Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 73 This work is licensed under a Creative Commons Attribution 4.0 International License. The Homomorphism of Cubic bipolar ideals of a KU-semigroup Abstract The idea of a homomorphism of a cubic set of a KU-semigroup is studied and the concept of the product between two cubic sets is defined. And then, a new cubic bipolar fuzzy set in this structure is discussed, and some important results are achieved. Also, the product of cubic subsets is discussed and some theorems are proved. 2010 AMS Classification: 06F35, 03G25, 08A72 . Key words: A KU-semigroup, a cubic sub KU-semigroup, a cubic bipolar k-ideal, homomorphism. 1.Introduction In 2009, Prabpayak and Leerawat [1,2] studied a new algebra called a KU-algebra. They introduced homomorphism in a KU-algebra and discussed some recent results. After that, Mostafa et al. [3, 4] studied the concepts of fuzzy KU-ideals and an interval value fuzzy KU- ideals. In [5] Kareem and Hasan presented the structure of a KU-semigroup and introduced some ideals of this structure. After that, they introduced the fuzzy ideals of this structure in [6]. In [7] Kareem and Talib gave the concept of an interval value fuzzy some ideal in KU- semigroup. Jun et al. [8, 9] introduced the concept of cubic subalgebras/ideals in BCK/BCI-algebras. Yaqoob et al [10] presented a cubic KU-algebra and discussed a few interesting theorems. This work studied the idea of a homomorphism of a cubic set of a KU-semigroup, and a new cubic bipolar fuzzy set in this structure is defined, and some important results are achieved. Also, the product of cubic subsets was discussed, and a few theorems were proved. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/35.1.2801 Article history: Received 22, June 2021, Accepted 4,October 2021, Published in January 2022. Wisam K. Awad aa.ww21@yahoo.com Department of Mathematics, Ibn- Al-Haitham College of Education, University of Baghdad. Fatema F. Kareem fa_sa20072000@yahoo.com Department of Mathematics, Ibn-Al- Haitham College of Education, University of Baghdad. https://creativecommons.org/licenses/by/4.0/ mailto:aa.ww21@yahoo.com mailto:fa_sa20072000@yahoo.com Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 74 2. Basic concepts We introduce some definitions, propositions and theorems of KU-algebra and KU- semigroup in this part. Definition2.1[1]. A KU-algebra (ℵ,∗ ,0) is satisfied the following conditions, for allα, β, δ ∈ ℵ, (ku1) (𝛼 ∗ 𝛽) ∗ [(𝛽 ∗ δ) ∗ (𝛼 ∗ δ)] = 0 (ku2)𝛼 ∗ 0 = 0 (ku3) 0∗ 𝛼 = 𝛼 (ku4)𝛼 ∗ 𝛽 = 0 and𝛽 ∗ 𝛼 implies 𝛼 = 𝛽 and (ku5)𝛼 ∗ 𝛼 = 0 The relation ≤ on a KU-algebra ℵ is define 𝛼 ≤ 𝛽 ⟺ 𝛽 ∗ 𝛼 = 0. Example 2.2 [1].The following table define the binary operation ∗ on the set ℵ = {0, 𝑎, 𝑏, 𝑐} Then (ℵ,∗ ,0) is a KU-algebra. Theorem2.3[2]. The following axioms are satisfying, in a KU-algebraℵ. For all α, β, δ ∈ ℵ, (1) if 𝛼 ≤ 𝛽 imply𝛽 ∗ δ ≤ 𝛼 ∗ δ (2)𝛼 ∗ (𝛽 ∗ δ) = 𝛽 ∗ (𝛼 ∗ δ) (3)((𝛽 ∗ 𝛼) ∗ 𝛼) ≤ 𝛽 Definition2.4[5]. The set ℵ ≠ 𝜑 and two binary operations ∗,∘ with a constant~0 is named a KU-semigroup if ~~ (I) The triple (ℵ,∗ ,0)isˑa KU-algebra (II) the ordered pair (ℵ,∘)is aˑsemigroup (III) α, β, δ ∈ ℵ,, α ∘ (β ∗ δ) = (α ∘ β) ∗ (α ∘ δ)and(α ∗ β) ∘ δ = (α ∘ δ) ∗ (β ∘ δ). Example 2.5[5]. If ℵ = {0,1,2,3} is ˑa set and two binary operations ∗ and ∘ are defined by the following. Thenˑ(ℵ,∗,∘ ,0) is aˑKU-semigroup. " Definition 2.6[5]. A non-empty subset A of ℵ is named AˑsubKU-semigroup if it is satisfied 𝛼 ∗ 𝛽, 𝛼 ∘ 𝛽 ∈ 𝛢, for all𝛼,𝛽 ∈ 𝛢. Definition2.7[5]. A non-empty subset 𝐼ℵ is called an S-ideal of ℵ, if (i) 0 ∈ 𝛪 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 75 (ii) 𝛼 ∗ 𝛽 ∈ 𝛪 and 𝛼 ∈ 𝛪 imply 𝛽 ∈ 𝛪.ˑˑ (iii)∀𝛼 ∈ ℵ, 𝒆 ∈ 𝐼, we have 𝛼 ∘ 𝒆 ∈ 𝐼 and 𝒆 ∘ 𝛼 ∈ 𝐼. Definition2.8[5]. The set 𝜑 ≠ 𝐴ℵ is named a k-ideal ofℵ, if~ i) 0 ∈ 𝛪 ii) ∀𝛼,𝛽, δ ∈ ℵ, (𝛼 ∗ (𝛽 ∗ δ)) ∈ 𝐼, 𝛽 ∈ 𝛪imply𝛼 ∗ δ ∈ 𝛪. iii) ∀𝛼 ∈ ℵ~,𝒆 ∈ Α,~we have~𝛼 ∘ 𝒆 ∈ Α and 𝒆 ∘ 𝛼 ∈ Α~. Definition2.9[5]. Letˑℵ and ℵ′be two KU-semigroups. A mapping ˑ𝑓: ℵ → ℵ′is called a KU- semigroup homomorphism if𝑓(𝛼 ∗ 𝛽) = 𝑓𝛼 ∗ 𝑓(𝛽)and𝑓(𝛼 𝛽) = 𝑓(𝛼) 𝑓(𝛽) for all𝛼, 𝛽 ∈ ℵ. The set {𝛼 ∈ ℵ: 𝑓(𝛼) = 0}is called the kernel of𝑓 and denoted by𝑘𝑒𝑟𝑓 Moreover, the set { 𝑓(𝛼) ∈ ℵ′ ∶ 𝛼 ∈ ℵ} is called the image of 𝑓 and denoted by 𝑖𝑚𝑓. We recall that a cubic bipolar valued fuzzy subset in [11] as follows: Definition 2.10[11]. Let ℵ be a non-empty set. A cubic bipolar set in a set ℵ is the structureΩ = {〈𝛼, 𝜇Ω +(𝛼), 𝜇Ω −(𝛼), 𝜆Ω + (𝛼), 𝜆Ω − (𝛼): 𝛼 ∈ ℵ〉} where 𝑁(𝛼) = {�̃�Ω +(𝛼), 𝜇Ω −(𝛼)} is calledinterval valued bipolar fuzzy set and 𝐾(𝛼) = {𝜆Ω + (𝛼), 𝜆Ω − (𝛼)} is a bipolar fuzzy set. Consider 𝜇Ω +: ℵ → 𝐷[0,1] such that 𝜇Ω +(𝛼) = [𝛿ΩL + (𝛼), 𝛿ΩU + (𝛼)] and 𝜇Ω −: ℵ → 𝐷[−1,0] such that 𝜇Ω −(𝛼) = [𝛿ΩL − (𝛼), 𝛿ΩU − (𝛼)] , also 𝜆Ω + : ℵ → [0,1], and 𝜆Ω − : ℵ → [−1,0]it follows that Ω = {< 𝛼, {[𝛿ΩL + (𝛼), 𝛿ΩU + (𝛼)], [𝛿ΩL − (𝛼), 𝛿ΩU − (𝛼)]}, 𝜆Ω + (𝛼), 𝜆Ω − (𝛼)} >: 𝛼 ∈ ℵ} It is a cubicbipolar set and can be denoted by Ω = 〈𝑁, 𝐾〉. Definition2.11[11]. A Cubic bipolar set Ω = 〈𝑁, 𝐾〉 is named a cubic bipolar sub KU- semigroup if: ∀𝛼, 𝛽 ∈ ℵ, (1) 𝜇Ω +(𝛼 ∗ 𝛽) ≥ 𝑟𝑚𝑖𝑛 {𝜇Ω +(𝛼)ˑ, 𝜇Ω +(𝛽)}, 𝜇Ω −(𝛼 ∗ 𝛽) ≤ 𝑟𝑚𝑎𝑥 {𝜇Ω −(𝛼)ˑ, 𝜇Ω −(𝛽)} 𝜆Ω + (𝛼 ∗ 𝛽) ≥ 𝑚𝑖𝑛 {𝜆Ω + (𝛼)ˑ, 𝜆Ω + (𝛽)}, 𝜆Ω − (𝛼 ∗ 𝛽) ≤ 𝑚𝑎𝑥 {𝜆Ω − (𝛼)ˑ, 𝜆Ω − (𝛽)}, (2) 𝜇Ω +(𝛼 ∘ 𝛽) ≥ 𝑟𝑚𝑖𝑛 {𝜇Ω +(𝛼)ˑ, 𝜇Ω +(𝛽)}, 𝜇Ω −(𝛼 ∘ 𝛽) ≤ 𝑟𝑚𝑎𝑥 {𝜇Ω −(𝛼)ˑ, 𝜇Ω −(𝛽)} 𝜆Ω + (𝛼 ∘ 𝛽) ≥ 𝑚𝑖𝑛 {𝜆Ω + (𝛼)ˑ, 𝜆Ω + (𝛽)}, 𝜆Ω − (𝛼 ∘ 𝛽) ≤ 𝑚𝑎𝑥 {𝜆Ω − (𝛼)ˑ, 𝜆Ω − (𝛽)}, Example2.12[11]. If ℵ = {0,1,2,3} is ˑa set and two binary operations ∗ and ∘ are define by the following. Then(ℵ,∗,∘ ,0) ˑ is a KU-semigroup. DefineΩ = 〈𝑁, 𝐾〉 as follows 𝑁(𝛼) = { {[−0.2, −0.5], [0.1,0.9]} 𝑖𝑓 𝛼 = {0,1} {[−0.6, −0.2], [0.2,0.5]} 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , 𝜆Ω + (𝛼) = { 0.5 𝑖𝑓 𝛼 = {0,1} 0.3 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝜆Ω − (𝛼) = { −0.6 𝑖𝑓 𝛼 = {0,1} −0.3 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 then Ω = 〈𝑁, 𝐾〉is a cubic