47 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Cubic Bipolar Fuzzy Ideals with Thresholds (α, β), ( 𝛚 ,𝛝) of a Semigroup in  

KU-algebra 

 

  

                                                                                                            

Abstract  

In this paper, we introduce the concept of cubic bipolar-fuzzy ideals with thresholds (α,β),(ω,ϑ) of 

a semigroup in KU-algebra as a generalization of sets and in short (CBF). Firstly, a (CBF) sub-

KU-semigroup with a threshold (α,β),(ω,ϑ) and some results in this notion are achieved. Also, 

(cubic bipolar fuzzy ideals and cubic bipolar fuzzy k-ideals) with thresholds (α,β),(ω ,ϑ) are defined 

and some properties of these ideals are given. Relations between a (CBF).sub algebra and-a (CBF) 

ideal are proved. A few characterizations of a (CBF) k-ideal with thresholds (α, β), (ω,ϑ) are 

discussed. Finally, we proved that a (CBF) k-ideal and a (CBF) ideal with thresholds (α, β), (ω,ϑ) 

of a KU-semi group are equivalent relations. 

Keywords: A KU-semigroup, cubic k-ideal, cubic bipolar fuzzy k-ideal with thresholds (α, β), 

(ω,ϑ).      

1. Introduction   

     The fuzzy sets were introduced by Zadeh [1] in 1956; after that, many authors applied this 

concept in different mathematics fields. Mostafa [2, 3] studied the notion of fuzzy KU-ideals of 

KU-algebras and Generalizations of Fuzzy sets, which are called bipolar- fuzzy n-fold KU-ideals.  

Jun [4- 6] studied the notion of a cubic set as a generalization of fuzzy set and interval-valued 

fuzzy set. Kareem and Hasan[7,8] defined the cubic ideals of a KU-semigroup and a 

homomorphism of a cubic set in this structure. Bipolar–valued fuzzy sets are extensions of fuzzy 

sets whose membership degree range is enlarged from the interval [0,1] to [-1,1]. Kareem and 

Article history: Received,21, November,2021, Accepted,25, January, 2022, Published in April  2022. 

 

 

 

Ibn Al Haitham Journal for Pure and Applied Sciences 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/35.2.2724 

Omniat Adnan Hasan 

Department of Mathematics, College of Education 

for Pure Sciences, Ibn Al Haitham,University of 

Baghdad, Iraq 

umniyatadnan@gmail.com 
 

 

 

 

Fatema F. Kareem 

Department of Mathematics, College of 

Education for Pure Sciences, Ibn Al 

Haitham,University of Baghdad, Iraq 

fatma.f.k@ihcoedu.uobaghdad.edu.iq 

https://creativecommons.org/licenses/by/4.0/
mailto:umniyatadnan@gmail.com
mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

48 

Hassan[9] and Kareem and Awad [10] defined the concepts of bipolar fuzzy k-ideals and cubic 

bipolar ideals in KU-semigroup respectively, also Kareem and Abed [11] presented the idea of -

bipolar fuzzy-k-ideals with a threshold of KU--semigroup. 

The paper aims to introduce a cubic bipolar fuzzy k-ideals with thresholds (α,β),(ω,ϑ) of KU-semi 

group and discuss some relations between a cubic bipolar fuzzy k-ideal with thresholds (α,β), (ω,ϑ) 

and a bipolar fuzzy k-ideal. 

2. Basic concepts 

Definition(1)[12]. Algebra(ℵ,∗ ,0) is a set ℵ ,and  a binary operation  ∗  which is satisfies the 
following  ,for all χ,𝛾,τ ∈ ℵ 

(ku1)(χ ∗ 𝛾) ∗ [(𝛾 ∗ τ) ∗ (χ ∗ τ)] = 0 

(ku2) χ ∗0 = 0   

(ku3) 0∗ χ = χ 

(ku4) χ ∗ 𝛾 = 𝛾 ∗ χ = 0   and   𝛾 ∗ χ  implies  χ = 𝛾 

(ku5) χ ∗ χ = 0. 

We can define a  binary operation  ≤ on  ℵ is defined by  χ ≤ 𝛾 ⟺ 𝛾 ∗ χ = 0. It follows that 

(ℵ,≤)is a~ partially ordered set~.  

Theorem(2)[2]. In a KU-algebra (ℵ,∗ ,0 ) ∀
 
 χ , 𝛾 ,τ ∈ ℵ, then the following holds 

 (1) χ ≤ 𝛾   𝑖𝑚𝑝𝑙𝑦  𝛾 ∗  τ ≤ χ ∗ τ 

 (2) χ ∗ (𝛾 ∗  τ) = 𝛾 ∗ (χ ∗ τ) 

(3) 𝛾 ∗ χ ≤ χ ,also   (𝛾 ∗ χ) ∗ χ ≤ 𝛾 

 

Definition(3)[2]. A non-empty subset I of a KU-algebra ℵ is namedˑˑan ˑideal if ˑfor anyχ ,𝛾 ∈

ℵ, then  

(1)  0 ∈ 𝐼 

(2) If  χ ∗ 𝛾 ∈ 𝐼  implies that  𝛾 ∈ 𝐼  . 

Definition(4)[2]. A non-empty subset I of a KU-algebra ℵ is namedˑˑa  KU-ideal if  

(1)  0 ∈ 𝐼 

(2) If  χ ∗ (𝛾 ∗ 𝜏) ∈ 𝐼  , and 𝛾 ∈ 𝐼  imply that  χ ∗ 𝜏 ∈ 𝐼  . 

