Fuzzy Zorn’s lemma with applications

We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring R, and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.


§1 Introduction
Zorn's lemma is a useful result to appear in proofs of some non-constructive esistence theorems throughout many mathematical branches. In 1933 Artin and Chevalley first referred to the principle as Zorns lemma. Especially, the equivalences of axioms of choice, Zorn's lemma, well-ordering principle and comparability principle were discussed ( [4,8,7]).
The theory of fuzzy sets which was introduced by Zadeh ( [16]) was applied to the branches of pure and applied mathematics. The study of fuzzy relations was started by Zadeh ([17]) in 1971. In [17], the author introduced the concept of fuzzy relation, defined the notion of equivalence, and gave the concept of fuzzy orderings. Fuzzy orderings have broad utility. They can be applied, for example, when expressing our preferences with a set of alternatives. Since then many notions and results from the theory of ordered sets have been extended to the fuzzy ordered sets. In [15], Venugopalan introduced a definition of fuzzy ordered set (foset) (P, µ) and presented an example on the set of positive integers. He extended this concept to obtain a fuzzy lattice in which he defined a (fuzzy) relation as a generalization of equivalence. The (FO2) for all x, y ∈ X, µ R (x, y) > 0 and µ R (y, x) > 0 imply x = y, (Antisymmetric) (FO3) for all x, y, z ∈ X, µ R (x, z) ≥ ∨ y∈X (µ R (x, y) ∧ µ R (y, z)). (Transitive) Let X be a set and µ R be a fuzzy order on X. A pair (X, µ R ) is called a fuzzy ordered set. If A is a subset of X, then we call (A, µ R ) a fuzzy ordered subset of X (for shortly we call A is a fuzzy ordered subset of X). Definition 2.2. Let (X, µ R ) be a fuzzy ordered set and A be a fuzzy ordered subset of X.
(1) The fuzzy order µ R is said to be total if for all x, y ∈ X we have either µ R (x, y) = 1 or µ R (y, x) = 1.
(2) If the fuzzy order µ R is total on A, then A is called a fuzzy chain.
(3) An element x ∈ A is called fuzzy maximal element of A if there is no y(̸ = x) in A for which µ R (x, y) = 1. (4) An x ∈ X satisfying µ R (y, x) = 1 for all y ∈ A is called fuzzy upper bound of A. (5) An x ∈ A satisfying µ R (y, x) = 1 for all y ∈ A is called fuzzy greatest element of A.
Similarly, we can define fuzzy lower bound, fuzzy minimal and least elements of A. Definition 2.3. Let X be a set and µ, λ be two fuzzy sets on X.
(3) If µ(x) = 0 for some x ∈ X, we call µ is a proper fuzzy set.
It is well-known the result about famous axioms in set theory.
Theorem 2.4. ( [2,8,7]) The following statements are equivalent: (1)(Axiom of choice) If {X i } i∈I is a family of nonempty sets, then ∏ i∈I X i is also nonempty. (2)(Zorn's lemma) Let P be a partially ordered set. If every chain in P has an upper bound, then X has a maximal element.
(3)(Well-ordering principle) Every set can be well-ordered. (4)(Comparability principle) Given any two sets X and Y , there exists either a bijection between X and a subset of Y , or a bijection between Y and a subset of X. §3 The fuzzy axioms of choice, fuzzy Zorn's lemma and fuzzy well-ordering principle In this section, we introduce the fuzzy axioms of choice, the fuzzy Zorn's lemma and the fuzzy well-ordering principle.
Fuzzy axioms of choice (FAC): Let X be a set. Given a family {µ i∈I } of non-zero fuzzy sets on X, one could choice a family {λ i∈I } of fuzzy points, with λ i ∈ µ i , for each i ∈ I.
Fuzzy Zorn's lemma (FZL): Let (X, µ R ) be a fuzzy ordered set. If every fuzzy chain in (X, µ R ) has a fuzzy upper bound, then X has a fuzzy maximal element.
