F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.175837.1 Research Article Articles Some Results of Fermatean Fuzzy Set on Subalgebras and Ideals of Bn-Algebras

[version 1; peer review: 1 approved, 1 approved with reservations]

 Derso Derebew Conceptualization Investigation Methodology Supervision Validation Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-2431-2802 a 1 Tefera Gerima Conceptualization Formal Analysis Methodology Resources Supervision Validation Visualization Writing – Review & Editing https://orcid.org/0000-0002-9033-560X 2 Assen Teshome Eshetu Conceptualization Data Curation Formal Analysis Methodology Writing – Original Draft Preparation Writing – Review & Editing 2 Mathematics, Woldia University, Woldia, Amhara, 400, Ethiopia Mathematics, Wollo University, Dessie, Amhara, Ethiopia nderebew@gmail.com

No competing interests were disclosed.

 2 2 2026 2026 15 163 24 1 2026 Copyright: © 2026 Derso D et al. 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Background

Classical fuzzy sets have been generalized to better model uncertainty, leading to developments such as intuitionistic, pythagorean, and fermatean fuzzy sets. fermatean fuzzy sets (ffss), characterized by the cube sum constraint on membership and non-membership degrees, provide a more flexible framework for handling complex uncertainty. bn-algebras are important algebraic structures with applications in logic and information theory.

 Methods

This study integrates fermatean fuzzy sets with bn-algebras by introducing the notions of fermatean fuzzy subalgebras and fermatean fuzzy ideals. level cuts of fermatean fuzzy sets are defined and analyzed, and algebraic techniques are employed to investigate their structural properties.

 Results

It is shown that fermatean fuzzy level subalgebras and fermatean fuzzy level ideals correspond to classical subalgebras and ideals of bn-algebras. several characterizations and closure properties are established, supported by illustrative examples and rigorous proofs.

 Conclusions

The results demonstrate that fermatean fuzzy sets significantly enrich the theory of bn-algebras by accommodating higher degrees of uncertainty. this framework provides a solid theoretical foundation for further studies and potential applications in algebraic logic and uncertainty-based systems.

 Fermatean fuzzy set level set fermatean subalgebra Fermatean fuzzy subalgebra Fermatean ideal Fermatean homomorphism No funding Agency The author(s) declared that no grants were involved in supporting this work. 1. Introduction

The concept of fuzzy sets (fs), introduced by zadeh, 14 and has revolutionized the handling of uncertainty by allowing elements to have varying degrees of membership. this foundational idea was extended by atanassov 3 with intuitionistic fuzzy sets (IFS), which added a degree of non-membership and yager 13 further advanced this field with pythagorean fuzzy sets (PFS), where the square sum of membership and non-membership degrees is less than or equal to one. Adak, a.k., nilkaumal, and bb arman 1 introduced fermatean fuzzy semi-prime order semi groups, and the concept of fermatean fuzzy set (FFS) was naturalized by senapati.t and yager, 11 anas and et al. 2 introduced the concepts of the direct product of sets that address the importance of fermatean neutrosophic elements. y., komori 8 introduced varieties of BCC-Algebra, the relation other algebraic structures, and j.neggers and H.S.kim 9 introduced B-algebras, which are related to several generalizations of BCK-algebras, such as BCIci-algebras, BCH-algebras, BCC-algebras, BH-algebras, and 10 D-algebras. in addition, Kim, C.B., and Km, hH.s 6 discussed the concepts of BN-algebras with different structures, and grzegorz dymek and andrzej walendziak extend the concepts of ideals of BN-algebra to fuzzy ideals of BN-algebra 4 with different properties. In this paper, we initiate the concept of a fermatean fuzzy set on the ideals of BN-algebras and study its application. we state and prove some theorems discussed in the fermatean fuzzy set on the ideals of BN-algebras and applications. we also extend the notions of an ideal and a normal ideal in a fermatean fuzzy set on the ideals of bn-algebras.

 2. Preliminaries

1 Let X be a universe of study. A fermatean fuzzy set F in X is an object

F = { ( x , δ F ( x ) , θ F ( x ) ) | x ∈ X } , where δ F : X → [ 0 , 1 ] and θ F : X → [ 0 , 1 ] , θ F ( x ) = 1 − δ F ( x ) satisfy the following criteria:

0 ≤ δ F ( x ) ³ + θ F ( x ) ³ ≤ 1 for all x ∈ X , where δ F ( x ) and θ F ( x ) represent degree membership value and non-degree membership value of an element in F .

13 The pythagorean fuzzy set defined on a nonempty set X is of the form P = { ( x , θ P ( x ) , δ P ( x ) ) | x ∈ X } , where θ P ( x ) and δ P ( x ) are the degree membership and non-degree membership functions from X to [0, 1] and

0 ≤ ( θ P ( x ) ) ² + ( δ P ( x ) ) ² ≤ 1 for all x ∈ X .

11 Let X be a universal set. A fermatean fuzzy set (FF) in X is F = { ( x , θ F ( x ) , δ F ( x ) ) | x ∈ X } with 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 for all x ∈ X , where θ F : X → [ 0 , 1 ] and δ F : X → [ 0 , 1 ] represent the degree membership and non-degree membership functions, respectively.

