Multipolar Intuitionistic Fuzzy Ideal in B-Algebras


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 293-301 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: November 17, 2021 Reviewed: December 10, 2021 Accepted: January 20, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i1.14003       

Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo*, Noor Hidayat, Vira Hari Krisnawati 

Department of Mathematics, University of Brawijaya, Malang, Indonesia 
 

*Corresponding Author 
Email: amigo.royyan@yahoo.com*  

ABSTRACT 

 

B-algebra is an algebraic structure which combine some properties from 𝐵𝐶𝐾-algebras and        
𝐵𝐶𝐼-algebras. Some researchers have investigated the concept of multipolar fuzzy ideals in 
𝐵𝐶𝐾/𝐵𝐶𝐼-algebras and multipolar intuitionistic fuzzy set in 𝐵-algebras. In this paper, we construct 
a new structure which is called a multipolar intuitionistic fuzzy ideal in 𝐵-algebras.  This structure 
is a combination of three structures such as multipolar fuzzy ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras, fuzzy 
𝐵-subalgebras in 𝐵-algebras, and multipolar intuitionistic fuzzy 𝐵-algebras. We investigated and 
proved some characterizes of the multipolar intuitionistic fuzzy ideal, such as a necessary 
condition and sufficient condition.  

Keywords: B-algebras; multipolar fuzzy ideal; multipolar intuitionistic fuzzy set; multipolar 
intuitionistic fuzzy ideal 

 

INTRODUCTION 

Zadeh [1] introduced a new idea, namely a fuzzy set as a non-empty set with a 
degree of membership whose value in interval [0,1] in 1965. The degree of membership 
of each member of the set is determined by the membership function. That notion from 
Zadeh became the basis for further researchers to develop fuzzy concepts in various fields 
such as graph theory, data analysis, decision making, and so on. 

A simple example of an algebraic structure is a group. Not only groups,                    
𝐵𝐶𝐾-algebras, 𝐵𝐶𝐼-algebras and 𝐵-algebras are also other examples of algebraic 
structures. Imai and Iseki [2] proposed the notion a new algebraic structure called       
𝐵𝐶𝐾-algebras in 1966. 𝐵𝐶𝐾-algebras is an important class of algebraic structure which is 
constructed from two different fragments, set theory and propositional calculus. In the 
same year, Iseki [3] continued his research to propose the notion of 𝐵𝐶𝐼-algebras which 
is generalization from 𝐵𝐶𝐾-algebras. A new idea about algebraic structure is called 𝐵-
algebras which satisfies some properties from 𝐵𝐶𝐾-algebras and 𝐵𝐶𝐼-algebras was 
proposed by Neggers and Kim in [4]. They also investigated its properties. 

Zhang [5] introduced the concepts of bipolar fuzzy sets which is the extension of 
fuzzy set. Meng [6] studied about fuzzy implicative ideals in 𝐵𝐶𝐾-algebras in 1997. 
Moreover, Muhiuddin and Al-Kadi [7] introduced bipolar fuzzy implicative ideals in    
𝐵𝐶𝐾-algebras. They discussed about the relationship between a bipolar fuzzy ideal and 
bipolar fuzzy implicative ideal. Furthermore, Chen et al. [8] introduced the concepts of 
multipolar fuzzy sets which is the extension of bipolar fuzzy set. Kang et al. [9] proposed 

http://dx.doi.org/10.18860/ca.v7i1.14003
mailto:amigo.royyan@yahoo.com


Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo 294 

the concepts about multipolar intuitionistic fuzzy set with finite degree and its application 
in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. 

In 1999, Attanasov [10] introduced the new notion about intuitionistic fuzzy set. 
Jun et al. [11] defined fuzzy 𝐵-algebras. Then, Al-Masarwah and Ahmad [12] discussed 
about multipolar fuzzy ideals in 𝐵𝐶𝐾/𝐵𝐶𝐼-algebras. Ahn and Bang [13] studied fuzzy                           
𝐵-subalgebras in 𝐵-algebras. Recently, Borzooei et al. [14] proposed the concept about 
multipolar intuitionistic fuzzy 𝐵-algebras and some properties. They constructed a simple    
multipolar fuzzy set. Then, they also discussed about multipolar intuitionistic fuzzy 
subalgebras of 𝐵-algebras. 
 In this paper, we construct a new structure which is called a multipolar 
intuitionistic fuzzy ideal in 𝐵-algebras.  This structure is a combination of three structures 
which are the results of research by Al-Masarwah and Ahmad [12], Ahn and Bang [13], 
and Borzooei et al. [14]. Next, we investigated and proved some necessary condition and 
sufficient condition of the multipolar intuitionistic fuzzy ideal.  
 

