Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

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Normal BE-Algebras 

 

M. Bala Prabhakar1*, M. Sambasiva Rao2, S. Kalesha Vali3 and G. V. Ramana4. 

1Associate Professor, Department of Mathematics, Aditya University, Surampalem, Kakinada, Andhra Pradesh, India-

533437.  Mail: prabhakar_mb@yahoo.co.in 
2Professor, Department of Mathematics, MVGR College of Engineering(A), Chintalavalasa, Vizianagaram, Andhra 

Pradesh, India-535005. Mail: mssraomaths35@rediffmail.com 
3Professor, Department of Engineering Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh, India-530003. 

Mail: valijntuv@gmail.com 
4Associate Professor, Department of Mathematics, Aditya University, Surampalem, Kakinada, Andhra Pradesh., India-

533437. Mail: ramanaginjala9@gmail.com 
*Corresponding & Main Author. 

 

Article History: 

Received: 28-10-2024 

Revised:12-11-2024 

Accepted:19-12-2024 

Abstract: In this present article we propounded the idea of normal BE-algebras, derived 

some significant properties of normal BE-algebras and obtained a set of equivalent 

conditions indicating when a normal BE-algebra assumes the characteristics of an 

involutory BE-algebra. Some sufficient conditions for a BE-algebra to become a normal 

BE-algebra are derived. Also, congruence relation is introduced on a normal BE-algebra. 

Keywords: BE-algebra; congruence; involutory BE-algebra; normal BE-algebra; 

transitive BE-algebra. 

2020 Mathematics Subject Classification: 03G25. 

 

1. Introduction. 

In [11], H. S. Kim and Y. H. Kim were introduced the theory of BE-algebras. BE-algebras was made 

familiar to extend the class of BCK-algebras of K. Iseki and S. Tanaka [10]. In [1], S.S. Ahn and Y. 

H. Kim studied some properties of filters of BE-algebras and by B. L. Meng in [12]. Some relationships 

between congruence relations and normal filters of a BE-algebras was discussed by A. Walendziak in 

[14]. In [13], P. Sun investigated homomorphism theorems via dual ideals of BCK-algebras. In [9], Z. 

ciloglu and Y. Ceven introduced the notion of commutative and bounded BE-algebras. In [8], R. 

Borzooei et al. introduced the notion of involutory BE-algebras. In [2], M. Bala Prabhakar, S. K. Vali 

and M. Sambasiva Rao were introduced the idea of Closed and Dense elements of BE-algebras. Also, 

these authors were introduced the concepts of Ideals of transitive BE-algebras in [3], Semi Maximal 

Ideals of BE-algebras in [4], Maximal Ideals of transitive BE-algebras in [5], Prime Ideals of transitive 

BE-algebras in [6] and Generalized Lower sets of transitive BE-algebras in [7]. 

In this work, the concept of normal BE-algebras is introduced, derived some properties and equivalent 

conditions. A congruence is introduced on a normal BE-algebra.  

2. Preliminary Results. 

This section outlines a combination of definitions and results, formerly sourced from existing papers 

for the readers convenience.  

 



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Definition 1.1. [11] An algebra (𝑋,∗, 1)of type (2, 0) is called a BE-algebra, if it satisfies the following 

properties: 

(1)   𝑥 ∗ 𝑥 = 1, 

(2)   𝑥 ∗ 1 = 1, 

(3)   1 ∗ 𝑥 = 𝑥, 

(4)   𝑥 ∗ (𝑦 ∗ 𝑧) = 𝑦 ∗ (𝑥 ∗ 𝑧) for all 𝑥, 𝑦, 𝑧 ∈ 𝑋. 

A BE-algebra 𝑋 is called transitive if 𝑦 ∗ 𝑧 ≤ (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑧)for all 𝑥, 𝑦, 𝑧 ∈ 𝑋.  

Every self-distributive BE-algebra is transitive.  

We introduce a relation ≤ on a BE-algebra 𝑋 by 𝑥 ≤ 𝑦 if and only if 𝑥 ∗ 𝑦 = 1 for all for all 𝑥, 𝑦 ∈ 𝑋. 

Theorem 1.2. [12] Let X be a transitive BE-algebra and 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then 

(1) 1 ≤ 𝑥 implies 𝑥 = 1, 

(2) 𝑦 ≤ 𝑧 implies 𝑥 ∗ 𝑦 ≤ 𝑥 ∗ 𝑧 and 𝑧 ∗ 𝑥 ≤ 𝑦 ∗ 𝑥. 

