Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

428 
https://internationalpubls.com 

Neutrosophic Fuzzy 𝜶 Composition  

 

R. Vijayalakshmi1*, R. Rushma2, B. Kumaraswamy Achari3, K. Hemabala4, Nainaru 

Tarakaramu5 

1Department of Mathematics, Sri Venkateswara College of Engineering (Autonomous), Karkambadi Road, Tirupati-

517507, A.P, India 

2Department of Mathematics, Government Degree College, Vedurukuppam-517569, Chittoor District, A.P, India  

3Department of Mathematics, Academic Consultant, Sri Venkateswara University, Tiruppati-517102, A.P, India 

4Department of Mathematics, Sree Rama Engineering College (Autonomous), Karkambadi Road, Tirupati-517507, A.P, 

India 

5Department of Mathematics, School of Liberal Arts and Sciences, Mohan Babu University, Sree Sainath Nagar, 

Tirupati-517102, A.P, India   

*Corresponding Author: vijayalakshmi.r@svce.edu.in 
 
 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

We discussed the methods for finding the optimal (maximum and minimum) solution of 

inverse problem of Neutrosophic fuzzy relation equation of the form R ◦ Q = T where for 

both cases R and Q are kept unknown interchangeably using alpha operator.  

 Keywords: Fuzzy set, Neutrosophic fuzzy set, alpha operator, composition. 

1. Introduction 

In 1965, Zadeh[1] introduced the concept of fuzzy set which takes the values between 0 &1. Later 

many authors developed different algebras , concepts and results in fuzzy sets. In vast theories we are 

studied few literature on algebraic structures. Uzair Ahmed developed Optimal Solution of Fuzzy 

Relation Equations. In fuzzy set only true values are included. But in some situations indeterminacy 

and falsity included. To counter this situation. Smarandache introduced and developed concepts of 

Neutrosophic sets. Later many authors start working on NS sets. Hemabala Srinivasa Kumar[14-16] 

developed algebrac structures in Neutrosophic multi fuzzy sets. In this paper  particularly emphasize 

finding the maximum solution of   Neutrosophic fuzzy relation equations through the  use of  alpha 

operator and inverse relations. 

2. Preliminaries 

Fuzzy Set 2.1 Fuzzy set was first defined by Lofti A.Zadeh[1] in 1965. He defined a fuzzy set as a 

collection of objects with membership values in the interval [0,1]. These membership values represent 

the grades of membership with the properties and distinct features of the collection.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

429 
https://internationalpubls.com 

Fuzzy sets are mathematically defined as follows: “A subset A of universe X with the membership 

function µ(x) which may take any value in the interval [0,1] is called fuzzy set”.  

  𝜇𝐴 : X → [0,1] is called a fuzzy subset of X.                  

Neutrosophic fuzzy set 2.2 

Let X be a non empty set. A neutrosophic fuzzy set 𝒩on X can be defined as follows 

𝒩={< 𝑥,(𝓉𝒩(𝑥), 𝒾𝒩(𝑥), 𝒻𝒩(𝑥) >/𝑥 ∈  X} 

Where 𝓉𝒩(𝑥) : X →[0,1]                

           𝒾𝒩(𝑥) ∶ X →[0,1] 

           𝒻𝒩(𝑥) ∶ X →[0,1] 

0 ≤ sup𝓉𝒩(𝑥) + 𝑠𝑢𝑝𝒾𝒩(𝑥) + sup𝒻𝒩(𝑥) ≤ 3    

𝓉𝒩(𝑥) is the truth membership value, 𝒾𝒩(𝑥) is the indeterminacy membership value and 𝒻𝒩(𝑥)falsity 

membership value.  