METHODS  

 By using literary study and analogical related concepts from [12], [13] and [14], 
we propose the terminology of multipolar intuitionistic fuzzy ideal in 𝐵-algebras. We start 
to describe the structure of 𝐵-algebra, fuzzy 𝐵-algebra, and multipolar intuitionistic fuzzy 
sets. Each structure is given its definition, examples, and some of its properties. 

 
Definition 2.1 [15] 𝐵-algebra is a nonempty set 𝑋 with 0 as identity element (right) and 
a binary operation ∗ satisfying the following axioms for all 𝑥, 𝑦, 𝑧 ∈ 𝑋:  

i. 𝑥 ∗ 𝑥 = 0.  
ii. 𝑥 ∗ 0 = 𝑥. 

iii. (𝑥 ∗ 𝑦) ∗ 𝑧 = 𝑥 ∗ (𝑧 ∗ (0 ∗ 𝑦)). 

 
For all 𝑥, 𝑦 ∈ 𝑋, we define a partial ordering relation " ≤ " on 𝑋 by 𝑥 ≤ 𝑦 if and only 

if 𝑥 ∗ 𝑦 = 0 ([14]). 
 

Example 2.2 [15] Let 𝑋 = {0, 𝑎, 𝑏, 𝑐} be a set with Cayley table as follows: 
 
                                       Table 1: Cayley table for (𝑋;∗ ,0). 

∗ 𝟎 𝒂 𝒃 𝒄 
𝟎 0 0 𝑏 𝑏 
𝒂 𝑎 0 𝑐 𝑏 
𝒃 𝑏 𝑏 0 0 
𝒄 𝑐 𝑏 𝑎 0 

 
Then, (𝑋;∗ ,0) is a 𝐵-algebra. 

 
Example 2.3 [15] Let (ℤ; −,0) with ′′ − ′′ be a substraction operation of integers ℤ. Then, 
(ℤ; −,0) is a 𝐵-algebra.  

 
Example 2.4 Let (ℝ+ − {0};∗ ,1) with ′′ ∗ ′′ be a binary operation of ℝ+ − {0} defined by 

  

𝑥 ∗ 𝑦 =
𝑥

𝑦
 . 

 



Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo 295 

Then, (ℝ+ − {0};∗ ,1) is a 𝐵-algebra. 
 

Proposition 2.5 [16] If (𝑋;∗ ,0) is a 𝐵-algebra, then for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies the following 
conditions. 
i. (𝑥 ∗ 𝑦) ∗ (0 ∗ 𝑦) = 𝑥. 

ii. 𝑥 ∗ (𝑦 ∗ 𝑧) = (𝑥 ∗ (0 ∗ 𝑧)) ∗ 𝑦. 

iii. If 𝑥 ∗ 𝑦 = 0 then 𝑥 = 𝑦. 
iv. 0 ∗ (0 ∗ 𝑥) = 𝑥. 
v. (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧) = 𝑥 ∗ 𝑦. 

vi. 0 ∗ (𝑥 ∗ 𝑦) = 𝑦 ∗ 𝑥. 
vii. 𝑥 ∗ 𝑦 = 0 if and only if 𝑦 ∗ 𝑥 = 0. 

viii. If 0 ∗ 𝑥 = 0 then 𝑋 contains only 0. 
 

Definition 2.6 [16] A 𝐵-algebra (𝑋;∗ ,0) is called commutative 𝐵-algebra if for all 𝑥, 𝑦 ∈ 𝑋 
satisfies: 

 
𝑥 ∗ (0 ∗ 𝑦) = 𝑦 ∗ (0 ∗ 𝑥). 

  
Example 2.7 Let (ℤ; −,0) with ′′ − ′′ be a substraction operation of integers ℤ. Then, 
(ℤ; −,0) is a commutative 𝐵-algebra.  

