T-Fuzzy Structure on JU-Algebra

TL;DR

This paper studies T-fuzzy structures in JU-algebras, focusing on their properties and behavior in Cartesian products.

Contribution

The paper introduces and characterizes idempotent T-fuzzy JU algebras and their closure under Cartesian products.

Findings

Idempotent T-fuzzy JU algebras have unique structural characteristics.

Cartesian products of T-fuzzy JU-subalgebras and JU-ideals remain T-fuzzy JU-subalgebras and JU-ideals.

The study extends the theoretical foundations of fuzzy algebra.

Abstract

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.

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.

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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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) = $[eqn]$

Lemma 2. ^ 18 ^ Let $[eqn]$ and $[eqn]$ 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

$$[eqn]$$

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 1234 2 2122 3 1213 4 1211

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,

$$[eqn]$$

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 1234 2 1141 3 1111 4 1441

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. $[eqn]$

$[eqn]$ Let x, y ∈ U( M: γ). Then, M _ ϖ _ ( x) ≥ γ and M _ ϖ _ ( y) ≥ γ. We have

$$[eqn]$$

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

$$[eqn]$$

Take β = $[eqn]$ [ M _ ϖ _ ( z 0 ◦ z 1) + { M _ ϖ _ ( z 0), M _ ϖ _ ( z 1)}] $[eqn]$ → 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 _ < γ _ j _ whenever 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 _ ( $[eqn]$ ) = γ _ m _ , where $[eqn]$ = S _ m / S _ m−1 for m = 1, 2, 3, n and $[eqn]$ = 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 _ ( $[eqn]$ ) = γ _ m _, where m = 1, 2, 3,…, n. Let x, y ∈ X and let x, y ∈ $[eqn]$ then M _ ϖ _ ( x) = γ _ m _ = M _ ϖ _ ( y) and x ◦ y ∈ $[eqn]$ . Now

$$[eqn]$$

Suppose that x ∈ $[eqn]$ and y ∈ $[eqn]$ 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 ∈ $[eqn]$ . Thus

$$[eqn]$$

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

$$[eqn]$$

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

$$[eqn]$$

Case 2, If x, y ∉ S, then

$$[eqn]$$

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)

$$[eqn]$$

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 123456 2 113356 3 111256 4 111156 5 555511 6 112111

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,

$$[eqn]$$

And,

$$[eqn]$$

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 $[eqn]$

$[eqn]$

$[eqn]$ And,Let x, x ◦ y ∈ J implies that X _ J _ ( x) = 1 = X _ J _ ( x ◦ y). Now $[eqn]$

$[eqn]$

$[eqn]$

$[eqn]$

$[eqn]$ → 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 $[eqn]$ 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 $[eqn]$ Case 3: if x, x ◦ y ∉ J then X _ J _ ( x) = 0 = X _ J _ ( x ◦ y). Thus $[eqn]$

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

$$[eqn]$$

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

$$[eqn]$$ $$[eqn]$$

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

$$[eqn]$$

Take β = $[eqn]$ [ M _ ϖ _ ( z 1) + { M _ ϖ _ ( z 0), M _ ϖ _ ( z 0 ◦ z 1)}]

$$[eqn]$$ $$[eqn]$$

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

$$[eqn]$$

→ M _ ϖ _ ( y) ≥ M _ ϖ _ (1) but M _ ϖ _ ( y) ≤ M _ ϖ _ (1) by definition 9(a) $[eqn]$

$[eqn]$

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

$$[eqn]$$

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,

$$[eqn]$$

Case 2. If x, x ◦ y ∈ $[eqn]$ J _ n _, then y ∈ $[eqn]$ J _ n _.Thus M _ ϖ _ ( y) = 1 = T{ M _ ϖ _ ( x), M _ ϖ _ ( x ◦ y)}.Case 3. If x ∉ $[eqn]$ J _ n _ and x ◦ y ∈ $[eqn]$ J _ n _, then there exists a positive integer p such that

$$[eqn]$$

It follows that y ∈, then M _ ϖ _ ( y) ≥ $[eqn]$ = T{ M _ ϖ _ ( x), M _ ϖ _ ( x ◦ y)}. Or, if x ∈ $[eqn]$ J _ n _ and x ◦ y ∉ $[eqn]$ 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) ≥ $[eqn]$ = 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

$$[eqn]$$

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

$$[eqn]$$

We went to show that ϖ is a T-fuzzy JU-ideal of X. Since 1 ∈ J _ n , ∀ n, M _ ϖ _ (1) ≥ $[eqn]$ = 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 $[eqn]$ 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) ≥ $[eqn]$ ≥ $[eqn]$ ≥ 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) ≥ $[eqn]$ ≥ $[eqn]$ ≥ 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 $[eqn]$ = ( X × Y, $[eqn]$ ) , is defined by $[eqn]$ ( 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

$$[eqn]$$

Theorem 18. Let $[eqn]$ be the finite family of JU-algebras and X = $[eqn]$ 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 = $[eqn]$ ϖ _ j _ is defined by M _ ϖ _ ( x 1, x 2, …, x _ n _) = $[eqn]$ ( 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 = $[eqn]$ X _ j _. Then

$$[eqn]$$

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 [ ϖ1 · ϖ2] ( x) = T ( M _ ϖ 1 _ ( x), M _ ϖ 2 _ ( x))∀ x ∈ X .

Also [ ϖ 1 _ · _ ϖ 2] = $[eqn]$

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 . ϖ2]′ is a T-fuzzy JU-subalgebra of X.

Proof. Let x, y ∈ X, then

$$[eqn]$$

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

$$[eqn]$$

And, let ( x 1, x 2), ( y 1, y 2) ∈ X × Y. Then

$$[eqn]$$

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.

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