T-Fuzzy Structure on JU-Algebra

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We investigated the properties of the T-fuzzy structures within this algebraic framework. Our work focuses on characterizing idempotent T-fuzzy JU algebras and analyzing their behavior in Cartesian products. Method We begin by defining T-fuzzy JU-subalgebras and JU-ideals using T-norm operations. Next, we examine the structural properties of these algebraic systems through a theoretical analysis. We then studied the idempotent cases to identify their distinctive features. Finally, we prove the closure properties by constructing Cartesian products of these fuzzy structures. Conclusion Our analysis demonstrated that idempotent T-fuzzy JU algebras possess unique structural characteristics. Furthermore, we establish that the Cartesian product of the two T-fuzzy JU-subalgebras remains a T-fuzzy JU-subalgebra, and similarly for JU-ideals. These findings extend the theoretical foundations of fuzzy algebra and suggest its potential applications in related mathematical fields. 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Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] Selamawit Hunie Gelaw https://orcid.org/0009-0003-4977-7056 1,2 , Berhanu Assaye Alaba 1 , Mihret Alamneh Taye 1 Selamawit Hunie Gelaw https://orcid.org/0009-0003-4977-7056 1,2 , Berhanu Assaye Alaba 1 , Mihret Alamneh Taye 1 PUBLISHED 23 Sep 2025 Author details Author details 1 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 2 Department of Mathematics, Injibara University, Injibara, Amhara, Ethiopia Selamawit Hunie Gelaw Roles: Formal Analysis, Methodology, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization, Supervision, Writing – Review & Editing Mihret Alamneh Taye Roles: Conceptualization, Supervision, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS Abstract Introduction This study explored the application of T-norms in fuzzy algebra, specifically by examining JU-subalgebras and JU-ideals derived from crisp JU-algebras. We investigated the properties of the T-fuzzy structures within this algebraic framework. Our work focuses on characterizing idempotent T-fuzzy JU algebras and analyzing their behavior in Cartesian products. Method We begin by defining T-fuzzy JU-subalgebras and JU-ideals using T-norm operations. Next, we examine the structural properties of these algebraic systems through a theoretical analysis. We then studied the idempotent cases to identify their distinctive features. Finally, we prove the closure properties by constructing Cartesian products of these fuzzy structures. Conclusion Our analysis demonstrated that idempotent T-fuzzy JU algebras possess unique structural characteristics. Furthermore, we establish that the Cartesian product of the two T-fuzzy JU-subalgebras remains a T-fuzzy JU-subalgebra, and similarly for JU-ideals. These findings extend the theoretical foundations of fuzzy algebra and suggest its potential applications in related mathematical fields. READ ALL READ LESS Keywords JU-algebra, T-norm,T-fuzzy JU-subalgebra, T-Fuzzy JU-Ideals, Cartesian product Corresponding Author(s) Selamawit Hunie Gelaw ( [email protected] ) Close Corresponding author: Selamawit Hunie Gelaw Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Hunie Gelaw S et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Hunie Gelaw S, Alaba BA and Taye MA. T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.12688/f1000research.165402.1 ) First published: 23 Sep 2025, 14 :971 ( https://doi.org/10.12688/f1000research.165402.1 ) Latest published: 23 Sep 2025, 14 :971 ( https://doi.org/10.12688/f1000research.165402.1 ) 1. Introduction BCK and BCI-algebras are two important classes of logical algebras introduced in Ref. 1 – 3 . The class of BCK-algebra is an appropriate subclass of BCI-algebras. The concept of JU algebras was first proposed in Ref. 4 in 2020. Subsequently, in 2022 5 the subalgebras and ideals of the JU-algebras are investigated. The introduction of fuzzy sets 6 opens the door to looking at things in different dimensions, and Ref. 7 applied this concept to group theory and introduced a fuzzy subgroup, leading to the fuzzification of different algebraic structures. The triangular norm (T-norm) first introduced in Ref. 8 . He developed generalized triangular inequalities for the statistical metric space. The T-norm is a fundamental function for solving a problem that arises from the multiple-valued logic fuzzy set theory. Following Ref. 9 introducing of T-norms into the area of fuzzy logic, numerous researchers have combined the notions of fuzzy sets and T-norms into different algebraic structures such as BCI, 10 , 11 KU, 12 , 13 BG, 8 and TM. 14 The JU algebra is a crisp set. The combination of fuzzy set and T-norm concepts in JU-algebra offers several advantages that make them more effective in handling uncertainty, imprecision, and complex logical relationships. This motivated us to extend the notion of T-fuzzy JU-subalgebras and T-fuzzy JU-ideals of JU-algebras and investigate the results. Furthermore, this study discusses the characteristics of the idempotent T-fuzzy JU algebras. We also prove that if the T-fuzzy JU-ideal has a finite image, then every descending chain of JU-ideals converges at a finite step, and every ascending chain of JU-ideals converges at a finite step if and only if the set of values of any T-fuzzy JU ideals is a well-ordered subset of [0 , 1]. Moreover, the Cartesian product of any two T-fuzzy JU-subalgebras and T-fuzzy JU-ideals of a JU-algebra also preserves its fuzzy counterpart. 2. Preliminaries Definition 1. 