Properties 2.3:          

 Let X be a non empty set and 𝒩and ℳneutrosophic fuzzy sets of X. Then  

1. 𝒩 ∪ ℳ = {< x, 𝓉𝒩 ∪ ℳ (𝑥), 𝒾𝒩 ∪ ℳ (𝑥), 𝒻𝒩 ∪ ℳ (𝑥) >/𝑥 ∈ 𝑋 },  

    where 𝓉𝒩 ∪ ℳ (𝑥) = max (𝓉𝒩(𝑥), 𝓉 ℳ (𝑥)) 

               𝒾𝒩 ∪ ℳ (𝑥) = min (𝒾𝒩 (𝑥), 𝒾 ℳ (𝑥)) 

               𝒻𝒩 ∪ ℳ (𝑥) = min (𝒻𝒩(𝑥), 𝒻 ℳ (𝑥)), 

 2. 𝒩 ∩ ℳ = {< x, 𝓉𝒩 ∩ ℳ (𝑥), 𝒾𝒩 ∩ ℳ (𝑥), 𝒻𝒩 ∩ ℳ (𝑥) >/𝑥 ∈ 𝑋 },  

     where 𝓉𝒩 ∩ ℳ (𝑥) = min (𝓉𝒩 (𝑥), 𝓉 ℳ (𝑥)) 

               𝒾𝒩 ∩ ℳ (𝑥)  = max (𝒾𝒩  (𝑥), 𝒾 ℳ (𝑥)) 

              𝒻𝒩 ∩ ℳ (𝑥)  = max (𝒻𝒩  (𝑥), 𝒻 ℳ (𝑥)), 

Definition 2.4: A neutrosophic set 𝒩 is contained in another neutrosophic set ℳ if  

𝓉𝒩 (𝑥) ≤ 𝓉 ℳ (𝑥), 𝒾𝒩  (𝑥) ≥ 𝒾 ℳ (𝑥)  ,𝒻𝒩  (𝑥) ≥ 𝒻 ℳ (𝑥)     

 

Definition 2.5: Two neutrosophic sets 𝒩 and ℳ are said to be equal if  

𝓉𝒩 (𝑥) = 𝓉 ℳ (𝑥), 𝒾𝒩  (𝑥) = 𝒾 ℳ (𝑥)  ,𝒻𝒩  (𝑥) = 𝒻 ℳ (𝑥)    



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

430 
https://internationalpubls.com 

Definition 2.6: 

 A lattice is a partially ordered set (poset) L in which any two elements “x” and “y” have a greatest 

lower bound (inf) denoted by x ∧ y = min(x, y) and a least upper bound (sup) denoted by x ∨ y = 

max(x, y).  

Definition 2.7: 

A brouwerian lattice is a lattice L [2] in which for any given elements “a” and “b”, the set of all x ∈ L 

such that a ∨ x ≤ b contains a greatest element, denoted “a α b”, the relative pseudo complement of a 

in b. 

Definition 2.8: 

For any given a and b in lattice L ∈  [0,1], α - operator is defined as 

a α b = {
1 𝑖𝑓 𝑎 ≤ 𝑏
𝑏 𝑖𝑓 𝑎 > 𝑏

 

It is also called Sanchez operator 

Example 1: 

For different values of a and b , α - operator will defned as: 

 0.8α0.5 = 0.5, 0.5α0.6 = 1 , 0.3α0.3 = 1, 0.6α0.3 = 0.3 

Properties of α - Operator 2.9: 

1. If b = 0 then “a α b” will be given as 

a α 0 = {
1 𝑖𝑓 𝑎 = 𝑏
𝑏 𝑖𝑓 𝑎 > 𝑏

 

2. If a = 0 then “a α b” will be given as: 

0 α b = 1hh 

3. If b = 1 then “a α b” will be given as: 

a α 1 = 1 

4. If a = 1 then “a α b” will be given as: 

1 α b =b 

      5. α - operator is not commutative. 

a α b ≠ b α a (Not Commutative) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

431 
https://internationalpubls.com 

      6. α - operator is not associative. 

 a α (b α c) ≠ (a α b) α c (Not Associative) 

3. Neutrosophic composition of the α – operator type 

In this section we will discuss the methods of finding the maximal solution of neutrosophic fuzzy 

relation equation with respect to unknowns R and Q. We will discuss the methods for finding the 

maximal “Q” and maximal “R” respectively for NFRE of the form R ◦ Q = T. Here “∇” denotes the 

maximal solution. 