Definition 1.3. [11] A non-empty subset F of a BE-algebra X is called a filter of X if, for all 𝑥, 𝑦 ∈ 𝑋, 

it satisfies the following properties: 

(1)1 ∈ 𝐹, 

(2) 𝑥 ∈ 𝐹and 𝑥 ∗ 𝑦 ∈ 𝐹 imply that 𝑦 ∈ 𝐹. 

Definition 1.4. [9] A BE-algebra 𝑋 is called bounded BE-algebra, if there exist an element 0 satisfying 

0 ≤ 𝑥(𝑜𝑟 0 ∗ 𝑥 = 1) for all 𝑥 ∈ 𝑋. 

Define an unary operation 𝑁 on 𝑋 by 𝑥𝑁 = 𝑥 ∗ 0 for all 𝑥 ∈ 𝑋.  Clearly, 0𝑁 = 1and 1𝑁 = 0. 

Theorem 1.5. [9] Let 𝑋 be a transitive BE-algebra and 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then 

(1) 0𝑁 = 1and 1𝑁 = 0, 

(2) 𝑥 ≤ 𝑥𝑁𝑁,  

(3) 𝑥 ∗ 𝑦𝑁 = 𝑦 ∗ 𝑥𝑁. 

Definition 1.6. [8] An element 𝑥 of a BE-algebra 𝑋 is called involutory element if 𝑥𝑁𝑁 = 𝑥. If every 

element of a BE-algebra 𝑋 is involutory, then 𝑋 is called an involutory BE-algebra. 

Definition 1.7. [2] An element 𝑎 of a BE-algebra 𝑋 is called closed element if 𝑎𝑁𝑁 = 𝑎. 

We denote the set 𝐶(𝑋) = {𝑎 ∈ 𝑋/𝑎𝑁𝑁 = 𝑎}, is the set of all closed elements of a BE-algebra 𝑋. 

Definition 1.8. [7] Let 𝑋 be a bounded BE-algebra. ∅ ≠ 𝑆 ⊆ 𝑋 is called a bounded subalgebra if 𝑆 is 

closed under the operations ∗ and 𝑁. In particular (𝑥𝑁 ∗ 𝑦𝑁)𝑁 ∈ 𝑆 whenever 𝑥, 𝑦 ∈ 𝑆. 

Lemma 1.9. [3] Let 𝑋 be a transitive BE-algebra. For any 𝑥, 𝑦, 𝑧 ∈ 𝑋, we have: 

(1) 𝑥𝑁𝑁𝑁 ≤ 𝑥𝑁, 

(2) 𝑥 ∗ 𝑦 ≤ 𝑦𝑁 ∗ 𝑥𝑁, 

(3) 𝑥 ∗ 𝑦𝑁 ≤ 𝑥𝑁𝑁 ∗ 𝑦𝑁, 

(4) (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 ≤ 𝑥 ∗ 𝑦𝑁𝑁, 



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(5) (𝑥𝑁 ∗ 𝑦𝑁)𝑁𝑁 ≤ 𝑥𝑁 ∗ 𝑦𝑁, 

(6) 𝑥 ≤ 𝑦 ⇒ 𝑦𝑁 ≤ 𝑥𝑁, 

(7) 𝑥 ≤ 𝑦 ⇒ 𝑦 ∗ 𝑧𝑁 ≤ 𝑥 ∗ 𝑧𝑁. 

3. Main Results (Normal BE-Algebras). 

In this section, the concept of Normal BE-Algebras is introduced. Some properties of normal BE-

algebras are studied. Some sufficient conditions for a BE-algebra to become a normal BE-algebra are 

derived. 

Definition 2.1. A bounded BE-algebra (𝑋, ∗, 0, 1) is said to be a normal BE-algebra, if it satisfies the 

following properties for all 𝑥, 𝑦 ∈ 𝑋: 

(N1) 𝑥𝑁𝑁𝑁 = 𝑥𝑁, 

(N2) (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = 𝑥 ∗ 𝑦𝑁𝑁. 

Proposition 2.2. Let (𝑋, ∗, 0, 1) be a normal BE-algebra. For any 𝑥, 𝑦 ∈ 𝑋, the following properties 

hold: 

(1) 𝑥 ∗ 𝑦𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁, 

(2) 𝑥 ∗ 𝑦𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁, 

(3) (𝑥 ∗ 𝑦𝑁)𝑁𝑁 = 𝑥 ∗ 𝑦𝑁. 