1 Let X be a universe of study. A fermatean fuzzy set F in X is an object F = { ( x , δ F ( x ) , θ F ( x ) ) | x ∈ X } , where δ F : X → [ 0 , 1 ] and θ F : X → [ 0 , 1 ] , θ F ( x ) = 1 − δ F ( x ) satisfy the following criteria: 0 ≤ δ F ( x ) ³ + θ F ( x ) ³ ≤ 1 for all x ∈ X , where δ F ( x ) and θ F ( x ) represent degree membership value and non-degree membership value of an element in F .

13 Algebra ( X , ∗ , 0 ) of type (2, 0) is called a BN-algebra if for all x , y , z ∈ X the following identities hold:

x ∗ x = 0

x ∗ 0 = x

( x ∗ y ) ∗ z = ( 0 ∗ z ) ∗ ( y ∗ x ) . Let ( X , ∗ , 0 ) be a BN-algebra. We define binary relation ≤ on X by x ≤ y if and only if x ∗ y = 0 . for any x ∈ X , if x ≤ 0 , then x = 0 .

Let ℝ be the set of real numbers and let r = (ℝ, *, 0) be the algebra with the operation * defined by x ∗ y = { x if y = 0 y if x = 0 0 , otherwise then r is a BN-algebra.

5 If ( X , ∗ , 0 ) is a BN-algebra then for all x , y ∈ X

0 ∗ ( 0 ∗ x ) = x

0 ∗ ( x ∗ y ) = y ∗ x

y ∗ x = ( 0 ∗ x ) ∗ ( 0 ∗ y )

x ∗ y = 0 ⇒ y ∗ x = 0

0 ∗ x = 0 ∗ y ⇒ x = y .

7 If BCI-algebra satisfies the condition 0 ∗ x = x it becomes a BCK-algebra. An algebra ( X , ∗ , 0 ) satisfies the condition:

x ∗ x = 0

x ∗ 0 = x

( x ∗ y ) ∗ z = x ∗ ( z ∗ ( 0 ∗ y ) ) for all x , y ∈ X as a generalization of BCK-algebra.

8 Algebra ( X , ∗ , 0 ) is said to be a BM-algebra if it satisfies the following axioms for all x , y , z ∈ X :

x ∗ 0 = x

( z ∗ x ) ∗ ( z ∗ y ) = y ∗ x

12 Algebra ( X , ∗ , 0 ) of type (2, 0) is called a BF-algebra if it satisfies the following axioms for all x , y ∈ X :

x ∗ x = 0

x ∗ 0 = x

0 ∗ ( x ∗ y ) = y ∗ x

1 Let X be a universe under discussion. a fermatean fuzzy set F in X is an object F = { ( x , δ F ( x ) , θ F ( x ) ) | x ∈ X } where δ F ( x ) : X → [ 0 , 1 ] , θ F ( x ) : X → [ 0 , 1 ] and θ F ( x ) = 1 − δ F ( x ) and the following condition holds: 0 ≤ ( δ F ( x ) ) ³ + ( θ F ( x ) ) ³ ≤ 1 for all x ∈ X .

3. Main results 3.1 Fermatean BN-Sub algebra of BN-Algebras

Let F be a fermatean fuzzy set (FFS) in X . then f is called a fermatean fuzzy BN-subalgebra of x if the following conditions hold for all x , y ∈ X :

δ F ( x ∗ y ) ≥ δ F ( x ) ∧ δ F ( y )

θ F ( x ∗ y ) ≤ θ F ( x ) ∨ θ F ( y )

where θ F ( x ) = 1 − δ F ( x ) , δ F : X → [ 0 , 1 ] , θ F : X → [ 0 , 1 ] for all x ∈ X , and 0 ≤ ( δ F ( x ) ) ³ + ( θ F ( x ) ) ³ ≤ 1 .

Let x = {0, 1, 2, 3, 4} be a set, and let * be defined by Table 1. The operation table defining the BN-algebra structure used in the following example is given in Table 1.

Then ( X , ∗ , 0 ) is a BN-algebra. Define δ F : X → [ 0 , 1 ] by δ F ( 0 ) = 0.9 , δ F ( 1 ) = 0.5 = δ F ( 3 ) , δ F ( 4 ) = 0.6 = δ F ( 2 ) .

Define θ F ( x ) = 1 − δ F ( x ) . Then θ F ( 0 ) = 0.1 , θ F ( 1 ) = 0.5 , θ F ( 2 ) = 0.4 , θ F ( 3 ) = 0.5 , θ F ( 4 ) = 0.4 .

Let S = { 0 , 1 } ⊆ X . Then S is a subalgebra of the BN-algebra X .

We have δ F ( 0 ∗ 1 ) = δ F ( 1 ) ≥ 0.9 ∧ 0.5 = δ F ( 0 ) ∧ δ F ( 1 ) implies δ F ( 0 ∗ 1 ) ≥ δ F ( 0 ) ∧ δ F ( 1 ) .