4 A JU-algebra is an algebra of ( X , ◦, 1) of type (2, 0) with a binary operation ◦ and a fixed element 1 , if it holds, ∀ x, y, z ∈ X: (a) ( x ◦ y ) ◦ [( y ◦ z ) ◦ ( x ◦ z )] = 1 , (b) 1 ◦ x = x , (c) x ◦ y = 1 and y ◦ x = 1 → x = y , ∀ x , y ∈ X . In what follows, let ( X , ◦, 1) denote a JU-algebra unless otherwise specified. For brevity, we also refer to call X JU-algebra. Lemma 1. 4 , 15 If X is a JU-algebra, then x ◦ x = 1 for any x ∈ X . Definition 2. 5 A nonempty subset S of a JU-algebra X is called JU-subalgebra of X, if x ◦ y ∈ S , ∀ x , y ∈ S . Definition 3. 5 A nonempty subset J of a JU-algebra X is said to be a JU-ideals of X if it satisfies: (a) 1 ∈ J (b) x , x ◦ y ∈ J → y ∈ J , ∀ x , y ∈ X . Definition 4. 16 , 17 A fuzzy set ϖ in a set X is a pair ( X , M ϖ ) , where the function M ϖ : X → [0, 1] is called the memebership function of ϖ . For γ ∈ [0, 1] , the set U ( M ϖ ; γ ) = { x ∈ X | M ϖ ≥ γ } is called an upper level subset of ϖ . Definition 5. 14 A triangular norm(T-norm) is a function T : [0, 1] × [0, 1] → [0, 1] that satisfies the following conditions: (a) T ( x , 1) = x (b) T ( x , y ) = T ( y , x ) (c) T ( x , T ( y , z )) = T ( T ( x , y ), z ) (d) T ( x , y ) ≤ T ( x , z ) whenever y ≤ z , ∀ x , y , z ∈ [0, 1] Examples of T-norm are: (a) Lukasiewicz T-norm T L u k ( x , y ) = m a x { x + y − 1, 0}, ∀ x , y ∈ [0, 1] (b) Minimum T-norm T m i n ( x , y ) = m i n ( x , y ), ∀ x , y ∈ [0, 1] (c) Product T-norm T p ( x , y ) = x . y , ∀ x , y ∈ [0, 1] (d) Drastic T-norm T D ( x , y ) = { y if x = 1 x if y = 1 , ∀ x , y ∈ [ 0 , 1 ] 0 otherwise Lemma 2. 18 Let T and T ′ be T-norms. Then T ′ ( T ( p , q ), T ( r , s )) = T ( T ′ ( p , r ), T ′ ( q , s )) Definition 6. 19 In a JU-algebra ( X , ◦, 1) , an element x ∈ X is idempotent if x ◦ x = x . Definition 7. 14 Let T be a T-norm, denoted by T i d e m the set of all idempotent with respect to T, That is, T i d e m = { T ( x , x ) = x , for some x ∈ [0, 1]} . A fuzzy set ϖ in X is said to be an idempotent T-fuzzy set if Im ( M ϖ ) ⊆ T i d e m . 3. T-fuzzy JU-subalgebra of JU-algebra In this section, we introduce the notion of T-fuzzy JU-subalgebra and discuss some of their properties. Definition 8. Let ϖ = ( X , M ϖ ) be a fuzzy set in X. Then the set ϖ is a T-fuzzy JU-subalgebra with the binary operation ◦ if M ϖ ( x ◦ y ) ≥ T { M ϖ ( x ) , M ϖ ( y ) } , ∀ x , y ∈ X . Example 1. Let X = {1, 2, 3, 4} in which ◦ is defined by the following Cayley table: “See ( Table 1 ) 4 is a JU-algebra”. Let T : [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T ( x , y ) = max { x + y − 1, 0} , ∀ x , y ∈ [0, 1] . Define a fuzzy set ϖ in X by M ϖ (1) = 0.8 , M ϖ (2) = 0.6 , M ϖ (3) = 0.5 and M ϖ (4) = 0.3 . By routine calculation ϖ is a T-fuzzy JU-subalgebra of X. Table 1. T-fuzzy JU-subalgebra of a JU-algbera X . ◦ 1 2 3 4 1 1 2 3 4 2 2 1 2 2 3 1 2 1 3 4 1 2 1 1 Theorem 3. If ϖ is an idempotent T-fuzzy JU-subalgebra of X, then M ϖ (1) ≥ M ϖ ( x ). Proof. Suppose ϖ is an idempotent T-fuzzy JU-subalgebra. Since M ϖ (1) = M ϖ ( x ◦ x ) ≥ T { M ϖ ( x ), M ϖ ( x )} = M ϖ ( x ). Theorem 4. The intersection of any two T-fuzzy JU-subalgebras of X is also a T-fuzzy JU-subalgebra of X. Proof. Suppose ϖ 1 = ( X , M ϖ 1 ) and ϖ 2 = ( X , M ϖ 2 ) are two T-fuzzy JU-subalgebras of X and ∀ x , y ∈ X . Then, M ϖ 1 ∩ ϖ 2 ( x ◦ y ) = min { M ϖ 1 ( x ◦ y ) , M ϖ 2 ( x ◦ y ) } ≥ min { T { M ϖ 1 ( x ) , M ϖ 1 ( y ) } , T { M ϖ 2 ( x ) , M ϖ 2 ( y ) } } ≥ T { min { M ϖ 1 ( x ) , M ϖ 2 ( x ) } , min { M ϖ 1 ( y ) , M ϖ 2 ( y ) } } = T { M ϖ 1 ∩ ϖ 2 ( x ) , M ϖ 1 ∩ ϖ 2 ( y ) } Corollary 5. Let { j : j ∈ Ω} be a family of T-fuzzy JU-subalgebras of X. Then ∩ j ∈Ω ϖ j is also T-fuzzy JU-subalgebra of X, where ∩ j ∈Ω ϖ j = {( x , i n f j ∈Ω M ϖ j ) | x ∈ X }. Remark 1. The union of any two T-fuzzy JU-subalgebras of a JU-algebra X may not be a T-fuzzy JU-subalgebra of X. Example 2. Let X = {1, 2, 3, 4} in which ◦ is defined by the following Cayley table: “( See Table 2 ) 4 is a JU-algebra”. Let T : [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T ( x , y ) = min { x , y } , ∀ x , y ∈ [0, 1] . Let us define the fuzzy set ϖ 1 and ϖ 2 in X by Table 2. Union of two T-fuzzy JU-subalgebra of a JU-algbra X . ◦ 1 2 3 4 1 1 2 3 4 2 1 1 4 1 3 1 1 1 1 4 1 4 4 1 M ϖ 1 (1) = 0.7 , M ϖ 1 (2) = 0.4 , M ϖ 1 (3) = 0.2 , M ϖ 1 (4) = 0.1 and M ϖ 2 (1) = 0.8 , M ϖ 2 (2) = 0.6 , M ϖ 2 (3) = 0.3 and M ϖ 2 (4) = 0.2 . Then, M ϖ 1 ∪ ϖ 2 (2 ◦ 3) = max { M ϖ 1 (2 ◦ 3), M ϖ 2 (2 ◦ 3)} = 0.2 (*) And, M ϖ 1 ∪ ϖ 2 (2 ◦ 3) = max { M ϖ 1 (2 ◦ 3), M ϖ 2 (2 ◦ 3)} ≥ max { T ( M ϖ 1 (2), M ϖ 1 (3)), T ( M ϖ 2 (2), M ϖ 2 (3))} = T { max { M ϖ 1 (2), M ϖ 2 (2)}, max { M ϖ 1 (3), M ϖ 2 (3)}} = 0.3 (**) From (∗) and (∗∗) we get 0.2 ≥ 0.3 which is false. Theorem 6. Let S be a nonempty subest of a JU-algebra X. Then the characteristics function X S is a T-fuzzy JU-subalgebras of X if and only if S is a subalgebra of X. Proof. Suppose X S is a T-fuzzy JU-subalgebras of X and S ≠ ∅. Let x , y ∈ S implies that X S ( x ) = 1 = X S ( y ). Now X J ( x ◦ y ) ≥ T { X S ( x ), X S ( y )} = T {1, 1} = 1 → X S ( x ◦ y ) ≥ 1 but X S ( x ◦ y ) ≤ 1 → X S ( x ◦ y ) = 1 → x ◦ y ∈ S → S is the subalgebra of X . Conversely, suppose S is a JU-subalgebra of X . We need to show that X S is a T-fuzzy JU-subalgebra of X . Now consider the following cases: Case 1: if x , y ∈ X where x ◦ y ∈ S then X S ( x ◦ y ) = 1 ≥ T { X S ( x ), X S ( y )} Case 2: if x ∈ S and y ∉ (or x ∉ S and y ∈ S ), then X S ( x ) = 1, X S ( y ) = 0. Thus X S ( x ◦ y ) ≥ 0 = T {1, 0} = T {0, 1} = T { X S ( x ), X S ( y )} Case 3: Suppose x , y ∉ S then X S ( x ) = 0 = X S ( y ). Thus X S ( x ◦ y ) ≥ 0 = T {0, 0} = T { X S ( x ), X S ( y )} Theorem 7. Let U ( M ϖ : γ ) be nonempty and ∀ γ ∈ [0, 1] . A fuzzy subset ϖ of a JU-algebra X is T-fuzzy JU-subalgebra