bipolar sub KU-semigroup ofℵˑ. Definition 2.13[11]. A cubic bipolar set Ω = 〈𝑁, 𝐾〉in ℵis called a cubic bipolar ideal of ℵif, ∀𝛼 , 𝛽 ∈ ℵ (BC1) 𝜇Ω +(0) ≥ 𝜇Ω +(𝛼), 𝜆Ω + (0) ≥ 𝜆Ω + (𝛼) and 𝜇Ω −(0) ≤ 𝜇Ω −(𝛼), 𝜆Ω − (0) ≤ 𝜆Ω − (𝛼)   Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 76 (BC2) 𝜇Ω +(𝛽) ≥ 𝑟𝑚𝑖𝑛{𝜇Ω +(𝛼 ∗ 𝛽), 𝜇Ω +(𝛼)} , 𝜇Ω −(𝛽) ≤ 𝑟𝑚𝑎𝑥{𝜇Ω −(𝛼 ∗ 𝛽), 𝜇Ω −(𝛼)}and 𝜆Ω + (𝛽) ≥ 𝑚𝑖𝑛 {𝜆Ω + (𝛼 ∗ 𝛽)ˑ, 𝜆Ω + (𝛼)}, 𝜆Ω − (𝛽) ≤ 𝑚𝑎𝑥 {𝜆Ω − (𝛼 ∗ 𝛽)ˑ, 𝜆Ω − (𝛼)}, (BC3)𝜇Ω +(𝛼 ∘ 𝛽) ≥ 𝑟𝑚𝑖𝑛 {𝜇Ω +(𝛼)ˑ, 𝜇Ω +(𝛽)}, 𝜇Ω −(𝛼 ∘ 𝛽) ≤ 𝑟𝑚𝑎𝑥 {�̃�Ω −(𝛼)ˑ, 𝜇Ω −(𝛽)} and 𝜆Ω + (𝛼 ∘ 𝛽) ≥ 𝑚𝑖𝑛 {𝜆Ω + (𝛼)ˑ, 𝜆Ω + (𝛽)}, 𝜆Ω − (𝛼 ∘ 𝛽) ≤ 𝑚𝑎𝑥 {𝜆Ω − (𝛼)ˑ, 𝜆Ω − (𝛽)}. Example 2.14[11]. If ℵ = {0,1,2} is ˑa set and two binary operations ∗ and ∘ are defined by the following. Then(ℵ,∗,∘ ,0) ˑ is a KU-semigroup. DefineΩ = 〈𝑁, 𝐾〉 as follows 𝑁(𝛼) = { {[−0.3, −0.1], [0.1,0.8]} 𝑖𝑓 𝛼 = 0 {[−0.7, −0.3], [0.4,0.6]} 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , 𝜆Ω + (𝛼) = { 0.9 𝑖𝑓 𝛼 = 0 0.4 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝜆Ω − (𝛼) = { −0.8 𝑖𝑓 𝛼 = 0 −0.3 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 We can easily prove thatΩ = 〈𝑁, 𝐾〉is a cubic bipolarideal ofℵˑ. Definition 2.15[11]. A cubicbipolar set Ω = 〈𝑁, 𝐾〉in ℵis called a cubic bipolark-ideal of ℵ if, ∀𝛼 , 𝛽, δ ∈ ℵ (a) 𝜇Ω +(0) ≥ 𝜇Ω +(𝛼), 𝜆Ω + (0) ≥ 𝜆Ω + (𝛼) and 𝜇Ω −(0) ≤ 𝜇Ω −(𝛼), 𝜆Ω − (0) ≤ 𝜆Ω − (𝛼). (b) 𝜇Ω +(𝛼 ∗ δ) ≥ 𝑟𝑚𝑖𝑛{𝜇Ω +((𝛼 ∗ (𝛽 ∗ δ)), 𝜇Ω +(𝛽)}, 𝜇Ω −(𝛼 ∗ δ) ≤ 𝑟𝑚𝑎𝑥{𝜇Ω −((𝛼 ∗ (𝛽 ∗ δ)), 𝜇Ω −(𝛽)}and 𝜆Ω + (𝛼 ∗ δ) ≥ 𝑚𝑖𝑛 {𝜆Ω + ((𝛼 ∗ (𝛽 ∗ δ))ˑ, 𝜆Ω + (𝛽)}, 𝜆Ω − (𝛼 ∗ δ) ≤ 𝑚𝑎𝑥 {𝜆Ω − ((𝛼 ∗ (𝛽 ∗ δ))ˑ, 𝜆Ω − (𝛽)}, (c)𝜇Ω +(𝛼 ∘ 𝛽) ≥ 𝑟𝑚𝑖𝑛 {𝜇Ω +(𝛼)ˑ, 𝜇Ω +(𝛽)}, 𝜇Ω −(𝛼 ∘ 𝛽) ≤ 𝑟𝑚𝑎𝑥 {�̃�Ω −(𝛼)ˑ, 𝜇Ω −(𝛽)} 𝜆Ω + (𝛼 ∘ 𝛽) ≥ 𝑚𝑖𝑛 {𝜆Ω + (𝛼)ˑ, 𝜆Ω + (𝛽)}, 𝜆Ω − (𝛼 ∘ 𝛽) ≤ 𝑚𝑎𝑥 {𝜆Ω − (𝛼)ˑ, 𝜆Ω − (𝛽)}. 3. A Cubic Bipolark-ideal underHomomorphism We study some definitions of homomorphism; the product of cubic bipolar k-ideals and a cubic bipolar ideal. Also some theorems are discussed. Definition 3.1. For any 𝛼 ∈ ℵ. We define a new cubicbipolar fuzzy set Ω𝑓 = (𝛼, 𝜇𝑓 +, 𝜇𝑓 −, 𝜆𝑓 −, 𝜆𝑓 +)in ℵ by 𝜇𝑓 −(𝛼) = 𝜇−(𝑓(𝛼)) and𝜇𝑓 +(𝛼) = 𝜇+(𝑓(𝛼)),𝜆𝑓 −(𝛼) = 𝜆 −(𝑓(𝛼)) and𝜆𝑓 +(𝛼) = 𝜆+(𝑓(𝛼)),where𝑓: ℵ → ℵ′ is a KU-semigroup homomorphism. For short Ω𝑓 = (𝛼, 𝜇𝑓 +, 𝜇𝑓 −, 𝜆𝑓 −, 𝜆𝑓 +) is written Ω𝑓 and a cubic bipolar is ACB. Example3.2. In Example2.14, we have ℵ′ = {0′, 𝑎, 𝑏} is a set and 𝑓: ℵ → ℵ′ is mapping such that 𝑓(𝜒) = 𝜒′ with two tables ∗ 0 1 2 0 0 1 2 1 0 0 1 2 0 1 0 ∘ 0 1 2 0 0 0 0 1 0 1 0 2 0 0 2 ∗ 0′ a b 0′ 0′ a b a 0′ 0′ b b 0′ b 0′ ∘ 0′ a b 0′ 0′ 0′ 0′ a 0′ a 0′ b 0′ 0′ a Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 77 Then 𝑓: ℵ → ℵ′ is a KU-semigroup homomorphism and 𝑁(𝜒′) = { {[−0.2, −0.1], [0.2,0.9]} 𝑖𝑓 𝜒 = 0 {[−0.8, −0.2], [0.3,0.5]} 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , 𝜆Ω + (𝑥) = { 0.6 𝑖𝑓 𝜒 = 0 0.2 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝜆Ω − (𝑥) = { −0.9 𝑖𝑓 𝜒 = 0 −0.4 𝑖𝑓 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 We have: 𝜇𝑓 −(0) = 𝜇−(𝑓(0)) = 