] = {ω} for all ω ∈ [0, 1]. For ω = 0 or 1, we shall denote [0, 0] by 0̌ and [1, 1] by 1̌. For ω̌ := [ωl,ωu] and ζ̌ := [ζl,ζu] in CS[0, 1], the operations “�", “=", “f", “g" are defined as follows: (1) ω̌ � ζ̌ if and only if ωl ≤ ζl and ωu ≤ ζu (2) ω̌ = ζ̌ if and only if ωl = ζl and ωu = ζu (3) ω̌ f ζ̌ = [(ωl ∧ζl), (ωu ∧ζu)] (4) ω̌ g ζ̌ = [(ωl ∨ζl), (ωu ∨ζu)]. If ω̌ � ζ̌, we mean ζ̌ � ω̌. Definition 2.3. [19] An interval valued fuzzy set (shortly, IVF set) of a non-empty set T is a function ω̌ : T → CS[0, 1]. Next, we shall give definitions of various types of IVF subsemigroups. Definition 2.4. [1] An IVF set ω̌ of a Γ-semigroup S is said to be an IVF subsemigroup of S if ω̌(eαf ) % ω̌(e) f ω̌(f ) for all e,f ∈S and α ∈ Γ. Definition 2.5. [1] An IVF set ω̌ of a semigroup S is said to be an IVF left (right) ideal of S if ω̌(eαf ) % ω̌(e) (ω̌(eαf ) % ω̌(e)) for all e,f ∈S and α ∈ Γ. An IVF subset ω̌ of S is called an IVF ideal of S if it is both an IVF left ideal and an IVF right ideal of S. Definition 2.6. [1] Let K be a subset of a non-empty set T . An interval valued characteristic function (shortly, IVCF) χ̌K of T is defined to be a function χ̌K : T → CS[0, 1] by χ̌K(e) =  1̌ if e ∈K, 0̌ if e /∈K for all e ∈T . For two IVF subsets ω̌ and ζ̌ of a non-empty set T , define (1) ω̌ v ζ̌ ⇔ ω̌(e) � ζ̌(e) for all e ∈T , 4 Int. J. Anal. Appl. (2022), 20:69 (2) ω̌ = ζ̌ ⇔ ω̌ v ζ̌ and ζ̌ v ω̌, (3) (ω̌ u ζ̌)(e) = ω̌(e) f ζ̌(e) for all e ∈T . (4) (ω̌ t ζ̌)(e) = ω̌(e) g ζ̌(e) for all e ∈T . Definition 2.7. [10] A cubic set (CB set) C of a non-empty set T is a structure of the form C = {〈e,ω̌(e),υ(r)〉 | e ∈T} and denoted by C = 〈ω̌,υ〉 where ω̌ is an IVF set and υ is a fuzzy set. In this case, we will use C(e) = 〈ω̌(e),υ(e)〉 = 〈[ωl(e),ωu(e)],υ(e)〉 for all e ∈T . Definition 2.8. [11] Let T be a semigroup and K be a non-empty set of T , the characteristic CB set of K in T is defined to be the structure ≥K = {〈e,ω̌λK(e),υλK(e)〉 : e ∈ T} which is briefly denoted by ≥K = 〈ω̌λK,υλK〉 where ω̌λK(e) =  1̌, if e ∈K, 0̌, if e /∈K and υλK(e) =  0, if e ∈K, 1, if e /∈K. Definition 2.9. [2] A CB set C = 〈ω̌,υ〉 of a Γ-semigroup S is called (1) a CB subsemigroup of S if ω̌(eαf ) % ω̌(e)f ω̌(f ) and υ(eαf ) ≤ υ(e)∨υ(f ) for all e,f ∈S and α ∈ Γ. (2) a CB left(right)ideal of S if ω̌(eαf ) % ω̌(f ) (ω̌eαf ) � ω̌(e)) and υ(eαf ) ≤ υ(f )(υ(eαf ) ≤ υ(e)) for all e,f ∈S and α ∈ Γ. A CB ideal of S if it is both a CB left ideal and a CB right ideal of S. For e ∈T , define Fe = {(y,z) ∈T ×T | e = yz}. Definition 2.10. [11] Let C = 〈ω̌,υ〉 and D = 〈ζ̌,ν〉 be two CB set in a semigroup S. Then the CB product of C and D is a structure C�D = {〈e, (ω̌�ζ̌)(e, (υ ·ν)(e)〉 : e ∈S} which is briefly denoted by C � D = 〈(ω̌�ζ̌), (υ · ν)〉 where ω̌�ζ̌ and υ · ν are defined as follows, respectively: (ω̌�ζ̌)(e) =  rsup(y,z)∈Fe{ω̌(y) f ζ̌(z)} if Fe 6= ∅, 0̌, if Fe = ∅, Int. J. Anal. Appl. (2022), 20:69 5 and (υ ·ν)(e) =  inf (y,z)∈Fe{υ(y) ∨ν(z)} if Fe 6= ∅, 1, if Fe = ∅. Definition 2.11. [11] For two CB st C = 〈ω̌,υ〉 and D = 〈ρ̌,τ〉 in a semigroup S, we define C v D ⇔ ω̌ - ρ̌ and υ ≥ τ Definition 2.12. [11] Let C = 〈ω̌,υ〉 and D = 〈ρ̌,τ〉 be two CB set in a semigroup S. Then the intersection of C and D denoted by CuD is the CB set CǔD = 〈ω̌ u ζ̌,υ ∨ν〉 where (C u D)(e) = ω̌(e) f ρ̌(e) and (υ ∨τ)(e) = υ(e) ∨τ(e) for all e ∈S. And union of C and D denoted by CtD is the CB set CťD = 〈ω̌ t ρ̌,υ∧τ〉 where (ω̌ t ρ̌)(e) = ω̌(e) g ρ̌(e) and (υ ∧τ)(e) = υ(e) ∧τ(e) for all e ∈S. 3. New Types of Cubic Ideals In this section, we define cubic fuzzy (α,β)-ideal and study basic properties of it. Definition 3.1. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ. Then C = 〈ω̌,υ〉 is called (1) a CB left α-ideal of S if ω̌(eαf ) % ω̌(f ) and υ(eαf ) ≤ υ(f ) for all e,f ∈S. (2) a CB right β-ideal of S if ω̌(eβf ) % ω̌(e) and υ(eβf ) ≤ υ(f ) for all e,f ∈S. (3) a CB (α,β)-ideal of S if it is both a CB left α-ideal and a CB right β-ideal of S. (4) a CB α-ideal of S if it is a CB (α,α)-ideal of S. Theorem 3.1. Let K be a nonempty subset of Γ-semigroup S. Then K is a left α-ideal (right β-ideal, (α,β)-ideal) of S if and only if ≥K = 〈ω̌λK,υλK〉 is a CB left α-ideal (right β-ideal, (α,β)-ideal) of S. Proof. Suppose that K is a left α-ideal of S and e,f ∈S. If f ∈K, then eαf ∈K. Thus ω̌λK(f ) = ω̌λK(eαf ) = 1̌ and υλK(f ) = υλK(eαf ) = 0. Hence ω̌λK(eαf ) % ω̌λK(f ) and υλK(eαf ) ≤ υλK(f ). If f /∈K, then eαf ∈K. Thus ω̌λK(f ) = 0̌, ω̌λK(eαf ) = 1̌ and υλK(f ) = 1,υλK(eαf ) = 0. Hence, ω̌λK(eαf ) % ω̌λK(f ) and υλK(eαf ) ≤ υλK(f ). Therefore ≥K = 〈ω̌λK,υλK〉 is a CB left α-ideal of S. Conversely, assume that ≥K = 〈ω̌λK,υλK〉 is a CB left α-ideal of S and e,f ∈S with f ∈K. Then ω̌λK(f ) = 1̌ and υλK(f ) = 0. By assumption, ω̌λK(eαf ) % ω̌λK(f ) and υλK(eαf ) ≤ υλK(f ). Thus, eαf ∈K. Hence, K is a left α-ideal of S. � 6 Int. J. Anal. Appl. (2022), 20:69 Theorem 3.2. The intersection and union of any two CB left α-ideals (right β-ideals, (α,β)-ideals) of a Γ-semigroup S is a CB left α-ideal (right β-ideal, (α,β)-ideal) of S. Proof. Let C = 〈ω̌,υ〉 and D = 〈ρ̌,τ〉 be CB left α-ideals of S and let e,f ∈S. Then (ω̌ u ρ̌)(eαf ) = ω̌(eαf ) f ρ̌(eαf ) % ω̌(f ) f ρ̌(f ) = (ω̌ u ρ̌)(f ) and (υ ∩τ)(eαf ) = υ(eαf ) ∨τ(eαf ) ≤ υ(v) ∨τ(f ) = (υ ∩τ)(f ). Similarly, (ω̌ t ρ̌)(eαf ) = ω̌(eαf ) g ρ̌(eαf ) % ω̌(f ) g ρ̌(f ) = (ω̌ u ρ̌)(f ) and (υ ∪τ)(eαf ) = υ(eαf ) ∧τ(eαf ) ≤ υ(v) ∧τ(f ) = (υ ∪τ)(f ). Thus, CǔD and CťD are CB left α-ideals of S. � Theorem 3.3. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and C(ň,m) = (ω̌ň,υm) be CB point with ω̌ň = {f ∈ S | ω̌ň(f ) % ň} and υm = {f ∈ S | υm(f ) ≤ m}. Then C = 〈ω̌,υ〉 is a CB left α-ideal (right β-ideal, (α,β)-ideal) of S if and only if C = 〈ω̌,υ〉 is a nonempty set and C(ň,m) is a left α-ideal (right β-ideal, (α,β)-ideal) of S for all (ň,m) ∈ (0, 1] × [0, 1). Proof. Suppose that C = 〈ω̌,υ〉 is a CB left α-ideal of S and α ∈ Γ. Then ω̌(eαf ) % ω̌(f ) and υ(eαf ) ≤ υ(f ) for all e,f ∈ S. Let (ň,m) ∈ (0, 1] × [0, 1) be such that C(ň,m) 6= ∅. Let f ∈ C(ň,m) and f ∈ S. Then ω̌(f ) % 1̌ and υ(f ) ≤ m. Thus ω̌(eαf ) % ω̌(f ) % ň and υ(eαf ) ≤ υ(f ) ≤ m so eαf ∈ C(ň,m). Hence, C(ň,m) = (ω̌ň,υm) is a left α-ideal of S. Conversely, assume that C(ň,m) = (ω̌ň,υm) is a left α-ideal of S if (ň,m) ∈ (0, 1] × [0, 1) and C(ň,m) 6= ∅. Let e,f ∈ S and ň = ω̌(f ),m = υ(f ). By assumption ω̌(f ) % ň and υ(f ) ≤ m. Then f ∈ C(ň,m). Thus C(ň,m) 6= ∅. Hence, C(ň,m) is a left α-ideal of S. Since f ∈ C(ň,m) and e ∈ S, we have eαf ∈ C(ň,m). Thus, ω̌(eαf ) ≥ ň = ω̌(f ) and υ(eαf ) ≤ m = υ(f ). Hence, C = 〈ω̌,υ〉 is a CB left α-ideal of S. � Next, we will define the (α,β)-product. For CB C = 〈ω̌,υ〉 and D = 〈ρ̌,τ〉, define product ω̌ ◦α ρ̌ and υ ◦α τ as follows: For e ∈S (ω̌�αρ̌)(e) =  rsup(y,z)∈Feα{ω̌(y) fα ρ̌(z)} if Feα 6= ∅, 0̌, if Feα = ∅, and (υ ·α τ)(e) =  inf (y,z)∈Fe{υ(y) ∨α τ(z)} if Feα 6= ∅, 1, if Feα = ∅. where Feα = {(y,z) ∈S×S | e = yz}, for e ∈S and α ∈ Γ. Int. J. Anal. Appl. (2022), 20:69 7 Next, we define CB (α,β)-bi-ideal and study basic properties of it. Definition 3.2. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ. Then C = 〈ω̌,υ〉 is called a CB (α,β)-bi-ideal of S if ω̌ ◦α ω̌λS ◦β ω̌ % ω̌ and υ ◦α υλS ◦β υ ≤ υ where ≥S = 〈ω̌λS,υλS〉 is CB set mapping every element of S to 〈1̌, 0〉. Theorem 3.4. Let K be a nonempty subset of Γ-semigroup S. Then K is an (α,β)-bi-ideal of S if and only if characteristic function ≥K = 〈ω̌λK,υλK〉 is a CB (α,β)-bi-ideal of S. Proof. Suppose that K is an (α,β)-bi-ideal of S and KαSβK⊆K. If e ∈KαSβK, then ω̌λK(e) = (ω̌λK ◦α ω̌λS ◦β ω̌λK)(e) = 1̌ and υλK(e) = (υλK ◦α υλS ◦β υλK)(e) = 0. Hence, (ω̌λK ◦α ω̌λS ◦β ω̌λK)(e) % ω̌λK(e) and (υλK ◦α υλS ◦β υλK) ≤ υλK(e) If e ∈KαSβK, then ω̌λK(e) = (ω̌λK ◦α ω̌λS ◦β ω̌λK)(e) = 0̌ and υλK(e) = (υλK ◦α υλS ◦β υλK)(e) = 1. Hence, (ω̌λK ◦α ω̌λS ◦β ω̌λK)(e) % ω̌λK(e) and (υλK ◦α υλS ◦β υλK) ≤ υλK(e). Therefore, ≥K = (ω̌λK,υλK) is a CB (α,β)-bi-ideal of S. Conversely, assume that ≥K = (ω̌λK,υλK) is a CB (α,β)-bi-ideal of S. and e ∈KαSβK. Then (ω̌ ◦α ω̌λK ◦β ω̌)(e) = 1̌ and (υ ◦α ≥S ◦β υ)(e) = 0. By assumption, (ω̌λK ◦α ω̌λS ◦β ω̌λK)(e) % ω̌λK(e) and (υλK ◦α υλS ◦β υλK) ≤ υλK(e). Thus, e ∈K. Hence, K is an (α,β)-bi-ideal of S. � Theorem 3.5. The intersection of any two CB (α,β)-bi-ideals of a Γ-semigroup S is a CB (α,β)-bi- ideal of S. Proof. Let C = 〈ω̌,υ〉 and D = 〈ρ̌,τ〉 be CB (α,β)-bi-ideals of S and e ∈S. Then ((ω̌ u ρ̌) ◦α ω̌λS ◦β (ω̌ u ρ̌))(e) % (ω̌ ◦α ω̌λS ◦β ω̌)(e) f (ρ̌◦α ω̌λS ◦β ρ̌)(e) % (ω̌ u ρ̌)(e) and ((υ ∩τ) ◦α υλS ◦β (υ ∩τ))(e) ≤ (υ ◦α υλS ◦β υ)(u) ∨ (τ ◦α υλS ◦β τ)(u) ≤ (υ ∩τ)(e). Thus, CǔD is a CB α-bi-ideals of S. � Next, we define CB (α,β)-quasi-ideal and study basic properties of it. Definition 3.3. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ. Then C = 〈ω̌,υ〉 is called a CB (α,β)-quasi-ideal of S if ω̌λS ◦α ω̌ u ω̌ ◦β ω̌λS - ω̌ and υλS ◦α υ ∪υ ◦β υλS ≥ υ. Theorem 3.6. If C = 〈ω̌,υ〉 and D = 〈ρ̌,η〉 is a CB left α-ideal and a CB right α-ideal of S respectively, then CǔD is a CB α-quasi-ideal of S. 8 Int. J. Anal. Appl. (2022), 20:69 Proof. Let C = 〈ω̌,υ〉 and D = 〈ρ̌,η〉 is a CB left α-ideal and a CB right α-ideal of S respectively. Then ρ̌◦α ω̌ - ω̌λS ◦α ω̌ - ω̌ and ρ̌◦α ω̌ - ρ̌◦α ω̌λS - ρ̌. Thus, ρ̌◦α ω̌ - ω̌ ∩ ρ̌. So, ω̌λS ◦α (ω̌ u ρ̌) u (ω̌ u ρ̌) ◦α ω̌λS v ω̌λS ◦α (ω̌ u ρ̌) ◦α ω̌λS - ω̌ ∩ ρ̌. Thus, ω̌u ρ̌ is a CB α-quasi-ideal of S. Similarly, we can show that υ∩η is a CS α-quasi-ideal of S. Hence, CǔD is a CB α-quasi-ideal of S. � Theorem 3.7. Every CB (α,β)-quasi-ideal of Γ-semigroup S is intersection of a CB left α-ideal and a CB right β-ideal of S. Proof. Let C = 〈ω̌,υ〉 be a CB (α,β)-quasi-ideal of S. Consider ρ̌ = ω̌ t (ω̌λS ◦α ω̌) and τ = υ ∪ (υλS ◦α υ) where D = 〈ρ̌,τ〉, $̌ = ω̌ t (ω̌ ◦β ω̌λS ) and ν = υ ∪ (υ ◦β υλS ) where K = 〈$̌,ν〉. Then ω̌λS ◦α ρ̌ = ω̌λS ◦α (ω̌ t (ω̌λS ◦α ω̌)) = (ω̌λS ◦α ω̌) t (ω̌λS ◦α (ω̌λS ◦α ω̌)) = (ω̌λS ◦α ω̌) t ((≥S ◦α ω̌λS ) ◦α ω̌) = (ω̌λS ◦α ω̌) t (≥S ◦α ω̌) - ω̌ t (ω̌λS ◦α ω̌) = ρ̌. And $̌ ◦β ω̌λS = (ω̌ t (ω̌ ◦β ω̌λS )) ◦α ω̌λS = (ω̌ ◦α ω̌λS ) t (ω̌ ◦β ω̌λS ◦α ω̌λS ) = (ω̌ ◦α ω̌λS ) t ω̌ ◦β (ω̌λS ◦α ω̌λS ) = (ω̌ ◦α ω̌λS ) t (ω̌ ◦β ω̌λS ) - ω̌ t (ω̌ ◦β ω̌λS ) = $̌. Simlarly, we can show that υλS ◦α τ ≥ τ and ν ◦β υλS ≥ ν. Thus D = 〈ρ̌,τ〉 and K = 〈$̌,ν〉 is a CB left α-ideal and a CB right β-ideal of S. Now, ω̌ v (ω̌ t (ω̌λS ◦α ω̌)) u (ω̌ t (ω̌ ◦β ω̌λS )) = ρ̌u $̌ and ρ̌∩ $̌ = (ω̌ t (ω̌λS ◦α ω̌)) u (ω̌ t (ω̌ ◦β ω̌λS )) = ω̌ ∩ ((≥S ◦α ω̌) t (ω̌ ◦β ω̌λS )) - ω̌ u ω̌ = ω̌. Hence, ω̌ = ρ̌u $̌. Simlarly, we can show that υ = τ ∩ν. � Theorem 3.8. Let K be a nonempty subset of Γ-semigroup S. Then K is a (α,β)-quasi-ideal of S if and only if characteristic function ≥K = (ω̌λK,υλK) is a CB (α,β)-quasi-ideal of S. Proof. Suppose that K is a (α,β)-quasi-ideal of S and e ∈S. If f ∈ (SαK) ∩ (KβS), then e ∈K. Thus ω̌λK(e) = 1̌ and υλK(e) = 0. Hence ((ω̌λK ◦α ω̌λS ) f (ω̌λS ◦β ω̌λK))(f ) - ω̌λK(u) and ((υλK ◦α υλS ) ∨ (υλS ◦β υλK))(u) ≥ υλK(f ). If f /∈ (SαK) ∩ (KβS), then e ∈K. Thus ω̌λK(e) = 0̌ and υλK(e) = 1. Hence, ω̌λK))(f ) - ω̌λK(u) and ((υλK ◦α υλS ) ∨ (υλS ◦β υλK))(u) ≥ υλK(f ). Therefore ≥K = (ω̌λK,υλK) is a CB (α,β)-quasi-ideal of S. Int. J. Anal. Appl. (2022), 20:69 9 Conversely, assume that ≥K = (ω̌λK,υλK) is a CB (α,β)-quasi-ideal of S and f ∈ (SαK) ∩ (KβS). Then ((ω̌λK ◦α ω̌λS ) f (ω̌λS ◦β ω̌λK))(f ) = 1̌ and ((υλK ◦α υλS ) ∨ (υλS ◦β υλK))(f ) = 0. By assumption, ((ω̌λK ◦α ω̌λS ) f (ω̌λS ◦β ω̌λK))(f ) - ω̌λK(f ) and ((υλK ◦α υλS ) ∨ (υλS ◦β υλK))(f ) ≥ υλK(f ) Thus e ∈K. Hence, K is a (α,β)-quasi-ideal of S. � 4. New Types of Cubic Almost Ideals Definition 4.1. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ is said to be (1) a CB almost left α-ideal of S if (ω̌n ◦α ω̌) u ω̌ 6= 0̌ and (υm ◦α υ) ∪υ 6= 1. (2) a CB almost right β-ideal of S if (ω̌ ◦β ω̌n) u ω̌ 6= 0̌ and (υ ◦β υm) ∪υ 6= 1. (3) a CB almost (α,β)-ideal of S if it is both a CB almost left α-ideal and a CB almost right β-ideal of S. Theorem 4.1. If C = 〈ω̌,υ〉 is a CB almost left α-ideal (right β-ideal, (α,β)-ideal) of a Γ-semigroup S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D, then D = 〈ρ̌,τ〉 is a CB left almost α-ideal (right β-ideal, (α,β)-ideal) of S. Proof. Suppose that C = 〈ω̌,υ〉 is a CB almost left α-ideal of S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D. Then (ω̌n ◦α ω̌)u ω̌ 6= 0̌ and (υm ◦α υ)∪υ 6= 1. Thus (ω̌n ◦α ω̌)u ω̌ - (ρ̌n ◦α ρ̌)u ρ̌ 6= 0̌ and (υm ◦α υ) ∪υ ≥ (τm ◦α τ) ∪τ 6= 0. Hence, D = 〈ρ̌,τ〉 is a CB left almost α-ideal of S. � Theorem 4.2. Let K be a nonempty 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 characteristic function ≥K = (ω̌λK,υλK) is a CB 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 eαK∩K 6= ∅ for all e ∈ S. Thus there exists r ∈ eαK and r ∈K. So (ω̌n ◦α ω̌λK)(r) = ω̌λK(r) = 1̌ and (υm ◦α υλK)(r) = υλK(r) = 0. Hence, (ω̌n ◦α ω̌λK) u ω̌λK 6= 0̌ and (υm ◦α υλK) ∪υλK 6= 1. Therefore, ≥K = (ω̌λK,υλK) is a CB almost left α-ideal of S. Conversely, assume that ≥K = (ω̌λK,υλK) is a CB almost left α-ideal of S and e ∈ S. Then (ω̌n ◦α ω̌λK)u ω̌λK 6= 0̌ and (υm ◦α υλK)∪υλK 6= 1. Thus there exists r ∈S such that ((ω̌n ◦α ω̌λK)f ω̌λK)(r) 6= 0̌ and ((υm ◦α υλK) ∨υλK)(r) 6= 1. Hence, r ∈ eαK∩K implies eαK∩K 6= ∅. Therefore K is an almost left α-ideal of S. � Next, we review definition of supp(C) and we study properties between supp(ξ) and CB almost left α-ideal (right β-ideal, (α,β)-ideal) of Γ-semigroups. Let C = 〈ω̌,υ〉 be a CB set of a non-empty of S. Then the support of C instead of supp(C) = {e ∈S | ω̌(e) 6= 0̌ andυ(e) 6= 0}. 10 Int. J. Anal. Appl. (2022), 20:69 Theorem 4.3. Let C = 〈ω̌,υ〉 be a CB set of a non-empty of a Γ-semigroup S. Then C = 〈ω̌,υ〉 is a CB almost left α-ideal (right β-ideal, (α,β)-ideal) of S if and only if supp(C) is an almost left α-ideal (right β-ideal, (α,β)-ideal) of S. Proof. Let C = 〈ω̌,υ〉 be a CB almost left α-ideal of S and e ∈S. Then (ω̌n ◦α ω̌) u ω̌ 6= 0̌and (υm ◦α υλK) ∪ υλK 6= 1. Thus there exists r ∈ S such that ((ω̌n ◦α ω̌λK) f ω̌λK)(r) 6= 0̌ and ((υm ◦α υλK) ∨υλK)(r) 6= 1. So there exists k ∈ S such that r = uαk, ω̌(r) 6= 0̌, υ(r) 6= 0 and ω̌(k) 6= 0̌, υ(k) 6= 0. It implies that r,k ∈ supp(C). Thus, (ω̌n ◦α ω̌λsupp(C) )(r) 6= 0̌, (υm ◦α υλsupp(C) )(r) 6= 1 and ω̌λsupp(C) 6= 0̌, υλsupp(C) 6= 1. Hence, (ω̌m ◦α ω̌λsupp(C) ) u ω̌λsupp(C) 6= 0̌ and (υm ◦α υλsupp(C) ) ∪υλsupp(C) 6= 1. Therefore, supp(C) is a CB almost left α-ideal of S. This show that supp(C) is an almost left α-ideal of S. Conversely, let supp(C) be an almost left α-ideal of S. Then by Theorem 4.2, ≥supp(C) is a CB almost left α-ideal of S. Thus (ω̌m ◦α ω̌λsupp(C) ) u ω̌λsupp(C) 6= 0̌ and (υm ◦α υλsupp(C) ) ∪υλsupp(C) 6= 1. So there exists r ∈S such that ((ω̌m ◦α ω̌λsupp(C) ) f ω̌λsupp(C) )(r) 6= 0̌ and (υm ◦α υλsupp(C) ) ∨υλsupp(C) (r) 6= 1. It implies that ((ω̌n ◦α ω̌λsupp(C) ))(r) 6= 0̌, ((υm ◦α υλsupp(C) ))(r) 6= 1 and ω̌λsupp(C) (r) 6= 0̌, υλsupp(C) (r) 