Definition(5)[13]. An algebra-KU--semi group is-a structure containss- a nonempty- set ℵ -with 

two- binary operations ∗,∘ and a-constant 0 satisfying the following 

(I) The set ℵ with operation  ∗ and constant 0 isˑ KU-algebra 

(II) The set ℵ with operation ∘  is semigroup. 

 (III) χ ∘ (𝛾 ∗ τ) = (χ ∘ 𝛾) ∗ (χ ∘ τ),and(χ ∗ 𝛾)∘ τ = (χ ∘ τ) ∗ (𝛾 ∘ τ), for all χ , 𝛾,τ ∈ ℵ. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

49 

Definition(6)[13]. A non-empty subset A of  ℵ is called a sub-KU-semi group of ℵ if χ ∗ 𝛾 ∈ 𝑨  , 

and χ ∘ 𝛾 ∈ 𝑨 , for all  χ,𝛾 ∈ 𝑨   

Definition(7)[13]. In a KU-semi group (ℵ,∗,∘ ,0), the subset  𝜑 ≠ 𝐼 of ℵ  is said to be S ideal -, if 
- 

(i) It is an ideal in a KU-algebra 

(ii) χ ∘ 𝑎 ∈ 𝑰 , and 𝑎 ∘ χ ∈ 𝑰 , ∀ χ ∈ ℵ , 𝑎 ∈ 𝑰 

 

Definition(8)[13]. In KU-semigroup (ℵ,∗,∘ ,0), the subset  𝜑 ≠ 𝐴 of ℵ  is named a -k--ideal-, - if  - 

(i) It is a KU-ideal of ℵ  

(ii) χ ∘ 𝑎 ∈ 𝑰 , and 𝑎 ∘ χ ∈ 𝑰 , ∀ χ ∈ ℵ , 𝑎 ∈ 𝑰 

In this part , we recall some concepts of fuzzy logic 

A function μ: ℵ ⟶ [0,1]  is said to be a fuzzy set of a set ℵ,and the set 

𝑈(𝜇,𝑡) = {𝜒 ∈ ℵ ∶ 𝜇(𝜒) ≥ 𝑡}ˑis said to be a level set of  𝜇, for t, where 1 ≥ 𝑡 ≥ 0 

Now, an interval valued fuzzy set 𝜇 of ℵ  is defined as follows:  

Remark(9)[7-8].A function 𝜇: ℵ ⟶ 𝐷[0,1], where D[0,1] is a family of the closed sub ̂-

intervals ̂of[0, 1]. The level subset of 𝜇~is ~denoted ~by 𝜇�̃�and it is ~defined ~by 

𝜇�̃� ={𝜒 ∈ ℵ:𝜇(𝜒) ≥ �̃�}, for every ~[0,0] ≤ �̃� ≤ [1,1]. 

O. Hasan and F.Kareem [7-8] introduced the Cubic ideals of the KU-semigroup as follows: 

Definition(10)[7-8]. In the KU-semigroup(ℵ,∗,∘ ,0), a cubic set Θ is the form 

 Θ = {〈𝜒,𝜇Θ(𝜒),𝜆Θ(𝜒)〉:𝜒 ∈ ℵ}, such that 𝜆Θ(𝜒)"is a fuzzy" set and   𝜇Θ:ℵ → 𝐷[0,1] "is an" 

interval-valued", briefly Θ = 〈𝜇Θ,𝜆Θ〉.  

Definition(11)[7-8]. In  the KU-semigroup  (ℵ,∗,∘ ,0) a cubic set  Θ = 〈𝜇Θ,𝜆Θ〉in ℵ is named a 

cubic sub-KU-semigroup if: for all 𝜒,𝛾 ∈ ℵ, 

(1) �̃�Θ(𝜒 ∗ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {�̃�Θ(𝜒)ˑ, �̃�Θ(𝛾)},𝜆Θ(𝜒 ∗ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ(𝜒)ˑ,𝜆Θ(𝛾)} 

(2) �̃�Θ(𝜒 ∘ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {�̃�Θ(𝜒)ˑ, ˑ�̃�Θ(𝛾)}, 𝜆Θ(𝜒 ∘ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ(𝜒)ˑ, ˑ𝜆Θ(𝛾)}. 

 

Definition(12)[7-8]. The set Θ in ℵ is named a cubic ideal of a KU-semigroup 

(ℵ,∗,∘ ,0) if, ∀ 𝜒 ,𝛾 ∈  ℵ 

(CI1) 𝜇𝛩(0) ≥ 𝜇𝛩(𝜒) and 𝜆𝛩(0) ≤ 𝜆𝛩(𝜒), 

(CI2) μ̃Θ(γ) ≥ 𝑟𝑚𝑖𝑛{�̃�𝛩(𝜒 ∗ 𝛾),𝜇𝛩(𝜒)} ,   𝜆𝛩(𝛾) ≤ 𝑚𝑎𝑥 {λΘ(χ ∗ γ)ˑ,λΘ(χ)} 

(CI3) μ̃Θ(𝜒 ∘ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {�̃�Θ(𝜒), ˑ𝜇Θ(𝛾)}, 𝜆Θ(𝜒 ∘ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ(𝜒)ˑ, ˑ𝜆Θ(𝛾)}. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

50 

Example(13)[7-8]. Let ℵ = {0,1,2}  be a set. Define the operations ∗,∘ by the following tables. 

 

 

 

 

Then the structure (ℵ,∗,∘ ,0)
ˑ
is a KU-semi group. A cubic set  Θ = 〈𝜇Θ,𝜆Θ〉  is defined by: 

𝜇Θ(𝑥) = {
[0.4,0.8]           𝑖𝑓  𝜒ϵ {0,2}      

[0.1,0.3]      𝑖𝑓       𝜒 = 1        
and  𝜆Θ(𝑥) = {

0.1       𝑖𝑓  𝜒ϵ{0,2}        
0.3    𝑖𝑓       𝜒 = 1         

 

Then Θ = 〈𝜇Θ,𝜆Θ〉 is a cubic ideal of ℵ . 