Also θ F ( 0 ∗ 1 ) = θ F ( 1 ) ≤ 0.1 ∨ 0.5 = θ F ( 0 ) ∨ θ F ( 1 ) .

Hence δ F ( x ∗ y ) ≥ δ F ( x ) ∧ δ F ( y ) and θ F ( x ∗ y ) ≤ θ F ( x ) ∨ θ F ( y ) for all x , y ∈ X .

Thus, F = { x ∈ S : δ F ( x ) , θ F ( x ) } is a fermatean fuzzy subalgebra of X .

Fermatean fuzzy sub algebra.
* 0 1 2 3 4
0 0 1 2 2 4
1 1 0 3 2 3
2 2 4 0 1 2
3 3 3 2 0 0
4 4 2 3 3 0

Let X be a BN-algebra and s be a subalgebra of x. Then F = { x ∈ S : θ F ( x ) = 0 & δ F ( x ) = 1 } is a fermatean fuzzy subalgebra of x with 0 ≤ ( θ F ( x ) ) 3 + ( δ F ( x ) ) 3 ≤ 1 , θ F : X → [ 0 , 1 ] , δ F : X → [ 0 , 1 ] and θ F ( x ) = 1 − δ F ( x ) for x ∈ X .

Let S be a subalgebra of X . Then x , y ∈ S imply x ∗ y ∈ S . Since θ F ( 0 ) = 0 and δ F ( 0 ) = 1, it follows that 0 ∈ F . Hence F is nonempty.

Let x , y ∈ F . Then θ F ( x ) = 0 , δ F ( x ) = 1 and θ F ( y ) = 0 , δ F ( y ) = 1 . Since x , y ∈ F implies x , y ∈ S , it follows that x ∗ y ∈ S . We get θ F ( x ∗ y ) = 0 , δ F ( x ∗ y ) = 1 .

1 = δ F ( x ∗ y ) ≥ 1 ∧ 1 = δ F ( x ) ∧ δ F ( y ) . It follows δ F ( x ∗ y ) ≥ δ F ( x ) ∧ δ F ( y ) for all x , y ∈ S .

0 = θ F ( x ∗ y ) ≤ 0 ∨ 0 = θ F ( x ) ∨ θ F ( y ) . We get 0 = θ F ( x ∗ y ) ≤ θ F ( x ) ∨ θ F ( y ) for all x , y ∈ S .

Hence F is a fermatean fuzzy subalgebra of x. Moreover 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 for all x , y ∈ F .

Let X be a BN-algebra and S FF ( X ) be the set of all fermatean fuzzy subalgebras of X . Then for δ F , θ F ∈ S FF ( X ) we have:

U ( δ F , t ) = ( δ F ) t = { x ∈ S : δ F ( x ) ≥ t , t ∈ [ 0 , 1 ] }

L ( θ F , k ) = ( θ F ) k = { x ∈ S : θ F ( x ) ≤ k , k ∈ [ 0 , 1 ] }

are called level cuts of s. Here U ( δ F , t ) and L ( θ F , k ) are called the upper-level cut and lower-level cut of s respectively.

F = { x ∈ X , δ F ( x ) , θ F ( x ) } is a fermatean subalgebra of x if and only if U ( δ F , t ) and L ( θ F , k ) for k , t ∈ [ 0 , 1 ] are subalgebra of X .

Assume f is a fermatean fuzzy subalgebra of x. We need to prove U ( δ F , t ) and L ( θ F , k ) are subalgebra of X for k , t ∈ [ 0 , 1 ] .

Let x 0 ∈ X such that δ F ( x 0 ) ≥ t . Since δ F ( 0 ) = δ F ( x 0 ∗ x 0 ) = δ F ( x 0 ) ∧ δ F ( x 0 ) = δ F

( x 0 ) ≥ t , implies δ F ( 0 ) ≥ t and hence 0 ∈ U ( δ F , t ) . Therefore, U ( δ F , t ) is nonempty.

Let x , y ∈ U ( δ F , t ) . Then δ F ( x ) ≥ t and δ F ( y ) ≥ t . Put δ F ( x ) = t and δ F ( y ) = t . Now δ F ( x ∗ y ) ≥ δ F ( x ) ∧ δ F ( y ) = t ∧ t = t . It follows that δ F ( x ∗ y ) ≥ t . Hence x ∗ y ∈ U ( δ F , t ) , which implies U ( δ F , t ) is a subalgebra of X . Similar result holds for 0 < t 1 < t 2 ≤ 1 or 0 < t 2 < t 1 ≤ 1 .

Again let x 0 ∈ X such that θ F ( x _ 0 ) ≤ k , for δ F ( x _ 0 ) = 1 − θ F ( x _ 0 ) , k ∈ [ 0 , 1 ] . θ F ( 0 ) = θ F ( x 0 ∗ x 0 ) ≤ θ F ( x _ 0 ) ∨ θ F ( x _ 0 ) = θ F ( x _ 0 ) ≤ k . Imply that 0 ∈ L ( θ F , k ) which implies L ( θ F , k ) is non-empty.