 
Proposition 2.8 [16] If (𝑋;∗ ,0) is a commutative 𝐵-algebra, then for all 𝑥, 𝑦, 𝑧, 𝑡 ∈ 𝑋 
satisfies the following rules. 
i. (0 ∗ 𝑥) ∗ (0 ∗ 𝑦) = 𝑦 ∗ 𝑥. 

ii. (𝑧 ∗ 𝑦) ∗ (𝑧 ∗ 𝑥) = 𝑥 ∗ 𝑦. 
iii. (𝑥 ∗ 𝑦) ∗ 𝑧 = (𝑥 ∗ 𝑧) ∗ 𝑦. 
iv. (𝑥 ∗ (𝑥 ∗ 𝑦)) ∗ 𝑦 = 0. 

v. (𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑡) = (𝑡 ∗ 𝑧) ∗ (𝑦 ∗ 𝑥). 
 

Definition 2.9 [15] Let (𝑋;∗ ,0) be a 𝐵-algebra. A nonempty subset 𝐼 of 𝑋 is called ideal of 
𝑋 if it satisfies:  
i. 0 ∈ 𝐼, 

ii. for all 𝑥, 𝑦 ∈ 𝑋, if 𝑦 ∈ 𝐼 and 𝑥 ∗ 𝑦 ∈ 𝐼 then 𝑥 ∈ 𝐼. 
 

Example 2.10 [15] Let 𝐼 = ℤ+ ⋃ {0} be a subset of 𝐵-algebra (ℤ; −,0), then 𝐼 is ideal of ℤ. 
   
Let (𝑋;∗ ,0) be a 𝐵-algebra. A non empty subset 𝐼 of 𝑋 is called subalgebras                                  

(𝐵-subalgebras) of 𝑋 if for all 𝑥, 𝑦 ∈ 𝐼 satisfies 0 ∈ 𝐼 and 𝑥 ∗ 𝑦 ∈ 𝐼 ([15]). 
 

Definition 2.11 [11] Let (𝑋;∗ ,0) be a 𝐵-algebra. A fuzzy set 𝐴 in 𝑋 is called fuzzy                        
𝐵-algebra if it satisfies the inequality for all 𝑥, 𝑦 ∈ 𝑋, 
 

𝜇𝐴(𝑥 ∗ 𝑦) ≥ min{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}. 
 

Let (𝑋;∗ ,0) be a 𝐵-algebra. A fuzzy set 𝐴 in 𝑋 is called fuzzy ideal 𝐵-algebra ([17]) 
if it satisfies for all 𝑥, 𝑦 ∈ 𝑋, 

  
𝜇𝐴(0) ≥ 𝜇𝐴(𝑥), 

𝜇𝐴(𝑥) ≥ min{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)}. 



Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo 296 

   
A 𝐵-algebra (𝑋;∗ ,0) in the Example 2.2. If we define a fuzzy set 𝐴 in 𝑋 by          

𝜇𝐴(0) = 𝜇𝐴(𝑏) = 1 and 𝜇𝐴(𝑎) = 𝜇𝐴(𝑐) = 0.5, then 𝐴 is fuzzy ideal of 𝑋. Moreover, a             
𝐵-algebra (ℝ+ − {0};∗ ,1) in the Example 2.4, if we define a fuzzy set 𝐴 in ℝ+ − {0} by 

 

𝜇𝐴(𝑥) = {
1 𝑖𝑓 𝑥 = 1,

0.5 𝑖𝑓 𝑥 ≠ 1,
 

 
then 𝐴 is fuzzy ideal of ℝ+ − {0}. 

 
Let (𝑋;∗ ,0) be a 𝐵-algebra. A multipolar intuitionistic fuzzy set over 𝑋 is a mapping 
 

(ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])𝑚 

                       𝑥 ↦ (ℓ̂(𝑥), �̂�(𝑥)), 

 

where ℓ̂ ∶ 𝑋 → [0,1]𝑚 and �̂� ∶ 𝑋 → [0,1]𝑚 are multipolar fuzzy sets over 𝑋 which is 
satisfies the condition for all 𝑥 ∈ 𝑋, 

 

ℓ̂(𝑥) + �̂�(𝑥) ≤ 1 
 

where 𝜋𝑖 ∶ [0,1]
𝑚 → [0,1] such that 

 

(𝜋𝑖 ∘ ℓ̂)(𝑥) + (𝜋𝑖 ∘ �̂�)(𝑥) ≤ 1 

 
for 𝑖 = 1,2, … , 𝑚 (see [14]). 
 

RESULTS AND DISCUSSION 

 In this section, we will describe the structure of multipolar intuitionistic fuzzy 
ideal in 𝐵-algebras. The description begins with the definition of the new structure, then 
examples are given, and its properties are determined and proven. 
 