Definition(14)[7-8]. In  a KU-semigroup (ℵ,∗,∘ ,0), a cubic set Θ = 〈𝜇Θ,𝜆Θ〉 in ℵ is named a 

cubic k-ideal if  ∀ 𝜒 ,𝛾, τ ∈  ℵ 

(𝑪𝒌𝟏 )�̃�Θ(0)) ≥ 𝜇Θ(𝜒) , and 𝜆Θ(0) ≤ 𝜆Θ(𝑥) 

(𝑪𝒌𝟐)�̃�Θ(𝜒 ∗ τ) ≥ 𝑟𝑚𝑖𝑛{�̃�Θ(𝜒 ∗ (𝛾 ∗ τ)),𝜇Θ(𝛾)}, 

𝜆Θ(𝜒 ∗ τ) ≤ 𝑚𝑎𝑥 {𝜆Θ(𝜒 ∗ (𝛾 ∗ τ)),𝜆Θ(𝛾)} 

(𝑪𝒌𝟑)�̃�Θ(𝜒 ∘ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {�̃�Θ(𝜒)ˑ,𝜇Θ(𝛾)}   ,𝜆Θ(𝜒 ∘ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ(𝜒)ˑ, ˑ𝜆Θ(𝛾)}. 

In the following ,we recall some basic concepts of a bipolar fuzzy set.  

Definition(15)[9]. A bipolar fuzzy set   in a set ℵ is a form }:))(),(,{(    , 

where ]0,1[:)( 

 and ]1,0[:)( 


  are two fuzzy mappings. The two membership-

degrees )(


and )(


denote- the fulfillment -degree of- to the property corresponding of 

 and-the fulfillment-degree of  to some implicit- counter-property of-  , respectively.  

 

Kareem and Awad[10] introduced the cubic bipolar -ideals- of  a KU--semigroup- in -KU-algebra 

as- follows: -- 

Definition(16)[10]. Let ℵ be a non-empty set. A cubic bipolar set in a set ℵ is the structure Θ =

{〈χ,𝜇Θ
+(χ),𝜇Θ

−(χ),𝜆Θ
+(χ),𝜆Θ

−(χ):χ ∈ ℵ〉} is denoted as 

  Θ = 〈𝑁,𝐾〉, where  𝑁(χ) = {�̃�Θ
+(χ),𝜇Θ

−(χ)} is called interval-valued bipolar fuzzy set and  

𝐾(χ) = {𝜆Θ
+(χ),𝜆Θ

−(χ)} is a bipolar fuzzy set. Consider  𝜇Θ
+:ℵ → 𝐷[0,1] such that 𝜇Θ

+(χ) =

[𝜉ΘL
+ (χ),𝜉ΘU

+ (χ)] and  

 𝜇Θ
−:ℵ → 𝐷[−1,0] such that 𝜇Θ

−(χ) = [𝜉ΘL
− (χ),𝜉ΘU

− (χ)] , also  𝜆Θ
+:ℵ → [0,1] and 𝜆Θ

−:ℵ → [−1,0] it 

follows that  

∗ 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 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

51 

Θ = {< χ,{[𝜉ΘL
+ (χ),𝜉ΘU

+ (χ)],[𝜉ΘL
− (χ),𝜉ΘU

− (χ)]}, 𝜆Θ
+(χ),𝜆Θ

−(χ)} >:χ ∈ ℵ} 

Definition(17)[10]. A (CB) Θ = 〈𝑁,𝐾〉 in ℵ is named  a (CB) sub-KU-semigroup if: ∀𝜒,𝛾 ∈ ℵ, 

(1) 𝜇Θ
+(𝜒 ∗ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {𝜇Θ

+(𝜒)ˑ,𝜇Θ
+(𝛾)}, 𝜇Θ

−(𝜒 ∗ 𝛾) ≤ 𝑟𝑚𝑎𝑥 {𝜇Θ
−(𝜒)ˑ,𝜇Θ

−(𝛾)} 

    𝜆Θ
+(𝜒 ∗ 𝛾) ≥ 𝑚𝑖𝑛 {𝜆Θ

+(𝜒)ˑ,𝜆Θ
+(𝛾)}, 𝜆Θ

−(𝜒 ∗ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ
−(𝜒)ˑ,𝜆Θ

−(𝛾)},  

(2)  𝜇Θ
+(𝜒 ∘ 𝛾) ≥ 𝑟𝑚𝑖𝑛 {𝜇Θ

+(𝜒)ˑ,𝜇Θ
+(𝛾)}, 𝜇Θ

−(𝜒 ∘ 𝛾) ≤ 𝑟𝑚𝑎𝑥 {𝜇Θ
−(𝜒)ˑ,𝜇Θ

−(𝛾)} 

    𝜆Θ
+(𝜒 ∘ 𝛾) ≥ 𝑚𝑖𝑛 {𝜆Θ

+(𝜒)ˑ,𝜆Θ
+(𝛾)}, 𝜆Θ

−(𝜒 ∘ 𝛾) ≤ 𝑚𝑎𝑥 {𝜆Θ
−(𝜒)ˑ,𝜆Θ

−(𝛾)}, 

Example(18)[10]: The following table is Illustrates that the set ˑℵ = {0,1,2,3} with binary 

operations ∗ and  ∘ 

 

Then(ℵ,∗,∘ ,0)
ˑ
is a KU-semigroup. Define Θ = 〈𝑁,𝐾〉 as follows  

𝑀(𝑥) = {
{[−0.2,−0.5], [0.1,0.9]}      𝑖𝑓       𝜒 = {0,1}

{[−0.6,−0.2], [0.2,0.5]}      𝑖𝑓   𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  
,  

  𝜆Θ
+(𝑥) = {

0.5                   𝑖𝑓      𝜒 = {0,1}

  0.3                𝑖𝑓     𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒    
 𝜆Θ
−(𝑥) = {

−0.6                      𝑖𝑓      𝜒 = {0,1}

 −0.3                 𝑖𝑓     𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒    
 

And by applying definition 2.17, we can easily prove that Θ = 〈𝑁,𝐾〉 is a cubic bipolar sub KU-

semigroup of ℵˑ.
 