Let x , y ∈ L ( θ F , k ) then θ F ( x ) ≤ k and θ F ( y ) ≤ k . Put θ F ( x ) = k = θ F ( y ) now θ F ( x ∗ y ) ≤ θ F ( x ) ∨ θ F ( y ) = k ∨ k = k which imply θ F ( x ∗ y ) ≤ k we get x ∗ y ∈ L ( θ F , k ) therefore L ( θ F , k ) is a subalgebra of x a similar result holds for 0 < k 1 < k 2 ≤ 1 or 0 < k 2 < k 1 ≤ 1 .

Conversely, assume U ( δ F , t ) and L ( θ F , k ) are subalgebras of X . We must prove F = { x ∈ X , δ F ( x ) , θ F ( x ) } is a fermatean fuzzy subalgebra of X .

Let t , k ∈ [ 0 , 1 ] and because U ( δ F , t ) and L ( θ F , k ) are subalgebras, we have U ( δ F , t ) and L ( θ F , k ) nonempty. Let x 0 ∈ U ( δ F , t ) and x 0 ∈ L ( θ F , k ) then δ F ( x 0 ) ≥ t and θ F ( x 0 ) ≤ k .

Suppose δ F ( x ∗ y ) < δ F ( x ) ∧ δ F ( y ) then put β = ½ { δ F ( x ∗ y ) + δ F ( x ) ∧ δ F ( y ) } so that δ F ( x ∗ y ) < β < δ F ( x ) ∧ δ F ( y ) ≤ δ F ( x ∗ y ) implies that δ F ( x ∗ y ) < δ F ( x ∗ y ) which is a contradiction. Hence δ F ( x ∗ y ) ≥ δ F ( x ) ∧ δ F ( y ) for x , y ∈ X the cases x 0 ∈ U ( δ F , t ) and y 0 ∈ L ( θ F , k ) with x 0 ≠ y 0 followed by a similar argument.

Again, let x , y ∈ X such that θ F ( x ∗ y ) > θ F ( x ) ∨ θ F ( y ) . Then put γ = ½ { θ F ( x ∗ y ) + θ F ( x ) ∨ θ F ( y ) } then we have θ F ( x ∗ y ) > γ > θ F ( x ) ∨ θ F ( y ) ≥ θ F ( x ∗ y ) which implies that θ F ( x ∗ y ) > θ F ( x ∗ y ) which is not correct. Hence θ F ( x ∗ y ) ≤ θ F ( x ) ∨ θ F ( y ) for all x , y ∈ X the cases x 0 ∈ U ( δ F , t ) and y 0 ∈ L ( θ F , k ) with x 0 ≠ y 0 follow by a similar argument thus F = { x ∈ X , δ F ( x ) , θ F ( x ) } is a fermatean fuzzy subalgebra of X .

Let X be a BN-algebra and let s be a subalgebra of X . Then F = { x ∈ S : ( x , δ F ( x ) , θ F ( x ) ) } where θ F = 1 − δ F , δ F : X → [ 0 , 1 ] , θ F : X → [ 0 , 1 ] . Then f is a fermatean fuzzy subalgebra of x if x , y ∈ F ⇒ x ∗ y ∈ F .

Let x be a BN-algebra and let s be a subalgebra of x then F = { x ∈ S : θ F ( x ) = 0     & δ F ( x ) = 1 }

Is a fermatean fuzzy subalgebra of X with 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 , θ F : X → [ 0 , 1 ] , δ F : X → [ 0 , 1 ] and θ F ( x ) = 1 − δ F ( x ) for x ∈ X .

Let S be a subalgebra of X then x , y ∈ S imply x ∗ y ∈ S since θ F ( 0 ) = 0 and δ F ( 0 ) = 1 , it follows that 0 ∈ F hence F is nonempty.

Let x , y ∈ F then θ F ( x ) = 0 and δ F ( x ) = 1 and θ F ( y ) = 0 and δ F ( y ) = 1 since x , y ∈ F imply x , y ∈ S , it follows that x ∗ y ∈ S we get θ F ( x ∗ y ) = 0 , δ F ( x ∗ y ) = 1 .

0 = θ F ( x ∗ y ) ≥ 0 ∧ 0 = θ F ( x ) ∧ θ F ( y ) it follows θ F ( x ∗ y ) ≥ θ F ( x ) ∧ θ F ( y ) for all x , y ∈ S .

1 = δ F ( x ∗ y ) ≤ 1 ∨ 1 = δ F ( x ) ∨ δ F ( y ) we get δ F ( x ∗ y ) ≤ δ F ( x ) ∨ δ F ( y ) for all x , y ∈ S .

Hence F is a fermatean fuzzy subalgebra of x moreover 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 for all x , y ∈ F .

Let S be a subalgebra of X then ( δ F , θ F ) { ( t , k ) } is a fermatean level subset of s if and only if F = { x ∈ S : δ F ( x ) ≥ t , θ F ( x ) ≤ k , t , k ∈ [ 0 , 1 ] }

Is a fermatean fuzzy subalgebra of X here 0 ≤ ( δ F ( x ) ) ³ + ( θ F ( x ) ) ³ ≤ 1 for all x ∈ S .