Definition 3.1 Let (𝑋;∗ ,0) be a 𝐵-algebra. A multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over 

𝑋 is called multipolar intuitionistic fuzzy ideal in 𝑋 if it satisfies:  

i. (∀𝑥 ∈ 𝑋)(ℓ̂(0) ≥ ℓ̂(𝑥) and �̂�(0) ≤ �̂�(𝑥)) such that  

(𝜋𝑖 ∘ ℓ̂)(0) ≥ (𝜋𝑖 ∘ ℓ̂)(𝑥) and (𝜋𝑖 ∘ �̂�)(0) ≤ (𝜋𝑖 ∘ �̂�)(𝑥), 

ii. (∀𝑥, 𝑦 ∈ 𝑋)(ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)}) such that  

(𝜋𝑖 ∘ ℓ̂)(𝑥) ≥ inf{(𝜋𝑖 ∘ ℓ̂)(𝑥 ∗ 𝑦), (𝜋𝑖 ∘ ℓ̂)(𝑦)} and 
(𝜋𝑖 ∘ �̂�)(𝑥) ≤ sup{(𝜋𝑖 ∘ �̂�)(𝑥 ∗ 𝑦), (𝜋𝑖 ∘ �̂�)(𝑦)}, 

for 𝑖 = 1,2, … . , 𝑚. 
 

Example 3.2 Let (𝑋;∗ ,0) be a 𝐵-algebra in the Example 2.2. Given a multipolar 

intuitionistic fuzzy set (ℓ̂, �̂�) over 𝑋 by 

 

(ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])5, 

 



Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo 297 

𝑥 ↦ {
((0.7,0.3), (0.6,0.25), (0.7,0.15), (0.63,0.2), (0.8,0.18)) 𝑖𝑓 𝑥 ∈ {0, 𝑏},

((0.3,0.6), (0.4,0.5), (0.5,0.4), (0.2,0.7), (0.4,0.5)) 𝑖𝑓 𝑥 ∈ {𝑎, 𝑐}.
 

Then, (ℓ̂, �̂�) is 5-polar intuitionistic fuzzy ideal of 𝑋. 

 
Example 3.3 Let (ℝ+ − {0};∗ ,1) be a 𝐵-algebra in the Example 2.4. Given a multipolar 

intuitionistic fuzzy set (ℓ̂, �̂�) over ℝ+ − {0} by 

 
(ℓ̂, �̂�) ∶ 𝑋 → ([0,1] × [0,1])5, 

 

𝑥 ↦ {
((1,0), (1,0), (1,0), (1,0), (1,0)) 𝑖𝑓 𝑥 = 1,

((0.5,0.5), (0.4,0.4), (0.3,0.3), (0.2,0.2), (0.1,0.1)) 𝑖𝑓 𝑥 ≠ 1.
 

 

Then, (ℓ̂, �̂�) is 5-polar intuitionistic fuzzy ideal of ℝ+ − {0}. 

 

For any 𝜔 ∈ 𝑋 and multipolar intuitionistic fuzzy set (ℓ̂, �̂�) in 𝑋, we give the 

conditions for the set 𝐼(𝜔) to be an ideal of 𝑋 and its example.  
 

Theorem 3.4 Let (𝑋;∗ ,0) be a 𝐵-algebra and 𝑥 ∈ 𝑋. If (ℓ̂, �̂�) is a multipolar intuitionistic 

fuzzy ideal of 𝑋, then 𝐼(𝜔) is an ideal of 𝑋 where 
  

𝐼(𝜔) = {𝑥 ∈ 𝑋|ℓ̂(𝑥) ≥ ℓ̂(𝜔) 𝑎𝑛𝑑 �̂�(𝑥) ≤ �̂�(𝜔)}. 

 

Proof. Let (ℓ̂, �̂�) be a multipolar intuitionistic fuzzy ideal of 𝑋 where  

 
𝐼(𝜔) = {𝑥 ∈ 𝑋|ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ �̂�(𝜔)}. 

 
i. By using Definition 3.1 (i) we have that 

 

ℓ̂(0) ≥ ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(0) ≤ �̂�(𝑥) ≤ �̂�(𝜔). 
 

Hence, 0 ∈ 𝐼(𝜔).  
ii. Let 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ∗ 𝑦 ∈ 𝐼(𝜔) and 𝑦 ∈ 𝐼(𝜔). Then, 

 

ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂(𝜔) and �̂�(𝑥 ∗ 𝑦) ≤ �̂�(𝜔), 

ℓ̂(𝑦) ≥ ℓ̂(𝜔) and �̂�(𝑦) ≤ �̂�(𝜔). 
 