𝜇−(0′) = [−0.2, −0.1] and 𝜇𝑓 −(1) = 𝜇−(𝑓(1)) = 𝜇−(𝑎) = [−0.8, −0.2] and so on. Theorem 3.3. Let𝑓: ℵ → ℵ′ be a KU-semigroup homomorphism and onto mapping. Then Ω𝑓 is ACB k-ideal of ℵ ′if and only if Ω𝑓 is ACB k-ideal of ℵ. Proof. For any𝛼′ ∈ ℵ′ there exists𝛼 ∈ ℵ such that𝑓(𝛼) = 𝛼′, we have 𝜇𝑓 +(0) = 𝜇+(𝑓(0)) = 𝜇+(0′) ≥ 𝜇+(𝛼′) = 𝜇+(𝑓(𝛼)) = 𝜇𝑓 +(𝛼) And𝜇𝑓 −(0) = 𝜇−(𝑓(0)) = 𝜇−(0′) ≤ 𝜇−(𝛼′) = 𝜇−(𝑓(𝛼)) = 𝜇𝑓 −(𝛼). Also, 𝜆𝑓 +(0) = 𝜆+(𝑓(0)) = 𝜆+(0′) ≥ 𝜆+(𝛼) = 𝜆+(𝑓(𝛼)) = 𝜆𝑓 +(𝛼) And𝜆𝑓 −(0) = 𝜆−(𝑓(0)) = 𝜆−(0′) ≤ 𝜆−(𝛼′) = 𝜆−(𝑓(𝛼)) = 𝜆𝑓 −(𝛼). Let 𝛼, δ ∈ ℵ,𝛾′ ∈ ℵ′ then there exists 𝛽 ∈ ℵ such that 𝑓(𝛽) = 𝛽′. We have 𝜇𝑓 +(𝛼 ∗ δ) = 𝜇+(𝑓(𝛼 ∗ δ)) = 𝜇+(𝑓(𝛼) ∗ 𝑓(δ)) ≥ 𝑟𝑚𝑖𝑛 {𝜇+ (𝑓(𝛼) ∗ (𝛽′ ∗ 𝑓(δ)) , 𝜇+(𝛽)} = 𝑟𝑚𝑖𝑛 {�̃�+(𝑓(𝛼) ∗ (𝑓(𝛽) ∗ 𝑓(δ))}, 𝜇+(𝑓(𝛽)} = 𝑟𝑚𝑖𝑛 {�̃�𝑓 +(𝛼 ∗ (𝛽 ∗ δ)), 𝜇𝑓 +(𝛽)}. And 𝜇𝑓 −(𝛼 ∗ δ) = 𝜇−(𝑓(𝛼 ∗ δ)) = 𝜇−(𝑓(𝛼) ∗ 𝑓(δ)) ≤ 𝑟𝑚𝑎𝑥 {𝜇− (𝑓(𝛼) ∗ (𝛽′ ∗ 𝑓(δ)) , 𝜇−(𝛽′)} = 𝑟𝑚𝑎𝑥 {𝜇−(𝑓(𝛼) ∗ (𝑓(𝛽) ∗ 𝑓(δ))}, 𝜇−(𝑓(𝛽)} = 𝑟𝑚𝑎𝑥 {𝜇𝑓 −(𝛼 ∗ (𝛽 ∗ δ)), 𝜇𝑓 −(𝛽)}. Also, 𝜆𝑓 +(𝛼 ∗ δ) = 𝜆+(𝑓(𝛼 ∗ δ)) = 𝜆+(𝑓(𝛼) ∗ 𝑓(δ)) ≥ 𝑚𝑖𝑛 {𝜆+ (𝑓(𝛼) ∗ (𝛽′ ∗ 𝑓(δ)) , 𝜆+(𝛽′)} = 𝑚𝑖𝑛 {𝜆+(𝑓(𝛼) ∗ (𝑓(𝛽) ∗ 𝑓(δ))}, 𝜆+(𝑓(𝛽)} = 𝑚𝑖𝑛 {𝜆𝑓 +(𝛼 ∗ (𝛽 ∗ δ)), 𝜆𝑓 +(𝛽)}. And 𝜆𝑓 −(𝛼 ∗ δ) = 𝜆−(𝑓(𝛼 ∗ δ)) = 𝜆−(𝑓(𝛼) ∗ 𝑓(δ)) ≤ 𝑚𝑎𝑥 {𝜆− (𝑓(𝛼) ∗ (𝛽′ ∗ 𝑓(δ)) , 𝜆−(𝛽′)} = 𝑚𝑎𝑥 {𝜆−(𝑓(𝛼) ∗ (𝑓(𝛽) ∗ 𝑓(δ))}, 𝜆−(𝑓(𝛽)} = 𝑚𝑎𝑥 {𝜆𝑓 −(𝛼 ∗ (𝛽 ∗ δ)), 𝜆𝑓 −(𝛽)}. And the condition (c) is 𝜇𝑓 +(𝛼 ∘ 𝛽) = 𝜇+(𝑓(𝛼 ∘ 𝛽)) = 𝜇+(𝑓(𝛼) ∘ 𝑓(𝛽)) ≥ 𝑟𝑚𝑖𝑛 {𝜇+(𝑓(𝛼), 𝜇+(𝑓(𝛽)} = 𝑟𝑚𝑖𝑛{𝜇𝑓 +(𝛼), 𝜇𝑓 +(𝛽)} And 𝜇𝑓 −(𝛼 ∘ 𝛽) = 𝜇−(𝑓(𝛼 ∘ 𝛽)) = 𝜇−(𝑓(𝛼) ∘ 𝑓(𝛽)) ≤ 𝑟𝑚𝑎𝑥 {𝜇−(𝑓(𝛼), 𝜇−(𝑓(𝛽)} = 𝑟𝑚𝑎𝑥{𝜇𝑓 −(𝛼), 𝜇𝑓 −(𝛽)} Also, 𝜆𝑓 +(𝛼 ∘ 𝛽) = 𝜆+(𝑓(𝛼 ∘ 𝛽)) = 𝜆+(𝑓(𝛼) ∘ 𝑓(𝛽)) ≥ 𝑚𝑖𝑛 {𝜆+(𝑓(𝛼), 𝜆+(𝑓(𝛽)} = 𝑚𝑖𝑛{𝜆𝑓 +(𝛼), 𝜆𝑓 +(𝛽)} Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 78 And 𝜆𝑓 −(𝛼 ∘ 𝛽) = 𝜆−(𝑓(𝛼 ∘ 𝛽)) = 𝜆−(𝑓(𝛼) ∘ 𝑓(𝛽)) ≤ 𝑚𝑎𝑥 {𝜆−(𝑓(𝛼), 𝜆−(𝑓(𝛽)} = 𝑚𝑎𝑥{𝜆𝑓 −(𝛼), 𝜆𝑓 −(𝛽)} Conversely, since 𝑓: ℵ → ℵ′ is an onto mapping, then for any 𝛼, 𝛽, δ ∈ ℵ′. It follows that, there exists 𝒂, 𝒃, 𝒄 ∈ ℵsuch that 𝑓(𝒂) = 𝛼, 𝑓(𝒃) = 𝛽 and 𝑓(𝒄) = δ. We have 𝜇𝑓 +(𝛼 ∗ δ) = 𝜇+(𝑓(𝒂) ∗ 𝑓(𝒄))) = 𝜇+(𝑓(𝒂 ∗ 𝒄)) = 𝜇𝑓 +(𝒂 ∗ 𝒄) ≥ 𝑟𝑚𝑖𝑛 {�̃�𝑓 +(𝒂 ∗ (𝒃 ∗ 𝒄)), 𝜇𝑓 +(𝒃)} = 𝑟𝑚𝑖𝑛 {𝜇+(𝑓(𝒂) ∗ (𝑓(𝒃) ∗ 𝑓(𝒄))}, 𝜇+(𝑓(𝒃)} = 𝑟𝑚𝑖𝑛 {𝜇+(𝛼 ∗ (𝛽 ∗ δ)), 𝜇+(𝛽)}. And 𝜇𝑓 −(𝛼 ∗ δ) = 𝜇−(𝑓(𝒂) ∗ 𝑓(𝒄))) = 𝜇−(𝑓(𝒂 ∗ 𝒄)) = 𝜇𝑓 −(𝒂 ∗ 𝒄) ≤ 𝑟𝑚𝑎𝑥 {𝜇𝑓 −(𝒂 ∗ (𝒃 ∗ 𝒄)), 𝜇𝑓 −(𝒃)} = 𝑟𝑚𝑎𝑥 {�̃�−(𝑓(𝒂) ∗ (𝑓(𝒃) ∗ 𝑓(𝒄))}, 𝜇−(𝑓(𝒃)} = 𝑟𝑚𝑎𝑥 {�̃�−(𝛼 ∗ (𝛽 ∗ δ)), 𝜇−(𝛽)}. Also, 𝜆𝑓 +(𝛼 ∗ δ) = 𝜆+(𝑓(𝒂) ∗ 𝑓(𝒄))) = 𝜆+(𝑓(𝒂 ∗ 𝒄)) = 𝜆𝑓 +(𝒂 ∗ 𝒄) ≥ 𝑚𝑖𝑛 {𝜆𝑓 +(𝒂 ∗ (𝒃 ∗ 𝒄)), 𝜆𝑓 +(𝒃)} = 𝑚𝑖𝑛 {𝜆+(𝑓(𝒂) ∗ (𝑓(𝒃) ∗ 𝑓(𝒄))}, 𝜆+(𝑓(𝒃)} = 𝑚𝑖𝑛 {𝜆+(𝛼 ∗ (𝛽 ∗ δ)), 𝜆+(𝛽)}. And 𝜆𝑓 −(𝛼 ∗ δ) = 𝜆−(𝑓(𝒂) ∗ 𝑓(𝒄))) = 𝜆−(𝑓(𝒂 ∗ 𝒄)) = 𝜆𝑓 −(𝒂 ∗ 𝒄) ≤ 𝑚𝑎𝑥 {𝜆𝑓 −(𝒂 ∗ (𝒃 ∗ 𝒄)), 𝜆𝑓 −(𝒃)} = 𝑚𝑎𝑥 {𝜆−(𝑓(𝒂) ∗ (𝑓(𝒃) ∗ 𝑓(𝒄))}, 𝜆−(𝑓(𝒃)} = 𝑚𝑎𝑥 {𝜆−(𝛼 ∗ (𝛽 ∗ δ)), 𝜆−(𝛽)}. And the condition (c) is 𝜇𝑓 +(𝛼 ∘ 𝛽) = 𝜇+(𝑓(𝒂) ∘ 𝑓(𝒃))) ≥ 𝑟𝑚𝑖𝑛 {𝜇+(𝑓(𝒂)), 𝜇+(𝑓(𝒃)} = 𝑟𝑚𝑖𝑛{𝜇𝑓 +(𝛼), 𝜇𝑓 +(𝛽)} And 𝜇𝑓 −(𝛼 ∘ 𝛽) = 𝜇−(𝑓(𝒂) ∘ 𝑓(𝒃))) ≤ 𝑟𝑚𝑎𝑥 {𝜇−(𝑓(𝒂), 𝜇−(𝑓(𝒃)} = 𝑟𝑚𝑎𝑥{𝜇𝑓 −(𝛼), 𝜇𝑓 −(𝛽)} Also, 𝜆𝑓 +(𝛼 ∘ 𝛽) = 𝜆+(𝑓(𝒂) ∘ 𝑓(𝒃))) ≥ 𝑚𝑖𝑛 {𝜆+(𝑓(𝒂), 𝜆+(𝑓(𝒃)} = 𝑚𝑖𝑛{𝜆𝑓 +(𝛼), 𝜆𝑓 +(𝛽)} And 𝜆𝑓 −(𝛼 ∘ 𝛽) = 𝜆−(𝑓(𝒂) ∘ 