Definition 3.1:  

Consider two neutrosophic   fuzzy relations  R ⊆ X × Y and Q ⊆ Y × Z. Relationship between these 

two neutrosophic fuzzy relations when using @ composition is defined as 

R @ Q ⊆ X × Z with the membership function defined as: 

𝜇𝑅@𝑄(𝑥, 𝑧) = (∧ {𝜇𝑅𝑇(𝑥, 𝑦)𝛼𝜇𝑄𝑇(𝑦, 𝑧)},∨ {𝜇𝑅𝐼(𝑥, 𝑦)𝛼𝜇𝑄𝐼(𝑦, 𝑧)},∨ {𝜇𝑅𝐹(𝑥, 𝑦)𝛼𝜇𝑄𝐹(𝑦, 𝑧)})  

 ∀ x ∈ X, y ∈ Y and z ∈ Z 

Example 2: 

let X = {x1, x2, x3}, Y = {y1, y2, y3} and Z = {z1, z2, z3} 

Consider two fuzzy relations R ⊆ X × Y and Q ⊆ Y × Z which are given below respectively .We are 

to compute T ⊆ X × Z using “@” composition   

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

0.1,0.2,0.7 0,0,0 0.5,0.4,0.1

1,0,0 0.6,0.3,0.1 0.8,0.1,0.1

0.3,0.3,0.4 0.4,0.2,0.4 0.1,0.5,0.4

y y y

x

R x

x

 
 

=  
 
 

 

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

1,0,0 0.9,0,0.1 0.6,0.4,0

0.5,0.2,0.3 0,0,0 0.2,0.1,0.1

0,0,0.1 0.6,0.3,0.1 0.1,0.2,0.3

z z z

y

Q y

y

 
 

=  
 
 

 

 

 T(x, z) = 𝜇𝑅@𝑄(𝑥, 𝑧)  ∀ x ∈ X and z ∈ Z 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

432 
https://internationalpubls.com 

𝑅@𝑄= (∧{(0.1𝛼1),(0 𝛼0.5),(0.5 𝛼0)},∨ {(0.2𝛼0),(0 𝛼0.2),(0.4 𝛼0)}, ∨                    

             {(0.7𝛼0),(0 𝛼0.3),(0.1 𝛼0.1)}]=(0,1,1) 

𝑅@𝑄= (∧{(0.1𝛼0.9),(0 𝛼0),(0 𝛼0.6)},∨ {(0.2𝛼0),(0 𝛼0),(0.4 𝛼0.3)}, ∨                    

             {(0.7𝛼0.1),(0 𝛼0),(0 𝛼0.1)}]=(1,1,1) 

𝑅@𝑄= (∧{(0.1𝛼0.6),(0 𝛼0.2),(0 𝛼0.1)},∨ {(0.2𝛼0.4),(0 𝛼0.1),(0.4 𝛼0.2)}, ∨                    

             {(0.7𝛼0),(0 𝛼0.1),(0 𝛼0.3)}]=(0.1,1,1) 

𝑅@𝑄= (∧{(1𝛼1),(0.6 𝛼0.5),(0.8 𝛼0)},∨ {(0𝛼0.1),(0.3 𝛼0.2),(0.1 𝛼0)}, ∨                    

             {(0𝛼0),(0.1 𝛼0.3),(0.1 𝛼0.1)}]=(0.5,1,1) 

𝑅@𝑄= (∧{(1𝛼0.9),(0.6 𝛼0),(0.8 𝛼0.6)},∨ {(0𝛼0),(0.3 𝛼0),(0.1 𝛼0.3)}, ∨                    

             {(0𝛼0.1),(0 𝛼0),(0.1 𝛼0.1)}]=(0,1,1) 

𝑅@𝑄= (∧{(1𝛼0.6),(0.6 𝛼0.2),(0.8 𝛼0.1)},∨ {(0𝛼0.4),(0.3 𝛼0.1),(0.1 𝛼0.2)}, ∨                    

             {(0𝛼0),(0.1 𝛼0.1),(0.1 𝛼0.3)}]=(0.1,1,1) 