3. Cubic bipolar ideals of a KU-semi group with thresholds (𝜶, 𝜷), (𝝎 ,𝝑)   

In this part, the notion of cubic bipolar k-ideals with thresholds (𝛼, 𝛽), (𝜔 ,𝜗)  of a KU-semi 

group and some properties are defined. In the following, we denote a cubic bipolar fuzzy set by 

(CBF) ,and let   𝛼, 𝛽 ∈ 𝐷[0,1] ,𝑎𝑛𝑑,𝜔 ,𝜗 ∈ [0,1] ,𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  

 [0,0] < 𝛼 < 𝛽 < [1,1]        ,0 < 𝜔 <  𝜗 < 1, where  𝜔,𝜗 are arbitrary  values, and 

𝛼,𝛽, are arbitrary closed sub-intervals  

Definition(19). A (CBF) set  Θ = 〈𝑀,𝐿〉 is named a (CBF) sub-KU-semi group with thresholds  

(𝛼, 𝛽), (𝜔 ,𝜗) if ∀ 𝜒,𝛾 ∈ ℵ 

 (1)𝑚𝑖𝑛{𝜇Θ
−(𝜒 ∗ 𝛾),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(𝜒),𝜇Θ
−(𝛾),−𝛽} 

    𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝛾),𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝜇Θ
+(𝛾),𝛽} 

    𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∗ 𝛾),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),𝜆Θ
−(𝛾),−𝜗} 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

52 

    𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∗ 𝛾),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒),𝜆Θ
+(𝛾),𝜗}  

(2)𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒 ∘ 𝛾),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),𝜇Θ
−(𝛾),−𝛽}  

          𝑟𝑚𝑎𝑥{�̃�Θ
+(𝜒 ∘ 𝛾),𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝜇Θ
+(𝛾),𝛽}  

    𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∘ 𝛾),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),𝜆Θ
−(𝛾),−𝜗} 

            𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∘ 𝛾),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒),𝜆Θ
+(𝛾),𝜗} 

Remark(20). Every (CBF) sub-KU-semi group of ℵ is a (CBF) sub-KU-semigroup with 

thresholds (𝜶, 𝜷), (𝝎 ,𝝑) , but not converse as it is shown in the following example 

Example(21).Let ℵ = {0,1,2,3} be a set with two operations ∗ and  which are defined by the 

following tables.           

 

 

 

 

 

 

 

Then(ℵ,∗,∘ ,0)
ˑ
is a KU-semi group.Now, we define Θ = 〈𝑀,𝐿〉 by the next 

   

𝑀(𝑥) =  

{
 

 
[ −0.9,−0.8 ] , [0.8, 0.9  ]          𝑖𝑓    𝜒 = 0
[−0.8,−0.7], [0.7, 0.8]         𝑖𝑓     𝜒 = 1
[−0.6,−0.5] , [0.5, 0.6]        𝑖𝑓     𝜒 = 3
[−0.3,−0.2], [0.2 ,   0.3]        𝑖𝑓     𝜒 = 2

   

  𝐿(𝑥) =  {

−0.9,        0.9        𝑖𝑓   𝜒 = 0
−0.5,       0.6         𝑖𝑓  𝜒 = 1
−0.4 ,      0.5        𝑖𝑓   𝜒 = 3
−0.2 ,      0.2       𝑖𝑓   𝜒 = 2

   

And by applying definition (19), we can easily prove that Θ = 〈𝑀,𝐿〉 is a(CBF)sub KU-semi 

group with thresholds  (𝛼,𝛽) = ( [0.1,0.2], [0.2, 0.2]), and (𝜔,𝜗)=(0.1,0.2)  , but not a (CBF)sub 

KU-semi group  since 

𝜇Θ
+(1 ∗ 3) ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(1),𝜇Θ
+(3)} 

{𝜇Θ
+(2)} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ

+(1),𝜇Θ
+(3)} 

[0.2 ,0.3] ≥ 𝑟𝑚𝑖𝑛{[0.7,0.8], [0.5,0.6]} 

[0.2, 0.1] ≥ [0.5,0.6] ,which is incorrect phrase  

𝜇Θ
−(1 ∗ 3) ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(1)ˑ,𝜇Θ
−(3)} 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

53 

𝜇Θ
−(2) ≤ 𝑟𝑚𝑎𝑥{[−0.8 ,−0.7], [−0.6 ,−0.5]} 

[−0.3, −0.2≤ [−0.6 ,−0.5], which is the incorrect phrase, and 

𝜆Θ
+(1 ∗ 3) ≥ 𝑚𝑖𝑛{𝜆Θ

+(1)ˑ,𝜆Θ
+(3)} 

𝜆Θ
+(2) ≥ 𝑚𝑖𝑛{0.6ˑ,0.5} 

0.2 ≥ 0.5, it is  wrong  

𝜆Θ
−(1 ∗ 3) ≤ 𝑚𝑎𝑥 {𝜆Θ

−(1)ˑ,𝜆Θ
−(3)} 

𝜆Θ
−(2) ≤ 𝑚𝑎𝑥 {−0.5ˑ,−0.4} 

−0.2 ≤ −0.4, which is also wrong.      