Let ( δ F , θ F ) { ( t , k ) } be a level subset of S in X .

F is a fermatean fuzzy subalgebra.

Since 0 ∈ F , δ F ( 0 ) ≥ t and θ F ( 0 ) ≤ k to show closure put δ F ( 0 ) = 0 and θ F ( x ) = 0 ⇒ F is nonempty.

Let x , y ∈ F then δ F ( x ) ≥ t , θ F ( x ) ≤ k and δ F ( y ) ≥ t , θ F ( y ) ≤ k put δ F ( x ) = δ F ( y ) = t = t ∧ t = inf { t , t } = δ F ( x ) ∧ δ F ( y ) ≤ δ F ( x ∗ y ) ⇒ δ F ( x ∗ y ) ≥ t ⇒ x ∗ y ∈ F .

Let x , y ∈ F ⇒ θ F ( x ) ≤ k and θ F ( y ) ≤ k . Put k = θ F ( x ) = θ F ( y ) . k = sup { k , k } = sup { θ F ( x ) , θ F ( y ) } ≥ θ F ( x ∗ y ) ⇒ θ F ( x ∗ y ) ≤ k ⇒ x ∗ y ∈ F .

Hence f is a fermatean fuzzy subalgebra of X .

3.2 Fermatean Fuzzy Ideal of BN-Algebra

Let X be a BN Algebra. then fermatean fuzzy ideal of x is defined by:

δ F ( 0 ) ≥ δ F ( x ) , x ∈ X .

δ F ( x ) ≥ δ F ( x ∗ y ) ∧ δ F ( y ) .

θ F ( x ) ≤ θ F ( x ∗ y ) ∨ θ F ( y ) , for all x , y ∈ X .

 where δ F : X → [ 0 , 1 ] and θ F : X → [ 0 , 1 ] , θ F ( x ) = 1 − δ F ( x ) for all x ∈ X .

Let X = {0, a, b, c} and * be defined by Table 2.

Let 0 ≤ t 3 < t 2 < t 1 < 1 . Define a fuzzy subset δ F : X → [ 0 , 1 ] by δ F ( x ) = { t 1 if x = 0 t 2 if x = b t 3 if x ∈ { a , c } and θ F ( x ) = 1 − δ F ( x ) . we must show δ F and θ F satisfy the condition of the fermatean fuzzy ideal of X .

if x = b and y = a , we have δ F ( b ) = t 2 ≥ δ F ( b ∗ a ) ∧ δ F ( a ) hence it holds. δ F ( 0 ) = t 1 > t 2 > t 3 it follows that δ F ( 0 ) ≥ δ F ( x ) for all x ∈ X .

θ F ( b ) = 1 − t 2 ≤ ( 1 − t 2 ) ∨ ( 1 − t 3 ) = θ F ( b ) ∨ θ F ( b ∗ a ) . Imply θ F ( b ) = 1 − t 2 ≤ θ F ( b ) ∨ θ F ( b ∗ a ) .

If x = a , then θ F ( a ) = 1 − t 3 ≤ ( 1 − t 3 ) ∨ ( 1 − t 1 ) = θ F ( a ) ∨ θ F ( a ∗ a ) .

 Fermatean fuzzy ideal.
* 0 a b c
0 0 a b c
a a 0 a a
b b a 0 a
c c a a 0

For a fermatean fuzzy ideal I FF of x and any x , y ∈ X . if x ≤ y then

δ F ( x ) ≤ δ F ( y )

θ F ( y ) ≤ θ F ( x )

Let x , y ∈ X such that x ≤ y implies x ∗ y = 0 . it follows y ∗ x = 0 .

δ F ( y ) ≥ δ F ( y ∗ x ) ∧ δ F ( x ) = δ F ( 0 ) ∧ δ F ( x ) = δ F ( x ) . Hence δ F ( y ) ≥ δ F ( x ) .

θ F ( y ) = 1 − δ F ( y ) ≤ ( 1 − δ F ( y ∗ x ) ) ∨ ( 1 − δ F ( x ) ) = ( 1 − δ F ( 0 ) ) ∨ ( 1 − δ F ( x ) ) ≤ 1 − δ F ( x ) = θ F ( x ) . Hence θ F ( y ) ≤ θ F ( x ) for x ≤ y and x , y ∈ X .

Let I_FF be a fermatean fuzzy ideal of a BN-algebra x then θ F ( 0 ) ≤ θ F ( x ) for all x ∈ X , δ F : X → [ 0 , 1 ] and θ F ( x ) = 1 − δ F ( x ) with the property 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 .

Let I FF be a fermatean fuzzy ideal of a BN-algebra X and δ F : X → [ 0 , 1 ] be a degree membership function and θ F ( x ) = 1 − δ F ( x ) for all x ∈ X .

θ F ( 0 ) = 1 − δ F ( 0 ) ≤ 1 − δ F ( x ) = θ F ( x ) for all x ∈ X hence θ F ( 0 ) ≤ θ F ( x ) for all x ∈ X .