𝑅@𝑄= (∧{(0.3𝛼1),(0.4 𝛼0.5),(0.1 𝛼0)},∨ {(0.3𝛼0.1),(0.2 𝛼0.2),(0.5 𝛼0)}, ∨                    

             {(0.4𝛼0),(0.4 𝛼0.3),(0.4 𝛼0.1)}]=(0,1,0.3) 

𝑅@𝑄= (∧{(0.3𝛼0.9),(0.4 𝛼0),(0.1 𝛼0.6)},∨ {(0.3𝛼0),(0.2 𝛼0),(0.5 𝛼0.3)}, ∨                    

             {(0.4𝛼0.1),(0.4 𝛼0),(0.4 𝛼0.1)}]=(0,0,0.1) 

𝑅@𝑄= (∧{(0.3𝛼0.6),(0.4 𝛼0.2),(0.1 𝛼0.1)},∨ {(0.3𝛼0.4),(0.2 𝛼0.1),(0.5 𝛼0.2)}, ∨                    

             {(0.4𝛼0),(0.4 𝛼0.1),(0.4 𝛼0.3}]=(0.2,1,0.3) 

T(x, z) = 𝜇𝑅@𝑄(𝑥, 𝑧)=
( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

0,1,1 1,1,1 0.1,1,1

0.5,1,1 0,1,1 0.1,1,1

0,1,0.3 0,0,0.1 0.2,1,0.3

z z z

x

x

x

 
 
 
 
 

 

Lemma 3.2:  

If we have two fuzzy relations R ⊆ X × Y and Q ⊆ Y × Z then the following inclusion will hold: 𝜇𝑄
𝑇  

≤ 𝜇R−1@(R ◦ Q)
𝑇   , 𝜇R−1@(R ◦ Q)

𝐼 = 1 ,   𝜇R−1@(R ◦ Q)
𝐹 = 1  if R ⊆ Q or Q ⊆ R 

where “◦” denotes the max min composition and “@” is the composition made by α - operator. 

 Proof:  

Let A be the neutrosophic fuzzy set 

 Consider, A = R-1@(R ◦ Q) ⊆ Y × Z.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

433 
https://internationalpubls.com 

𝜇𝐴(𝑦, 𝑧) = {𝜇𝑅−1(𝑦, 𝑥)𝛼𝜇𝑅𝑜𝑄(𝑥, 𝑧)}              ∀ x ∈ X 

                  = {𝜇𝑅(𝑥, 𝑦)𝛼𝜇𝑅𝑜𝑄(𝑥, 𝑧)}              ∀ x ∈ X 

    = {𝜇𝑅
𝑇(𝑥, 𝑦)𝛼 (∨ (𝜇𝑅

𝑇(𝑥, 𝑡) ∧ 𝜇𝑄
𝑇(𝑡, 𝑧))) , 1,1}      since    R ⊆ Q and    α operator 

(𝜇𝐴
𝑇(𝑦, 𝑧), 𝜇𝐴

𝐼 (𝑦, 𝑧), 𝜇𝐴
𝐹(𝑦, 𝑧))     = {𝜇𝑅

𝑇(𝑥, 𝑦)𝛼((𝜇𝑅
𝑇(𝑥, 𝑦) ∧ 𝜇𝑄

𝑇(𝑦, 𝑧)) ∧∨𝑡≠𝑦 (∨ 𝜇𝑅
𝑇 ((𝑥, 𝑡) ∧

𝜇𝑄
𝑇(𝑡, 𝑧))) , 1,1}      

 𝜇𝐴
𝑇(𝑦, 𝑧)    = {𝜇𝑅

𝑇(𝑥, 𝑦)𝛼((𝜇𝑅
𝑇(𝑥, 𝑦) ∧ 𝜇𝑄

𝑇(𝑦, 𝑧)) ∧∨𝑡≠𝑦 (∨ 𝜇𝑅
𝑇 ((𝑥, 𝑡) ∧ 𝜇𝑄

𝑇(𝑡, 𝑧)))    

𝜇𝐴
𝐼 (𝑦, 𝑧)=1 

𝜇𝐴
𝐹(𝑦, 𝑧)=1 

 

    𝜇𝐴
𝑇(𝑦, 𝑧)    ≥{𝜇𝑅(𝑥, 𝑦)𝛼((𝜇𝑅(𝑥, 𝑦) ∧ 𝜇𝑄(𝑦, 𝑧))}  

since  a α (a ∧ b) ≥ b  

𝜇𝐴(𝑦, 𝑧)  ≥  𝜇𝑄(𝑦, 𝑧) 

 Example 3: 

Consider X = {x1, x2, x3}, Y = {y1, y2, y3} and Z = {z1, z2, z3}.  