Remark(22). If Θ = 〈𝑀,𝐿〉 is a (CBF) sub KU-semi group  with thresholds (α, β), (ω ,ϑ)  such 

that    α = [0,0],  β = [1,1, ] , ω = 0,  and   ϑ = 1 ,then  Θ = 〈𝑀,𝐿〉  is a (CBF) sub-KU-semi 

group of  ℵ. 

Proposition(23).If  Θ = 〈𝑀,𝐿〉 is a cubic bipolar sub-KU-semi group with thresholds (𝛼, 𝛽), 

(𝜔 ,𝜗) of  ℵ ,then for all 𝜒 ∈ ℵ 

(1) 𝑟𝑚𝑎𝑥{𝜇Θ
+(0), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒), 𝛽}  

  (2) 𝑟𝑚𝑖𝑛{𝜇Θ
−(0),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽}  

  (3) 𝑚𝑎𝑥{𝜆Θ
+(0), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒), 𝜗} 

(4 )𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗} 

 semi group-KU-sub bipolar is a cubicΘ = 〈𝑀,𝐿〉   and  since,       𝜒 ∗ 𝜒 = 0)5ku(by  Proof:   

with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ ,  

𝑟𝑚𝑎𝑥{𝜇Θ
+(0), 𝛼} = 𝑟𝑚𝑎𝑥{�̃�Θ

+(𝜒 ∗ 𝜒), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ
+(𝜒),𝜇Θ

+(𝜒), 𝛽} 

                                = 𝑟𝑚𝑖𝑛{�̃�Θ
+(𝜒),𝛽}, that is (1) 

𝑟𝑚𝑖𝑛{𝜇Θ
−(0),−𝛼} = 𝑟𝑚𝑖𝑛{𝜇Θ

−(𝜒 ∗ 𝜒),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ
−(𝜒),𝜇Θ

−(𝜒),−𝛽} 

                              = 𝑟𝑚𝑎𝑥{𝜇Θ
−(𝜒),−𝛽} ,that is (2)    

   𝑚𝑎𝑥{𝜆Θ
+(0), 𝜔} = 𝑚𝑎𝑥{𝜆Θ

+(𝜒 ∗ 𝜒),𝜔} ≥ 𝑚𝑖𝑛 {𝜆Θ
+(𝜒),𝜆Θ

+(𝜒), 𝜗} 

                                 = 𝑚𝑖𝑛 {𝜆Θ
+(𝜒) ,𝜗} ,that is (3) 

𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} = 𝑚𝑖𝑛{𝜆Θ

−(𝜒 ∗ 𝜒),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ
−(𝜒),𝜆Θ

−(𝜒),−𝜗} 

                             = 𝑚𝑎𝑥{𝜆Θ
−(𝜒),−𝜗},that is (4)  

Proposition(24).If  Θ = 〈𝑀,𝐿〉 is a (CBF) sub-KU-semi group with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  

ℵ ,then for all 𝜒 ∈ ℵ 

(1) 𝑟𝑚𝑎𝑥{𝜇Θ
+(0 ∘ 𝜒), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒), 𝛽}  

(2) 𝑟𝑚𝑖𝑛{𝜇Θ
−(0 ∘ 𝜒),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽}  

(3) 𝑚𝑎𝑥{𝜆Θ
+(0 ∘ 𝜒), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒), 𝜗} 

(4 )𝑚𝑖𝑛{𝜆Θ
−(0 ∘ 𝜒),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗} 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

54 

Proof: Since  Θ = 〈𝑀,𝐿〉 is a (CBF) sub-KU-semi group with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ , 

we have 

   𝑟𝑚𝑎𝑥{𝜇Θ
+(0 ∘ 𝜒), 𝛼} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ

+(0), 𝜇Θ
+(𝜒), 𝛽} = 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒), 𝛽 }, which is (1) 

                                 

 𝑟𝑚𝑖𝑛{𝜇Θ
−(0 ∘ 𝜒),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(0), 𝜇Θ
−(𝜒) − 𝛽} = 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽}, which is (2) 

𝑚𝑎𝑥{𝜆Θ
+(0 ∘ 𝜒),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(0),𝜆Θ
+(𝜒), 𝜗} = 𝑚𝑖𝑛 {𝜆Θ

+(𝜒), 𝜗}, which is (3)  

𝑚𝑖𝑛{𝜆Θ
−(0 ∘ 𝜒),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(0),𝜆Θ
−(𝜒),−𝜗} = 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗},which is (4)  

 

Definition(25). A (CBF) set Θ = 〈𝑀,𝐿〉 is named a (CBF) ideal of the KU-semi group with 

thresholds  (𝛼, 𝛽), (𝜔 ,𝜗) if ∀  𝜒,𝛾 ∈ ℵ 

(CBT1 ) 𝑟𝑚𝑖𝑛{�̃�Θ
−(0),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽} 

      𝑟𝑚𝑎𝑥{𝜇Θ
+(0), 𝛼} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒), 𝛽},and 

      𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗} 

      𝑚𝑎𝑥{𝜆Θ
+(0), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒), 𝜗}  

 (CBT2) 𝑟𝑚𝑖𝑛{�̃�Θ
−(𝛾),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒 ∗ 𝛾),𝜇Θ
−(𝜒),−𝛽} 

      𝑟𝑚𝑎𝑥{𝜇Θ
+(𝛾), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒 ∗ 𝛾),𝜇Θ
+(𝜒), 𝛽} 