𝑓(𝒃))) ≤ 𝑚𝑎𝑥 {𝜆−(𝑓(𝒂), 𝜆−(𝑓(𝒃)} = 𝑚𝑎𝑥{𝜆𝑓 −(𝛼), 𝜆𝑓 −(𝛽)} Therefore Ω𝑓 is ACB k-ideal of ℵ ′. In the following, we introduce the product of the cubic bipolar k-ideals and a cubic bipolar ideal as follows. Definition 3.4. Let Ω𝑓1 and Ω𝑓2 be two CB fuzzy sets of ℵ. The product Ω𝑓1 × Ω𝑓2 = ((𝛼, 𝛽), ; 𝜇1 − × 𝜇2 −, ; 𝜇1 +;×; 𝜇2 +, ; 𝜆1 −;×; 𝜆2 −, ; 𝜆1 +;×; 𝜆2 +) is defined by the following: (𝜇1 − × 𝜇2 −)(𝛼, 𝛽) = 𝑟𝑚𝑎𝑥 {; 𝜇1 −(𝛼), ; 𝜇2 −(𝛽)}, (𝜇1 +;×; 𝜇2 +)(𝛼, 𝛽) = 𝑟𝑚𝑖𝑛 {𝜇1 +(𝛼), 𝜇2 +(𝛽)} and (; 𝜆1 −;×; 𝜆2 −)(𝛼, 𝛽) = 𝑚𝑎𝑥 {; 𝜆1 −(𝛼), ; 𝜆2 −(𝛽)}, (; 𝜆1 +;×; 𝜆2 +)(𝛼, 𝛽) = 𝑚𝑖𝑛 {; 𝜆1 +(𝛼), ; 𝜆2 +(𝛽)}where ; 𝜇1 − ×; 𝜇2 −: ℵ × ℵ → [−1,0], 𝜇1 + × 𝜇2 +: ℵ × ℵ → [0,1] and 𝜆1 − × 𝜆2 −: ℵ × ℵ → [−1,0], 𝜆1 + × 𝜆2 +: ℵ × ℵ → [0,1], for all 𝛼, 𝛽 ∈ ℵ. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 79 Theorem 3.5. Let Ω𝑓1 and Ω𝑓2 be two CB k-ideals of ℵ, then Ω𝑓1 × Ω𝑓2 is ACB k-ideal of ℵ × ℵ. Proof. Let (𝛼, 𝛽) ∈ ℵ × ℵ, we have (𝜇1 + × 𝜇2 +)( 0, 0) = 𝑟𝑚𝑖𝑛{�̃�1 +(0), 𝜇2 +(0)} ≥ 𝑟𝑚𝑖𝑛{𝜇1 +(𝛼), 𝜇2 +(𝛽)} = (𝜇1 + × 𝜇2 +)(𝛼, 𝛽) and (𝜇1 − × 𝜇2 −)( 0, 0) = 𝑟𝑚𝑎𝑥{𝜇1 −(0), 𝜇2 −(0)} ≤ 𝑟𝑚𝑎𝑥{𝜇1 −(𝛼), 𝜇2 −(𝛽)} = (𝜇1 − × 𝜇2 −)(𝛼, 𝛽). Let (𝛼1, 𝛼2), (𝛽1, 𝛽2) and (δ1, δ2) ∈ ℵ × ℵ , then (�̃�1 + × 𝜇2 +)(𝛼1 ∗ δ1, 𝛼2 ∗ δ2) = 𝑟𝑚𝑖𝑛{𝜇1 +(𝛼1 ∗ 𝜏1), 𝜇2 +(𝛼2 ∗ δ2)} ≥ 𝑟𝑚𝑖𝑛{𝑟𝑚𝑖𝑛 {𝜇1 +(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜇1 +(𝛽1)} , 𝑟𝑚𝑖𝑛{𝜇2 +(𝛼2 ∗ (𝛽2 ∗ δ2)), 𝜇2 +(𝛽2)}} = 𝑟𝑚𝑖𝑛{𝑟𝑚𝑖𝑛 {𝜇1 +(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜇2 +(𝛼2 ∗ (𝛽2 ∗ δ2)}, 𝑟𝑚𝑖𝑛 {𝜇1 +(𝛽1), 𝜇2 +(𝛽2)} = 𝑟𝑚𝑖𝑛 {𝑟𝑚𝑖𝑛(�̃�1 + × 𝜇2 +) {(𝛼1 ∗ (𝛽1 ∗ δ1)), (𝛼2 ∗ (𝛽2 ∗ δ2))} , {(𝜇1 + × 𝜇2 +)(𝛽1, 𝛽2)}} And (�̃�1 − × 𝜇2 −)(𝛼1 ∗ δ1, 𝛼2 ∗ δ2) = 𝑟𝑚𝑎𝑥{𝜇1 −(𝛼1 ∗ δ1), 𝜇2 −(𝛼2 ∗ δ2)} ≤ 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥 {𝜇1 −(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜇1 −(𝛽1)} , 𝑟𝑚𝑎𝑥 {𝜇2 −(𝛼2 ∗ (𝛽2 ∗ δ2), 𝜇2 −(𝛽2)} = 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥 {�̃�1 −(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜇2 −(𝛼2 ∗ (𝛽2 ∗ δ2)}, 𝑟𝑚𝑎𝑥 {𝜇1 −(𝛽1), 𝜇2 −(𝛽2)} = 𝑟𝑚𝑎𝑥 {(�̃�1 − × 𝜇2 −) {(𝛼1 ∗ (𝛽1 ∗ δ1)), (𝛼2 ∗ (𝛽2 ∗ δ2))} , {(𝜇1 − × 𝜇2 −)(𝛽1, 𝛽2)}}. Also, (𝜆1 + × 𝜆2 +)(𝛼1 ∗ δ1, 𝛼2 ∗ δ2) = 𝑚𝑖𝑛{𝜆1 +(𝛼1 ∗ δ1), 𝜆2 +(𝛼2 ∗ δ2)} ≥ 𝑚𝑖𝑛{𝑚𝑖𝑛 {𝜆1 +(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜆1 +(𝛽1)} , 𝑚𝑖𝑛{𝜆2 +(𝛼2 ∗ (𝛽2 ∗ δ2)), 𝜆2 +(𝛽2)}} = 𝑚𝑖𝑛{𝑚𝑖𝑛 {𝜆1 +(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜆2 +(𝛼2 ∗ (𝛽2 ∗ δ2)}, 𝑚𝑖𝑛 {𝜆1 +(𝛽1), 𝜆2 +(𝛽2)} = 𝑚𝑖𝑛 {𝑚𝑖𝑛(𝜆1 + × 𝜆2 +) {(𝛼1 ∗ (𝛽1 ∗ δ1)), (𝛼2 ∗ (𝛽2 ∗ δ2))} , {(; 𝜆1 + ×; 𝜆2 +)(𝛽1, 𝛽2)}} And (; 𝜆1 − ×; 𝜆2 −)(𝛼1 ∗ δ1, 𝛼2 ∗ δ2) = 𝑚𝑎𝑥{𝜆1 −(𝛼1 ∗ δ1), 𝜆2 −(𝛼2 ∗ δ2)} ≤ 𝑚𝑎𝑥{𝑚𝑎𝑥 {𝜆1 −(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜆1 −(𝛽1)} , 𝑚𝑎𝑥 {𝜆2 −(𝛼2 ∗ (𝛽2 ∗ δ2), 𝜆2 −(𝛽2)} = 𝑚𝑎𝑥{𝑚𝑎𝑥 {𝜆1 −(𝛼1 ∗ (𝛽1 ∗ δ1)), 𝜆2 −(𝛼2 ∗ (𝛽2 ∗ δ2)}, 𝑚𝑎𝑥 {𝜆1 −(𝛽1), 𝜆2 −(𝛽2)} = 𝑚𝑎𝑥 {(𝜆1 − × 𝜆2 −) {(𝛼1 ∗ (𝛽1 ∗ δ1)), (𝛼2 ∗ (𝛽2 ∗ δ2))} , {(𝜆1 − × 𝜆2 −)(𝛽1, 𝛽2)}}. And (𝜇1 + × 𝜇2 +)(𝛼1 𝛽1, 𝛼2 𝛽2) = 𝑟𝑚𝑖𝑛{�̃�1 +(𝛼1 𝛽1), 𝜇2 +(𝛼2 𝛽2)} ≥ 𝑟𝑚𝑖𝑛{𝑟𝑚𝑖𝑛 {𝜇1 +(𝛼1), 𝜇1 +(𝛽1)} , 𝑟𝑚𝑖𝑛 {𝜇2 +(𝛼2), 𝜇2 +(𝛽2)} = 𝑟𝑚𝑖𝑛{𝑟𝑚𝑖𝑛 {𝜇1 +(𝛼1), 𝜇2 +(𝛼2)}, 𝑟𝑚𝑖𝑛 {𝜇1 +(𝛽1), 𝜇2 +(𝛽2)} = 𝑟𝑚𝑖𝑛{{(�̃�1 + × 𝜇2 +)(𝛼1, 𝛼2)}, {(�̃�1 + × 𝜇2 +)(𝛽1, 𝛽2)}} And (𝜇1 − × 𝜇2 −)(𝛼1 𝛽1, 𝛼2 𝛽2) = 𝑟𝑚𝑎𝑥{�̃�1 −(𝛼1 𝛽1), 𝜇2 −(𝛼2 𝛽2)} ≤ 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥{𝜇1 −(𝛼1), 𝜇1 −(𝛽1)} , 𝑟𝑚𝑎𝑥{𝜇2 −(𝛼2), 𝜇2 −(𝛽2)} = 𝑟𝑚𝑎𝑥{𝑟𝑚𝑎𝑥 {𝜇1 −(𝛼1), 𝜇2 −(𝛼2)}, 𝑟𝑚𝑎𝑥 {𝜇1 −(𝛽1), 𝜇2 −(𝛽2)} = 𝑟𝑚𝑎𝑥{{(�̃�1 − × 𝜇2 −)(𝛼1, 𝛼2)}, {(�̃�1 − × 𝜇2 −)(𝛽1, 𝛽2)}}. Also, (𝜆1 + × 𝜆2 +)(𝛼1 𝛽1, 𝛼2 𝛽2) = 𝑚𝑖𝑛{𝜆1 +(𝛼1 𝛽1), 𝜆2 +(𝛼2 𝛽2)} ≥ 𝑚𝑖𝑛{𝑚𝑖𝑛 {𝜆1 +(𝛼1), 𝜆1 +(𝛽1)} , 𝑚𝑖𝑛 {𝜆2 +(𝛼2), 𝜆2 +(𝛽2)} =             Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 80 𝑚𝑖𝑛{𝑚𝑖𝑛 {𝜆1 +(𝛼1), 𝜆2 +(𝛼2)}, 𝑚𝑖𝑛 {𝜆1 +(𝛽1), 𝜆2 +(𝛽2)} = 𝑚𝑖𝑛{{(𝜆1 + × 𝜆2 +)(𝛼1, 𝛼2)}, {(𝜆1 + × 𝜆2 +)(𝛽1, 𝛽2)}} And (𝜆1 − × 𝜆2 −)(𝛼1 𝛽1, 𝛼2 𝛽2) = 𝑚𝑎𝑥{𝜆1 −(𝛼1 𝛽1), 𝜆2 −(𝛼2 𝛽2)} ≤ 𝑚𝑎𝑥{𝑚𝑎𝑥{𝜆1 −(𝛼1), 𝜆1 −(𝛽1)} , 𝑚𝑎𝑥{𝜆2 −(𝛼2), 𝜆2 −(𝛽2)} = 𝑚𝑎𝑥{𝑚𝑎𝑥 {𝜆1 −(𝛼1), 𝜆2 −(𝛼2)}, 𝑚𝑎𝑥 {𝜆1 −(𝛽1), 𝜆2 −(𝛽2)} = 𝑚𝑎𝑥{{(𝜆1 − × 𝜆2 −)(𝛼1, 𝛼2)}, {(𝜆1 − × 𝜆2 −)(𝛽1, 𝛽2)}}. Then Ω𝑓1 × Ω𝑓2 is ACB k-ideal of ℵ × ℵ. Theorem 3.6.Let Ω𝑓1 and Ω𝑓2 be two CB ideal of KU-semigroupℵ, such that Ω𝑓1 × Ω𝑓2 is ACB ideal of ℵ × ℵ. We have (i) Either𝜇1 +(0) ≥ 𝜇1 +(𝛼),𝜇1 −(0) ≤ 𝜇1 −(𝛼)or𝜇2 +(0) ≥ 𝜇2 +(𝛽),𝜇2 −(0) ≤ 𝜇2 −(𝛽), also, 𝜆1 +(0) ≥ 𝜆1 +(𝛼),𝜆1 −(0) ≤ 𝜆1 −(𝛼) or𝜆2 +(0) ≥ 𝜆2 +(𝛽),𝜆2 −(0) ≤ 𝜆2 −(𝛽) for all 𝛼 , 𝛽 ∈ ℵ. (ii) If 𝜇1 +(0) ≥ 𝜇1 +(𝛼), 𝜇1 −(0) ≤ 𝜇1 −(𝛼) and 𝜆1 +(0) ≥ 𝜆1 +(𝛼),𝜆1 −(0) ≤ 𝜆1 −(𝛼)for all 𝛼 ∈ ℵ. Then either𝜇2 +(0) ≥ 𝜇1 +(𝛼),𝜇2 −(0) ≤ 𝜇1 −(𝛼) and 𝜆2 +(0) ≥ 𝜆1 +(𝛼),𝜆2 −(0) ≤ 𝜆1 −(𝛼)or 𝜇2 +(0) ≥ 𝜇2 +(𝛽),𝜇2 −(0) ≤ 𝜇2 −(𝛽)and 𝜆2 +(0) ≥ 𝜆2 +(𝛽),𝜆2 −(0) ≤ 𝜆2 −(𝛽)for all𝛼 , 𝛽 ∈ ℵ. (iii) If𝜇2 +(0) ≥ 𝜇2 +(𝛼), 𝜇2 −(0) ≤ 𝜇2 −(𝛼), and λ2 +(0) ≥ λ2 +(α),λ2 - (0) ≤ λ2 - (α),for all α ∈ ℵ , then either 𝜇1 +(0) ≥ 𝜇1 +(𝛼),𝜇1 −(0) ≤ 𝜇1 −(𝛼), and𝜆1 +(0) ≥ 𝜆1 +(𝛼),𝜆1 −(0) ≤ 𝜆1 −(𝛼)or𝜇1 +(0) ≥ 𝜇2 +(𝛼), 𝜇1 −(0) ≤ 𝜇2 −(𝛼) and 𝜆1 +(0) ≥ 𝜆2 +(𝛼),𝜆1 −(0) ≤ 𝜆2 −(𝛼)for all𝛼 ∈ ℵ. Proof. (i)Suppose that 𝜇1 +(0) ≥ 𝜇1 +(𝛼),𝜇1 −(0) ≤ 𝜇1 −(𝛼) and 𝜇2 +(0) ≥ 𝜇2 +(𝑦),𝜇2 −(0) ≤ 𝜇2 −(𝛽), also, 𝜆1 +(0) ≥ 𝜆1 +(𝛼),𝜆1 −(0) ≤ 𝜆1 −(𝛼) and𝜆2 +(0) ≥ 𝜆2 +(𝛽),𝜆2 −(0) ≤ 𝜆2 −(𝛽), for some 𝛼 , 𝛽 ∈ ℵ.Then (𝜇1 + × 𝜇2 +)(𝛼, 𝛽) = 𝑟𝑚𝑖𝑛{�̃�1 +(𝛼), 𝜇2 +(𝛽)} ≥ 𝑟𝑚𝑖𝑛{𝜇1 +(0), 𝜇2 +(0)} = (�̃�1 + × 𝜇2 +)(0,0) And (𝜇1 − × 𝜇2 −)(𝛼, 𝛽) = 𝑟𝑚𝑎𝑥{𝜇1 −(𝛼), 𝜇2 −(𝛽)} ≤ 𝑟𝑚𝑎𝑥{; 𝜇1 −(0), ; 𝜇2 −(0)} = (𝜇1 − ×; 𝜇2 −)(; 0, ; 0) Also, (; 𝜆1 + ×; 𝜆2 +)(𝛼, 𝛽) = 𝑚𝑖𝑛{𝜆1 +(𝛼), 𝜆2 +(𝛽)} ≥ 𝑚𝑖𝑛{; 𝜆1 +(0), ; 𝜆2 +(0)} = (; 𝜆1 + ×; 𝜆2 +)(0,0) And (𝜆1 − × 𝜆2 −)(𝛼, 𝛽) = 𝑚𝑎𝑥{𝜆1 −(𝛼), 𝜆2 −(𝛽)} ≤ 𝑚𝑎𝑥{𝜆1 −(0), 𝜆2 −(0)} = (𝜆1 − × 𝜆2 −)(0,0) , for all 𝛼 , 𝛽 ∈ ℵ. This is a contradiction. Therefore, either𝜇1 +(0) ≥ 𝜇1 +(𝛼),𝜇1 −(0) ≤ 𝜇1 −(𝛼) or𝜇2 +(0) ≥ 𝜇2 +(𝛽),𝜇2 −(0) ≤ 𝜇2 −(𝛽), also, 𝜆1 +(0) ≥ 𝜆1 +(𝛼),𝜆1 −(0) ≤ 𝜆1 −(𝛼) or𝜆2 +(0) ≥ 𝜆2 +(𝛽),𝜆2 −(0) ≤ 𝜆2 −(𝛽) for all 𝛼 , 𝛽 ∈ ℵ. (ii)Suppose that ; 𝜇2 +(0) ≤; 𝜇1 +(𝛼), ; 𝜇2 −(0) ≥; 𝜇1 −(𝛼) and 𝜇2 +(0) ≤ 𝜇2 +(𝛽),𝜇2 −(0) ≥ 𝜇2 −(𝛽) also,; 𝜆2 +(0); ≤; ; 𝜆1 +(𝛼);,; 𝜆2 −(0); ≥; 𝜆1 −(𝛼) and ; 𝜆2 +(0) ≤; 𝜆2 +(𝛽);,; 𝜆2 −(0); ≥; ; 𝜆2 −(𝛽), for all𝛼 , 𝛽 ∈ ℵ. Then (𝜇1 + × 𝜇2 +)(0,0) = 𝑟𝑚𝑖𝑛{𝜇1 +(0), 𝜇2 +(0)} = 𝜇2 +(0) And (𝜇1 + × 𝜇2 +)(𝛼, 𝛽) = 𝑟𝑚𝑖𝑛{�̃�1 +(𝛼), 𝜇2 +(𝛽)} ≥ {𝜇2 +(0), 𝜇2 +(0)} = 𝜇2 +(0) = (�̃�1 + × 𝜇2 +)(0,0) And (𝜇1 − × 𝜇2 −)(0,0) = 𝑟𝑚𝑎𝑥{𝜇1 −(0), 𝜇2 −(0)} = 𝜇2 −(0). (𝜇1 − × 𝜇2 −)(𝛼, 𝛽) = 𝑟𝑚𝑎𝑥{�̃�1 −(𝛼), 𝜇2 −(𝛽)} ≤ 𝑟𝑚𝑎𝑥{𝜇2 −(0), 𝜇2 −(0)} = 𝜇2 −(0) = (𝜇1 − × 𝜇2 −)(0,0)aaa     Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 81 Also, (; 𝜆1 + ×; 𝜆2 +; )(0,0) = 𝑚𝑖𝑛{; 𝜆1 +(0); , ; 𝜆2 +(0); } =; 𝜆2 +(0); And; ; (; 𝜆1 +;×; 𝜆2 +; )(𝛼, 𝛽) = 𝑚𝑖𝑛{; 𝜆1 +(𝛼); , ; 𝜆2 +(𝛽); } ≥ {; 𝜆2 +(0); , ; 𝜆2 +(0); } =; 𝜆2 +(0); = ; (; 𝜆1 + ×; 𝜆2 +)(0,0). And (; 𝜆1 − ×; 𝜆2 −; )(0,0) = 𝑚𝑎𝑥{; 𝜆1 −(0), ; 𝜆2 −(0)} =; 𝜆2 −(0) . (; 𝜆1 − ×; 𝜆2 −; )(𝛼, 𝛽) = 𝑚𝑎𝑥{; 𝜆1 −(𝛼), ; 𝜆2 −(𝛽)} ≤ 𝑚𝑎𝑥{; 𝜆2 −(0), ; 𝜆2 −(0)} =; 𝜆2 −(0) = (; 𝜆1 − ×; 𝜆2 −); (0,0) . This is a contradiction. Therefore, either 𝜇2 +(0) ≥ 𝜇1 +(𝜒),𝜇2 −(0) ≤ 𝜇1 −(𝛼) and 𝜆2 +(0) ≥ 𝜆1 +(𝛼),𝜆2 −(0) ≤ 𝜆1 −(𝑥)or 𝜇2 +(0) ≥ 𝜇2 +(𝛽),𝜇2 −(0) ≤ 𝜇2 −(𝛽)and 𝜆2 +(0) ≥ 𝜆2 +(𝛽),𝜆2 −(0) ≤ 𝜆2 −(𝛽). (iii)The proof is similar to (ii). The partial converse of Theorem (3.5) is the following. Theorem3.7. In a KU-semigroupℵ. If Ω𝑓1 × Ω𝑓2 is ACB ideal of ℵ × ℵ, then Ωf1 or Ωf2 is ACB ideal of ℵ. Proof. By use the Theorem (3.5) (i), without loss of generality we suppose that 𝜇2 +(0) ≥ 𝜇2 +(𝛼), 𝜇2 −(0) ≤ 𝜇2 −(𝛼), and λ2 +(0) ≥ λ2 +(α),λ2 - (0) ≤ λ2 - (α),for allα ∈ ℵ. It follows from Theorem (4.6)(iii) that either then either 𝜇1 +(0) ≥ 𝜇1 +(𝛼),𝜇1 −(0) ≤ 𝜇1 −(𝛼), and𝜆1 +(0) ≥ 𝜆1 +𝛼,𝜆1 −(0) ≤ 𝜆1 −(𝛼) or 𝜇1 +(0) ≥ 𝜇2 +(𝛼), ; �̃� 1 − (0); ≤; �̃� 2 − (𝛼); and ; 𝜆1 +(0); ≥; 𝜆2 +(𝛼),; 𝜆1 −(0); ≤; 𝜆2 −(𝛼). , for all α ∈ ℵ. Then (; �̃� 1 + ×; �̃� 2 + )(0, 𝛼) = 𝑟𝑚𝑖𝑛{; �̃� 1 + (0), ; �̃� 2 + (𝛼)} =; �̃� 2 + (𝛼) … . . (1) (; �̃� 1 − ×; �̃� 2 − )(0, 𝛼) = 𝑟𝑚𝑎𝑥{; �̃� 1 − (0), ; �̃� 2 − (𝛼)} =; �̃� 2 − (𝛼) … … (2) Also, (𝜆1 + × 𝜆2 +)(0, 𝛼) = 𝑚𝑖𝑛{𝜆1 +(0), 𝜆2 +(𝛼)} = 𝜆2 +(𝛼) … … (3) (𝜆1 − × 𝜆2 −)(0, 𝛼) = 𝑚𝑎𝑥{𝜆1 −(0), 𝜆2 −(𝛼)} = 𝜆2 −(𝛼) … . . (4) Since Ω𝑓1 × Ω𝑓2 is ACB ideal of  , then ; (; �̃� 1 + ×; �̃� 2 + )(𝛽 1 , 𝛽 2 ); ≥ 𝑟𝑚𝑖𝑛{(; �̃� 1 + ×; �̃� 2 + )((𝛼1, 𝛼2) ∗ (𝛽1, 𝛽2)), (�̃�1 + × �̃� 2 + )(𝛼1, 𝛼2)} = 𝑟𝑚𝑖𝑛{(�̃� 1 + × �̃� 2 + )(𝛼1 ∗ 𝛽1, 𝛼2 ∗ 𝛽2), (�̃�1 + × �̃� 2 + )(𝛼1, 𝛼2)} Put𝛼1 = 𝛽1 = 0 , then we have (�̃� 1 + × �̃� 2 + )(0, 𝛽 2 ) ≥ 𝑟𝑚𝑖𝑛{(; �̃� 1 + ×; �̃� 2 + )((0, 𝛼2) ∗ (0, 𝛽2)), (; �̃�1 + ×; �̃� 2 + )(0, 𝛼2)} = 𝑟𝑚𝑖𝑛{(; �̃� 1 + ×; �̃� 2 + )(0, 𝛼2 ∗ 𝛽2), (; �̃�1 + ×; �̃� 2 + )(0, 𝛼2)} and by equation (1), then 𝜇2 +(𝛽2) ≥ 𝑟𝑚𝑖𝑛{�̃�2 +(𝛼2 ∗ 𝛽2), 𝜇2 +(𝛼2)}. And (�̃� 1 − × �̃� 2 − )(𝛽 1 , 𝛽 2 ) ≤ 𝑟𝑚𝑎𝑥{(�̃� 1 − × �̃� 2 − )((𝛼1, 𝛼2) ∗ (𝛽1, 𝛽2)), (�̃�1 − × �̃� 2 − )(𝛼1, 𝛼2)} = 𝑟𝑚𝑎𝑥{(�̃� 1 − × �̃� 2 − )(𝛼1 ∗ 𝛽1, 𝛼2 ∗ 𝛽2), (�̃�1 − × �̃� 2 − )(𝛼1, 𝛼2)} Put 𝛼1 = 𝛽1 = 0 , then we have (�̃� 1 − × �̃� 2 − )(0, 𝛽 2 ) ≤ 𝑟𝑚𝑎𝑥{(; �̃� 1 − ×; �̃� 2 − ; )((0, 𝛼2) ∗ (0, 𝛽2)), (; �̃�1 − ×; �̃� 2 − ; )(0, 𝛼2)} = 𝑟𝑚𝑎𝑥{(; �̃� 1 − ×; �̃� 2 − ; )(0, 𝛼2 ∗ 𝛽2), (; �̃�1 − ×; �̃� 2 − ; )(0, 𝛼2)} Ibn Al-Haitham Jour. for Pure & Appl. Sci. 35(1)2022 82 and by using equation (2), we have 𝜇2 −(𝛽2) ≤ 