of X, if and only if U ( M ϖ : γ ) is subalgebra of X. Proof. Suppose that ϖ is T-fuzzy JU-subalgebra of X . Since X is JU-algebra, then 1 ∈ X implies that M ϖ (1) ≥ γ . → 1 ∈ U ( M : γ ) → U ( M ϖ : γ ) ≠ ∅ , γ ∈ [ 0 , 1 ] . Let x , y ∈ U ( M : γ ). Then, M ϖ ( x ) ≥ γ and M ϖ ( y ) ≥ γ . We have M ϖ ( x ◦ y ) ≥ T { M ϖ ( x ) , M ϖ ( y ) } ≥ γ This implies that x ◦ y ∈ U ( M ϖ : γ ) and hence U ( M ϖ : γ ) is a subalgebra of X . Conversely, suppose that U ( M ϖ : γ ) is a JU-subalgebra of X for any γ ∈ [0, 1] and U ( M ϖ : γ ) ≠ ∅. Assume that ϖ is not T-fuzzy JU-subalgebra of X . Then there exist some z 0 , z 1 ∈ X such that M ϖ ( z 0 ◦ z 1 ) < { M ϖ ( z 0 ) , M ϖ ( z 1 ) } Take β = 1 2 [ M ϖ ( z 0 ◦ z 1 ) + { M ϖ ( z 0 ), M ϖ ( z 1 )}] → M ϖ ( z 0 ◦ z 1 ) < β < { M ϖ ( z 0 ) , M ϖ ( z 1 ) } → z 0 ◦ z 1 ∉ ( M : β ), a contradiction, since U ( M ϖ : β ) is a subalgebra of X . Therefore, M ϖ ( z 0 ◦ z 1 ) ≥ { M ϖ ( z 0 ), M ϖ ( z 1 )} for any z 0 , z 1 ∈ X . Theorem 8. Let T be a T-norm and let ϖ = ( X , M ϖ ) be a fuzzy set in a JU-algebra of X with I ( M ϖ ) = { γ 1 , γ 2 , …, γ n } where γ i j . Assume that there exist an ascending chain of subalgebra S o ⊆ S 1 ⊆ S 2 ⊆, …, ⊆ S n = X of X such that M ϖ ( S ˜ m ) = γ m , where S ˜ m = S m / S m −1 for m = 1, 2, 3, n and S ˜ o = S o . Then ϖ is a T-fuzzy JU-subalgebra of X. Proof. Suppose that there exist an ascending chain of subalgebra S o ⊆ S 1 ⊆ S 2 ⊆, …, ⊆ S n = X of X such that M ϖ ( S ˜ m ) = γ m , where m = 1, 2, 3,…, n . Let x , y ∈ X and let x , y ∈ S ˜ m then M ϖ ( x ) = γ m = M ϖ ( y ) and x ◦ y ∈ S ˜ m . Now M ϖ ( x ◦ y ) ≥ γ m = min { M ϖ ( x ) , M ϖ ( y ) } ≥ T { M ϖ ( x ) , M ϖ ( y ) } Suppose that x ∈ S ˜ p and y ∈ S ˜ q for p ≠ q without loss of of generality we may assume that p > q . Then M ϖ ( x ) = γ p < γ q = M ϖ ( y ) and x ◦ y ∈ S ˜ . Thus M ϖ ( x ◦ y ) ≥ γ p = min { M ϖ ( x ) , M ϖ ( y ) } ≥ T { M ϖ ( x ) , M ϖ ( y ) } Hence ϖ is a T-fuzzy JU-subalgebra of X . Theorem 9. let S be a JU-subalgebra of X and ϖ be a fuzzy set in X given by M ϖ ( x ) = { p , if x ∈ S q , otherwise ∀ p , q ∈ [ 0 , 1 ] With p ≥ q . Then ϖ is a T L u k fuzzy JU-subalgebra of X . Particularly if p = 1 and q = 0 then ϖ is an idempotent T L u k fuzzy subalgebra X . Additionally, Im ( ϖ ) = S Proof. Let x , y ∈ X . Let us consider the following cases: Case 1, If x , y ∈ S , then T Luk ( M ϖ ( x ) , M ϖ ( y ) ) = T Luk ( p , p ) = ( 2 p − 1 , 0 ) = { 2 p − 1 , if p ≥ 1 / 2 0 , otherwise ≤ p = M ϖ ( x ◦ y ) Case 2, If x , y ∉ S , then T Luk ( M ϖ ( x ) , M ϖ ( y ) ) = T Luk ( q , q ) = ( 2 q − 1 , 0 ) = { 2 q − 1 , if q ≥ 1 / 2 0 , otherwise ≤ q = M ϖ ( x ◦ y ) Case 3, If x ∈ S and y ∉ S (or x ∉ S and y ∈ S ), then T L u k (( x ), M ϖ ( y )) = T L u k ( p , q ) = max ( p + q − 1 , 0 ) = { p + q − 1 , if p + q ≥ 1 0 , otherwise ≤ q = M ϖ ( x ◦ y ) Hence ϖ is T-fuzzy JU-subalgebra of X . Suppose that p = 1 and q = 0. Then T L u k ( p , p ) = m a x { p + p − 1, 0} = 1 = p and T L u k ( q , q ) = { q + q − 1, 0} = 0 = q . Thus p , q ∈ ( T i d e m ), and Im ( M ϖ ) = S . 4. T-fuzzy JU-Ideals of JU-algebra In this section, we introduce the notion of T-fuzzy JU-ideals in JU-algebra and discuss some of their properties. Definition 9. Let ϖ be a fuzzy set in X. Then the set ϖ is a T-fuzzy JU-ideal with the binary operation ◦ if it satisfies the following axioms: a) M ϖ (1) ≥ M ϖ ( x ) b) M ϖ ( y ) ≥ T { M ϖ ( x ), M ϖ ( x ◦ y )}, ∀ x , y ∈ X . Example 3. Let X = {1, 2, 3, 4, 5, 6} in which ◦ is defined by the following Cayley table” ( See Table 3 ) 4 is a JU-algebra”. Let T : [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T ( x , y ) = x . y , for all x , y ∈ [0, 1] . Define a fuzzy set ϖ in X by M ϖ (1) = 0.9 , M ϖ (2) = 0.7 = M ϖ (3) , M ϖ (4) = 0.5 , M ϖ (5) = 0.3 = M ϖ (6) . By routine calculation ϖ is a T-fuzzy JU-ideals of X. (i.e., M ϖ (4) = 0.5 ≥ 0.45 = T { M ϖ (1), M ϖ (1 ◦ 4)} = T {0.9, 0.5} = (0.9).(0.5)) Table 3. T-fuzzy JU-ideals of a JU-algebra X . ◦ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 1 1 3 3 5 6 3 1 1 1 2 5 6 4 1 1 1 1 5 6 5 5 5 5 5 1 1 6 1 1 2 1 1 1 Theorem 10. The intersection of any two T-fuzzy JU-ideals of X is also T-fuzzy JU-ideal of X. Proof. Suppose ϖ 1 = ( X , M ϖ 1 ) and ϖ 2 = ( X , M ϖ 2 ) are two T-fuzzy JU-ideals of X and ∀ x , y ∈ X . Then, M ϖ 1 ∩ ϖ 2 ( 1 ) = { M ϖ 1 ( 1 ) , M ϖ 2 ( 1 ) } ≥ { M ϖ 1 ( x ) , M ϖ 2 ( x ) } = M ϖ 1 ∩ ϖ 2 ( x ) And, M ϖ 1 ∩ ϖ 2 ( y ) = min { M ϖ 1 ( y ) , M ϖ 2 ( y ) } ≥ min { T { M ϖ 1 ( x ) , M ϖ 1 ( x ◦ y ) } , T { M ϖ 2 ( x ) , M ϖ 2 ( x ◦ y ) } } ≥ T { min { M ϖ 1 ( x ) , M ϖ 2 ( x ) } , min { M ϖ 1 ( x ◦ y ) , M ϖ 2 ( x ◦ y ) } = T { M ϖ 1 ∩ ϖ 2 ( x ) , M ϖ 1 ∩ ϖ 2 ( x ◦ y ) } Corollary 11. Let { ϖ j : j ∈ Ω} be a family of T-fuzzy JU-ideals of X. Then ∩ j ∈Ω ϖ j is also T-fuzzy JU-ideal of X, where ∩ j ∈Ω ϖ j = {( x , i n f j ∈Ω M ϖj )| x ∈ X }. Theorem 12. Let J be a nonempty sub of a JU-algebra X. Then the characteristics function X J is a T-fuzzy JU-ideals of X if and only if J is an ideal of X. Proof. Suppose X J is a T-fuzzy JU-ideals of X and J ≠ ∅. Let x ∈ J implies that X J ( x ) = 1. Hence, X J (1) = X J ( x ◦ x ) ≥ T { X J ( x ), X J ( x )} = T {1, 1} = 1 → X J ( 1 ) ≥ 1 but X J ( 1 ) ≤ 1 → X J ( 1 ) = 1 → 1 ∈ J And, Let x , x ◦ y ∈ J implies that X J ( x ) = 1 = X J ( x ◦ y ). Now X J ( y ) ≥ T { X J ( x ) , X J ( x ◦ y ) } = T { 1 , 1 } = 1 → X J ( y ) ≥ 1 but X J ( y ) ≤ 1 → X J ( y ) = 1 → y ∈ J → X J ( y ) = 1 → J is a JU-ideal of X . Conversely, suppose J is ideal of X . We need to show that X J is a T-fuzzy JU-ideal of X . Now consider the following cases: Case 1: if x , x ◦ y ∈ J where y ∈ J then X J ( y ) = 1 ≥ T { X J ( x ) , X J ( x ◦ y ) } Case 2: if x ∈ J and x ◦ y ∉ (or x ∉ J and x ◦ y ∈ J ), then X J ( x ) = 1, X J ( x ◦ y ) = 0. Thus X J ( y ) ≥ 0 = T { 1 , 0 } = T { 0 , 1 } = T { X J ( x ) , X J ( x ◦ y ) } Case 3: if x , x ◦ y ∉ J then X J ( x ) = 0 = X J ( x ◦ y ). Thus X J ( y ) ≥ 0 = T { 0 , 0 } = T { X J ( x ) , X J ( x ◦ y ) } Theorem 13. Let ( M ϖ : γ ) be nonempty and ∀ γ ∈ [0, 1] . A fuzzy subset ϖ of a JU-algebra X is T-fuzzy JU-ideal of X, if and only if U ( M ϖ : γ ) is JU-ideal of X. Proof. Suppose ϖ is a T-fuzzy JU-ideal of X . Let γ ∈ [0, 1] and ( M ϖ : γ ) ≠ ∅. Let x , x ◦ y ∈ ( M ϖ : γ ) implies that M ϖ ( x ) ≥ γ and M ϖ ( x ◦ y ) ≥ γ . Then M ϖ ( y ) ≥ T { M ϖ ( x ) , M ϖ ( x ◦ y ) } ≥ γ This implies that y ∈ ( M ϖ : γ ) and hence U ( M ϖ : γ ) is a JU-ideal of X . Conversely, suppose that ( M ϖ : γ ) is a JU-ideal of X for any γ ∈ [0, 1] and U ( M ϖ : γ ) ≠ ∅. Assume that ϖ is not T-fuzzy JU-ideal of X . Then there exist some z 0 ∈ X such that M ϖ ( 1 ) < M ϖ ( z 0 ) . Take β = 1 2 [ M ϖ ( 1 ) + M ϖ ( z 0 ) ] → M ϖ ( 1 ) < β < M ϖ ( z 0 ) z 0 ∈ U ( M : β ) and 1 ∉ U ( M ϖ : β ), which is contradict to our assumption that U ( M ϖ : β ) is a JU-ideal of X . Therefore, M ϖ (1) ≥ M ϖ ( x ), ∀ x ∈ X . And assume that z 0 , z 1 ∈ X such that M ϖ ( z 1 ) < T { M ϖ ( z 0 ) , M ϖ ( z 0 ◦ z 1 ) } Take β = 1 2 [ M ϖ ( z 1 ) + { M ϖ ( z 0 ), M ϖ ( z 0 ◦ z 1 )}] → M ϖ ( z 1 ) < β M ϖ ( z 1 ) and β < { M ϖ ( z 0 ) , M ϖ ( z 0 ◦ z 1 ) } z 1 ∉ U ( M : β ), a contradiction, since U ( M ϖ : β ) is a JU-ideal of X . Therefore, M ϖ ( z 1 ) ≥ { M ϖ ( z 0 ), M ϖ ( z 0 ◦ z 1 )} for any z 0 , z 1 ∈ X . Theorem 14. Let ϖ be an idempotent T-fuzzy JU-ideals of X. Then the set X M ϖ = { x ∈ X | M ϖ ( x ) = M ϖ (1)} is an ideal of X. Proof. Let ϖ be an idempotent T-fuzzy JU-ideals of X . Obviously, 1 ∈ X M ϖ . Let x , x ◦ y ∈ X M ϖ implies that M ϖ ( x ) = M ϖ (1) = M ϖ ( x ◦ y ). Now M ϖ ( y ) ≥ T { M ϖ ( x ) , M ϖ ( x ◦ y ) } = T { M ϖ ( 1 ) , M ϖ ( 1 ) } = M ϖ ( 1 ) → M ϖ ( y ) ≥ M ϖ (1) but M ϖ ( y ) ≤ M ϖ (1) by definition 9 (a) → M ϖ ( y ) = M ϖ ( 1 ) → y ∈ X M ϖ Theorem 15. If every T-fuzzy JU-ideal ϖ of X has a finite image, then every descending chain of JU-ideals of X converges at finite steps. Proof. Assume that there exists a strictly descending chain J 1 ⊋ J 2 ⊋ J 3 … of JU-ideal of X which does not converge at finite step. Define a fuzzy set ϖ in X by M ϖ ( x ) = { n − 1 n , if x ∈ J n \ J n + 1 1 , x ∈ ∩ n = 1 ∞ J n , where n ∈ N and J 1 = X . Since 1 ∈ J n , ∀ n, M ϖ (1) = 1 ≥ M ϖ ( x ), ∀ x ∈ X . And for any x , y ∈ X then by the above assumption consider the following cases. Case 1. If x ∈ J n \ J n +1 and x ◦ y ∈ J m \ J m +1 for n = 1, 2, 3, …; m = 1, 2, 3, … Without loss of generality, we may assume that n ≤ m . Then x and x ◦ y ∈ J n , (since J m ⊊ J n ). Thus y ∈ J n since J n is a JU-ideals of X . Hence, M ϖ ( y ) ≥ n − 1 n = T { M ϖ ( x ) , M ϖ ( x ◦ y ) } . Case 2. If x , x ◦ y ∈ ∩ n = 1 ∞ J n , then y ∈ ∩ n = 1 ∞ J n . Thus M ϖ ( y ) = 1 = T { M ϖ ( x ), M ϖ ( x ◦ y )}. Case 3. If x ∉ ∩ n = 1 ∞ J n and x ◦ y ∈ ∩ n = 1 ∞ J n , then there exists a positive integer p such that x ∈ J p \ J p + 1 . It follows that y ∈, then M ϖ ( y ) ≥ p − 1 p = T { M ϖ ( x ), M ϖ ( x ◦ y )}. Or, if x ∈ ∩ n = 1 ∞ J n and x ◦ y ∉ ∩ n = 1 ∞ J n , then there exists a positive integer q such that x ◦ y ∈ J q \ J q +1 . Implies that y ∈, then M ϖ ( y ) ≥ q − 1 q = T { M ϖ ( x ), M ϖ ( x ◦ y )}. Thus, ϖ is a T-fuzzy JU-ideals with an infinite number of different values, which is a contradiction. Theorem 16. Every ascending chain of JU-ideals of X terminates at finite step if and only if the set of values of any T-fuzzy JU-ideals is a well ordered subset of [0, 1] . Proof. Let ϖ be a T-fuzzy JU-ideals of X . Assume that the set of values of ϖ is not a well-ordered subset of [0, 1]. Then there exist a strictly decreasing sequence { γ n } such that M ϖ ( x n ) = γ n . This implies that ( M ϖ : γ 1 ) ⊊ U ( M ϖ : γ 2 ) ⊊ U ( M ϖ : γ 3 ) ⊊ … is strictly ascending chain of the JU-ideal of X that does not terminate. This is impossible. In contrast, assume that a strictly ascending chain (*) J 1 ⊊ J 2 ⊊ J 3 ⊊ … . , of the JU-ideals of X which does not converge at the finite step. Remember that J = ∪ n ∈ N J n is JU-ideals of X . Now let us define a fuzzy set ϖ in X by M ϖ ( x ) = { 1 t , if t = min { n ∈ N | x ∈ J n 0 , if x ∉ J n We went to show that ϖ is a T-fuzzy JU-ideal of X . Since 1 ∈ J n , ∀ n , M ϖ (1) ≥ 1 n = M ϖ ( x ), ∀ x ∈ X . And for any x , y ∈ X then by the above assumption consider the following cases. Case 1: if x , x ◦ y ∈ J n \ J n −1 for n = 2, 3, … then y ∈ J n \ J n −1 implies that M ϖ ( y ) ≥ 1 n = T { M ϖ ( x ) , M ϖ ( x ◦ y ) } Case 2: If x ∈ J n and x ◦ y ∈ J n \ J m and x ◦ y ∈ J n for any m < n . Since ϖ is a JU-ideal of X , then y ∈ J n .Thus M ϖ ( y ) ≥ 1 n ≥ 1 m + 1 ≥ M ϖ ( x ◦ y ). Or, If x ∈ J n \ J m and x ◦ y ∈ J n and x ∈ J n for any m < n . Since ϖ is a JU-ideal of X , then y ∈ J n .Thus M ϖ ( y ) ≥ 1 n ≥ 1 m + 1 ≥ M ϖ ( x ◦ y ). Therefore, ϖ is a T-fuzzy JU-ideals of X . Since the sequence (*) is not converge, ϖ has a strictly descending sequence of values. This contradicts that the value set of any T-fuzzy JU-ideal is well-ordered. 