6= 1. Thus there exists k ∈ S such that r = uαk, ω̌n(r) 6= 0̌, υn(r) 6= 0 and ω̌n(k) 6= 0̌, υm(k) 6= 0. Hence, (ω̌n ◦α ω̌) u ω̌ 6= 0 and (υm ◦α υ) ∪υ 6= 1. Therefore, C = 〈ω̌,υ〉 be a CB almost left α-ideal of S. � Definition 4.2. An almost ideal I of a Γ-semigroup S is called a minimal if for every almost ideal of J of S such that J ⊆I, we have J = I. Definition 4.3. A CB almost left α-ideal (right β-ideal, (α,β)-ideal) C = 〈ω̌,υ〉 of a Γ-semigroup S is minimal if for all CB almost left α-ideal (right β-ideal, (α,β)-ideal) D = 〈ρ̌,τ〉 of S such that D v C, then supp(D) = supp(C). Theorem 4.4. Let K be a nonempty 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 ≥K = (ω̌λK,υλK) is a minimal CB 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 4.2, ≥K = (ω̌λK,υλK) is a CB left α-ideal of S. Let D = 〈ρ̌,τ〉 be a CB left α-ideal of S such that D v C Then by Theorem 4.3, supp(D) is an almost left α-ideal of S. Thus supp(D) v supp(≥K) = K. By assumption, supp(D) = K = supp(≥K). Thus, ≥K = (ω̌λK,υλK) is a minimal CB almost left α-ideal of S. Conversely, suppose that ≥K = (ω̌λK,υλK) is a minimal CB almost left α-ideal of S. Then by Theorem 4.2, 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 4.2, ≥J = (ω̌λJ ,υλJ ) is a CS left α-ideal of S such that ≥J v ≥K. Thus, J = supp(≥J ) = supp(≥K) = K. Hence, K is a minimal almost left α-ideal of S. � Int. J. Anal. Appl. (2022), 20:69 11 Corollary 4.1. Let S be a Γ-semigroup Then S has no proper almost left α-ideal (right β-ideal, (α,β)-ideal) if and only if for any CB almost left α-ideal (right β-ideal, (α,β)-ideal) C = 〈ω̌,υ〉 of S, supp(C) = S. Next, we define CB almost (α,β)-quasi-ideals and we study properties of it. Definition 4.4. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ is said to be CB almost (α,β)-quasi-ideal of S if (ω̌ ◦α ω̌n) u (ω̌n ◦β ω̌) 6= 0̌ and (υ ◦α υm) ∨ (υm ◦β υ) 6= 1. Theorem 4.5. If C = 〈ω̌,υ〉 is a CB almost (α,β)-quasi-ideal of a Γ-semigroup S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D, then D = 〈ρ̌,τ〉 is a CB (α,β)-quasi-ideal of S. Proof. Suppose that C = 〈ω̌,υ〉 is a CB almost (α,β)-quasi-ideal of a Γ-semigroup S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D. Then (ω̌◦αω̌n)u(ω̌n◦βω̌) 6= 0̌ and (υ◦αυm)∪(υm◦βυ) 6= 1. Thus, (ω̌◦αω̌n)u(ω̌n◦β ω̌) - (ρ̌◦α ρ̌n)u(ρ̌n◦β ρ̌) 6= 0̌ and (υ◦αυm)∪(υm◦β υ) ≥ (τ◦ατm)∪(τm◦β τ) 6= 1. Hence, D = 〈ρ̌,τ〉 is a CB (α,β)-quasi-ideal of S. � Theorem 4.6. Let K be a nonempty subset of Γ-semigroup S. Then K is an almost (α,β)-quasi-ideal of S if and only if characteristic function ≥K = (ω̌λK,υλK) is a CB almost (α,β)-quasi-ideal of S. Proof. Suppose that K is an almost (α,β)-quasi-ideal of S. Then (Kαe) ∩ (eβK) ∩K 6= ∅ for all e ∈S. Thus there exists v ∈ (Kαe) ∩ (eβK) and v ∈K. So (ω̌n ◦α ω̌λK) f (ω̌λK ◦β ω̌n)(v) 6= 0̌ and (υm ◦α υλK) ∨ (υλK ◦β υm)(v) 6= 1. Hence, (ω̌n◦αω̌λK)u(ω̌λK◦β ω̌n) 6= 0̌ and (υm◦αυλK)∪(υλK◦βυm) 6= 1. Therefore, ≥K = (ω̌λK,υλK) is a CB almost (α,β)-quasi-ideal of S. Conversely, assume that ≥K = (ω̌λK,υλK) is a CB almost (α,β)-quasi-ideal of S and e ∈S. Then (ω̌n ◦α ω̌λK) u (ω̌λK ◦β ω̌n) 6= 0̌ and (υm ◦α υλK) ∪ (υλK ◦β υm) 6= 1. Thus there exists r ∈S such that (ω̌n◦αω̌λK)u(ω̌λK◦βω̌n)(r) 6= 0̌ and (υm◦αυλK)∪(υλK◦βυm)(r) 6= 1. Hence, r ∈ (Kαe) ∩ (eβK) ∩K implies (Kαe) ∩ (eβK) ∩K 6= ∅. Therefore, K is an almost (α,β)- quasi-ideal of S. � Next, we study properties between supp(C) and CB almost (α,β)-quasi-ideal of Γ-semigroups. Theorem 4.7. Let C = 〈ω̌,υ〉 is a CB sets of a non-empty of a Γ-semigroup S. Then C = 〈ω̌,υ〉 is a CB almost (α,β)-quasi-ideal of S if and only if supp(C) is an almost (α,β)-quasi-ideal of S. Proof. Let C = 〈ω̌,υ〉 be a CB almost (α,β)-quasi-ideal of S and e ∈S. Then (ω̌n ◦α ω̌) u (ω̌ ◦β ω̌n) 6= 0̌ and (υm ◦α υ) ∪ (υ ◦β υm) 6= 1. Thus there exists r ∈ S such that (ω̌n ◦α ω̌) f (ω̌ ◦β ω̌n)(r) 6= 0̌ and (υm ◦α υ) ∨ (υ ◦β υm)(r) 6= 1. So there exists k1,k2 ∈ S such that r = k1αu = uβk2, ω̌(r) 6= 0̌, υ(r) 6= 0 and ω̌(k1) 6= 0̌, υ(k1) 6= 0. It implies that r,k1,k2 ∈ supp(C). Thus ((ω̌λsupp(C) ◦α ω̌m) f (ω̌n ◦β ω̌λsupp(C) ))(r) 6= 0 and ω̌λsupp(C) 6= 0̌. Similalry ((υλsupp(C) ◦α υm) ∨ (υm ◦β υλsupp(C) ))(r) 6= 1 and 12 Int. J. Anal. Appl. (2022), 20:69 υλsupp(C) 6= 1. Hence, (ω̌λsupp(C) ◦α ω̌m) u (ω̌n ◦β ω̌λsupp(C) ) u ω̌λsupp(C) 6= 0̌ and (υλsupp(C) ◦α υm) t (υm ◦β υλsupp(C) ) 6= 1. Therefore, ≥supp(C) is a CB almost (α,β)-quasi-ideal of S. This show that supp(C) is an almost (α,β)-quasi-ideal of S. Conversely, let supp(C) be an almost (α,β)-quasi-ideal of S. Then by Theorem 4.6, supp(C) is a CB (α,β)-quasi-ideal of S. Thus (ω̌λsupp(C)◦αω̌n)u(ω̌n◦βω̌λsupp(C) )uω̌λsupp(C) 6= 0̌ and (υλsupp(C)◦αυm)∪ (υm◦βυλsupp(C) )∪υλsupp(C) 6= 1. So there exists r ∈S such that ((ω̌n◦αω̌λsupp(C) )f(ω̌λsupp(C)◦βω̌n))(r) 6= 0̌ and ((υm◦αυλsupp(C) )∨(υλsupp(C)◦βυm))(r) 6= 1. It implies that (ω̌n◦αω̌λsupp(C) )f(ω̌λsupp(C) )◦βω̌n)(r) 6= 0̌ and (υm◦αυλsupp(C) )∨(υλsupp(C) ◦βυm)(r) 6= 1. Thus there exist k1,k2 ∈S such that r = k1αe = eβk2, ω̌n(r) 6= 0̌, υm(r) 6= 1 and ω̌n(k) 6= 0̌, υm(k) 6= 1. Hence, (ω̌n ◦α ω̌) u (ω̌ ◦β ω̌n) 6= 0̌ and (υm ◦α υ) ∪ (υ ◦β υm) 6= 1. Therefore, C = 〈ω̌,υ〉 be a CB almost (α,β)-quasi-ideal of S. � Definition 4.5. A CB almost (α,β)-quasi-ideal C = 〈ω̌,υ〉 of a Γ-semigroup S is minimal if for all CS almost (α,β)-quasi-ideal D = 〈ρ̌,τ〉 of S such that D v C, then supp(D) = supp(C). Theorem 4.8. Let K be a nonempty subset of a Γ-semigroup S Then K is a minimal almost (α,β)- quasi-ideal if and only if ≥K = (ω̌λK,υλK) is a minimal CB almost (α,β)-quasi-ideal of S. Proof. Suppose that K is a minimal almost (α,β)-quasi-ideal of S. Then K is an almost (α,β)-quasi- ideal of S. Thus by Theorem 4.6, ≥K = (ω̌λK,υλK) is a CS (α,β)-quasi-ideal of S. Let D = 〈ρ̌,τ〉 be a CB (α,β)-quasi-ideal of S such that D v ≥K. Then by Theorem 4.7, supp(D) is an almost (α,β)-quasi-ideal of S. Thus supp(D) v supp(≥K) = K. By assumption, supp(D) = K = supp(≥K). Thus, ≥K = (ω̌λK,υλK) is a minimal CB almost (α,β)-quasi-ideal of S. Conversely, suppose that ≥K = (ω̌λK,υλK) is a minimal CB almost (α,β)-quasi-ideal of S. Then by Theorem 4.7, 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 4.7, ≥J = (ω̌λJ ,υλJ ) is a CB (α,β)-quasi-ideal of S such that ≥J ⊆≥K. Thus, J = supp(≥J ) = supp(≥K) = K. Hence, K is a minimal almost (α,β)-quasi-ideal of S. � Corollary 4.2. Let S be a Γ-semigroup Then S has no proper almost (α,β)-quasi-ideal if and only if for any CB almost (α,β)-quasi-ideal C = 〈ω̌,υ〉 of S, supp(C) = S. Next, we define CB almost (α,β)-bi-ideals and we study properties of it. Definition 4.6. Let C = 〈ω̌,υ〉 be a CB set of a Γ-semigroup S and α,β ∈ Γ is said to be CB almost (α,β)-bi-ideal of S if (ω̌ ◦α ω̌n ◦β ω̌) u ω̌ 6= 0̌ and (υ ◦α υm ◦β υ) ∪υ 6= 0. Theorem 4.9. If C = 〈ω̌,υ〉 is a CB almost (α,β)-bi-ideal of a Γ-semigroup S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D, then D = 〈ρ̌,τ〉 is a CB (α,β)-bi-ideal of S. Proof. Suppose that C = 〈ω̌,υ〉 is a CB almost (α,β)-bi-ideal of a Γ-semigroup S and D = 〈ρ̌,τ〉 is a CB set of S such that C v D. Then (ω̌ ◦α ω̌n ◦β ω̌) f ω̌ 6= 0̌ and (υ ◦α υm ◦β υ) ∨υ 6= 0. Thus, Int. J. Anal. Appl. (2022), 20:69 13 (ω̌◦α ω̌n ◦β ω̌)f ω̌ - (ρ̌◦α ρ̌n ◦β ρ̌)f ω̌ 6= 0̌ and (υ◦α υm ◦β υ)∨υ ≥ (τ ◦α τm ◦β τ)∨τ 6= 0. Hence, D = 〈ρ̌,τ〉 is a CB (α,β)-bi-ideal of S. � Theorem 4.10. Let K be a nonempty subset of Γ-semigroup S. Then K is an almost (α,β)-bi-ideal of S if and only if characteristic