In order to state the fuzzy well-ordering principle, we firstly give the definition of fuzzy well-ordered sets. Definition 3.1. A fuzzy ordered set (X, µ R ) is called a fuzzy well-ordered set if it is a totally fuzzy ordered set in which every non-empty subset has a fuzzy least element. Now we introduce the fuzzy well-ordering principle (FWOP): Every set X can be fuzzy well-ordered, i.e. for every set X, there exists a fuzzy binary relation µ R on X which makes it a fuzzy well-ordered set. Proof. Let fuzzy axiom of choice hold. Given a family of non-empty sets {X i } i∈I , then we define X = ∪ i∈I X i and define a family of fuzzy sets {µ i } i∈I for each i ∈ I by We can see that {µ i } i∈I is a non-zero fuzzy set on X for each i ∈ I. By use of the fuzzy axiom of choice, we can choice a family of fuzzy points Proof. Assume that Zorn's lemma holds. Let (X, µ R ) be a fuzzy ordered set, in which every fuzzy chain has a fuzzy upper bound in (X, µ R ). Define a binary relation " ≤ " by x ≤ y if µ R (x, y) = 1. Then we can check that (X, ≤) is an ordered set. If {a i } i∈I is a chain in (X, ≤) with a i ≤ a i+1 for each i ∈ I, then µ R (a i , a i+1 ) = 1 for each i ∈ I and thus it is a fuzzy chain. By hypothesis, {a i } i∈I has a fuzzy upper bound in (X, µ R ), denote it by a 0 . Then we have R(a i , a 0 ) = 1. By the definition of ≤, we have a i ≤ a 0 , which means that x 0 is a upper bound for {a i } i∈I in (X, ≤). Since Zorn's lemma holds, we have that (X, ≤) has a maximal element, denote it by x 0 . We claim that x 0 is a fuzzy maximal element in (X, µ R ). Otherwise, if x 0 is not a fuzzy maximal element in (X, µ R ), there exists x 1 ∈ X, such that x 1 ̸ = x 0 and R(x 0 , x 1 ) = 1. By the definition of ≤, we have x 0 ≤ x 1 but x 1 ̸ = x 0 , contradicting to that x 0 is a fuzzy maximal element in (X, µ R ). Proof. Assume that fuzzy Zorn's lemma holds. Let {µ i } i∈I be a family of non-zero fuzzy sets on a non-empty X. Let P be the "fuzzy partial functions" from I to ∪ i∈I µ i .
where for ϕ ∈ P we mean that ϕ ⊆ I × ∪ i∈I µ i such that for each i ∈ I, any element of ϕ with first component i has second component in µ i , and for each i, there is at most one such element. Note that P is non-empty because the empty fuzzy partial function is a member of it. Let P be partially ordered by fuzzy order µ R as follows: for all ϕ, ϕ ′ ∈ P Then we can check that (P, µ R ) is a fuzzy ordered set. Given any fuzzy chain {ϕ i } i∈Λ in P , we can verify that the union of Φ = ∪ i∈Λ ϕ i is a fuzzy upper bound to {ϕ i } i∈Λ in P . First we check that Φ ∈ P . Clearly Φ ⊆ I × ∪ i∈I µ i . For a ∈ Φ, then there is ϕ i such that a ∈ ϕ i . If the first component of a is i, then it's second component is in µ i and for such i, there is at most one such element. It follows that a ∈ P . Then we check that Φ is a fuzzy upper bound to {ϕ i } i∈Λ . For any ϕ i ∈ {ϕ i } i∈Λ , we have ϕ i ⊆ Φ and hence R(ϕ i , Φ) = 1 by the definition of R. It follows that Φ is a fuzzy upper bound to {ϕ i } i∈Λ . By the fuzzy Zorn's lemma, (P, µ R ) has a fuzzy maximal element ϕ. Therefore we can verify that such ϕ will be a fuzzy function, i.e, ϕ = {(i, λ i )} with λ i ∈ µ i , as required. In fact, if it is not a fuzzy function, then there is some a ∈ ϕ with the first component i 0 but it's second component is empty. Let ϕ 1 be a element in P , such that (i 0 , λ i0 ) is in ϕ 1 but for other i ∈ Λ the second component is same with one of ϕ, where λ i0 ∈ µ i0 . Therefore ϕ ⊆ ϕ 1 and hence µ R (ϕ, ϕ 1 ) = 1. This contradict to that ϕ is a fuzzy maximal element. 1, x ≤ y 0, otherwise. Then we can check that e is a fuzzy order on X. Since X is a well-ordered set, then it is a chain in (X, ≤) and hence is also a fuzzy chain in (X, e). Now we verify that (X, e) a fuzzy well-ordered set. Let A be a subset of X. Since X is a well-ordered set in (X, ≤), then there exists a least element x 0 ∈ A, that is, x 0 ≤ a for each a ∈ A. This means that e(x 0 , a) = 1 for each a ∈ A. It follows that x 0 is a fuzzy least element of A in (X, e), and thus (X, e) is a fuzzy well-ordered set.
(2)⇒(1) Let (FWOP) hold. Then for any set X, there exists a fuzzy binary relation e, such that (X, e) is a fuzzy well-ordered set. Define a binary relation ≤ by x ≤ y iff e(x, y) = 1, for all x, y ∈ X. Then we can check that ≤ is a order on X. Since X is a fuzzy chain in (X, e), then it is also a chain in (X, ≤). Let A ⊆ X. Since X is a fuzzy well-ordered set in (X, ≤), then there exists a least element x 1 ∈ A, that is, e(x 1 , a) = 1 for each a ∈ A. By the definition of ≤, we have x 1 ≤ a for each a ∈ A. This means that x 1 is a least element of A, and hence (X, ≤) is a well-ordered set.

