F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.160333.1 Research Article Articles Q-fuzzy structure on JU-algebra

[version 1; peer review: 1 approved, 2 approved with reservations]

 Gelaw Selamawit Hunie Conceptualization Data Curation Formal Analysis Investigation Methodology Writing – Original Draft Preparation https://orcid.org/0009-0003-4977-7056 a 1 Alaba Berhanu Assaye Conceptualization Methodology Supervision Validation Writing – Review & Editing 1 Taye Mihret Alamneh Conceptualization Methodology Supervision Validation Writing – Review & Editing 1 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, 79, Ethiopia ntazshu@gmail.com

No competing interests were disclosed.

 20 1 2025 2025 14 109 26 12 2024 Copyright: © 2025 Gelaw SH et al. 2025 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

JU-algebras, an important class in abstract algebra, are extended here by incorporating fuzzy set theory to handle uncertainty in algebraic structures. In this study, we apply the concept of Q-fuzzy sets to JU-subalgebras and JU-ideals in JU-algebra.

 Method

The study defines Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals as subsets of a JU-algebra. It also explores lower and upper level subsets of these fuzzy structures to analyze their properties. Additionally, the concepts of Doubt and Normal Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals are introduced, offering a way to deal with varying degrees of uncertainty and regularity in these algebraic structures. Supportive concepts relevant to this study are presented, along with illustrative examples.

 Conclusion

The study introduces and defines new types of fuzzy structures in JU-algebras, such as Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals, enhancing classical JU-algebra theory. It also examines key properties of these structures, including their lower and upper level subsets, and investigates specific cases like Doubt and Normal Q-fuzzy structures, paving the way for further exploration of fuzzy algebra in mathematical and applied contexts.

 JU-algebra Q-Fuzzy JU-subalgebra Q-Fuzzy JU- Ideal Doubt Q-Fuzzy JU-algebra Normal Q-Fuzzy JU-algebra Level subsets The author(s) declared that no grants were involved in supporting this work. Introduction

After fuzzy subsets were defined by Ref. 1, several researchers explored Q-fuzzy sets in many algebraic structures. In 2001, Ref. 2 introduced the concept of Q-fuzzy subalgebras within BCI/BCK-algebras. Subsequently, Ref. 3 explored Q-fuzzy sets and their level subsets in UP-algebras. Additionally, Ref. 4 presented the concept of Doubt fuzzy ideals in BF-algebras, while Ref. 5 discussed Doubt Fuzzy Ideals of B-Algebras. In 2018, Ref. 6 further contributed by introducing the normalization of fuzzy B-ideals in B-algebra. The concept of JU-algebras was first proposed by Ref. 7 in 2020, and in 2022, Ref. 8 investigated ideals and subalgebras of JU-algebras.

Fuzzy sets in general can be applied in the area of medicine, computer science, cyber security and cryptography and others. In medical diagnosis, symptoms often lack clear-cut boundaries, making it challenging to classify them strictly. For example, blood sugar levels in diabetes can vary over a range, and fuzzy sets help model these varying degrees of health conditions. Using fuzzy set membership functions, a blood glucose level of 160 mg/dL could be assigned a degree of membership, such as 0.2 for low levels, 0.5 for medium levels, and 1 for high levels, based on predefined thresholds.

Consider a Q-fuzzy JU-algebra as follows: Diabetes medical condition = { Low ( below 140 mg / dl , 0.2 ) , Medium ( around 160 mg / dl , 0.5 ) , high ( over 200 mg / dl , 1 ) }

We can see that membership degree 0.2 mean that the patient medical condition is pre-diabetes, while membership degree 1 means the medical condition of an individual is hypoglycemia. Which helps the doctors to precisely measure and make robust decisions in diagnosis.

Despite its various applications, JU-algebra still requires further development. This motivates us to introduce the notion of a Q-fuzzy JU-subalgebras and a Q-fuzzy JU-ideals of JU-algebra and proved some results. We proved that the lower and upper level subsets of a Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideal are also a JU-subalgebra or JU-ideals respectively. Additionally, we studied Doubt of Q-fuzzy JU-subalgebras and Doubt of Q-fuzzy JU-ideals of JU-algebra and investigated some of their properties. We proved that the Q-fuzzy subset of a JU-algebra is a Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideal if and only if its complement is a Doubt Q-fuzzy subalgebra and Doubt of Q-fuzzy ideal, respectively. Furthermore, we proved that a normal Q-fuzzy JU-subalgebras or Q-fuzzy JU-ideals generated by any other Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideals is also normal. We also studied the connection between Normal and Doubt of Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals of JU-algebra and obtained some important properties.

 Preliminaries

7 JU-algebra is an algebra ( X , ◦ , 1 ) of type ( 2 , 0 ) with a binary operation ◦ and a fixed element 1 if it holds , ∀ x , y , z ∈ X :

( ( x ◦ y ) ◦ [ ( y ◦ z ) ◦ ( x ◦ z ) ]

1 ◦ x = x

x ◦ y = 1 and y ◦ x = 1 ⇒ x = y , ∀ x , y ∈ X

In a JU-algebra X, we can define a partial ordering “≤” by Putting x ≤ y if and only y ◦ x = 1 .