5. Cartesian product of T-fuzzy JU-algebras In this section, the Cartesian products of the T-fuzzy JU-subalgebras and T-fuzzy JU-ideals of X are discussed and their properties are investigated. Definition 10. Let ϖ 1 = ( X , M ϖ 1 ) and ϖ 2 = ( Y , M ϖ 2 ) be two T-fuzzy JU-subalgebra of a JU-algebra X and Y respectively. The Cartesian product of ϖ 1 and ϖ 2 with respect to t-norm T denoted by M [ ϖ 1 × ϖ 2 ] T = ( X × Y , M [ ϖ 1 × ϖ 2 ] T ) , is defined by M [ ϖ 1 × ϖ 2 ] T ( x , y ) = T ( M ϖ 1 ( x ), M ϖ 2 ( y )) , ∀( x , y ) ∈ X × Y . Theorem 17. The Cartesian product of any two T-fuzzy JU-subalgebras of X is also a T-fuzzy JU-subalgebra of X. Proof. Suppose ϖ 1 = ( X , M ϖ 1 ) and ϖ 2 = ( Y , M ϖ 2 ) be two T-fuzzy JU-subalgebras of a JU-algebra X . Let ( x 1 , x 2 ), ( y 1 , y 2 ) ∈ X × Y . Then [ ϖ 1 × ϖ 2 ] T ( ( x 1 , x 2 ) ◦ ( y 1 , y 2 ) ) = M [ ϖ 1 × ϖ 2 ] T ( ( x 1 ◦ y 1 ) , ( x 2 ◦ y 2 ) ) = T ( M ϖ 1 ( x 1 ◦ y 1 ) , M ϖ 2 ( x 2 ◦ y 2 ) ) ≥ T ( T ( M ϖ 1 ( x 1 ) , M ϖ 1 ( y 1 ) ) , T ( M ϖ 2 ( x 2 ) , M ϖ 2 ( y 2 ) ) ) = T ( T ( M ϖ 1 ( x 1 ) , M ϖ 2 ( x 2 ) ) , T ( M ϖ 1 ( y 1 ) , M ϖ 2 ( y 2 ) ) ) = T ( M [ ϖ 1 × ϖ 2 ] T ( x 1 , x 2 ) , M [ ϖ 1 × ϖ 2 ] T ( y 1 , y 2 ) ) Theorem 18. Let { X j } j = 1 n be the finite family of JU-algebras and X = π j = 1 n X j the Cartesian product of JU-algebras of { X j } . Let ϖ j be a T-fuzzy JU-subalgebra of X, where 1 ≤ j ≤ n . Then ϖ = π j = 1 n ϖ j is defined by M ϖ ( x 1 , x 2 , …, x n ) = M ( π j = 1 n ϖ j ) T ( x 1 , x 2 , …, x n ) = T n ( M ϖ 1 ( x 1 ) , M ϖ 2 ( x 2 ) , …, M ϖ n ( x n ) ) is also T-fuzzy JU-subalgebra of X. Proof. Let x = ( x 1 , x 2 , …, x n ) and y = ( y 1 , y 2 , …, y n ) be any elements of X = π j = 1 n X j . Then M ϖ ( x ◦ y ) = M ϖ ( ( x 1 , x 2 , … , x n ) ◦ ( y 1 , y 2 , … , y n ) ) = M ϖ ( x 1 ◦ y 1 , x 2 ◦ y 2 , … , x n ◦ y n ) = T n ( M ϖ 1 ( x 1 ◦ y 1 ) , M ϖ 2 ( x 2 ◦ y 2 ) , … , M ϖ n ( x n ◦ y n ) ) ≥ T n ( T ( M ϖ 1 ( x 1 ) , M ϖ 1 ( y 1 ) ) , T ( M ϖ 2 ( x 2 ) , M ϖ 2 ( y 2 ) ) , … , T ( M ϖ n ( x n ) , M ϖ n ( y n ) ) ) = T ( T n ( M ϖ 1 ( x 1 ) , M ϖ 2 ( x 2 ) , … , M ϖ n ( x n ) ) , T n ( M ϖ 1 ( y 1 ) , M ϖ 2 ( y 2 ) , … , M ϖ n ( y n ) ) ) = T ( M ϖ ( x 1 , x 2 , … , x n ) , M ϖ ( y 1 , y 2 , … , y n ) ) = T ( M ϖ ( x ) , M ϖ ( y ) ) Definition 11. Let ϖ 1 and ϖ 2 be fuzzy sets in X and T be a t-norm. Then the T-fuzzy product of ϖ 1 and ϖ 2 denoted by [ ϖ 1 · ϖ 2 ] , is defined by M [ ϖ 1 · ϖ 2] ( x ) = T ( M ϖ 1 ( x ), M ϖ 2 ( x ))∀ x ∈ X . Also [ ϖ 1 · ϖ 2 ] = M [ ϖ 2 · ϖ 1 ] Theorem 19. Let ϖ 1 and ϖ 2 be T-fuzzy JU-subalgebras of X. If T ′ is a T-norm which dominates T, i.e. T ′ ( T ( p , q ), T ( r , s )) ≥ T ( T ′ ( p , r ), T ′ ( q , s )), ∀ p , q , r , s ∈ [0, 1] . Then the T ′ fuzzy product of ϖ 1 and ϖ 2 , [ ϖ 1 . ϖ 2]′ is a T-fuzzy JU-subalgebra of X. Proof. Let x , y ∈ X , then [ ϖ 1 · ϖ 2 ] ′ ( x ◦ y ) = T ′ ( M ϖ 1 ( x ◦ y ) , M ϖ 2 ( x ◦ y ) ) ≥ T ′ ( T ( M ϖ 1 ( x ) , M ϖ 1 ( y ) ) , T ( M ϖ 2 ( x ) , M ϖ 2 ( y ) ) ) ≥ T ( T ′ ( M ϖ 1 ( x ) , M ϖ 2 ( x ) ) , T ′ ( M ϖ 1 ( y ) , M ϖ 2 ( y ) ) ) ( by Lemma 2 ) = T ( [ ϖ 1 · ϖ 2 ] ′ ( x ) , M [ ϖ 1 · ϖ 2 ] ′ ( y ) ) Theorem 20. The Cartesian product of any two T-fuzzy JU-ideals of X is also a T-fuzzy JU-ideal of X. Proof. Suppose ϖ 1 = ( X , M ϖ 1 ) and ϖ 2 = ( Y , M ϖ 2 ) be two T-fuzzy JU-ideals of a JU-algebra X . Then let x , y ∈ X × Y . Now [ ϖ 1 × ϖ 2 ] T ( 1 , 1 ) = T ( M ϖ 1 ( 1 ) , M ϖ 2 ( 1 ) ) ≥ T ( M ϖ 1 ( x ) , M ϖ 2 ( y ) ) = [ ϖ 1 × ϖ 2 ] T ( x , y ) And, let ( x 1 , x 2 ), ( y 1 , y 2 ) ∈ X × Y . Then [ ϖ 1 × ϖ 2 ] T ( y 1 , y 2 ) = T ( M ϖ 1 ( y 1 ) , M ϖ 2 ( y 2 ) ) ≥ T ( T ( M ϖ 1 ( x 1 ) , M ϖ 1 ( x 1 ◦ y 1 ) ) , T ( M ϖ 2 ( x 2 ) , M ϖ 2 ( x 2 ◦ y 2 ) ) ) = T ( T ( M ϖ 1 ( x 1 ) , M ϖ 2 ( x 2 ) ) , T ( M ϖ 1 ( x 1 ◦ y 1 ) , M ϖ 2 ( x 2 ◦ y 2 ) ) ) = T { [ ϖ 1 × ϖ 2 ] T ( x 1 , x 2 ) , M [ ϖ 1 × ϖ 2 ] T ( ( x 1 ◦ y 1 ) , ( x 2 ◦ y 2 ) ) } = T { [ ϖ 1 × ϖ 2 ] T ( x 1 , x 2 ) , M [ ϖ 1 × ϖ 2 ] T ( ( x 1 ◦ x 2 ) , ( y 1 ◦ y 2 ) ) } 6. Conclusion In this study, we introduced the concepts of T-fuzzy JU-subalgebras and T-fuzzy JU-ideas of JU-algebras and obtained important results. The characteristics of idempotent T-fuzzy JU-algebra were discussed. We prove that if every T-fuzzy JU-ideal has a finite image, then every descending chain of JU-ideal converges at finite steps and every ascending chain of JU-ideals converges at a finite step if and only if the set of values of any T-fuzzy JU-ideals is a well ordered subset of [0, 1]. Moreover, the Cartesian product of any two T-fuzzy JU-subalgebras and T-fuzzy JU-ideals of a JU-algebra are also T-fuzzy JU-subalgebra and T-fuzzy JU-ideal respectively. This introduction of the T-norm in fuzzy JU-algebras opens the door to more effective modeling of real-world problems involving uncertainty and potential topics to develop its study in the future, such as a derivatives, bipolar forms, and interval values. Declarations Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. Data availability All data analyzed during this study are included in the manuscript. References 1. Imai Y: Iséki K. On axiom systems of propositional calculi. XIV. Proc. Jpn. Acad. 1966; 42 (1): 19–22. 2. Iséki K: An introduction to the theory of BCK-algebras. Math Japon. 1978; 23 (1): 1–26. 3. Iseki K: On BCI-algebras. Math. Seminar Notes. 1980; 8 : 235–236. 4. Ansari MA, Haider A, Koam A: On JU-algebras and p-closure ideals. Int. J. Math. Comput. Sci. 2020; 15 (1): 135–154. 5. Romano DA: Concept of JU-filters in JU-algebras. An. Univ. Oradea, Fasc. Mat. 2022; 29 (1): 47–55. 6. Zadeh LA: Fuzzy sets. Inf. Control. 1965; 8 : 338–353. Publisher Full Text 7. Rosenfeld A: Fuzzy groups. J. Math. Anal. Appl. 1971; 35 (3): 512–517. Publisher Full Text 8. Senapati T, Bhowmik M, Pal M: Triangular norm based fuzzy BG-algebras. Afr. Mat. 2016; 27 (1): 187–199. Publisher Full Text 9. Höhle U: Probabilistic uniformization of fuzzy topologies. Fuzzy Sets Syst. 1978; 1 (4): 311–332. Publisher Full Text 10. Rasuli R: T-fuzzy subalgebras of BCI-algebras. Int. J. Open Problems Compt. Math. 2023; 16 (1): 55–72. 11. Akram M, Dar K, et al. : T-fuzzy ideals in BCI-algebras. Int. J. Math. Math. Sci. 2005; 2005 (12): 1899–1907. Publisher Full Text 12. Senapati T: T-fuzzy KU-subalgebras of KU-algebras. Ann. Fuzzy Math. Inform. 