By using Definition 3.1 (ii), we have  
  

ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} ≤ �̂�(𝜔), 

 
such that 

  

ℓ̂(𝑥) ≥ ℓ̂(𝜔) and �̂�(𝑥) ≤ �̂�(𝜔). 
 

Hence, 𝑥 ∈ 𝐼(𝜔). 
Therefore, 𝐼(𝜔) is an ideal of 𝑋.            ∎ 
 



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Royyan Amigo 298 

Example 3.5 Let (𝑋;∗ ,0) be a 𝐵-algebra in the Example 2.2. Given a multipolar 

intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋 in the Example 3.2 where 

𝐼(𝑏) = {0, 𝑏|ℓ̂(0) ≥ ℓ̂(𝑏) and �̂�(0) ≤ �̂�(𝑏), ℓ̂(𝑏) ≥ ℓ̂(𝑏) and �̂�(𝑏) ≤ �̂�(𝑏)}. 
 

Then, 𝐼(𝑏) is an ideal of 𝑋.  
 
Next, we discuss some properties of multipolar intuitionistic fuzzy ideal in                 

𝐵-algebras. 
 

Proposition 3.6 Let (𝑋;∗ ,0) be  a 𝐵-algebra. Every multipolar intuitionistic fuzzy ideal 

(ℓ̂, �̂�) over 𝑋 satisfies the following implication for all 𝑥, 𝑦 ∈ 𝑋,  

 

if 𝑥 ≤ 𝑦 then ℓ̂(𝑥) ≥ ℓ̂(𝑦) and �̂�(𝑥) ≤ �̂�(𝑦). 
 

Proof. Let 𝑥, 𝑦 ∈ 𝑋 such that 𝑥 ≤ 𝑦. So, 𝑥 ∗ 𝑦 = 0. By using Definition 3.1 (i) and (ii), we 
have that 

  
ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} = inf{ℓ̂(0), ℓ̂(𝑦)} = ℓ̂(𝑦) 

and 
�̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} = sup{�̂�(0), �̂�(𝑦)} = �̂�(𝑦). 

 
∎  

 
Proposition 3.7 Let (𝑋;∗ ,0) be a commutative 𝐵-algebra. For any multipolar intuitionistic 

fuzzy ideal (ℓ̂, �̂�) over 𝑋, if for all 𝑥, 𝑦 ∈ 𝑋 satisfies 

 
ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) 𝑎𝑛𝑑 �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦), 

 
then for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies 

 
ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 𝑎𝑛𝑑 �̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). 

  
Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 such that 

 

((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧 ≤ (𝑥 ∗ 𝑦) ∗ 𝑧. 

 
By using Proposition 2.5 and 2.8, we have that 

 

((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧 = ((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧 ≤ (𝑥 ∗ 𝑦) ∗ 𝑧. 

 
From Proposition 3.6, we have 
 

ℓ̂ (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 

and 

�̂� (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). 

 



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Royyan Amigo 299 

 
 
So, from Proposition 2.8, we get 

 

ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) = ℓ̂ ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) 

                                 ≥ ℓ̂ (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) 

        ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 

and 

�̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) = �̂� ((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) 

       ≤ �̂� (((𝑥 ∗ (𝑦 ∗ 𝑧)) ∗ 𝑧) ∗ 𝑧) 

       ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧). 

 
∎ 

  
Proposition 3.8 Let (𝑋;∗ ,0) be a 𝐵-algebra. For any multipolar intuitionistic fuzzy ideal 

(ℓ̂, �̂�) over 𝑋, if  for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 satisfies 

 
ℓ̂((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧) 𝑎𝑛𝑑 �̂�((𝑥 ∗ 𝑧) ∗ (𝑦 ∗ 𝑧)) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑧), 

 
then for all 𝑥, 𝑦 ∈ 𝑋 satisfies 

 
ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) 𝑎𝑛𝑑 �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦). 

   
Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋. If 𝑧 is replaced by 𝑦 on the assumption, then by using Definition 
2.1 (i) and (ii) we have 

 
ℓ̂(𝑥 ∗ 𝑦) = ℓ̂((𝑥 ∗ 𝑦) ∗ 0) = ℓ̂((𝑥 ∗ 𝑦) ∗ (𝑦 ∗ 𝑦)) = ℓ̂(𝑥 ∗ 𝑦) ≥ ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑦) 

and 
�̂�(𝑥 ∗ 𝑦) = �̂�((𝑥 ∗ 𝑦) ∗ 0) = �̂�((𝑥 ∗ 𝑦) ∗ (𝑦 ∗ 𝑦)) = �̂�(𝑥 ∗ 𝑦) ≤ �̂�((𝑥 ∗ 𝑦) ∗ 𝑦). 