7, 9 If X is a JU-algebra, then the following hold for any x , y , z ∈ X :

x ≤ y implies y ◦ z ≤ x ◦ z

x ≤ y and y ≤ x ⇒ x = y

x ◦ x = 1

z ◦ ( y ◦ x ) = y ◦ ( z ◦ x )

( x ◦ [ ( x ◦ y ) ◦ y ] ) = 1

8 A nonempty subset S of a JU-algebra X is called a subalgebra of X if x ◦ y ∈ S , ∀ x , y ∈ S .

8 A nonempty subset J of a JU-algebra X is said to be a JU-ideal of X if it satisfies:

1 ∈ J

x , x ◦ y ∈ J ⇒ y ∈ J , ∀ x , y ∈ X .

3 A Q-fuzzy set in a nonempty set X (or a Q-fuzzy subset of X) is an arbitrary function η : X × Q → [ 0 , 1 ] , where Q is a nonempty set and [0, 1] is the unit segment of the real line.

3 let η be a Q-fuzzy set in X, ∀ t ∈ [ 0 , 1 ] .The set U ( η ; t , q ) = { x ∈ X | η ( x ) ≥ t } is called an upper-level subset of η. And the set L ( η ; t , q ) = { x ∈ X | η ( x ) ≤ t } is called a lower-level subset of η.

10 If η be a Q-fuzzy set in a set X. Then the complement denoted by ηc , is the Q-fuzzy subset of X given by ηc ( x , q ) = 1 − η ( x , q ) , ∀ x ∈ X and q ∈ Q .

4 let µ be a fuzzy subset of a JU-algebra X and α ∈ [ 0 , 1 ] . The fuzzy sets µ α and µ α of X are defined as follows:

µ α = min { µ ( x ) , α } and µ α = max { µ ( x ) , 1 − α } , ∀ x ∈ X . It is evident that µ 1 = μ , μ 0 = 0 , and μ 1 = μ , μ 0 = μ

6 A fuzzy set μ of a JU-algebra X is said to be normal if there exists x ∈ X such that µ ( x ) = 1 .

Q-Fuzzy subalgebras of a JU-algebra

In this section we discuss and investigate a new notion of Q-fuzzy JU-subalgebras in JU-algebras and study their related properties.

Let X be a JU-algebra, a Q-fuzzy set η in X is called Q-fuzzy JU-subalgebra of X if for any x , y ∈ X η ( x ◦ y , q ) ≥ min { η ( x , q ) , η ( y , q ) } , ∀ q ∈ Q .

Let X = { 1 , 2 , 3 , 4 } in which ◦ is defined by the following table ( Table 1 ). 7 Clearly, ( X , ◦ , 1 ) is a JU-algebra. Let Q = { q 1 , q 2 , q 3 } and we define a Q-fuzzy set η : Q × X → [ 0 , 1 ] : by Routine calculation gives that η is a Q-fuzzy JU-subalgebra of X ( Table 2 ).
◦ 1 2 3 4
1 1 2 3 4
2 1 1 4 1
3 1 1 1 1
4 1 4 4 1
◦ q 1 q 2 q 3
1 0.9 0.7 0.8
2 0.6 0.5 0.6
3 0.4 0.3 0.5
4 0.5 0.4 0.7

If η is a Q-fuzzy JU-subalgebra of a JU-algebra X. Then η ( 1 , q ) ≥ η ( x , q ) , ∀ x ∈ X and ∀ q ∈ Q .

Suppose η be a Q-fuzzy JU-subalgebra of a JU-algebra X.

Since, η ( 1 , q ) = η ( x ◦ x , q ) ............... (Lemma no. 1 c)

      ≥ min { η ( x , q ) , η ( x , q ) }

      = η ( x , q )

Let η be a Q-fuzzy set in a JU-algebra X, ∀ t ∈ [ 0 , 1 ] . The set η = { x ∈ X | η ( x , q ) ≥ t } is called an upper-level subset of η. And the set η t = { x ∈ X | η ( x ) ≤ t } is called a lower-level subset of η.

Clearly , ηt ∪ ηt = X for any t ∈ [ 0 , 1 ] if t 1 ≤ t 2 , then η t 1 ⊆ η t 2 but if t 2 ≤ t 1 , then η t 2 ⊆ η t 1 .

A Q-fuzzy set η of a JU-algebra X is a Q-fuzzy subalgebra if and only if for every t ∈ [ 0 , 1 ] , q ∈ Q , and ηt is a JU-subalgebra of X.

Proof: Suppose that η is a Q-fuzzy subalgebra of a JU-algebra X and η t ≠ ∅ . For any x , y ∈ η t such that η ( x , q ) ≤ t and η ( y , q ) ≤ t implies that η ( x ◦ y , q ) ≤ min { η ( x , q ) , η ( y , q ) } ≤ min { t , t } = t ⇒ x ◦ y ∈ η t

Hence, η t is a subalgebra of X.

Conversely, if η t is a subalgebra of X. Let x, y ∈ η t and there exist t 1 , t 2 ∈ [ 0 , 1 ] such that η ( x , q ) ≤ t 1 and η ( y , q ) ≤ t 2 .

Take, = min { t 1 , t 2 } . By assumption η t is a subalgebra of X, we have   x ◦ y ∈ η t then η ( x ◦ y ) ≤ t = min { η ( x , q ) , η ( y , q ) } η is a Q-fuzzy JU-subalgebra of X.