The application of the fuzzy Zorn's lemma for fuzzy ideals in rings
Firstly by use of the fuzzy Zorn's lemma, we will prove that in a ring R, any fuzzy ideal of R is contained in a maximal fuzzy ideal.
We denote the set of all fuzzy ideals of a ring R by F I(R). In the following we give a relation between fuzzy ideals and ideals in rings. For a fuzzy set

Proposition 4.3. Let R be a ring and {A i } i∈I be a family of fuzzy ideals of R such that
Similarly we can prove that for all x ∈ R and y ∈ J t , we have xy ∈ J t . By Proposition 4.2, we get that J is a fuzzy ideal of R.  Proof. Let F I(µ) be the set of proper fuzzy ideals in R containing µ. We define a fuzzy order e on F I(µ) by e(λ, ν) = 1 iff λ ⊆ ν for all λ, ν ∈ F I(µ). Let C be a fuzzy chain in F I(µ), that is, C is just a string of fuzzy ideals, · · · µ 1 ⊆ µ 2 ⊆ · · · . Now we show that C has a fuzzy upper bound in F I(µ).
It follows that J(1) = 0, and thus J is proper. We get J ∈ F I(µ). Note that µ i ⊆ J and hence e(µ i , J) = 1 for each µ i ∈ C. This shows that J is a fuzzy upper bound of C in F I(µ). By the fuzzy Zorn's lemma, F I(µ) has a fuzzy maximal element, say it as ν. We can get that ν is a fuzzy maximal ideal of R. Otherwise, if there is a proper fuzzy ideal λ such that ν ⊆ λ and ν ̸ = λ, then λ ∈ F I(µ) and e(ν, λ) = 1, and so ν is not a fuzzy maximal element of F I(µ), a contradiction. Proof. Let R be a nonzero ring. Define a fuzzy set A 0 as follwing: Since R is a nonzero ring, then there is a ∈ R and a ̸ = 0 such that A 0 (a) = 0. This shows that A 0 is a proper fuzzy ideal. From Theorem 4.5, A 0 is contained in a maximal fuzzy ideal M . It follows that R contains a maximal fuzzy ideal M . Now, we discuss some properties of fuzzy ideals generated by fuzzy subsets, which will be used in the following discussions.
By use of the above representation of fuzzy ideal generated by a fuzzy subset, we give a representation of fuzzy ideal generated by a fuzzy point.
In the following, by use of fuzzy Zron's lemma, we discuss the connection between fuzzy prime ideals and the fuzzy nilpotent elements in the rings. First we recall notion of nilpotent elements and an important theorem concerns nilpotent elements in commutative rings.  For fuzzy subsets A, B of a set X, define A · B, A + B and A * as follows: for any x ∈ X, We denote AB = A · B and A 1 = A, A n = A n−1 A for n > 1. Firstly, we give a property of A n for a fuzzy subset A of R.
Proof. We can directly check that the above statements hold.  Proof. We can directly check that the above statements hold. Definition 4.15. ( [11]) Let µ P be a fuzzy ideal of a ring R and |Imµ P | > 1. If for any fuzzy points λ and ν, λν ⊆ µ P implies λ ⊆ µ P or µ ⊆ µ P , then µ P is called a fuzzy prime ideal.
We denote the set of all fuzzy prime ideals of R by F P I(R).