 
∎  

 
Based on Proposition 3.7 and Proposition 3.8, we get the following corollary. 

 
Corollary If we assume that 𝑋 is a commutative 𝐵-algebra, then the statements in  
Proposition 3.7 and Proposition 3.8 are equivalent.  
 

Furthermore, we also give another condition of multipolar intuitionistic fuzzy ideal 
in 𝐵-algebras such that make this following proposition. 

 
Proposition 3.9 Let (𝑋;∗ ,0) be a 𝐵-algebra. A multipolar intuitionistic fuzzy set (ℓ̂, �̂�) over 

𝑋 is a multipolar intuitionistic fuzzy ideal (ℓ̂, �̂�) over 𝑋 if and only if for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, 
(𝑥 ∗ 𝑦) ∗ 𝑧 = 0 implies 

 
ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑧)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑧)}. 



Multipolar Intuitionistic Fuzzy Ideal in B-Algebras 

Royyan Amigo 300 

 
Proof. We assume that (ℓ̂, �̂�) is a multipolar intuitionistic fuzzy ideal over 𝑋. Let 𝑥, 𝑦, 𝑧 ∈ 𝑋 

such that (𝑥 ∗ 𝑦) ∗ 𝑧 = 0. So, 𝑥 ∗ 𝑦 ≤ 𝑧. By using Definition 3.1 (i) and (ii), we have 
 
ℓ̂(𝑥) ≥ inf{ℓ̂(𝑥 ∗ 𝑦), ℓ̂(𝑦)} 

  ≥ inf{inf{ℓ̂((𝑥 ∗ 𝑦) ∗ 𝑧), ℓ̂(𝑧)} , ℓ̂(𝑦)} = inf{inf{ℓ̂(0), ℓ̂(𝑧)} , ℓ̂(𝑦)} = inf{ℓ̂(𝑦), ℓ̂(𝑧)} 

 
and 

 
�̂�(𝑥) ≤ sup{�̂�(𝑥 ∗ 𝑦), �̂�(𝑦)} 

          ≤ sup{sup{�̂�((𝑥 ∗ 𝑦) ∗ 𝑧), �̂�(𝑧)} , �̂�(𝑦)} = sup{sup{�̂�(0), �̂�(𝑧)} , �̂�(𝑦)} = sup{�̂�(𝑦), �̂�(𝑧)}. 

 
Conversely, we assume for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, (𝑥 ∗ 𝑦) ∗ 𝑧 = 0.  Then 

 
ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑧)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑧)}. 

 
Let 𝑥 ∈ 𝑋. By using Definition 2.1 (ii) and Definition 2.11, we have 

  

ℓ̂(0) = ℓ̂(𝑥 ∗ 𝑥) ≥ inf{ℓ̂(𝑥), ℓ̂(𝑥)} = ℓ̂(𝑥) 

and 
�̂�(0) = �̂�(𝑥 ∗ 𝑥) ≤ sup{�̂�(𝑥), �̂�(𝑥)} = �̂�(𝑥). 

 
Then, let 𝑥, 𝑦 ∈ 𝑋. By using Definition 2.1 (i), we have (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑦) = 0 such that 

  
ℓ̂(𝑥) ≥ inf{ℓ̂(𝑦), ℓ̂(𝑥 ∗ 𝑦)} and �̂�(𝑥) ≤ sup{�̂�(𝑦), �̂�(𝑥 ∗ 𝑦)}. 

 
Hence, (ℓ̂, �̂�) is a multipolar intuitionistic fuzzy ideal over 𝑋.    

                 ∎ 
 

CONCLUSIONS 

In this paper, we apply the terminology of multipolar intuitionistic fuzzy ideal in                      
𝐵-algebras and investigate some properties. We also explain the conditions for a 
multipolar intuitionistic fuzzy set to be a multipolar intuitionistic fuzzy ideal and give 
some examples. These definitions and main results can be applied with similarly in other 
algebraic structure such as 𝐵𝐺-algebras, 𝐵𝐹-algebras and 𝐵𝐷-algebras.  

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