𝑟𝑚𝑎𝑥{�̃�2 −(𝛼2 ∗ 𝛽2), 𝜇2 −(𝛼2)}. Also, ; (; 𝜆1 + ×; 𝜆2 +; )(𝛽1, 𝛽2) ≥ 𝑚𝑖𝑛{; (; 𝜆1 + ×; 𝜆2 +; )((𝛼1, 𝛼2) ∗ (𝛽1, 𝛽2)), (; 𝜆1 + ×; 𝜆2 + )(𝛼1, 𝛼2)} = 𝑚𝑖𝑛{(; 𝜆1 + ×; 𝜆2 +)(𝛼1 ∗ 𝛽1, 𝛼2 ∗ 𝛽2), (; 𝜆1 + ×; 𝜆2 +)(𝛼1, 𝛼2)} Put 𝛼1 = 𝛽1 = 0, then we have (: 𝜆1 + ×: 𝜆2 +; )(0, 𝛽 2 ) ≥ 𝑚𝑖𝑛{(: 𝜆1 + ×: 𝜆2 +; )((0, 𝛼2) ∗ (0, 𝛽2)), (: 𝜆1 + ×: 𝜆2 +; )(0, 𝛼2)} = 𝑚𝑖𝑛{(: 𝜆1 + ×: 𝜆2 +)(0, 𝛼2 ∗ 𝛽2), (: 𝜆1 + ×: 𝜆2 +; )(0, 𝛼2)} and by using equation (3), we have 𝜆2 +(𝛽2) ≥ 𝑚𝑖𝑛{𝜆2 +(𝛼2 ∗ 𝛽2), 𝜆2 +(𝛼2)}. And (; 𝜆1 − ×; 𝜆2 −)(𝛽 1 , 𝛽 2 ) ≤ 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −)((𝛼1, 𝛼2) ∗ (𝛽1, 𝛽2)), (; 𝜆1 − ×; 𝜆2 −)(𝛼1, 𝛼2)} = 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −)(𝛼1 ∗ 𝛽1, 𝛼2 ∗ 𝛽2), (; 𝜆1 − ×; 𝜆2 −)(𝛼1, 𝛼2)} Put𝛼1 = 𝛽1 = 0 , then we have (; 𝜆1 − ×; 𝜆2 −)(0, 𝛽 2 ) ≤ 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −; )((0, 𝛼2) ∗ (0, 𝛽2)), (; 𝜆1 − ×; 𝜆2 −; )(0, 𝛼2)} = 𝑚𝑎𝑥{(; 𝜆1 − × ; 𝜆2 −; )(0, 𝛼2 ∗ 𝛽2), (; 𝜆1 − ×; 𝜆2 −; )(0, 𝛼2)}and by using equation (4), we have 𝜆2 −(𝛽2) ≤ 𝑚𝑎𝑥{𝜆2 −(𝛼2 ∗ 𝛽2), 𝜆2 −(𝛼2)}. And the condition (BC3) is (�̃� 1 + × �̃� 2 + )((𝛼1, 𝛼2) ∘ (𝛽1, 𝛽2) ≥ 𝑟𝑚𝑖𝑛{(; �̃�1 + ×; �̃� 2 + )(𝛼1, 𝛼2), (; �̃�1 + ×; �̃� 2 + )(𝛽 1 , 𝛽 2 )}(; �̃� 1 + × ; �̃� 2 + )(𝛼1 ∘ 𝛽1, 𝛼2 ∘ 𝛽2) ≥ 𝑟𝑚𝑖𝑛{(; �̃�1 + ×; �̃� 2 + )(𝛼1, 𝛼2), (; �̃�1 + ×; �̃� 2 + )(𝛽 1 , 𝛽 2 )} Put 𝛼1 = 𝛽1 = 0, then we have (; �̃� 1 + ×; �̃� 2 + ; )(0, 𝛼2 ∘ 𝛽2) ≥ 𝑟𝑚𝑖𝑛{(; �̃�1 + ×; �̃� 2 + ; )(0, 𝑥2), (; �̃�1 + ×; �̃� 2 + ; )(0, 𝛽 2 )} and by using equation (1), we have 𝜇2 +(𝛼2 ∘ 𝛽2) ≥ 𝑟𝑚𝑖𝑛{�̃�2 +(𝛼2), 𝜇2 +(𝛽2)} And (�̃� 1 − × �̃� 2 − )((𝛼1, 𝛼2) ∘ (𝛽1, 𝛽2) ≤ 𝑟𝑚𝑎𝑥{(; �̃�1 − ×; �̃� 2 − )(𝛼1, 𝛼2), (; �̃�1 − ×; �̃� 2 − )(𝛽 1 , 𝛽 2 )} (; �̃� 1 − ×; �̃� 2 − ; )(𝛼1 ∘ 𝛽1, 𝛼2 ∘ 𝛽2) ≤ 𝑟𝑚𝑎𝑥{(; �̃�1 − ×; �̃� 2 − )(𝛼1, 𝛼2); , (; �̃�1 − ×; �̃� 2 − )(; 𝛽 1 , 𝛽 2 ; )} Put 𝛼1 = 𝛽1 = 0, then we have (; �̃� 1 − ×; �̃� 2 − ; )(0, ; 𝛼2 ∘ 𝛽2) ≤ 𝑟𝑚𝑎𝑥{(; �̃�1 − ;×; �̃� 2 − ; )(0, 𝛼2), (; �̃�1 − ×; �̃� 2 − ; )(0, 𝛽 2 )} And by using equation (2), we have 𝜇2 −(𝛼2 ∘ 𝛽2) ≤ 𝑟𝑚𝑎𝑥{�̃�2 −(𝛼2), 𝜇2 −(𝛽2)} Also, we have (𝜆1 + × 𝜆2 +)((𝛼1, 𝛼2) ∘ (𝛽1, 𝛽2) ≥ 𝑚𝑖𝑛{(; 𝜆1 + ×; 𝜆2 +)(𝛼1, 𝛼2), (; 𝜆1 + ×; 𝜆2 +)(𝛽 1 , 𝛽 2 )} (; 𝜆1 + ×; 𝜆2 +)(𝛼1 ∘ 𝛽1, 𝛼2 ∘ 𝛽2) ≥ 𝑚𝑖𝑛{(; 𝜆1 + ×; 𝜆2 +)(𝛼1, 𝛼2), (; 𝜆1 + ×; 𝜆2 +)(𝛽 1 , 𝛽 2 )} Put 𝛼1 = 𝛽1 = 0, then we have (; 𝜆1 + ×; 𝜆2 +)(0, 𝛼2 ∘ 𝛽2) ≥ 𝑚𝑖𝑛{(; 𝜆1 + ×; 𝜆2 +)(0, 𝛼2), (; 𝜆1 + ×; 𝜆2 +)(0, 𝛽 2 )} and by using equation (3), we have 𝜆2 + (𝛼2 ∘ 𝛽2) ≥ 𝑚𝑖𝑛{𝜆2 +(𝛼2), 𝜆2 +(𝛽2)} And (𝜆1 − × 𝜆2 −)((𝛼1, 𝛼2) ∘ (𝛽1, 𝛽2) ≤ 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −; )(𝛼1, 𝛼2), (; 𝜆1 − ×; 𝜆2 −)(; 𝛽 1 , 𝛽 2 ; )} (; 𝜆1 − ×; 𝜆2 −; )(𝛼1 ∘ 𝛽1, 𝛼2 ∘ 𝛽2) ≤ 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −; )(𝛼1, 𝛼2), (; 𝜆1 − ×; 𝜆2 −; )(𝛽 1 , 𝛽 2 )} Put 𝛼1 = 𝛽1 = 0, then we have (; 𝜆1 − ×; 𝜆2 −)(0, 𝛼2 ∘ 𝛽2) ≤ 𝑚𝑎𝑥{(; 𝜆1 − ×; 𝜆2 −; )(0, 𝛼2), (; 𝜆1 − ×; 𝜆2 −; )(0, 𝛽 2 )} and by using equation (4), we have 𝜆2 − (𝛼2 ∘ 𝛽2) ≤ 𝑚𝑎𝑥{𝜆2 −(𝛼2), 𝜆2 −(𝛽2)} Ibn Al-Haitham Jour. for Pure & Appl. 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