Proof. (1). Let 𝑥, 𝑦 ∈ 𝑋. Then 𝑥𝑁𝑁 ∗ 𝑦𝑁 = 𝑥𝑁𝑁 ∗ (𝑦 ∗ 0) 

=  𝑦 ∗ (𝑥𝑁𝑁 ∗ 0) 

= 𝑦 ∗ 𝑥𝑁𝑁𝑁 

= 𝑦 ∗ 𝑥𝑁 

= 𝑦 ∗ (𝑥 ∗ 0) 

= 𝑥 ∗ (𝑦 ∗ 0) 

= 𝑥 ∗ 𝑦𝑁. 

(2). Let 𝑥, 𝑦 ∈ 𝑋. Then by (1), we have 𝑥 ∗ 𝑦𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁. Replace 𝑦 by 𝑦𝑁, we get      

𝑥 ∗ 𝑦𝑁𝑁 = 𝑥 ∗ (𝑦𝑁)𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁. 

 (3). Let 𝑥, 𝑦 ∈ 𝑋. Since 𝑋 is normal, we get that 

(𝑥 ∗ 𝑦𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦𝑁𝑁𝑁)𝑁𝑁 = 𝑥 ∗ 𝑦𝑁𝑁𝑁 = 𝑥 ∗ 𝑦𝑁. 

Theorem 2.3. Let 𝑋 be a BE-algebra which satisfies the following conditions: 

(1) 𝑥𝑁𝑁𝑁 = 𝑥𝑁, 

(2) (𝑥 ∗ 𝑦)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 for all 𝑥, 𝑦 ∈ 𝑋. Then 𝑋 is a normal BE-algebra. 

Proof. Let 𝑥, 𝑦 ∈ 𝑋. Suppose 𝑋 is satisfying the above two conditions. Then 

(𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = 𝑦𝑁𝑁𝑁 ∗ 𝑥𝑁𝑁𝑁 = 𝑦𝑁 ∗ 𝑥𝑁 = 𝑥 ∗ 𝑦𝑁𝑁. 

Therefore, 𝑋 is a normal BE-algebra. 



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Proposition 2.4. Every involutory BE-algebra is a normal BE-algebra. 

Proof. Assume that 𝑋 is an involutory BE-algebra. Let 𝑥 ∈ 𝑋. Then 𝑥𝑁𝑁 = 𝑥.  

Hence 𝑥𝑁𝑁𝑁 = (𝑥𝑁𝑁)𝑁 = 𝑥𝑁.   

Again, let 𝑥, 𝑦 ∈ 𝑋. Then  (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦)𝑁𝑁 = 𝑥 ∗ 𝑦 = 𝑥 ∗ 𝑦𝑁𝑁.  

Therefore, 𝑋 is a normal BE-algebra. 

Example 2.5. Let 𝑋 = {1, 𝑎, 𝑏, 𝑐, 𝑑, 0} be a set. Define a binary operation ∗ on 𝑋 as follows: 

* 1 a b c d 0 

1 1 a b c d 0 

a 1 1 a c c d 

b 1 1 1 c c c 

c 1 a b 1 a b 

d 1 1 a 1 1 a 

0 1 1 1 1 1 1 

                                            

Clearly, (𝑋, ∗, 0, 1) is a bounded BE-algebra. It is easy to observe that X is an involutory 

BE-algebra and a normal BE-algebra too. 

The converse of the above proposition is not true. i.e. A normal BE-algebra need not be involutory. 

For this consider the following example: 

Example 2.6. Let 𝑋 = {1, 𝑎, 𝑏, 𝑐, 0} be a set. Define a binary operation ∗ on 𝑋 as follows: 

                                                  

 

 

 

 

 

Clearly, (𝑋, ∗, 0, 1) is a normal BE-algebra. However, 𝑋 is not an involutory BE-algebra because of 

𝑎𝑁𝑁 = 0𝑁 = 1, 𝑏𝑁𝑁 = 0𝑁 = 1 & 𝑐𝑁𝑁 = 0𝑁 = 1. 

In the following theorem, we derive a set of equivalent conditions for a normal BE-algebra to become 

an involutory BE-algebra. 

Theorem 2.7. Let (𝑋, ∗, 0, 1) is a normal BE-algebra. Then the following conditions are equivalent: 

(1) 𝑋 is involutory; 

(2) for any 𝑥, 𝑦 ∈ 𝑋, 𝑥𝑁 = 𝑦𝑁 implies 𝑥 = 𝑦; 

(3) for all 𝑥 ∈ 𝑋; (𝑥 ∗ 0) ∗ 0 = (0 ∗ 𝑥)𝑥. 