A Q-fuzzy set η of a JU-algebra X is a Q-fuzzy subalgebra if and only if for every t ∈ [ 0 , 1 ] and q ∈ Q , ηt is a JU-subalgebra of X.

It is straight forward by theorem 3.

A Q-fuzzy set η of a JU-algebra X is called a Doubt Q-fuzzy JU-subalgebra of X if ∀ x , y ∈ X and q ∈ Q η ( x + y , q ) ≤ max { η ( x q ) η ( y , q ) }

Let X = { 1 , 2 , 3 , 4 } in which ◦ is defined by the following table ( Table 3 ). 7
◦ 1 2 3 4
1 1 2 3 4
2 2 1 2 2
3 1 2 1 3
4 1 2 1 1

Clearly, ( X , ◦ , 1 ) is a JU-algebra. Let Q = { q 1 , q 2 } and we define a Q-fuzzy set η : Q × X → [ 0 , 1 ] by:

Routine calculation gives that η is a Doubt Q-fuzzy JU-subalgebra of X ( Table 4 ).
◦ q 1 q 2
1 0.3 0.1
2 0.5 0.6
3 0.8 0.7
4 0.9 0.8

If η is a Doubt Q-fuzzy JU-subalgebra of a JU-algebra X. Then η ( 1 , q ) ≤ η ( x , q ) , ∀ x ∈ X and ∀ q ∈ Q .

Suppose η be a Doubt Q-fuzzy JU-subalgebra of a JU- algebra X.

Since, ( 1 , q ) = η ( x ◦ x , q ) .................. (Lemma no. 1 c)

      ≥ max { η ( x , q ) , η ( x , q ) }

      = η ( x , q )

For every subalgebra of a JU-algebra X, it can be realized as a level subalgebra of some Doubt Q-fuzzy JU-subalgebra of X.

Let X be a subalgebra of a JU-algebra X, and let η be a Q-fuzzy set in X defined by: η ( x , q ) = { t , if x ∈ X and q ∈ Q 0 , if x ∉ X and q ∈ Q

Where t ∈ [ 0 , 1 ] is fixed. Clearly, ηt ∪ ηt = X .

We went to prove that η is a Doubt Q-fuzzy subalgebra of X.

If x , y ∈ X and q ∈ Q then x ◦ y ∈ X η ( x , q ) = η ( x ◦ y , q ) = t η ( x ◦ y , q ) ≤ max { η ( x , q ) , η ( y , q ) }

If x , y ∉ X and q ∈ Q then η ( x , q ) = η ( y , q ) = 0

η ( x ◦ y , q ) ≤ max { η ( x , q ) , η ( y , q ) } = 0

If at most one of x , y ∈ X and q ∈ Q then at least one of η ( x , q ) and η ( y , q ) is equal to t. Thus max { η ( x , q ) , η ( y , q ) } = t ⇒ ( x ◦ y , q ) ≤ t .

A Q-fuzzy subset η of a JU-algebra X is a Q-fuzzy subalgebra of X if and only if its complement η c is a Doubt Q-fuzzy subalgebra of X.

Let η be a Q-fuzzy subalgebra of X and let x, y ∈ X.

Then η c ( x ◦ y , q ) ≤ 1 − min { η ( x , q ) , η ( y , q ) } = max { 1 − η ( x , q ) , 1 − η ( y , q ) } = max η c ( x , q ) , η c ( y , q )

Conversely, suppose is a Doubt Q-fuzzy subalgebra of X. 1 − η ( x ◦ y , q ) = η c ( x ◦ y , q ) ≤ max { η c ( x , q ) , η c ( y , q ) } = max { 1 − η ( x , q ) , 1 − η ( y , q ) } = 1 − min { η ( x , q ) , η ( y , q ) } ⇒ η ( x ◦ y , q ) ≥ min { η ( x , q ) , η ( y , q ) }

A Q-fuzzy JU-subalgebra η of X is said to be normal if there exists x ∈ X such that η ( x , q ) = 1

Consider X = { 1 , 2 , 3 , 4 , 5 } , we construct the following table.

Clearly, ( X, ◦, 1) is a JU-algebra ( Table 5 ). 7
◦ 1 2 3 4 5
1 1 2 3 4 5
2 1 1 3 4 5
3 1 2 1 4 4
4 1 1 3 1 3
5 1 1 1 1 1

Let Q = { q 1 , q 2 } , given a Q-fuzzy set η : Q × X → [ 0 , 1 ] then it gives η is a normal Q-fuzzy JU-subalgebra of X( Table 6 ).
◦ q 1 q 2
1 1 1
2 0.8 0.9
3 0.7 0.6
4 0.5 0.4
5 0.3 0.1

If η is a normal Q-fuzzy JU-subalgebra of a JU-algebra X, then η ( 1 , q ) = 1

Suppose η be normal Q-fuzzy JU-subalgebra of a JU-algebra X.

Since, η ( 1 , q ) ≥ η ( x , q ) = 1 .......................( Lemma 2 )

⇒ η ( 1 , q ) ≥ 1 B ut, η ( 1 , q ) ≤ 1

Therefore, η ( 1 , q ) = 1 .