Proposition 4.17. Let µ P be a fuzzy prime ideal of a ring R. Then for any fuzzy ideal
Proof. Note that (A + µ P )(x) ≥ A(x) ∧ µ P (0) for all x ∈ R. By Proposition 4.16(2), we get µ P (0) = 1, and thus (A + µ P )( Proposition 4.18. Let R be a ring and A be a fuzzy set of R. Define a fuzzy point x A for x ∈ R as follows:   (2) λ n is also a fuzzy point of R with supp{λ n } = x n 0 , for any n ∈ N; Proof. We can directly check that the above statements hold.
where R is the set of all real numbers.
Then (R 2×2 , +, ·, 0) forms a commutative ring, where 0 denotes zero matrix, + and · are additions and multiplications of matrix, respectively. It is easy to see that B =

Theorem 4.21. The intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R, that is, ∩{µ
Proof. Take a fuzzy nilpotent element λ with supp{λ} = x 0 and a fuzzy prime ideal µ P in R. Then we have that λ n (0) > 0 for some n ≥ 1. By Proposition 4.20(2), λ n is also a fuzzy point of R with supp{λ n } = x n 0 = 0, and hence, λ n (x) = 0 for any x ̸ = 0. It follows from Proposition 4. 16(2) that λ n ⊆ µ P . Since µ P is prime, we get λ ⊆ µ P , and so, ∪{λ | λ ∈ F N (R)} ⊆ µ P . It follows that ∪{λ|λ ∈ F N (R)} ⊆ ∩{µ P | µ P ∈ F P I(R)}. Now we prove the inverse inclusion relation. Let λ be a fuzzy point of R such that λ ⊆ µ P for all fuzzy prime ideals. We claim that λ is a fuzzy nilpotent element. Otherwise, if λ is not a fuzzy nilpotent element, then λ n (0) = 0 for each n ≥ 1. Define a set S as follows: Consider the fuzzy ideal A 0 given in the proof of Corollary 4.6. Since λ is not a fuzzy nilpotent element, then supp{λ n } ̸ = {0}, and so A 0 ∈ S. This shows that S is nonempty. We define a fuzzy relation on S by the following: for all A, B ∈ S, Then we can check that A = ∪ i∈I A i is a fuzzy upper bound of C in S. By Proposition 4.3, A is a fuzzy idea of R. In order to prove A ∈ S, it is only to check λ n / ∈ A, for all n ≥ 1. If it is not, then λ n ∈ A for some n ∈ N. By Definition 2.3(2), we have that So that we get that λ n / ∈ A, for all n ≥ 1. This shows that A is a fuzzy upper bound of S, and thus the condition of fuzzy Zorn's lemma holds in S. By the fuzzy Zorn's lemma, there is a maximal element in S, say M . In the following, we prove that M is a fuzzy prime ideal. First we claim that | ImM |> 1.  (1). From the above arguments, we get x n+m 0 ∈ supp{M }, that is λ n+m ∈ M . It is a contradiction to M ∈ S. This shows that M is a fuzzy prime ideal. Since M ∈ S, we have λ / ∈ M , and hence we get a contradiction to the choice of λ. Therefore, if a fuzzy point λ ⊆ µ P for all prime fuzzy ideals, then λ must be a fuzzy nilpotent element. Let F = ∩{µ P | µ P ∈ F P I(R)}. By Proposition 4.18, we have F = ∪{x F | x ∈ R}, where x F is a fuzzy point and x F ⊆ F . It follows that x F ⊆ µ P , for each fuzzy prime ideal µ P . By the above arguments, we get x F is a fuzzy nilpotent element, and so or F ⊆ ∪{λ | λ ∈ F N (R)}. Therefore the inverse inclusion relation holds. We complete the proof.
From the following example, we can see that in Theorem 4.21, if fuzzy prime ideals are replaced by fuzzy ideals, it does not hold.