* 1 a b c 0 

1 1 a b c 0 

a 1 1 b b 0 

b 1 a 1 a 0 

c 1 1 1 1 0 

0 1 1 1 1 1 



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Proof. (1) ⇒ (2) Assume that 𝑋 is involutory BE-algebra. Let 𝑥, 𝑦 ∈ 𝑋 be such that 𝑥𝑁 = 𝑦𝑁. Then 

𝑥𝑁𝑁 = 𝑦𝑁𝑁 and hence 𝑥 = 𝑦. 

(2) ⇒ (3) Assume the condition (2). Let 𝑥, 𝑦 ∈ 𝑋. Since ((𝑥 ∗ 0) ∗ 0)𝑁 = (𝑥𝑁𝑁)𝑁 = 𝑥𝑁𝑁𝑁 = 𝑥𝑁.  

Then by (2), we get (𝑥 ∗ 0) ∗ 0 = 𝑥 = 1 ∗ 𝑥 = (0 ∗ 𝑥) ∗ 𝑥. 

(3) ⇒ (1) Assume the condition (3). Let 𝑥 ∈ 𝑋. Then (𝑥 ∗ 0) ∗ 0 = (0 ∗ 𝑥)𝑥. Hence 

𝑥𝑁𝑁 = (𝑥𝑁)𝑁 = (𝑥 ∗ 0) ∗ 0 = (0 ∗ 𝑥)𝑥 = 1 ∗ 𝑥 = 𝑥. Therefore 𝑋 is involutory. 

Theorem 2.8. A normal BE-algebra 𝑋 satisfies the property, (𝑥 ∗ 𝑦)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 if and only if 
(𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦)𝑁𝑁 for all 𝑥, 𝑦 ∈ 𝑋. 

Proof. Let 𝑋 be a normal BE-algebra. Assume that (𝑥 ∗ 𝑦)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 for all 𝑥, 𝑦 ∈ 𝑋. Then 

we have 

(𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 

= 𝑥 ∗ 𝑦𝑁𝑁 

= 𝑦𝑁 ∗ 𝑥𝑁 

= 𝑦𝑁 ∗ 𝑥𝑁𝑁𝑁 

= 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 

= (𝑥 ∗ 𝑦)𝑁𝑁. 

Conversely, assume the condition (𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦)𝑁𝑁for all 𝑥, 𝑦 ∈ 𝑋. For any 𝑥, 𝑦 ∈ 𝑋, we 

get 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 = (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦)𝑁𝑁. 

 

      In the following theorem, we derive a set of equivalent conditions for a transitive BE-algebra to 

become normal. 

Proposition 2.9. Let 𝑋 be a transitive BE-algebra which satisfies the property, 𝑥𝑁𝑁𝑁 = 𝑥𝑁 for all 

𝑥, 𝑦 ∈ 𝑋. Then the following conditions are equivalent: 

(1) 𝑋 is a normal BE-algebra. 

(2) for all 𝑥, 𝑦 ∈ 𝑋, (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁; 

(3) for all 𝑥, 𝑦 ∈ 𝑋, (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁; 

(4) for all 𝑥, 𝑦 ∈ 𝑋, (𝑥𝑁 ∗ 𝑦𝑁)𝑁𝑁 = 𝑥𝑁 ∗ 𝑦𝑁. 

Proof. (1) ⇔ (2) Assume that 𝑋 is a normal BE-algebra. Let 𝑥, 𝑦 ∈ 𝑋. Then by the property (N2), we 

get (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁. 

Conversely, assume the condition (2). Let 𝑥, 𝑦 ∈ 𝑋. Then 

(𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 
= (𝑦𝑁 ∗ 𝑥𝑁𝑁𝑁)𝑁𝑁 
= (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 
= 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 
= 𝑦𝑁 ∗ 𝑥𝑁𝑁𝑁 
= 𝑦𝑁 ∗ 𝑥𝑁 
= 𝑥 ∗ 𝑦𝑁𝑁. 



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Hence 𝑋 is a normal BE-algebra, which proves the condition (1). 

(2) ⇒ (3) Assume that the condition (2) holds. Let 𝑥, 𝑦 ∈ 𝑋. Then 

 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 = (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁𝑁𝑁)𝑁𝑁 = (𝑦𝑁 ∗ 𝑥𝑁)𝑁𝑁 = (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 

Therefore (𝑥 ∗ 𝑦𝑁𝑁)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 for all 𝑥, 𝑦 ∈ 𝑋. 

 (3) ⇒ (4) Assume that the condition (3) holds. Let 𝑥, 𝑦 ∈ 𝑋. Then by (3), we get 

(𝑥𝑁 ∗ 𝑦𝑁)𝑁𝑁 = (𝑦 ∗ 𝑥𝑁𝑁)𝑁𝑁 = 𝑦𝑁𝑁 ∗ 𝑥𝑁𝑁 = 𝑥𝑁 ∗ 𝑦𝑁𝑁𝑁 = 𝑥𝑁 ∗ 𝑦𝑁. 