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

0.5,0.9,0.8 0.6,0.3,0.2 0.4,0.5,0.1

0.3,0.2,0.3 0.2,0.7,0.6 0.8,0.2,0.2

1,0.3,0.2 0.0,0.7,0.6 0.6,0.3,0.2

y y y

x

R x

x

 
 

=  
 
 

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

0.7,0.8,0.6 0.8,0.2,0.1 0.5,0.4,0

0.5,0.1,0.2 0.4,0.5,0.5 0.9,0.1,0.1

1,0.2,0.1 0.3,0.6,0.5 0.8,0.2,0.1

z z z

y

Q y

y

 
 

=  
 
 

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

1

2

3

0.5,0.9,0.8 0.3,0.2,0.3 1,0.3,0.2

0.6,0.3,0.2 0.2,0.7,0.6 0.0,0.7,0.6

0.4,0.5,0.1 0.8,0.2,0.2 0.6,0.3,0.2

x x x

y

R y

y

−

 
 

=  
 
 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

434 
https://internationalpubls.com 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

0.5,0.3,0.2 0.5,0.5,0.5 0.6,0.3,0.2

0.8,0.2,0.2 0.3,0.2,0.3 0.8,0.2,0.2

0.7,0.3,0.2 0.8,0.3,0.2 0.6,0.3,0.2

z z z

x

RoQ x

x

 
 

=  
 
 

 

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1

2

3

1

0.7,1,1 0.8,1,1 0.5,1,1

0.5,1,1 0.5,1,1 0.6,1,1

1,1,1 0.3,1,1 1,1

@

,1

R

z z

y

yoQ

y

R

z

−

 
 

=  
 
 

 

So this example shows that 𝜇𝑄
𝑇  ≤ 𝜇R−1@(R ◦ Q)

𝑇   , 𝜇R−1@(R ◦ Q)
𝐼 = 1 ,  

  𝜇R−1@(R ◦ Q)
𝐹 = 1 Hence it clearly satisfies the lemma  

 Lemma 3.3: 

Assume that we have two fuzzy relations R ⊆ X × Y and T ⊆ X × Z then the following inclusion holds: 

R ◦ (R −1@T) ⊂ T where “◦” denotes the max-min composition and “@” is the composition of α - 

operator. The proof of this lemma is analogous to the proof of lemma  

Lemma 3.4: 

Consider we have two fuzzy relations R ⊆ X × Y and Q ⊆ Y × Z then the following inclusion holds:  

R ⊆ (Q@ (R ◦ Q) −1) −1  

 Lemma 3.5: 

Consider we have two fuzzy relations Q ⊆ Y × Z and T ⊆ X × Z then the following inclusion holds: 

 (Q@T −1) −1 ◦ Q ⊂ T  

Theorem 3.6: 

Let R ⊆ X × Y and T ⊆ X × Z be the two fuzzy relations, S(Q) be the set of fuzzy relations Q ∈ Y × 

Z such that R ◦ Q = T. S(Q) = {Q ∈ Y × Z | R ◦ Q = T} ≠ 𝜑, if and only if R−1@T ∈ S(Q) then “R−1@T” 

is the the greatest element in S(Q).  