For a Q-fuzzy JU-subalgebra of η of X, we can generate a normal fuzzy subalgebra of X which contains η.

Proof: Let η be a Q-fuzzy JU-subalgebra of X. Define a Q-fuzzy subset η ∗ of X as: η ∗ ( x , q ) = η ( x , q ) + η c ( 1 , q , ) , ∀ x ∈ X .

Let x , y ∈ X , η ∗ ( x ◦ y , q ) = η ( x ◦ y , q ) + η c ( 1 , q ) ≥ min { η ( x , q ) , η ( y , q ) } + η c ( 1 , q ) ) = min { η ( x , q ) + η c ( 1 , q ) , η ( y , q ) + η c ( 1 , q ) } = min { η ∗ ( x , q ) , η ∗ ( y , q ) }

And η ∗ ( 1 , q ) = η ( 1 , q ) + η c ( 1 , q ) = η ( 1 , q ) + 1 − η ( 1 , q ) = 1

⇒ is normal Q-fuzzy JU-algebra of X. Clearly, η ⊆ η ∗ thus η ∗ is a normal JU-subalgebra of X which contains η.

If η itself is normal then η = η ∗ If η is a Q-fuzzy JU-subalgebra of X then ( η ∗ ) ∗ = η ∗

It is straight forward by theorem 9 .

Let η be a Q-fuzzy JU-subalgebra of X. If η contains a normal Q-fuzzy JU-subalgebra of X generated by any other Q-fuzzy JU-subalgebra of X then η is normal.

Let λ be a Q-fuzzy JU-subalgebra of X. by theorem 9 and λ ∗ is a normal Q-fuzzy JU-subalgebra of X. i.e. λ ∗ = 1 by Lemma 8 . Then let η is a Q-fuzzy JU-subalgebra of X such that λ ∗ ⊆ η then η ( x , q ) ≥ λ ∗ ( x , q ) , ∀ x ∈ X

Put x = 1 ⇒ η (1, q) ≥ λ ∗ = 1

⇒ η ( 1 , q ) ≥ 1 . Hence η is normal.

Let η be a Q-fuzzy JU-subalgebra of X and let f : [ 1 , η ( 1 , q ) ] → f [ 0 , 1 ] be an increasing function. Define a Q-fuzzy set η : X → [ 0 , 1 ] by η = f ( η ( x , q ) ) , ∀ x ∈ X then

η f is a Q-fuzzy JU-subalgebra of X.

If f ( η ( 1 , q ) ) = 1 then η f is normal

If f ( t ) ≥ t , ∀ t ∈ [ 1 , η ( 1 , q ) ] then η ⊆ η f

η f ( x ◦ y , q ) = f ( η ( x ◦ y , q ) ) ≥ f ( min { η ( x , q ) , η ( y , q ) } ) = min { f ( η ( x , q ) ) , f ( η ( y , q ) ) } = min η f ( x , q ) , η f ( y , q )

⇒ η f is a Q-fuzzy subalgebra of X.

If f ( η ( 1 , q ) ) = 1 ⇒ η f ( 1 , q ) = 1

⇒ η f is normal

Let f ( t ) ≥ t , ∀ t ∈ [ 1 , η ( 1 , q ) ]

Then ⇒ η f ( x , q ) = f ( η ( x , q ) ) ≥ η ( x , q ) , ∀ x f ∈ X

Hence η ⊆ η f .

Let η be a Doubt Q-fuzzy JU-subalgebra of X and let g : [ 1 , η ( 1 , q ) ] → [ 0 , 1 ] be a decreasing function. Define a Q-fuzzy set η : X → [ 0 , 1 ] by η g = g ( η ( x , q ) ) , ∀ x ∈ X then

η g is a Doubt Q-fuzzy JU-subalgebra of X.

If g ( η ( 1 , q ) ) = 1 then ηg is normal

If g ( t ) ≤ t , ∀ t ∈ [ 1 , η ( 1 , q ) ] then η g ⊆ η

Suppose η be a Doubt Q-fuzzy JU-subalgebra of X. Then = max { g ( η ( x , q ) ) , g ( η ( y , q ) ) } = max { η g ( x , q ) , η g ( y , q ) } = η g ( x ◦ y , q ) = ( η ( x ◦ y , q ) ) ≤ g ( max { η ( x , q ) , η ( y , q ) } )

⇒ η g is a Doubt Q-fuzzy subalgebra of X.

If g ( η ( 1 , q ) ) = 1 ⇒ η g ( 1 , q ) = 1

⇒ η g is normal.

let g ( t ) ≤ t , ∀ t ∈ [ 1 , η ( 1 , q ) ]

Then η g ( x , q ) = g ( η ( x , q ) ) ≤ η ( x , q ) , ∀ x ∈ X

Hence η g ⊆ η

Q-Fuzzy Ideal of JU-algebras

In this section we discuss and investigate new notion of Q-fuzzy JU-ideals in JU-algebras and study their related properties.

Let X be a JU-algebra, a Q-fuzzy set η in X is called Q-fuzzy JU-ideal of X if it satisfies the following conditions:

η ( 1 , q ) ≥ η ( x , q )

η ( y , q ) ≥ min { η ( x , q ) , η ( x ◦ y , q ) } , ∀ x , y ∈ X and ∀ q ∈ Q .