Let I FF be a fermatean fuzzy ideal of X and J = I FF then χ J ( x ) = { 1 if x ∈ J 0 if x ∉ J . then χ J is a fuzzy ideal of X .

Let j be a fermatean fuzzy ideal of x and χ J ( x ) = { 1 if x ∈ J 0 if x ∉ J

Let χ J = δ F then we have δ F ( x ) = 1 ≥ δ F ( x ) for all x ∈ J and 0 ∈ J but δ J ( x ) = 1 it follows that 0 ∈ χ J . δ F ( 0 ) = δ F ( x ) for all x ∈ χ J it follows that δ F ( 0 ) ≥ δ F ( x ) for all x ∈ χ J .

Let x ∗ y ∈ J and y ∈ J then δ F ( x ∗ y ) = 1 and δ F ( y ) = 1 we get δ F ( x ) ≥ δ F ( x ∗ y ) ∧ δ F ( y ) = 1 ∧ 1 = 1 imply δ F ( x ) ≥ 1 but δ F ( x ) ≤ 1 so that δ F ( x ) = 1 hence x ∈ χ J and x ∈ J consequently δ F ( x ) ≥ δ F ( x ∗ y ) ∧ δ F ( y ) for x , y ∈ J therefore χ J is a fuzzy ideal of X .

3.3 Fermatean level ideal of BN-Algebra

Let X be a BN-algebra and let I FF ( X ) be the set of fermatean fuzzy ideals of X then for δ F ∈ I FF ( X ) for δ F &   θ F ∈ I FF ( X ) we have:

U ( δ F , t ) = { x ∈ X : δ F ( x ) ≥ t , t ∈ [ 0 , 1 ] }

L ( θ F , k ) = { x ∈ X : θ F ( x ) ≤ k , k ∈ [ 0 , 1 ] } are called fermatean level ideal of X and U ( δ F , t ) and L ( θ F , k ) are called upper level ideal and lower-level ideal respectively.

Let X be a BN-algebra and δ F , θ F ∈ I FF ( X ) , θ F ( x ) = 1 − δ F ( x ) if and only if the non-empty level subsets U ( δ F , t ) and L ( θ F , k ) , t , k ∈ [ 0 , 1 ] are ideals of X .

Let δ F , θ F ∈ I FF ( X ) and t , k ∈ [ 0 , 1 ] with θ F ( x ) = 1 − δ F ( x ) . we need to prove U ( δ F , t ) and L ( θ F , k ) are ideals of X .

Let x 0 ∈ X such that δ F ( x 0 ) ≥ t , t ∈ [ 0 , 1 ] since δ F , θ F ∈ I FF ( X ) we have δ F ( 0 ) ≥ δ F ( x 0 ) ≥ t implies 0 ∈ U ( δ F , t ) .

Again, for x 0 ∈ x, such that θ F ( x 0 ) ≤ k and θ F ( 0 ) ≤ θ F ( x 0 ) ≤ k it follows that θ F ( 0 ) ≤ k hence 0 ∈ L ( θ F , k ) .

Let x , y ∈ X and x ∗ y , y ∈ U ( δ F , t ) , t ∈ [ 0 , 1 ] , imply δ F ( x ∗ y ) ≥ t and δ F ( y ) ≥ t since δ F ∈ I FF ( X ) , δ F ( x ) ≥ δ F ( x ∗ y ) ∧ δ F ( y ) ≥ t . δ F ( x ) ≥ t it follows that x ∈ U ( δ F , t ) .

Again let x , y ∈ X such that x ∗ y , y ∈ L ( θ F , k ) then we have θ F ( x ∗ y ) ≤ k and θ F ( y ) ≤ k since 0 ≤ ( θ F ( x ) ) ³ + ( δ F ( x ) ) ³ ≤ 1 for all x ∈ X and θ F ∈ I FF ( X ) we get θ F ( x ) ≤ θ F ( x ∗ y ) ∨ θ F ( y ) ≤ k it follows that θ F ( x ) ≤ k hence x ∈ L ( θ F , k ) therefore U ( δ F , t ) and L ( θ F , k ) , t , k ∈ [ 0 , 1 ] are ideals of X .

Conversely, suppose U ( δ F , t ) and L ( θ F , k ) , t , k ∈ [ 0 , 1 ] are ideals of x we must prove I FF ( X ) is fermatean fuzzy ideal of X for each t ∈ [ 0 , 1 ] , U ( δ F , t ) is non-empty if x 0 ∈ X such that δ F ( 0 ) δ F ( x 0 ) = s ∈ [ 0 , 1 ] then U ( δ F , s ) ≠ ∅ by assumption U ( δ F , s ) is an ideal of x hence 0 ∈ U ( δ F , s ) so that δ F ( 0 ) ≥ s which is a contradiction hence δ F ( 0 ) ≥ δ _ F ( x _ 0 ) ≥ s ∈ [ 0 , 1 ] , x 0 ∈ X .