Theorem 3.7:  

Let R ⊆ X × Y and T ⊆ X × Z be the two fuzzy relations, the set of fuzzy relations Q ∈ Y × Z such 

that R ◦ Q ⊆ T contains a greatest element R−1@T. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

435 
https://internationalpubls.com 

Proof: 

Let S(Q) = {Q ∈ (Y × Z) | R ◦ Q ⊆ T}  ≠ φ.  

let Q ⊆ S(Q): R ◦ Q = T 

 then we have R−1@ (R ◦ Q) ⊆ R−1@T, 

 but Q ⊂ R−1@ (R ◦ Q)  

then it shows  

Q ⊂ R−1@T  

we have R−1@T ∈ S(Q).  

Then it shows that R−1@T ∈ S(Q). 

then R−1@T will be the greatest element in S(Q). 

Hence R−1@T be the greatest element in S(Q). 

 Then Q ∇ = R −1@ T   which is the maximum relation “Q” satisfying the equation R ◦ Q = T 

4. Conclusion:  

Our findings suggest that the identification of an optimal solution among various alternatives in a given 

problem can be achieved through the utilization of nonlinear fuzzy regression equation. After 

conducting extensive calculations using NFREs in the context of our proposed scenarios in civil 

engineering, we have been able to discern the most favourable outcome among the presented options. 

The application of fuzzy relation operations played a crucial role in assessing and determining the best 

project outcome. A potential avenue for future research could involve optimizing the influence factors 

rather than solely focusing on identifying the best project. 

Refrences 

[1]  L.A.Zadeh, Fuzzy sets, Information and control, 8(1965), 338-353. 

 [2] Sanchez.E., “Resolution fo composite fuzzy relation equations.inf and control.” vol. 30, 1976., pp. 

48–58. 

 [3] G. M. M. S. G. N. G. B. .R.(red), Sanchez E. Solutions in composite fuzzy relation equations : 

application to medical diagnosis in brouwerian logic., Amsterdam:North- Holland, 1977.  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

436 
https://internationalpubls.com 

[4] Sanchez.E., “Functional relations and fuzzy relational equations,” in IEEE Faculty de 

Medicine.Marsylia, France, 2002.  

[5] L. C. .Paul.Wang, “Fuzzy relation equations(ll):the branch -point-solutions and the categorized 

minimal solutions.” springer, 2006. 

 [6] J. G. B. C. W.C.Amaral, “Hierarchical fuzzy relational models:linguistic interpretation,” in IEEE, 

University of Campinas(Unicamp).Brazil. 2002.  

[7] T. Kiseliova, “A theoretical comparison of disco and cadiag-ll-likesystems for medical diagnoses,” 

vol. 42. New York: John Wiley & Sons, 2006, pp. 723–748. 

 [8] Y.-K. L. Leh Luoh, Wen-June Wang, “New algorithms for solving fuzzy relation equations,” may 

3, 2002. 

 [9] L. N. Martin.St.epni.cka, Bernard De Baets, “Arthmetic fuzzy models,” in IEEE Tranasactions on 

Fuzzy Systems, University of Ostrava, 2010.  

[10] S.Sapana and D. .A.Tamilarasi, “Fuzzy relations equations in preventin neuropathy diabetic,” vol. 

2, no. 4. Recent Trends in Engineering, nov 2009. 

 [11] S. Jain and K. Lachhwani, “Multiobjective programming problem with fuzzy relatinal equations,” 

2009.  

[12] P.N.Smith, Application of Fuzzy Relation Equations in Transport Project Evalution .Department 

of geographical sciences and planning. The university of Queenland st.Lucia,Queenland., 

Australia, 1999. 55 56 Bibliography  

[13] Ayyub.B.M., “Fuzzy sets in civil engineering,” in Fuzzy Sets and Systems,40, Killarney, Ireland, 

1991, pp. 491–5. 

[14] K. Hemabala, Srinivasa kumar, Cartesian Product of multi 𝐿 fuzzy ideals of Г near ring, Advances 

in Mathematics: Scientific Journal 9(2020), no.7, 5273-5281. 

[15] K. Hemabala, Srinivasa kumar, Anti Neutrosophic multi fuzzy ideals of Г near ring, Neutrosophic 

sets and systems, Vol48(2022). 

[16] K. Hemabala, Srinivasa kumar, Neutrosophic multi fuzzy ideals of Г near ring,Neutrosophic sets 

and systems, Vol49(2022).