(See table 3.1 from example 3 ) consider X = { 1 , 2 , 3 , 4 } , and let Q = { q 1 , q 2 } , given a Q-fuzzy set η : Q × X → [ 0 , 1 ] by Routine calculation gives that η is a Q-fuzzy JU-ideal of X ( Table 7 ).

Every Q-fuzzy JU-ideals of a JU-algebra X is order reversing.

Let η be a Q-fuzzy JU-ideals of a JU-algebra X and for any x , y ∈ X be such that x ≤ y , then y ◦ x = 1 η ( x , q ) ≥ min { η ( y ◦ x , q ) , η ( y , q ) } , ∀ q ϵ Q = min { η ( 1 , q ) , η ( y , q ) } = η ( y , q )

If η is a Q-fuzzy JU-ideals of X, then η (( x ◦ y) ◦ y, q) ≥ η( x, q).

Since ( x◦ [( x◦y) ◦y]) = 1 .......................... (Lemma no.1e)

Now, η ( ( x ◦ y ) ◦ y , q ) ≥ min { η ( x , q ) , η ( x ◦ [ ( x ◦ y ) ◦ y ] ) } , ∀ q ∈ Q = min { η ( x , q ) , η ( 1 , q ) } = η ( x , q )

A Q-fuzzy ideal of a JU- algebra X is Q-fuzzy JU-subalgebra when η ( x ◦ ( x ◦ y ) , q ) ≥ η ( y , q ) , for any x , y ∈ X and ∀ q ∈ Q .

Suppose η is a Q-fuzzy ideal of a JU- algebra X and η ( x ◦ ( x ◦ y ) , q ) ≥ η ( y , q ) . Then for any x , y ∈ X and ∀ q ∈ Q . η ( x ◦ y , q ) ≥ min { η ( x , q ) , η ( x ◦ ( x ◦ y ) , q ) } ≥ min { η ( x , q ) , η ( y , q ) } .

Let η be a Q-fuzzy set in a JU-algebra X and let t ∈ [ 0 , 1 ] . Then η is a Q-fuzzy JU-ideal of X if and only if the upper level subset η t is a JU-ideal of X.

Suppose η be a Q-fuzzy JU-ideal in a JU-algebra X and η t ≠ ∅

Let x , x ◦ y ∈ η t and ∀ q ∈ Q implies that η ( x , q ) ≥ t and η ( x ◦ y , q ) ≥ t then η ( y , q ) ≥ min { η ( x , q ) , η ( x ◦ y , q ) } ≥ min { t , t } = t

This implies that y ∈ η t and hence η t is a JU-ideal of X.

Conversely, assume that the upper level subset η t is a JU-ideal of X.

Further, let we assume that there exist some x 0 ∈ X such that η ( 1 , q ) < η ( x 0 , q )

Take s = 1 2 [ η ( 1 , q ) ∗ η ( x 0 , q ) ] ⇒ η ( 1 , q ) < s < η ( x 0 , q )

x 0 ∈ η s and 1 ∉ η s , which is contradict to our assumption that η s is a JU-ideal of X.

Therefore, η ( 1 , q ) ≥ η ( x , q ) , ∀ x ∈ X .

And

assume that x 0 , y 0 ∈ X such that η ( y 0 , q ) < min { η ( x 0 , q ) , η ( x 0 ◦ y 0 , q ) }

Take s = 1 2 [ η ( y 0 , q ) + min { η ( x 0 , q ) , η ( x 0 ◦ y 0 , q ) } ] ⇒ η ( y 0 , q ) < s < min { η ( x 0 , q ) , η ( x 0 ◦ y 0 , q ) } ⇒ s > η ( y 0 , q ) and s < min { η ( x 0 , q ) , η ( x 0 ◦ y 0 ) , q } ⇒ s > η ( y 0 , q ) , s < η ( x 0 , q ) , and   s < η ( x 0 ◦ y 0 , q )

y 0 ∉ η s , a contradiction, since η s is a JU-ideal of X. Therefore, η ( y 0 , q ) ≥ min { η ( x 0 , q ) , η ( x 0 ◦ y 0 ) , q } for any x , y ∈ X and ∀ q ∈ Q .

For a Q-fuzzy JU-ideal of η of X, we can generate a normal fuzzy ideal of X which contains η.

Let η be a Q-fuzzy JU-ideal of X. Define a Q-fuzzy subset η ∗ of X as: η ∗ ( x , q ) = η ( x , q ) + η c ( 1 , q ) , ∀ x ∈ X .

Let x , y ∈ X , η ∗ ( 1 , q ) = η ( 1 , q ) + η c ( 1 , q )

≥ η ( x , q ) + η c ( 1 , q ) = η ∗ ( x , q ) And η ∗ ( y , q ) = η ( y , q ) + η c ( 1 , q ) ≥ min { η ( x , q ) , η ( x ◦ y , q ) } + η c ( 1 , q ) = min { η ( x , q ) + η c ( 1 , q ) , η ( x ◦ y , q ) + η c ( 1 , q ) } = min { η ∗ ( x , q ) , η ∗ ( x ◦ y , q ) }   Also  η ∗ ( 1 , q ) = η ( 1 , q ) + η c ( 1 , q ) = η ( 1 , q ) + 1 − η ( 1 , q ) = 1

⇒ η ∗ is normal Q-fuzzy JU-algebra of X.