(4) ⇒ (2) Assume the condition (4) holds. Let 𝑥, 𝑦 ∈ 𝑋. Then by (4), we get (𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁)𝑁𝑁 =
𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁.  Hence condition (2) is derived. 

 

Theorem 2.10. Let (𝑋, ∗, 0, 1) be a normal BE-algebra. Then for any 𝑥, 𝑦 ∈ 𝑋, define a binary 

relation 𝜃 on 𝑋 as (𝑥, 𝑦) ∈ 𝜃 if and only if 𝑥𝑁 = 𝑦𝑁. Then 𝜃 is an equivalence relation on 𝑋 and for 

any 𝑎 ∈ 𝑋, the following are hold: 

(1) The element 𝑎𝑁𝑁 is the greatest element in the class [𝑎]𝜃 where[𝑎]𝜃 = {𝑏 ∈ 𝑋/(𝑎, 𝑏) ∈ 𝜃}; 

(2) The class [𝑎]𝜃 contains just one element from 𝐶(𝑋) which is 𝑎𝑁𝑁. 

Proof. Clearly 𝜃 is an equivalence relation on 𝑋. 

(1). Let 𝑎 ∈ 𝑋. Since 𝑋 is normal, we get 𝑎𝑁𝑁𝑁 = 𝑎𝑁.  Hence (𝑎, 𝑎𝑁𝑁) ∈ 𝜃, which means 𝑎𝑁𝑁 ∈
[𝑎]𝜃. Let 𝑥 ∈ [𝑎]𝜃. Then 𝑥𝑁 = 𝑎𝑁. Since 𝑥 ≤ 𝑥𝑁𝑁 = 𝑎𝑁𝑁, we get that 𝑎𝑁𝑁 is the greatest element 

in the class [𝑎]𝜃. 

(2). Let 𝑏 ∈ 𝐶(𝑋) such that 𝑏 ∈ [𝑎]𝜃. Then 𝑏𝑁 = 𝑎𝑁. Hence 𝑏 = 𝑏𝑁𝑁 = 𝑎𝑁𝑁. 

Therefore, the class [𝑎]𝜃 contains one element from 𝐶(𝑋) which is 𝑎𝑁𝑁. 

Theorem 2.11. Let (𝑋, ∗, 0, 1) be a normal BE-algebra which satisfies the condition (𝑥 ∗ 𝑦)𝑁𝑁 =
𝑥𝑁𝑁 ∗ 𝑦𝑁𝑁 for all 𝑥, 𝑦 ∈ 𝑋.Then we get the following: 

(1) 𝜃 is a congruence on 𝑋, 

(2) 𝐶(𝑋)is a retract of 𝑋. 

Proof. (1). Let (𝑥, 𝑦) ∈ 𝜃 and (𝑧, 𝑤) ∈ 𝜃 for 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝑋. Then we get 𝑥𝑁 = 𝑦𝑁 and 𝑧𝑁 = 𝑤𝑁. 

 Now (𝑥 ∗ 𝑧)𝑁𝑁 = 𝑥𝑁𝑁 ∗ 𝑧𝑁𝑁 = 𝑦𝑁𝑁 ∗ 𝑤𝑁𝑁 = (𝑦 ∗ 𝑤)𝑁𝑁. 

Hence (𝑥 ∗ 𝑧, 𝑦 ∗ 𝑤) ∈ 𝜃. Therefore 𝜃 is a congruence on 𝑋. 

(2). By Proposition 2.9(2), 𝐶(𝑋) is a subalgebra of 𝑋. Define 𝜑: 𝑋 → 𝑋 by 𝜑(𝑥) = 𝑥𝑁𝑁 for all𝑥 ∈ 𝑋. 

Then we get 𝜑(𝑥) = 𝑥 for all 𝑥 ∈ 𝐶(𝑋). Clearly 𝜑(𝑥) = 𝑥𝑁𝑁 ∈ 𝐶(𝑋) for all 𝑥 ∈ 𝑋 − 𝐶(𝑋). 

Therefore 𝐶(𝑋) is a retract of 𝑋. 

Acknowledgement: The authors would like to thank the Editor and Referees for their valuable 

suggestions to improve this work. The authors also thank their respective college managements for 

their continuous support and constant encouragement to carried out the research work. 

Conflict of Interest: The authors have no conflict of interest related to this publication. 

 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 7s (2025) 

 

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