Fuzzy Tychonoff theorem and the applications of fuzzy Zorn's lemma
In this subsection, we discuss the fuzzy version of Tychonoff Theorem and give it's proof by fuzzy Zorn's lemma. Firstly we present Tychonoff Theorem.
Tychonoff Theorem. ([14]) For an arbitrary family of compact topological spaces {X i } i∈I , the product space ∏ i∈I X i with the product topology is compact.

Definition 4.22.
A set F µ consisting of fuzzy subsets of a set X is called a fuzzy filter on X, if the following conditions are satisfied: for any fuzzy subsets U, V of X, if U ∈ F µ and U ⊆ V , then V ∈ F µ , (3) for any U, V ∈ F µ , we have U ∩ V ∈ F µ . Definition 4.23. A fuzzy filter F µ on a set X is called a fuzzy ultrafilter or a fuzzy maximal filter, if F µ is maximal with respect to the inclusion, that is, any fuzzy filter including F µ must be equal to F µ . Proposition 4.24. Let X, Y be two sets and f : X → Y be a map.
) and (f −1 (U ) ∩ Proposition 4.26. Let X, Y be two sets, and f : X → Y be a map. For a fuzzy filter F µ on X and a fuzzy filter G µ on Y such that f −1 (V ) ̸ = 0 for all V ∈ G µ , we have the following equivalence: Proof. The equivalence follows since we have: This is essentially the Galois connection between the power fuzzy subsets P (F (X)) and P (F (Y )) induced by the inverse image map f −1 : F (Y ) → F (X). Proposition 4.27. Let X, Y be two sets and f : X → Y be a onto map. Let F µ and G µ be fuzzy filters on X and Y , respectively. If f (F µ ) ⊆ G µ , then there exists a fuzzy filter H µ on X such that F µ ⊆ H µ and G µ = f (H µ ).

Proof. For any
Let H µ be the smallest fuzzy filter on X including F µ ∪ f −1 (G µ ), namely, Conversely we have For any U ∈ F µ , V ∈ G µ , and Z ∈ F (Y ), we have the following: Proposition 4.28. Let X, Y be two sets and f : X → Y be a onto map. For any fuzzy ultrafilter F µ on X, f (F µ ) is also a fuzzy ultrafilter on Y .
Proof. It follows from Proposition 4.27 that f (F µ ) is also a fuzzy filter on Y . If f (F µ ) is not maximal, then there is a fuzzy filter G µ on Y such that f (F µ ) ⊆ G µ and f (F µ ) ̸ = G µ . From Proposition 4.27, there is a fuzzy filter H µ on X, such that F µ ⊆ H µ and G µ = f (H µ ). Since f (F µ ) ̸ = G µ , we have F µ ̸ = H µ , contradicting to the maximality of F µ . Definition 4.29. A fuzzy topological space X µ is a set equipped with a map N : F P (X) → P (F (X)), which satisfies the following conditions: (1) N (λ) is a fuzzy filter on X for all λ ∈ F P (X), (2) For all λ ∈ F P (X), λ belongs to the intersection ∩N (λ). That is, λ ∈ U for all U ∈ N (λ), (3) For all λ ∈ F P (X) and U ∈ N (λ), there exists an element V ∈ N (λ) such that V ⊆ U and U ∈ N (ν) for each ν ∈ V .
An element of N (λ) is called a neighborhood of λ.
Definition 4.30. Let X µ be a fuzzy topological space with the neighborhood filter N (λ) for λ ∈ F P (X). A fuzzy filter F µ on X is said to converge to an element λ of F P (X), and we Definition 4.31. A fuzzy topological space X µ is called compact if for any filter F µ on X, there exists a fuzzy filter G µ on X and an element λ of F P (X) such that F µ ⊆ G µ and G → λ.
By Definitions 4.30 and 4.31, we have following corollary.
Corollary 4.32. Any fuzzy ultrafilter on a compact space converges.

Definition 4.33.
For an arbitrary family of sets {x i } i∈I , the product set is defined as An element of the product set is called a choice map of {X i } i∈I . For each index i ∈ I, the projection map pr i from the product X = ∏ i∈I X i onto X i is defined by pr i (x) = x(i) for all x ∈ X. Definition 4.34. For an arbitrary family of fuzzy topological spaces {X µi } i∈I , the product fuzzy topology on the product set X = ∏ i∈I X i is the weakest topology making the projections {pr i : X → X µi } i∈I continuous. Namely, the neighborhood filter of λ ∈ X is the smallest fuzzy filter including the union: ∪ i∈I {pr −1 i (U ) | U ∈ N (pr i (λ))}, or ∪ i∈I pr −1 i (N (pr i (λ))).