function ≥K = (ω̌λK,υλK) is a CB almost (α,β)-bi-ideal of S. Proof. Suppose that K is an almost (α,β)-bi-ideal of S. Then KαeβK∩K 6= ∅. for all e ∈S. Thus there exists f ∈ KαeβK and f ∈ K. So (ω̌λK ◦α ω̌n ◦β ω̌λK)(f ) = ω̌λK(f ) = 1̌ and (υλK ◦α υm ◦β υλK)(f ) = υλK(f ) = 0. Hence, (ω̌λK ◦α ω̌n ◦β ω̌λK) u ω̌λK 6= 0̌ and (υλK ◦α υm ◦β υλK) ∨υλK 6= 1. Therefore, ≥K = (ω̌λK,υλK) is a CB almost (α,β)-bi-ideal of S. Conversely, assume that ≥K = (ω̌λK,υλK) is a CB almost (α,β)-bi-ideal of S and u ∈ S. Then (ω̌λK ◦α ω̌n ◦β ω̌λK) u ω̌λK 6= 0 and (υλK ◦α υm ◦β υλK) ∪υλK 6= 1. Thus there exists e ∈S such that ((ω̌λK ◦α ω̌n ◦β ω̌λK) f ω̌λK)(e) 6= 0 and ((υλK ◦α υm◦β υλK)∨υλK)(e) 6= 1. Hence, r ∈KαeβK∩K implies that KαeβK∩K 6= ∅. Therefore, K is an almost (α,β)-bi-ideal of S. � Next, we study properties between supp(C) and CB almost (α,β)-bi-ideal of Γ-semigroups. Theorem 4.11. Let C = 〈ω̌,υ〉 be a fuzzy sets of a non-empty of a Γ-semigroup S. Then C = 〈ω̌,υ〉 is a CB almost (α,β)-bi-ideal of S if and only if supp(C) is an almost (α,β)-bi-ideal of S. Proof. Let C = 〈ω̌,υ〉 is a CB almost (α,β)-bi-ideal of S and u ∈S. Then (ω̌◦α ω̌m ◦β ω̌) u ω̌ 6= 0̌ and (υ◦α υn ◦β υ) ∪υ 6= 1. Thus there exists r ∈S such that ((ω̌◦α ω̌m ◦β ω̌) f ω̌)(r) 6= 0 and ((υ ◦α υn ◦β υ) ∨υ)(r) 6= 0. So there exists k1,k2 ∈ S such that r = k1αβk2, ω̌(r) 6= 0̌, υ(r) 6= 1. It implies that r,k1,k2 ∈ supp(C). Thus (ω̌λsupp(C) ◦α ω̌m ◦β ω̌λsupp(C) )(r) 6= 0 and ≥̌supp(C) 6= 0̌. Similarly (υλsupp(C) ◦α υm ◦β υλsupp(C) )(r) 6= 0 and υλsupp(C) 6= 0. Hence, (ω̌λsupp(C)◦αω̌m◦βω̌λsupp(C)uω̌λsupp(C) 6= 0̌ and (υλsupp(C)◦αυm◦βυλsupp(C) )∨υλsupp(C) ) 6= 0. Therefore ≥supp(C) is a CB almost (α,β)-bi-ideal of S. This show that supp(C) is an almost (α,β)-bi-ideal of S. Conversely, let supp(C) is an almost (α,β)-bi-ideal of S. Then by Theorem 4.10, ≥supp(C) is a CB almost (α,β)-bi-ideal of S. Thus (ω̌λsupp(C) ◦α ω̌m ◦β ω̌λsupp(C) ) u ω̌λsupp(C) 6= 0̌ and (υλsupp(C) ◦α υn ◦β υλsupp(C) )∪υλsupp(C) 6= 0. So there exists r ∈S such that ((ω̌λsupp(C) ◦α ω̌m◦β ω̌λsupp(C) )fω̌λsupp(C) )(r) 6= 0̌ and ((υλsupp(C) ◦α υn ◦β υλsupp(C) ) ∨υλsupp(C) )(r) 6= 0. It implies that (ω̌λsupp(C) ◦α ω̌m ◦β ω̌λsupp(C) )(r) 6= 0̌ and ω̌λsupp(C) (r) 6= 0̌. Similarly (υλsupp(C)◦αυn◦βυλsupp(C) )(r) 6= 0 and ≥ n supp(r) 6= 0. Thus there exist k1,k2 ∈S such that r = k1αuβk2, ω̌(r) 6= 0, υ(r) 6= 0 and ω̌(k) 6= 0, υ(k) 6= 0. Hence, (ω̌◦αω̌m◦βω̌)uω̌ 6= 0̌ and (υ◦αυn◦βυ)∪υ 6= 0. Therefore, C = 〈ω̌,υ〉 is a CB almost (α,β)-bi-ideal of S. � Definition 4.7. A CB almost (α,β)-bi-ideal C = 〈ω̌,υ〉 of a Γ-semigroup S is minimal if for all CB almost (α,β)-bi-ideal D = 〈ρ̌,τ〉 of S such that D v C, then supp(D) = supp(C). 14 Int. J. Anal. Appl. (2022), 20:69 Theorem 4.12. Let K be a nonempty subset of a Γ-semigroup S Then K is a minimal almost (α,β)- bi-ideal if and only if ≥K = (ω̌λK,υλK) is a minimal CB 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)-bi-ideal of S. Thus by Theorem 4.10, ≥K = (ω̌λK,υλK) is a CB (α,β)-bi-ideal of S. Let D = 〈ρ̌,τ〉 be a CB (α,β)-bi-ideal of S such that D v ≥K. Then by Theorem 4.11, supp(D) is an almost (α,β)- bi-ideal of S. Thus, supp(D) v supp(≥K) = K. By assumption, supp(D) = K = supp(≥K). Thus, ≥K = (ω̌λK,υλK) is a minimal CB almost (α,β)-bi-ideal of S. Conversely, suppose that ≥K = (ω̌λK,υλK) is a minimal CB almost (α,β)-bi-ideal of S. Then by Theorem 4.10, 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 4.10, ≥J = (ω̌λJ ,υλJ ) is a CB (α,β)-bi-ideal of S such that ≥J ⊆≥K. Thus, J = supp(≥J ) = supp(≥K) = K. Hence, K is a minimal almost (α,β)-bi-ideal of S. � Corollary 4.3. Let S be a Γ-semigroup Then S has no proper almost (α,β)-bi-ideal if and only if for any CB almost (α,β)-bi-ideal C = 〈ω̌,υ〉 of S, supp(C) = S. 5. Conclusion In this article, we give the concept of a new cubic ideals and cubic almost ideals in a Γ-semigroups. We study properites of new cubic ideals and cubic almost ideals. We hope that the study of this topic are useful mathematical tools. In the future we study a new hesitant fuzzy ideal and hesitant fuzzy almost ideals in semigroups or algebric system. Acknowledgements: This research project was supported by the thailand science research and inno- vation 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. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] S. Bashir, A. Sarwar, Characterizations of Γ-Semigroups by the Properties of Their Interval Valued T-Fuzzy Ideals, Ann. Fuzzy Math. Inform. 9 (2015), 441-461. [2] V. Chinnadaurai, K. Bharathivelan, Cubic Ideal of Γ-Semigroups, Int. J. Current Res. Modern Educ. 1 (2016), 138-150. [3] R. Chinram, On Quasi-Gamma-Ideals in Gamma-Semigroups, ScienceAsia. 32 (2006), 351-353. https://doi.org/ 10.2306/scienceasia1513-1874.2006.32.351. [4] N. Deetae, P. Khamrod, Q-Cubic Bi-Quasi Ideals of Semigroups, Glob. J. Pure Appl. Math. 16 (2020), 553-566. [5] T. Gaketem, Cubic (1, 2)-Ideals in Semigroups, TWMS J. Appl. Eng. Math. 12 (2022), 1271-1282. [6] T. Gaketem, Cubic Interior Ideals in Semigroups, Azerbaijan J. Math. 10 (2020), 85-104. [7] T. Gaketem, A. Iampan, Cubic Filters of Semigroups, Appl. Sci. 24 (2022), 131-141. https://doi.org/10.2306/scienceasia1513-1874.2006.32.351 https://doi.org/10.2306/scienceasia1513-1874.2006.32.351 Int. J. Anal. Appl. (2022), 20:69 15 [8] A. Hadi, A. Khan, Cubic Generalized Bi-Ideal in Semigroups, Discuss. Math., Gen. Algebra Appl. 36 (2016), 131- 146. https://doi.org/10.7151/dmgaa.1254. [9] A. Iampan, Note on Bi-Ideal in Γ-Semigroups, Int. J. Algebra. 34 (2009), 181-188. [10] Y.B. Jun, C.S. Kim, K.O. Yang, Cubic Sets, Ann. Fuzzy Math. Inform. 4 (2012), 83-98. [11] Y.B. Junand, A. Khan, Cubic Ideals in Semigroups, Honam Math. J. 35 (2013), 607-623. https://doi.org/10. 5831/HMJ.2013.35.4.607. [12] N. Kuroki, Fuzzy Bi-Ideals in Semigroup, Comment. Math. Univ. St. Paul. 28 (1980), 17–21. https://doi.org/ 10.14992/00010265. [13] P. Khamrot, T. Gaketem, Cubic bi-quasi ideals of semigroups, J. Discrete Math. Sci. Cryptography. 24 (2021), 1113-1126. https://doi.org/10.1080/09720529.2021.1889777. [14] P. Kummoon, T. Changphas, Bi-Bases of Γ-Semigroups, Thai J. Math. Special Issue (2017), 75-86. [15] D. Mandal, Characterizations of Γ-Semirings by Their Cubic Ideals, Int. J. Math. Comput. Sci. 13 (2019), 150-156. [16] Al. Narayanan, T. Manikantan, Interval-Valued Fuzzy Ideals Generated by an Interval-Valued Fuzzy Subset in Semigroups, J. Appl. Math. Comput. 20 (2006), 455–464. https://doi.org/10.1007/bf02831952. [17] A. Simuen, A. Iampan, R. Chinram, A Novel of Ideals and Fuzzy Ideals of Γ-Semigroups, J. Math. 2021 (2021), 6638299. https://doi.org/10.1155/2021/6638299. [18] L.A. Zadeh, Fuzzy Sets, Inform. Control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [19] L.A. Zadeh, The Concept of a Linguistic Variable and Its Application to Approximate Reasoning–I, Inform. Sci. 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90036-5. https://doi.org/10.7151/dmgaa.1254 https://doi.org/10.5831/HMJ.2013.35.4.607 https://doi.org/10.5831/HMJ.2013.35.4.607 https://doi.org/10.14992/00010265 https://doi.org/10.14992/00010265 https://doi.org/10.1080/09720529.2021.1889777 https://doi.org/10.1007/bf02831952 https://doi.org/10.1155/2021/6638299 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/0020-0255(75)90036-5 1. Introduction 2. Preliminaries 3. New Types of Cubic Ideals 4. New Types of Cubic Almost Ideals 5. Conclusion References