Let x , y ∈ X , such that δ F ( x ) < δ F ( x ∗ y ) ∧ δ F ( y ) . Let β = ½ { δ F ( x ) + ( δ F ( x ∗ y ) ∧ δ F ( y ) ) } such that δ F ( x ) < β < δ F ( x ∗ y ) ∧ δ F ( y ) ≤ δ F ( x ∗ y ) and β < δ_f(y) hence x ∗ y , y ∈ U ( δ F , β ) . But x ∉ U ( δ F , β ) this is impossible as U ( δ F , t ) is an ideal of X to show θ F is a fermatean fuzzy ideal is simply taking the reverse of the proof done for δ F . This completes the proof.

4. Conclusions

This paper has addressed a significant gap in the study of algebraic structures by exploring the application of fermatean fuzzy sets (FFS) to the ideals and sub-algebras of BN-algebras. the findings confirm that fermatean fuzzy sets enrich the understanding of BN-algebras by accommodating imprecision more effectively, establishing new structural properties, and broadening the scope of algebraic studies under uncertainty. this makes them a valuable tool not only for theoretical mathematics but also for applications in computer science, decision-making systems, and information processing.

 Data availability

No datasets were generated or analyzed during this study. All results are derived analytically, and all supporting information is fully contained within the manuscript.

 Acknowledgements

The authors acknowledge the referees for their excellent academic support.

 References Adak AK Nilkamal Barman N : Fermatean Fuzzy Semi-Prime Ordered Semi-Groups. Topol. Algebra Appl. 2023;11(1). Al-Masarwah A Kaviyarasu M Ainetaie K : Fermatean Neutrosophic INK-Algebras. European Journal of Pure and Applied Mathematics. 2014;17(2):1113–1128. Atanassov KT : Intuitionistic Fuzzy Sets. Fuzzy Sets Syst. 1986;20(1):87–96. Dymek G Walendziak A : Fuzzy Ideals of BN-Algebras. The Scientific World Journal Hindawi. 2015. 10.1155/2015/925040 Jun YB Roh EH Kim HS : On BH-Algebras. Scientiae Mathematicae. 1998;1(3):347–354. Kim CB Kim HS : On BN-Algebras. Kyungpook Mathematical Journal. 2013;53(2):175–182. Kim CB Kim HS : On BN-Algebras. Kyungpook Mathematical Journal. 2013;53(2):175–184. Komori Y : The Variety Generated by BCC-Algebras is Finitely Based. Shizuoka University. 1983;17:13–16. Neggers J Kim HS : On B-Algebras. Mat. Vesn. 2002;54(1–2):21–29. Neggers J Kim HS : On D-Algebras. Mathematical Slovaca. 1999;49(1):19–26. Senapati T Yager RR : Fermatean Fuzzy Set. J. Ambient. Intell. Humaniz. Comput. 2020;11:663–674. Walendziak A : On BF-Algebras. Mathematical Slovaca. 2007;57(2):119–128. Yager RR : Pythagorean Fuzzy Subsets. 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). IEEE;2013; pp.57–61. Zadeh LA : Fuzzy Sets. Inf. Control. 1965;8:338–353. 10.5256/f1000research.193852.r472221 Reviewer response for version 1 Kider Jehad R. 1 Referee https://orcid.org/0000-0002-3173-5654 University of Technology, Baghdad, Iraq

Competing interests: No competing interests were disclosed.

 3 4 2026 Copyright: © 2026 Kider JR 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

Thank you for sending me to review this paper.

1.The main question addressed by this research, is can we state and prove some theorems discussed in the fermatean fuzzy set on the ideals of BN-algebras and applications. we also can we extend the notions of an ideal and a normal ideal in a fermatean fuzzy set on the ideals of Bn-algebras.

2.Yes, this research is in the relevant to the field as well as it addresses a specific gap in the field because it is shown that fermatean fuzzy level subalgebras and fermatean fuzzy level ideals correspond to classical subalgebras and ideals of Bn-algebras. several characterizations and closure properties are established, supported by illustrative examples and rigorous proofs.

3. In this research the authors results demonstrate that fermatean fuzzy sets significantly enrich the theory of Bn-algebras by accommodating higher degrees of uncertainty.

4. Generally, the paper is well written and has a good structureas well as the theoretical framework is built systematically, starting from definitions to illustrative examples and theorems so that I do not suggest any specific improvements should the authors consider regarding the methodology.

5. The conclusions are consistent with the evidence and the arguments presented as well as they address the main question posed. The results presented in this paper

 provides a solid theoretical foundation for further studies and potential applications in algebraic logic and uncertainty-based systems.

6. The references are appropriate.

7.The research contains tables which are good but no figures.

Taking the above into consideration, I recommend the paper for indexing in your journal.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Fuzzy Metric Spaces, Fuzzy Normed Spaces, Fuzzy Inner Product Spaces.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

 10.5256/f1000research.193852.r459976 Reviewer response for version 1 Kalavath Anjaneyulu Naik 1 Referee https://orcid.org/0009-0006-2625-1301 Acharya Nagarjuna University, Guntur, Andhrapradesh, India

Competing interests: No competing interests were disclosed.