Clearly η ⊆ η ∗ thus η ∗ is a normal JU-ideal of X which contains η.

If η itself is normal then η = η ∗ . And if η is a Q-fuzzy JU-ideal of X then ( η ∗ ) ∗ = η ∗

It is straight forward by theorem 22.

Let η is a Q-fuzzy JU-ideal of X. If η contains a normal Q-fuzzy JU-ideal of X generated by any other Q-fuzzy JU- ideal of X then η is normal.

Let λ be a Q-fuzzy JU- ideal of X. By theorem 22 λ ∗ is a normal Q-fuzzy JU-ideal of X. i.e., λ ∗ = 1 by Lemma 21

Let η is a Q-fuzzy JU-ideal of X such that λ ∗ ⊆ η then η ( x , q ) ≥ λ ∗ ( x , q ) , ∀ x ∈ X Put x = 1 ⇒ η ( 1 , q ) ≥ λ ∗ = 1

⇒ η ( 1 , q ) ≥ 1 . but, η ( 1 , q ) ≤ 1

Therefore, η ( 1 , q ) = 1 .

Let η be a Q-fuzzy JU- ideal of X and let f : [ 1 , η ( 1 , q ) ] → [ 0 , 1 ] be an increasing function. Define a Q-f fuzzy set η f : X → [ 0 , 1 ] by η = f ( η ( x , q ) ) , ∀ x ∈ X then

η f is a Q-fuzzy JU- ideal of X.

If f ( η ( 1 , q ) ) = 1 then ηf is normal

If f ( t ) ≥ t , ∀ t ∈ [ 1 , η ( 1 , q ) ] then η ⊆ ηf

η f ( 1 , q ) = f ( η ( 1 , q ) ) ≥ f ( η ( x , q ) ) = η f ( x , q )

Also η f ( y , q ) = f ( η ( y , q ) ) ≥ f ( min { η ( x , q ) , η ( x ◦ y , q ) } ) = min { f ( η ( x , q ) ) , f ( η ( x ◦ y , q ) ) } = min { η f ( x , q ) , η f ( x ◦ y , q ) }

⇒ η f is a Q-fuzzy JU-ideal of X.

If f ( η ( 1 , q ) ) = 1 ⇒ η f ( 1 , q ) = 1

⇒ η f is normal

Let f ( t ) ≥ t , ∀ t ∈ [ 1 , η ( 1 , q ) ]

Then η f ( x , q ) = f ( η ( x , q ) )

≥ η ( x , q ) , ∀ x ∈ X

Hence η ⊆ η f .

Let η be a Doubt Q-fuzzy JU- ideal of X and let g : [ 1 , η ( 1 , q ) ] g → [ 0 , 1 ] be a decreasing function. Define g a Q-fuzzy set η : X → [ 0 , 1 ] by η = g ( η ( x , q ) ) , ∀ x ∈ X then

η g is a Doubt Q-fuzzy JU-ideal of X.

If g ( η ( 1 , q ) ) = 1 then η g is normal

g ( t ) ≤ t , ∀ t ∈ [ 1 , η ( 1 , q ) ] then η g ⊆ η

Suppose η is a Doubt Q-fuzzy JU-ideal of X. Then η g ( 1 , q ) = g ( η ( 1 , q ) ) ≤ g ( η ( x , q ) ) = η g ( x , q )

Also η g ( y , q ) = g ( η ( y , q ) ) ≤ g ( max { η ( x , q ) , η ( x ◦ y , q ) } ) = max { g ( η ( x , q ) ) , g ( η ( x ◦ y , q ) ) } = max { η g ( x , q ) , η g ( x ◦ y , q ) }

⇒ η g is a Doubt Q-fuzzy ideal of X.

If g ( η ( 1 , q ) ) = 1 ⇒ η g ( 1 , q ) = 1

⇒ η g is normal

let g ( t ) ≤ t , ∀ t ∈ [ 1 , η ( 1 , q ) ] t hen η g ( x , q ) = g ( η ( x , q ) ) ≤ η ( x , q ) , ∀ x ∈ X

Hence η g ⊆ η

◦ q 1 q 2
1 0.6 0.8
2 0.3 0.5
3 0.2 0.4
4 0.2 0.1
Conclusions

In this study, we discussed the Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals of JU-algebra and investigated several related properties. We proved that the lower and upper level subsets of a Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideal are also a JU-subalgebra or JU-ideals. Additionally, we studied Doubt of Q-fuzzy JU-subalgebras and Doubt of Q-fuzzy JU-ideals of JU-algebra and investigated some of their properties. We proved that the Q-fuzzy subset of a JU-algebra is a Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideal if and only if its complement is a Doubt Q-fuzzy subalgebra and Doubt of Q-fuzzy ideal, respectively. Furthermore, we proved that a normal Q-fuzzy JU-subalgebras or Q-fuzzy JU-ideals generated by any other Q-fuzzy JU-subalgebra or Q-fuzzy JU-ideals is also normal. We also studied the connection between Normal and Doubt of Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals of JU-algebra and obtained some important properties. The findings of this research are expected to enrich the theory of Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals, based on Q-fuzzy sets, and will serve as a base for further study on this concept.