      𝑚𝑖𝑛{𝜆Θ
−(𝛾),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒 ∗ 𝛾),𝜆Θ
−(𝜒),−𝜗} 

      𝑚𝑎𝑥{𝜆Θ
+(𝛾), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒 ∗ 𝛾),𝜆Θ
+(𝜒),   𝜗}  

 (CBT3)𝑟𝑚𝑖𝑛{�̃�Θ
−(𝜒 ∘ 𝛾),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),𝜇Θ
−(𝛾),−𝛽}  

      𝑟𝑚𝑎𝑥{�̃�Θ
+(𝜒 ∘ 𝛾), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝜇Θ
+(𝛾), 𝛽}  

𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∘ 𝛾),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),𝜆Θ
−(𝛾),−𝜗} 

        𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∘ 𝛾), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒),𝜆Θ
+(𝛾),   𝜗} 

Example(26).The following table Illustrates the set ˑℵ = {0,1,2} with binary operations ∗ and  ∘ 

   

 

 

 

 

 

Then(ℵ,∗,∘ ,0)
ˑ
is a KU-semigroup. Define Θ = 〈𝑀,𝐿〉 as follows: 

𝑀(𝜒) = {

[−0.8,−0.7] , [0.6,0.8]         𝑖𝑓  𝜒 = 0 
[−0.6,−0.5] , [0.4,0.6]         𝑖𝑓  𝜒 = 1 
[−0.4,−0.3] , [0.3,0.2]         𝑖𝑓  𝜒 = 2 

 

∗ 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 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

55 

 

𝐿(𝜒){

−0.6 ,     0.8                     𝑖𝑓  𝜒 = 0   
−0.5 ,     0.6                     𝑖𝑓  𝜒 = 1   
−0.3 ,     0.3                     𝑖𝑓  𝜒 = 2   

 

We can show that Θ = 〈𝑀,𝐿〉 is a (CBF) ideal with thresholds ([0.1, 0.1],  [0.3,0.2]) and (0.4, 

0.2) of ℵ 

Definition(27). A (CBF)set Θ = 〈𝑀,𝐿〉 is named a (CBF)k-ideal of KU-semigroup with 

thresholds  (𝛼, 𝛽), (𝜔 ,𝜗) if ∀ 𝜒,𝛾,𝜏 ∈ ℵ 

 (CBҠ1)𝑟𝑚𝑖𝑛{�̃�Θ
−(0),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽} 

      𝑟𝑚𝑎𝑥{𝜇Θ
+(0), 𝛼} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒), 𝛽} 

      𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗} 

      𝑚𝑎𝑥{𝜆Θ
+(0), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒), 𝜗}  

(CBҠ2) 𝑟𝑚𝑖𝑛{�̃�Θ
−(𝜒 ∗ 𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜇Θ
−(𝛾),−𝛽} 

      𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝜏), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
+(𝛾), 𝛽} 

      𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∗ 𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
−(𝛾),−𝜗} 

      𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∗ 𝜏), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
+(𝛾), 𝜗}  

(CBҠ3)𝑟𝑚𝑖𝑛{�̃�Θ
−(𝜒 ∘ 𝛾),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),𝜇Θ
−(𝛾),−𝛽}  

      𝑟𝑚𝑎𝑥{�̃�Θ
+(𝜒 ∘ 𝛾), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝜇Θ
+(𝛾), 𝛽}  

𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∘ 𝛾),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),𝜆Θ
−(𝛾),−𝜗} 

        𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∘ 𝛾), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒),𝜆Θ
+(𝛾), 𝜗} 

Lemma(28). Every (CBF) k-ideal of ℵ is a (CBF) k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ  

 then let,     ℵideal of-k )CBF( is a  Θ = 〈𝑀,𝐿〉Suppose that Proof: 

𝑟𝑚𝑎𝑥{𝜇Θ
+(0),𝛼} < 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝛽},and 𝛼 < 𝛽 it follows that 𝜇Θ
+(0) < 𝜇Θ

+(𝜒). But that is a 

contradiction, since Θ is a(CBF) k-ideal of ℵ  ,   

𝑟𝑚𝑎𝑥{𝜇Θ
+(0),𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝜒),𝛽},  

also let   𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} > 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗},  and 𝜔 < 𝜗 , it follows that 

𝜆Θ
−(0) > 𝜆Θ

−(𝜒); this is a contradiction since Θ is a(CBF) k-ideal of ℵ  . this means that  

𝑚𝑖𝑛{𝜆Θ
−(0),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒),−𝜗}, in the same way, we can prove  

𝑟𝑚𝑖𝑛{𝜇Θ
−(0),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒),−𝛽}, and  𝑚𝑎𝑥{𝜆Θ
+(0),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒),𝜗} 

 Again , assume that    

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝜏),𝛼} < 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
+(𝛾),𝛽}, and  𝛼 < 𝛽 it follows that 

𝜇Θ
+(𝜒 ∗ 𝜏) < 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
+(𝛾)} , which is a contradiction, so  



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

56 

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝜏),𝛼} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
+(𝛾),𝛽},  

Also let 

𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∗ 𝜏),−𝜔} > 𝑚𝑎𝑥{𝜆Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
−(𝛾),−𝜗} , and  𝜔 < 𝜗, so 

𝜆Θ
−(𝜒 ∗ 𝜏) > 𝑚𝑎𝑥{𝜆Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
−(𝛾)}  , which is a contradiction. That is   𝑚𝑖𝑛{𝜆Θ

−(𝜒 ∗

𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ
−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ

−(𝛾),−𝜗} 

In the same way, we get  

𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒 ∗ 𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜇Θ
−(𝛾),−𝛽} 

𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∗ 𝜏),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
+(𝛾),𝜗},and the condition (CBҠ3) 

Then,Θ = 〈𝑀,𝐿〉is a (CBF) k-ideal with thresholds (𝛼, 𝛽),(𝜔 ,𝜗) of  ℵ. 