2015; 10 (2): 261–270. 13. Senapati T: T-fuzzy KU-ideals of KU-algebras. Afr. Mat. 2018; 29 (3): 591–600. Publisher Full Text 14. Thomas J, Indhira K, Chandrasekaran V: T-normed fuzzy TM-subalgebra of TM- algebras. Int. J. Comput. Intell. Syst. 2019; 12 (2): 706–712. Publisher Full Text 15. Ral RI, Flores V, AL.: A Binary Block Code Generated by JU-Algebras. Asian Research. J. Math. 2023; 19 (9): 58–67. Publisher Full Text 16. Ougen X: Fuzzy BCK-algebras. Math Japonica. 1991; 36 : 935–942. 17. Das PS: Fuzzy groups and level subgroups. J. Math. Analy. Applic. 1981; 84 (1): 264–269. Publisher Full Text 18. Hadzic O, Pap E: Fixed point theory in probabilistic metric spaces. Springer Science & Business Media; 2013; Vol. 536 . . 19. Walsh A: Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce. Middlesex University; 2012. Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 23 Sep 2025 ADD YOUR COMMENT Comment Author details Author details 1 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia 2 Department of Mathematics, Injibara University, Injibara, Amhara, Ethiopia Selamawit Hunie Gelaw Roles: Formal Analysis, Methodology, Writing – Original Draft Preparation, Writing – Review & Editing Berhanu Assaye Alaba Roles: Conceptualization, Supervision, Writing – Review & Editing Mihret Alamneh Taye Roles: Conceptualization, Supervision, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 23 Sep 2025, 14:971 https://doi.org/10.12688/f1000research.165402.1 Copyright © 2025 Hunie Gelaw S et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Hunie Gelaw S, Alaba BA and Taye MA. T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.12688/f1000research.165402.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 23 Sep 2025 Views 0 Cite How to cite this report: Derseh BL. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421232 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421232 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 17 Oct 2025 Beza Lamesgin Derseh , Debre Markos University, Debre Markos, Amhara, Ethiopia Approved VIEWS 0 https://doi.org/10.5256/f1000research.182032.r421232 After carefully reading the manuscript thoroughly, I have found that the manuscript introduces a new concept of T-fuzzy structures in JU-algebra , presenting key theoretical findings that enhance the understanding of these structures by extending existing algebraic frameworks through innovative approaches. The ... Continue reading READ ALL After carefully reading the manuscript thoroughly, I have found that the manuscript introduces a new concept of T-fuzzy structures in JU-algebra , presenting key theoretical findings that enhance the understanding of these structures by extending existing algebraic frameworks through innovative approaches. The study explores the application of T-norms in fuzzy algebra, focusing on JU-subalgebras and JU-ideals derived from JU-algebras. It establishes definitions for T-fuzzy JU-subalgebras and JU-ideals based on T-norm operations, examines their structural characteristics, and identifies the distinctive features of idempotent T-fuzzy JU-algebras, including their behavior under Cartesian products. Generally, the study broadens the theoretical base of fuzzy algebra, provides novel perspectives, and underscores its potential applications in advancing mathematical research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy algebra, Intuitionistic fuzzy structures, Bipolar Fuzzy 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Derseh BL. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421232 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421232 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Kider JR. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421229 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421229 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 17 Oct 2025 Jehad R. Kider , University of Technology, Baghdad, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.182032.r421229 REVIEW REPORT Title: T-Fuzzy Structure on JU-Algebra Thank you for selecting me for review of this manuscript. 1.Generally, the paper is well written and has a good structure. 2. The aim of this paper is ... Continue reading READ ALL REVIEW REPORT Title: T-Fuzzy Structure on JU-Algebra Thank you for selecting me for review of this manuscript. 1.Generally, the paper is well written and has a good structure. 2. The aim of this paper is to offer a study explored the application of T-norms in fuzzy algebra, specifically by examining JU-subalgebras and JU-ideals derived from crisp JU-algebras. We investigated the properties of the T-fuzzy structures within this algebraic framework. 3.The theoretical framework is built systematically, starting from definitions to illustrative examples and theorems. 4.Correct the line in case 2 of Theorem 6, Case 2: if x ∈ S and y ∉ (or x ∉ S and y ∈ S), it must be x ∈ S and y ∉ S(or x ∉ S and y ∈ S). 5.The work appears to be original and free of plagiarism, and no ethical concerns are noted. 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy metric spaces, Fuzzy norned 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Kider JR. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421229 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421229 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Senapati T. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r417907 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-417907 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 30 Sep 2025 Tapan Senapati , Southwest University, Chongqing, Chongqing, China Approved VIEWS 0 https://doi.org/10.5256/f1000research.182032.r417907 This paper explores the application of T-norms in the context of fuzzy algebra, focusing on JU-algebras. Specifically, the authors investigate T-fuzzy JU-subalgebras and T-fuzzy JU-ideals derived from crisp JU-algebras, aiming to expand the theoretical framework of fuzzy algebra. By characterizing ... Continue reading READ ALL This paper explores the application of T-norms in the context of fuzzy algebra, focusing on JU-algebras. Specifically, the authors investigate T-fuzzy JU-subalgebras and T-fuzzy JU-ideals derived from crisp JU-algebras, aiming to expand the theoretical framework of fuzzy algebra. By characterizing idempotent T-fuzzy JU-algebras and analyzing their behavior in Cartesian products, the authors provide a valuable contribution to the field. The study primarily aims to enrich the understanding of the structural properties of these algebraic systems, ultimately enhancing their practical applications. The concepts introduced in this study serve as foundational for further research in areas that involve uncertainty and fuzzy logic. Key Points for Enhancing Paper Quality: Some of the definitions (such as T-fuzzy JU-subalgebra and T-fuzzy JU-ideal) are a bit dense, which may cause confusion for readers unfamiliar with the topic. A more detailed explanation or simpler formulations could be beneficial, especially for the introductory sections. While the paper includes a few examples, these could be expanded or illustrated more comprehensively. Visual aids, such as diagrams or flowcharts, would help readers better grasp complex concepts like the Cartesian product of T-fuzzy JU-subalgebras. The literature review section could be strengthened by providing more specific methods used in this paper and those employed in similar studies. Add these papers to the references and cite them in the introduction:Atanassov’s intuitionistic fuzzy bi-normed KU-subalgebras of a KU-algebra, Atanassov’s intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra. The conclusion can be strongly revised. This reviewer strongly suggests improving the flow of the conclusion section. Start with a brief explanation of the paper's goal (like the abstract), but make sure that the conclusion differs from the abstract. Provide the main findings/claims. Explain the numerical findings of the simulations. Clearly explain the significant findings and why your paper is important. The introduction must clearly describe the motivation for this work and the background of the available techniques. In principle, the selected approach and theory must be properly defined, though some improvements are necessary. Please briefly explain how the references in the introduction relate to the approach you are suggesting. The overall structure is logical but could be more organized for smoother transitions between sections. A brief summary or overview at the beginning of each section would guide the reader through the paper more effectively. While the language is generally clear, some sections would benefit from minor revisions for grammatical consistency. Specifically, tightening sentence structure in complex mathematical descriptions could improve readability. The authors provide a rigorous theoretical exploration of T-fuzzy JU-algebras and their applications. Their findings, particularly regarding the idempotent T-fuzzy JU-algebras and Cartesian products, are valuable contributions to the field of fuzzy algebra. With minor revisions to improve clarity, organization, and illustrative content, this paper could make a significant impact in the mathematical community and beyond. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fuzzy algebra, Artificial Intelligence 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Senapati T. Reviewer Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r417907 ) The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-417907 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 23 Sep 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 Version 1 23 Sep 25 read read read Tapan Senapati , Southwest University, Chongqing, China Jehad R. Kider , University of Technology, Baghdad, Iraq Beza Lamesgin Derseh , Debre Markos University, Debre Markos, Ethiopia Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Derseh B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 17 Oct 2025 | for Version 1 Beza Lamesgin Derseh , Debre Markos University, Debre Markos, Amhara, Ethiopia 0 Views copyright © 2025 Derseh B. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions After carefully reading the manuscript thoroughly, I have found that the manuscript introduces a new concept of T-fuzzy structures in JU-algebra , presenting key theoretical findings that enhance the understanding of these structures by extending existing algebraic frameworks through innovative approaches. The study explores the application of T-norms in fuzzy algebra, focusing on JU-subalgebras and JU-ideals derived from JU-algebras. It establishes definitions for T-fuzzy JU-subalgebras and JU-ideals based on T-norm operations, examines their structural characteristics, and identifies the distinctive features of idempotent T-fuzzy JU-algebras, including their behavior under Cartesian products. Generally, the study broadens the theoretical base of fuzzy algebra, provides novel perspectives, and underscores its potential applications in advancing mathematical research. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy algebra, Intuitionistic fuzzy structures, Bipolar Fuzzy 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. reply Respond to this report Responses (0) Derseh BL. Peer Review Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421232) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421232 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Kider J. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 17 Oct 2025 | for Version 1 Jehad R. Kider , University of Technology, Baghdad, Iraq 0 Views copyright © 2025 Kider J. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions REVIEW REPORT Title: T-Fuzzy Structure on JU-Algebra Thank you for selecting me for review of this manuscript. 1.Generally, the paper is well written and has a good structure. 2. The aim of this paper is to offer a study explored the application of T-norms in fuzzy algebra, specifically by examining JU-subalgebras and JU-ideals derived from crisp JU-algebras. We investigated the properties of the T-fuzzy structures within this algebraic framework. 3.The theoretical framework is built systematically, starting from definitions to illustrative examples and theorems. 4.Correct the line in case 2 of Theorem 6, Case 2: if x ∈ S and y ∉ (or x ∉ S and y ∈ S), it must be x ∈ S and y ∉ S(or x ∉ S and y ∈ S). 5.The work appears to be original and free of plagiarism, and no ethical concerns are noted. 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 Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy metric spaces, Fuzzy norned 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. reply Respond to this report Responses (0) Kider JR. Peer Review Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r421229) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-971/v1#referee-response-421229 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Senapati T. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 30 Sep 2025 | for Version 1 Tapan Senapati , Southwest University, Chongqing, Chongqing, China 0 Views copyright © 2025 Senapati T. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions This paper explores the application of T-norms in the context of fuzzy algebra, focusing on JU-algebras. Specifically, the authors investigate T-fuzzy JU-subalgebras and T-fuzzy JU-ideals derived from crisp JU-algebras, aiming to expand the theoretical framework of fuzzy algebra. By characterizing idempotent T-fuzzy JU-algebras and analyzing their behavior in Cartesian products, the authors provide a valuable contribution to the field. The study primarily aims to enrich the understanding of the structural properties of these algebraic systems, ultimately enhancing their practical applications. The concepts introduced in this study serve as foundational for further research in areas that involve uncertainty and fuzzy logic. Key Points for Enhancing Paper Quality: Some of the definitions (such as T-fuzzy JU-subalgebra and T-fuzzy JU-ideal) are a bit dense, which may cause confusion for readers unfamiliar with the topic. A more detailed explanation or simpler formulations could be beneficial, especially for the introductory sections. While the paper includes a few examples, these could be expanded or illustrated more comprehensively. Visual aids, such as diagrams or flowcharts, would help readers better grasp complex concepts like the Cartesian product of T-fuzzy JU-subalgebras. The literature review section could be strengthened by providing more specific methods used in this paper and those employed in similar studies. Add these papers to the references and cite them in the introduction:Atanassov’s intuitionistic fuzzy bi-normed KU-subalgebras of a KU-algebra, Atanassov’s intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra. The conclusion can be strongly revised. This reviewer strongly suggests improving the flow of the conclusion section. Start with a brief explanation of the paper's goal (like the abstract), but make sure that the conclusion differs from the abstract. Provide the main findings/claims. Explain the numerical findings of the simulations. Clearly explain the significant findings and why your paper is important. The introduction must clearly describe the motivation for this work and the background of the available techniques. In principle, the selected approach and theory must be properly defined, though some improvements are necessary. Please briefly explain how the references in the introduction relate to the approach you are suggesting. The overall structure is logical but could be more organized for smoother transitions between sections. A brief summary or overview at the beginning of each section would guide the reader through the paper more effectively. While the language is generally clear, some sections would benefit from minor revisions for grammatical consistency. Specifically, tightening sentence structure in complex mathematical descriptions could improve readability. The authors provide a rigorous theoretical exploration of T-fuzzy JU-algebras and their applications. Their findings, particularly regarding the idempotent T-fuzzy JU-algebras and Cartesian products, are valuable contributions to the field of fuzzy algebra. With minor revisions to improve clarity, organization, and illustrative content, this paper could make a significant impact in the mathematical community and beyond. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? 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 Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Fuzzy algebra, Artificial Intelligence 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. reply Respond to this report Responses (0) Senapati T. Peer Review Report For: T-Fuzzy Structure on JU-Algebra [version 1; peer review: 3 approved] . F1000Research 2025, 14 :971 ( https://doi.org/10.5256/f1000research.182032.r417907) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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