 Author contributions

Selamawit Hunie Gelaw: Conceptualization, data curation, formal analysis, Investigation, Methodology, and original draft preparation. Birhanu Assaye Alaba and Mihret Alamneh Taye: Conceptualization, Methodology, supervision, Validation, writing review and editing. All authors have read and agreed to the published version of the manuscript.

 Ethics approval and consent to participate

Not applicable.

 Consent for publication

Not applicable.

 Data availability

No data is associated with this article.

 References Zadeh LA : Fuzzy sets. Inf. Control. 1965;8:338–353. 10.1016/S0019-9958(65)90241-X Âùò Ó : É-FUZZY SUBALGEBRAS OF BCK/BCI-ALGEBRAS. 2001. Tanamoon K Sripaeng S Iampan A : Q-fuzzy sets in UP-algebras. Songklanakarin J. Sci. Technol. 2018;40(1):9–29. Barbhuiya S : Doubt fuzzy ideals of BF-algebra. IOSR J. Mathe. (IOSR-JM). 2014;10(2):65–70. 10.9790/5728-10276570 Prabpayak C : Doubt Fuzzy Ideals of B-Algebras. Appl. Mech. Mater. 2017;866:410–413. 10.4028/www.scientific.net/AMM.866.410 Prasanna APM Ismail Mohideen S : Doubt Fuzzy B-Ideals in B-Algebra. Proceedings in International Conference on Mathematical Computer Engineering. Nov-2018. Ansari MA Haider A Koam A : On JU-algebras and p-closure ideals. Int. J. Math. Comput. Sci. 2020;15(1):135–154. Da R : CONCEPT OF FILTERS IN JU-ALGEBRAS. Annals of Oradea University-Mathematics Fascicola. 2022 Jan 1;19(1). Ral RIVFA : A Binary Block Code Generated by JU-Algebras. Asian Res. J. Math. 2023 Jun 30;19(9):58–67. 10.9734/arjom/2023/v19i9699 Muthuraj RSM Muthuraman MS Sitharselvam PM : Anti Q-Fuzzy BG–Ideals in BG–Algebra. Int. J. Comput. Appl. 2010 Aug;4(11):27–31. 10.5120/867-1217 10.5256/f1000research.176217.r368443 Reviewer response for version 1 Olgun Necati 1 Referee https://orcid.org/0000-0001-6683-126X Gaziantep University, Gaziantep, Turkey

Competing interests: No competing interests were disclosed.

 25 3 2025 Copyright: © 2025 Olgun N 2025 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

JU-algebras is an important class in abstract algebra.

In the article JU-algebras have extended by incorporating fuzzy set theory to handle uncertainty in algebraic structures.  In this study, authors have applied the concept of Q-fuzzy sets to JU subalgebras and JU-ideals in JU-algebra and defined Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals as subsets of a JU-algebra also explored lower and upper level subsets of these fuzzy structures to analyze their properties.

Additionally, the concepts of Doubt and Normal Q-fuzzy JU-subalgebras and Q-fuzzy JU ideals are introduced, offering a way to deal with varying degrees of uncertainty and regularity in these algebraic structures.

​​​​​The paper is original and has no errors from a mathematical point of view.

However, it needs some minor corrections. For example, there are definitions at the beginning of the paper but no relevant examples. There is no information about the definition of fuzzy set. Again, there is no definition and result from references 1 and 2. The number of references in the article is very low. You can look at the studies done in recent years and add them to the reference section.

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?

Partly

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

Yes

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:

Abstract Algebra, Sets, Groups , Rings, Modules, Fuzzy sets

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.176217.r369357 Reviewer response for version 1 Ansari Moin A 1 Referee https://orcid.org/0000-0002-1175-9704 Jazan University, Jazan, Jazan, Saudi Arabia

Competing interests: No competing interests were disclosed.

 24 3 2025 Copyright: © 2025 Ansari MA 2025 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

The authors have extended the concept of JU-algebras in fuzzy set theory to handle uncertainty in algebraic structures. JU-algebra is a class of logical algebras. Authors have studied Q-fuzzy sets to JU-subalgebras and JU-ideals in JU-algebra. 

The paper is well-written and the results are mathematically true. This work can be considered as a contribution to the literature of logical algebras. However, the following revision is required.

1. The study is based on JU-algebras. A separate paragraph for past and recent studies that have been done on JU-algebras must be provided in the introduction part possibly the second paragraph. This paragraph should include works based on JU-algebras specifically recent works on it. 

2. Provide an example for each definition starting from Definition 1 to Definition 8 to make the study interesting for readers.

3. In the conclusion part, write the possible directions for further studies that can be done based on your current work.

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?

Yes

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

Yes

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:

Algebras including logical and classical with its applications in different fields of science and technology

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.

 10.5256/f1000research.176217.r361133 Reviewer response for version 1 Şanlıbaba İbrahim 1 Referee Nevşehir Hacı Bektaş Veli University, Nevşehir, Turkey

Competing interests: No competing interests were disclosed.