 

Proposition(29). Let Θ = 〈𝑀,𝐿〉 be a cubic bipolar k-ideal with thresholds  (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ  

if  𝜒 ≤ 𝛾 , then  

(a) 𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝛾),−𝛽},  𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝛾), 𝛽}   

 (b) 𝑚𝑖𝑛{𝜆Θ
−(𝜒),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝛾),−𝜗}   ,   𝑚𝑎𝑥{𝜆Θ
+(𝜒), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝛾), 𝜗}  

 

Proof: Since  𝜒 ≤ 𝛾 ,then  𝛾 ∗ 𝜒 = 0 ,and by (ku3) 0 ∗ 𝜒 = 𝜒  

 Since Θ = 〈𝑀,𝐿〉 is a (CB) k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of ℵ ,we get 

𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒),−𝛼} = 𝑟𝑚𝑖𝑛{𝜇Θ

−(0 ∗ 𝜒),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ
−(0 ∗ (𝛾 ∗ 𝜒)),𝜇Θ

−(𝛾),−𝛽}

= 𝑟𝑚𝑎𝑥{𝜇Θ
−(0 ∗ 0),𝜇Θ

−(𝛾),−𝛽} 

                = 𝑟𝑚𝑎𝑥{𝜇Θ
−(0),𝜇Θ

−(𝛾),−𝛽}  

                  = 𝑟𝑚𝑎𝑥{𝜇Θ
−(𝛾),−𝛽} 

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒),𝛼} = 𝑟𝑚𝑎𝑥{𝜇Θ

+(0 ∗ 𝜒),𝛼} ≥ 𝑟𝑚𝑖𝑛{�̃�Θ
+(0 ∗ (𝛾 ∗ 𝜒)),𝜇Θ

+(𝛾),𝛽} 

                     = 𝑟𝑚𝑖𝑛{�̃�Θ
+(0 ∗ 0),𝜇Θ

+(𝛾), 𝛽}  

                     = 𝑟𝑚𝑖𝑛{�̃�Θ
+(0),𝜇Θ

+(𝛾), 𝛽} 

                      = 𝑟𝑚𝑖𝑛{�̃�Θ
+(𝛾), 𝛽},which is (a) ,     And 

𝑚𝑖𝑛{𝜆Θ
−(𝜒),−𝜔} = 𝑚𝑖𝑛{𝜆Θ

−(0 ∗ 𝜒),−𝜔}                                                          

                            ≤ 𝑚𝑎𝑥{𝜆Θ
−(0 ∗ (𝛾 ∗ 𝜒)),𝜆Θ

−(𝛾),−𝜗}   

                            = 𝑚𝑎𝑥{𝜆Θ
−(0 ∗ 0),𝜆Θ

−(𝛾),−𝜗}   

                            = 𝑚𝑎𝑥{𝜆Θ
−(0),𝜆Θ

−(𝛾),−𝜗} 

                            = 𝑚𝑎𝑥{𝜆Θ
−(𝛾),−𝜗}                               

𝑚𝑎𝑥{𝜆Θ
+(𝜒),𝜔} = 𝑚𝑎𝑥{𝜆Θ

+(0 ∗ 𝜒),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ
+(0 ∗ (𝛾 ∗ 𝜒)),𝜆Θ

+(𝛾),𝜗} 

                                                     = 𝑚𝑖𝑛{𝜆Θ
+(0 ∗ 0),𝜆Θ

+(𝛾), 𝜗} 

                                                              = 𝑚𝑖𝑛{𝜆Θ
+(0),𝜆Θ

+(𝛾), 𝜗} 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

57 

                                                              = 𝑚𝑖𝑛{𝜆Θ
+(𝛾), 𝜗},which is(b)  

Theorem(30).Let Θ = 〈𝑀,𝐿〉 be a cubic bipolar fuzzy set of a KUsemigroup(ℵ,∗,∘ ,0)  then, Θ is 

a (CBF) k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ if and only if  it is a (CBF)-ideal with 

thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ . 

Proof: ⇒ Let Θ = 〈𝑀,𝐿〉 be a cubic bipolar k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ ,if we put 

𝜒 = 0  in (CBҠ2) , we get   

𝑟𝑚𝑖𝑛{𝜇Θ
−(0 ∗ 𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(0 ∗ (𝛾 ∗ 𝜏)),𝜇Θ
−(𝛾),−𝛽} is  

𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{�̃�Θ

−(𝛾 ∗ 𝜏),𝜇Θ
−(𝛾),−𝛽}  

,also       𝑟𝑚𝑎𝑥{�̃�Θ
+(0 ∗ 𝜏),𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(0 ∗ (𝛾 ∗ 𝜏),𝜇Θ
−(𝛾), 𝛽} is 

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜏), 𝛼} ≥ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝛾 ∗ 𝜏),𝜇Θ
−(𝛾), 𝛽} ,and 

𝑚𝑖𝑛{𝜆Θ
−(0 ∗ 𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(0 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
−(𝛾),−𝜗} is  

𝑚𝑖𝑛{𝜆Θ
−(𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝛾 ∗ 𝜏)),𝜆Θ
−(𝛾),−𝜗},  

Also    𝑚𝑎𝑥{𝜆Θ
+(0 ∗ 𝜏), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(0 ∗ (𝛾 ∗ 𝜏)),𝜆Θ
+(𝛾), 𝜗 }  