 9 3 2026 Copyright: © 2026 Kalavath AN 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

Review Recommendations:

Clarity and Organization: The paper is easy to follow, with a clear abstract, background, methods, results, and conclusions. But some sections could use more examples to help readers less familiar with fermatean fuzzy sets and bn-algebras.

Technical Depth and Novelty: The study adds a lot by to bn-algebras by using fermatean fuzzy sets, which handles higher degrees of uncertainty. It new to apply fermatean fuzzy sets to bn-algebras, and it could be really useful for dealing with complex uncertainty in information theory.

 Methodology: The method involves defining and looking at fermatean fuzzy subalgebras and ideals in bn-algebras using algebraic techniques. It’s thorough and supported by illustrative examples and proofs. Linking the definitions and theorems to existing research would make the method stronger.

Results and Implications: The results show that fermatean fuzzy sets add a lot to bn-algebras. This is big for areas like algebraic logic and systems that deals with uncertainty, and could be used in computer science, decision-making, and information processing.

Future Work: The authors say more research could be done in areas like algebraic logic and systems that deal with uncertainty. If the authors added more details, it would give other clear path to follow for future research, like exploring new uses or digging deeper into theory.

 References and Citations: The paper mentions key research (like Zadeh, Atanassov, Senapati et al., Komori, Neggers, H.S.Kim, Yager). Checking these references are right and complete is important. Also, talking about how this work fits into what’s happening in research now could make it more impactful.  

Accessibility: The paper is technically wise but making it easier for more people to understand (like graduate students or researchers from other fields) would be good. They could add examples at the start or a list of key terms like “fermatean fuzzy set,” “bn-algebra,” and “level cut.”

Definitions 2.1, 2.4, and 2.11 seem to be similar or identical. Please check for typographical errors or clarify the distinction between these definitions if they're meant to be different.

References 6 and 7 appear to be duplicates; reference 7 is the correct one. Please remove or correct reference 6.

 Potential Areas for Improvement:

Enhance clarity with additional illustrative examples for complex concepts.

Discuss limitations of the current study.

Suggest specific future research directions beyond general applications.

Ensure all technical terms are well-defined for a general readership.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Partly

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Fuzzy Algebraic Structures, Hyper Algebraic Structures, Intuitionistic Fuzzy Algebraic Structures, Neutrosophic Fuzzy Algebraic Structures, Soft Sets, Rough Sets and Their Decision-Making Systems in Different algebraic structures.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Tefera Gerima Mathematics, Wollo University, Dessie, Amhara, Ethiopia

Competing interests: No competing interests were disclosed.

 12 3 2026

Response for comment"

This article has many diversified Applications in related area of uncertainty.  In addition, this research can be extended to BCK-algebra, BGG-algebra, BG-algebra, KU-algebra and other related algebraic structures.

 Derso Derebew Mathematics, Woldia University, Weldiya, Amhara, Ethiopia

Competing interests: No competing interests were disclosed.

 6 4 2026

Author Response to Reviewer Comments

Reviewer: Anjaneyulu Naik Kalavath

1. Regarding Future Work

Author’s Response: We appreciate the reviewer's suggestion. Fermatean Fuzzy Sets (FFS) offer a robust framework for handling uncertainty in decision-making. We have updated the conclusion to highlight that the proposed concepts of Fermatean BN-subalgebras and Fermatean ideals can be extended to other algebraic structures, including B-algebras, BCK/BCC-algebras, and BG-algebras, as well as other related mathematical frameworks.

2. References and Citations

Author’s Response: We thank the reviewer for the careful check of our references. We have re-evaluated references [6] and [7]. In accordance with the reviewer’s suggestion, reference [6] has been removed, and the citations in the main text have been adjusted accordingly to ensure accuracy.

3. Accessibility and Preliminaries

Author’s Response: This paper presents original research on Fermatean Fuzzy BN-algebras. To maintain a concise focus on our new contributions, we have assumed the reader possesses a foundational understanding of BN-algebras and Fermatean Fuzzy Set theory. However, to ensure the paper remains self-contained, we have clearly stated the essential definitions of BN-algebras and FFS in the Preliminaries section. We believe this provides sufficient context for the reader without redundant elaboration.

4. Revisions to Definitions (2.1, 2.4, and 2.11)

Author’s Response: We agree with the reviewer’s observation regarding the redundancy in these definitions. These errors occurred during the final editing stage. We have made the following corrections:

Definition 2.1: Replaced with a more fundamental definition of Fermatean Fuzzy Sets from reference [3].

Definition 2.11: This definition has been removed to avoid redundancy. We thank the reviewer for their keen eye and for helping us improve the technical rigor of the manuscript.

 5. Potential Areas for Improvement and Limitations

Author’s Response: We have addressed the study's limitations in the revised manuscript. Specifically:

Literature Gap: We acknowledge that there is currently limited literature regarding the application of Fermatean Fuzzy Sets specifically within algebraic structures, which constrained our comparative analysis.

Future Scope: Because FFS can model more complex uncertainties than standard Intuitionistic Fuzzy Sets, we have suggested that future research should focus on its application in algebraic logic, decision theory, and control theory. ---