 5 2 2025 Copyright: © 2025 Şanlıbaba İ 2025 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 Report on the paper:

" Q-fuzzy structure on JU-algebra "

I gave the manuscript a thorough reading. The paper presents and specifies new kinds of fuzzy structures, in JU - algebras, to improve on classical JU-algebra theory, such as Q-fuzzy JU - subalgebras and Q-fuzzy JU-ideals. In order to open the door for more research into fuzzy algebra in mathematical and applied contexts, it also looks at important characteristics of these structures, such as their lower and upper level subsets, and looks at particular cases like Doubt and Normal Q-fuzzy structures.

The outcomes are positive and satisfying, and the study appears to follow accepted scientific practices. The argument in the paper was constructed on a solid foundation of theory, concept, or other ideas. Additionally, the study or intellectual effort equivalent on which the publication is based was well planned. The technique is correctly applied. The article's substance is accurately and succinctly described in the abstract, and the conclusions and interpretations are supported by the data. But I wish to draw attention to these problems in the manuscript:

 

The motivation and originality of the article should be mentioned in the introduction section.

The novelty of the paper should be better emphasized in the introduction section.

Authors should better explain the advantages of the present method over the other numerical methods.

Conclusion part of the manuscript can be developed and more explanation about the results of the examples can be added. (Maybe the advantages of the present methods could be given in this section).

Please check entire manuscript carefully for grammatical errors and typos.

The importance of the design carried out in this manuscript can be explained better than other important studies published in this field. I recommend the authors to review other recently developed works. Şanlıbaba İ, et al., 2024 (Ref 1) and Şanlıbaba İ., 2025 (Ref 2), Şanlıbaba İ., 2022 (Ref 3)

Finally, please check the entire manuscript carefully about the above mentioned suggestions.

Last of all, this manuscript does have significant numerical methods content. The topic content of this paper is consistent with the goals of your journal. So my recommendation is  acceptance of the manuscript after the major changes. It will hopefully add value to the existing literature.

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?

Yes

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

Yes

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 set, fuzzy logic, Decision makin support, Artificial intelligient, Fuzzy soft set.

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.

References : A novel method for decision-making approach using multi-fuzzy soft systems with applications in analyzing the ımpact of two distinct drug categories. Computational and Applied Mathematics .2024;43(5) : 10.1007/s40314-024-02791-7 10.1007/s40314-024-02791-7 : Uncredibility with fuzzy variables in uncertainty and its use in decision support in making sense of heart sounds. Applied Soft Computing .2025;169: 10.1016/j.asoc.2024.112641 10.1016/j.asoc.2024.112641 : Similarity measurement of fuzzy entropies of respiratory sounds and risk measurement according to credibility distributions. Soft Computing .2022;26(19) : 10.1007/s00500-022-07415-y 10007-10017 10.1007/s00500-022-07415-y Gelaw Selamawit Bahir Dar University, Ethiopia

Competing interests: No competing interst

 5 2 2025

I sincerely appreciate and thank you for your insightful suggestions and constructive comments, which have been invaluable in improving the quality of our work. 

 Herewith below  my response to reviewer comments

1. The motivation and originality of the article should be mentioned in the introduction section

In the introduction, the motivation and originality is presented under the last paragraph. To explain clearly, JU-algebras have not been thoroughly explored in the context of Q-fuzzy sets, specifically with respect to JU-subalgebras and JU-ideals.  This gap motivates us to conduct this study, where we introduce the novel concepts of Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals to extend the theoretical understanding and practical applications of JU-algebras.

2. The novelty of the paper should be better emphasized in the introduction section.

The novelty of the paper is mentioned in the introduction, basically we extends Q-fuzzy sets to JU-algebras, offering new insights and algebraic properties that were previously unexplored. NOTE:  JU- algebra is a crisp set.

3.  Authors should better explain the advantages of the present method over the other numerical methods.

The advantages of the present method is explained in the introduction with example how Q-fuzzy JU-subalgebras and Q-fuzzy JU-ideals allow for a more nuanced representation of fuzzy relationships, making them particularly well-suited for applications in fields like medicine, where conditions often do not have clear-cut boundaries. Furthermore, the inclusion of Doubt fuzzy sets enhances the robustness of the method, allowing for better handling of uncertain data.

4.  Conclusion part of the manuscript can be developed and more explanation about the results of the examples can be added. (Maybe the advantages of the present methods could be given in this section).

Yes we can explain more and our examples demonstrate how the method can be applied to real-world scenarios, but our focus is to highlight the theoretical concepts of  Q-fuzzy JU- sub algebra and Q- fuzzy JU- ideals of JU-algebra . Since we present examples for the theoretical illustration purpose only  so that adding results under conclusion is not as such important 

6. The importance of the design carried out in this manuscript can be explained better than other important studies published in this field. I recommend the authors to review other recently developed works. Şanlıbaba İ, et al., 2024 (Ref 1) and Şanlıbaba İ., 2025 (Ref 2), Şanlıbaba İ., 2022 (Ref 3)"

 I appreciate your nice work and the material you shared.  Yes this novel concepts of Q-fuzzy JU-sub algebras and Q-fuzzy JU-ideals demonstrates not only extends the theoretical frameworks of JU-algebras but also insights and invites other researchers work on practical applications in area like medical diagnostics, where precise and adaptable fuzzy models are critical for robust decision-making. However, at this time bringing and introducing this new theoretical concepts should come first.