𝑚𝑎𝑥{𝜆Θ
+(𝜏), 𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝛾 ∗ 𝜏),𝜆Θ
+(𝛾), 𝜗},the other conditions (CBT1), (CBT3) are holds from 

the definition of (CBF)k-ideal; therefore Θ = 〈𝑀,𝐿〉is a (CB)-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗)  

of  ℵ 

⇐ Let Θ = 〈𝑀,𝐿〉 be a cubic bipolar ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of  ℵ , 

 By (CBT2) 𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒 ∗ 𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝛾 ∗ (𝜒 ∗ 𝜏),𝜇Θ
−(𝛾),−𝛽}, also  

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝜏), 𝛼} ≤ 𝑟𝑚𝑖𝑛{𝜇Θ

+(𝛾 ∗ (𝜒 ∗ 𝜏),𝜇Θ
+(𝛾), 𝛽},  

𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∗ 𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝛾 ∗ (𝜒 ∗ 𝜏),𝜆Θ
−(𝛾),−𝜗}, also 

𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∗ 𝜏),𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝛾 ∗ (𝜒 ∗ 𝜏),𝜆Θ
+(𝛾),𝜗} 

Applying theorem 2 (2) to the previous four steps ,we obtain 

𝑟𝑚𝑖𝑛{𝜇Θ
−(𝜒 ∗ 𝜏),−𝛼} ≤ 𝑟𝑚𝑎𝑥{𝜇Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
−(𝛾),−𝛽},  

𝑟𝑚𝑎𝑥{𝜇Θ
+(𝜒 ∗ 𝜏),𝛼} ≤ 𝑟𝑚𝑖𝑛{�̃�Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜇Θ
+(𝛾), 𝛽}  , and 

𝑚𝑖𝑛{𝜆Θ
−(𝜒 ∗ 𝜏),−𝜔} ≤ 𝑚𝑎𝑥{𝜆Θ

−(𝜒 ∗ (𝛾 ∗ 𝜏),𝜆Θ
−(𝛾),−𝜗},  

𝑚𝑎𝑥{𝜆Θ
+(𝜒 ∗ 𝜏) ,𝜔} ≥ 𝑚𝑖𝑛{𝜆Θ

+(𝜒 ∗ (𝛾 ∗ 𝜏),𝜆Θ
+(𝛾), 𝜗}, which is a (CBF) k-ideal, 

 The remaining two conditions (CBҠ1),(CBҠ3) are holds from the definition of (CBF)-ideal . 

4.Conclusion 

     During this work,   we present the definitions of the cubic bipolar sub-KU-semigroup with 

thresholds (𝛼, 𝛽), (𝜔 ,𝜗) and cubic bipolar k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗)  of  ℵ. The 

relationship among these types of ideals and some properties are studied, We obtained the 

following result: every (CBF) sub-KU-semi group of ℵ  is a (CBF) sub-KU-semi group with 

thresholds (𝛼, 𝛽), (𝜔 ,𝜗) of ℵ ,but the converse is not true. Finally, we proved that a cubic bipolar 

fuzzy k-ideal with thresholds (𝛼, 𝛽), (𝜔 ,𝜗) and a cubic bipolar fuzzy ideal with thresholds (𝛼, 𝛽), 

(𝜔 ,𝜗) of a KU-semi group are equivalents. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

58 

 References 

1. Zadeh, L.A. Fuzzy Sets, Inform and Control, 1965, 8, 338-353. 

2. Mostafa, S.M.; Abd-Elnaby, M.A.; Yousef, M.M.M.  Fuzzy ideals of KU-Algebras, Int. 

Math, Forum, 2011, 6(63), 3139-3149. 

3. Mostafa, S. M.; Kareem, F. F.bipolar fuzzy N-fold KU-ideals of KU-algebras, 

Mathematica Aeterna, 2014, 4, 633-650.  

4. Jun, Y. B.; Kim, C. S.; Kang, M. S.  Cubic subalgebras and ideals of BCK/BCI-algebras, 

Far East Journal of Mathematical Sciences, 2010, 2 (44) , 239–250. 

5. Jun, Y. B.; Kim, C. S.; Kang, J. G. Cubic q-ideals of BCIalgebras, Ann. Fuzzy 

Math.Inform. 2011, 1 (1), 25-34. 

6. Jun, Y. B.; Kim, C. S.; Yang, K. O.  Cubic sets, Ann. Fuzzy Math. Inform. 2012,4 (1) 83-

98. 

7. Kareem, F. F.; Hasan, O. A.  Cubic ideals of semigroup in KU-algebra, Journal of 

Physics: Conference Series 1804 (2021) 012018 IOP Publishing 

8. Kareem, F. F.; Hasan, O. A.  The Homomorphism of a cubic set of a semigroup in a KU-

algebra, Journal of Physics, 1879 (2021) 022119 IOP Publishing. 

9. Kareem, F.F.; Hasan, E. R.  Bipolar fuzzy k-ideals in KU-semigroups, Journal of New 

Theory, 2019,9, 71-78. 

10. K Awad, Wisam K., and Fatema F. Kareem. The Homomorphism of Cubic bipolar ideals 

of a KU-semigroup. Ibn AL-Haitham Journal For Pure and Applied Sciences,  2022, 

35(1)., 73-83. 

11. Kareem, F.F.; Abed, M. M. Generalizations of Fuzzy k-ideals in a KU-algebra with 

Semigroup, Journal of Physics, 2021, 1879,022108 IOP Publishing.  

12. Prabpayak, C. Leerawat, U. On ideals and congruence in KU-algebras, scientia Magna, 

2009, 5(1), 54-57. 

13. Kareem, F.F.; Hasan, E. R.  On KU-semigroups, International Journal of Science and 

Nature. 2018, 9 (1), 79-84.