Lemma 4.35.
For any fuzzy filter F µ on the product space X = ∏ i∈I X i , and an element λ of F P (X), the following two conditions are equivalent: Proof. The lemma follows from Definition 4.29 and Proposition 4.26.
In following, we prove Tychonoff theorem as a direct application of Zorns lemma. The idea is to construct an ultrafilter and an element of the product space simultaneously by taking a suitable partially ordered set. Theorem 4.36. For an arbitrary family of compact fuzzy topological spaces {X µi } i∈I , any fuzzy filter on the product space ∏ i∈I X µi with the product topology is included in a convergent fuzzy ultrafilter.
Proof. Let F µ be a fuzzy filter on the product set ∏ i∈I X µi . Let P be the set of pairs (G µ , x), where G µ is a fuzzy filter on ∏ i∈I X µi including F µ , and x : J → ∪ i∈I X µi is a map with J ⊆ I satisfying x(j) ∈ X µj and pr j (G µ ) → x(j) for all j ∈ J. If we define a fuzzy binary relation R µ on P by: x ⊆ y 0, otherwise Then we can check that R µ is a fuzzy order on P . Now we check that the fuzzy ordered set (P, R µ ) satisfies the assumption of fuzzy Zorns lemma. Namely, (F µ , 0) ∈ P , and for any nonempty fuzzy chain C ⊆ P , define H µ = ∪ (Gµ,x)∈C {G µ } and y = ∪ (Gµ,x)∈C {x}. Then we can check that (H µ , y) ∈ P . Clearly (H µ , y) is a fuzzy upper bound for C in (P, R µ ). By the fuzzy Zorn's lemma, a maximal element (G µ , x) of P exists. Note that if G µ is included in a fuzzy filter H µ , then R µ ((G µ , x), (H µ , x))) = 1. Since (G µ , x) is maximal, we have (G µ , x) = (H µ , x), and hence G µ must be equal to H µ . Thus G µ is a fuzzy ultrafilter. If x : J → ∪ i∈I X µi with J ̸ = I, then there is an element i ∈ I with i / ∈ J. Since pr i (G µ ) is a fuzzy ultrafilter by Proposition 4.28, and X µi is compact, pr i (G µ ) converges to an element p of X µi by Corollary 4.32. This implies that the pair (G µ , x) ∪ {i, p} is an element of P and is strictly bigger than (G µ , x), contradicting its maximality. Thus J = I and therefore, x is an element of the product space ∏ i∈I X µi . As pr i (G µ ) → pr i (x) for all i ∈ I, the fuzzy ultrafilter G µ converges to x, by Lemma 4.35.

Theorem 4.37. (Fuzzy Tychonoff Theorem)
For an arbitrary family of compact fuzzy topological spaces {X µi } i∈I , the product space ∏ i∈I X µi with the product fuzzy topology is compact.
Proof. Since any fuzzy filter on ∏ i∈I X µi is included in a convergent fuzzy filter, the product space is compact, by Definition 4.31. §5 Conclusion As well-known, there are some interesting and profound statements and results, such as, axiom of choice, Zorn's lemma, well-ordering principle and comparability principle. It is important to discuss the fuzzy versions of them. In this paper, by use of Zadeh's fuzzy order, we introduce the fuzzy axiom of choice, fuzzy Zorn's lemma and fuzzy well-ordering principle, and prove that they are equivalent. As an application of fuzzy Zorn's lemma, we prove that every proper fuzzy ideal of a ring is contained in a maximal fuzzy ideal. Moreover we give the fuzzy version of Tychonoff Theorem. By use of the fuzzy Zorn's lemma, we prove that fuzzy Tychonoff Theorem. But as a equivalent statement to axiom of choice, it's fuzzy version have not introduced in this paper. In the future, we will consider giving the fuzzy version of the comparability principle and discuss the relations between it and other fuzzy versions. Moreover we can consider the applications of the fuzzy axiom of choice, for example, we can discuss the existence of the bases of fuzzy victor spaces by the fuzzy axiom of choice.
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