In this paper, based on a complete residuated lattice, we introduce the concept of fuzzy cut-stable maps and discuss the relationship between fuzzy cut-stable maps and some special maps. We also obtain the extension property of fuzzy cut-stable maps. Furthermore, it is proved that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps.
The completion of a partially ordered set (a poset) [8, 16, 8, 16] originated from Dedekind’s pinoeer construction of the real line by cuts of rational numbers [9]. Later, MacNeille [22] generalized Dedekind’s construction and introduced the famous completion by cuts for arbitrary posets, which was known as the Dedekind-MacNeille completion (also called the normal completion or the completion by cuts). After that, many researchers have been devoted to studying it[1, 11].
Quantitative domain theory (QDT for short) [12–15, 28], which refines ordinary domain theory [16] by replacing the qualitative notion of approximation by a quantitative one of degree of approximation, has undergone active research in the past three decades, and forms a new branch of domain theory. The appearance of Quantitative domain theory promotes the development of the completion theory, and many researchers have been committed to studying it. Wagner [28] introduced the enriched Dedekind-MacNeille completion for an enriched category. Blohlvek [5] described the Dedekind-MacNeille completion for an L-ordered set as an application of the theory of concept lattices in fuzzy setting. Xie, Zhang and Fan [29] built and characterized the Dedekind-MacNeille completions for fuzzy posets. Wang and Zhao obtained a categorical characterization of the Dedekind-MacNeille completions for an L-ordered set [27]. Motivated by the studying of the previous works, we show that the category of fuzzy complete (continuous) lattices will be a full reflective subcategory of the category of fuzzy (precontinuous) posets when suitable morphisms aredefined.
The outline of this article is the following: In Section 2, we recall some basic definitions and results used in the rest of this paper. In Section 3, we introduce the notions of fuzzy lower (upper) cut-continuous maps, fuzzy lower (upper) cut-stable maps, fuzzy residuated (residual) maps and fuzzy cut-preserving maps. The relationships among these maps are also discussed. In Section 4, we obtain the extension property of fuzzy cut-stable maps on fuzzy posets, and prove that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps.
Preliminaries
We refer to [2] for general category theory, to [17, 19, 20, 17, 19, 20] for fuzzy set theory, and to [8, 16] for lattice theory.
Definition 2.1. (See [4, 25]) A residuated lattice is an algebraic structure (L ; ∧ , ∨ , * , → , 0, 1) of type (2, 2, 2, 2, 0, 0) such that
(L ; ∧ , ∨ , 0, 1) is a bounded lattice with the least element 0 and the greatest element 1;
(L ; * , 1) is a commutative monoid with the identity 1;
satisfies the adjointness property, i . e . , ∀ x, y, z ∈ L, x * y ≤ ziffx ≤ y → z .
A residuated lattice is called complete if the underlying lattice is complete.
Theorem 2.2. (See [6, 18]) LetLbe a residuated lattice. Then
a* b ≤ a ∧ b ;
a = 1→ a ;
a≤ b ⇔ a → b = 1 ;
a→ (b → c) = (a * b) → c ;
a→ b ≥ b ;
When L is complete,
a* (⋁ ibi) = ⋁ i (a * bi) ;
a→ (⋀ ibi) = ⋀ i (a → bi) ;
(⋁ ibi) → a = ⋀ i (bi → a) .
In this paper, if there is no further statement, L always denotes a complete residuated lattice.
Definition 2.3. (See [4, 12]) A fuzzy poset is a pair (X, e) such that X is a set and e : X × X ⟶ L is a map (called a fuzzy partial order over X), which satisfies for every x, y, z ∈ X,
e (x, x) =1 (reflexivity);
e (x, y) * e (y, z) ≤ e (x, z) (transitivity);
e (x, y) = e (y, z) =1 implies x = y (antisymmetry).
For a set X, LX denotes the set of all fuzzy subsets of X, that is, the set of all maps from X to L. For λ ∈ L, the constant map with the value λ is denoted by λX.
Example 2.4. (1) (The canonical fuzzy partial order on L) Define eL : L × L → L by eL (x, y) = x → y for all x, y ∈ L . Then (L, eL) is a fuzzy poset.
(2) Let X be a set. For A, B ∈ LX, the subsethood degree [17] sub (A, B) of A in B is defined by sub (A, B) = ⋀ x∈X1.2 (A (x) → B (x) 1.2) . Then (LX, sub) is a fuzzy poset.
(3) Let (X, e) be a fuzzy poset. Then ≤e = {(x, y) ∈ X × X ∣ e (x, y) =1} is a classical (partial) order on X.
Definition 2.5. (See [32–34]) Let (X, e) be a fuzzy poset and A ∈ LX. An element x0 ∈ X is called a join (meet) of A, in symbols x0 = ⊔ A (x0 = ⊓ A), if for all x ∈ X,
A (x)≤ e (x, x0) (A (x) ≤ e (x0, x)) ;
⋀y∈X (A (y) → e (y, x)) ≤ e (x0, x) (⋀ y∈X (A (y)→e (y, x)) ≤ e (x, x0)) .
Theorem 2.6. (See [32–34]) Let (X, e) be a fuzzy poset andA ∈ LX. Thenx0 = ⊔ A (x0 = ⊓ A) iffforallx ∈ X, e (x0, x) = ⋀ y∈Y (A (y) → e (y, x))(e (x, x0) = ⋀ y∈Y (A (y) → e (x, y))).
Definition 2.7. (See [32–34]) A fuzzy poset (X, e) is called a fuzzy complete lattice if for all A ∈ LX, ⊔ A and ⊓A exist.
Proposition 2.8. (See [32–34]) Let (X, e) be a fuzzy poset. Then the following statements are equivalent:
(X, e) is a fuzzy complete lattice;
for any A ∈ LX, ⊔ A exists;
for any A ∈ LX, ⊓ A exists.
Definition 2.9. (See [30]) Let (P, e) be a fuzzy complete lattice. ∀ x ∈ P, define ⇓x ∈ LP as follows
If ∀ x ∈ P, x = ⊔ ⇓ x, then (P, e) will be called a continuous lattice.
For all x ∈ X, if ⇓x (x) =1, then we call x a compact element in P. The set of all compact elements in P is denoted by K (P).
Definition 2.10. (See [30]) Let (P, e) be a fuzzy complete lattice. Define kx : P → L by ∀ y ∈ P,
If ⨆kx = x, then we call X an algebraic lattice. Suppose f is a map between fuzzy posets (P, eP) and (Q, eQ). Then f is said to be a fuzzy order-preserving (order-embedding) map if eP (x, y)≤eQ (f (x) , f (y)) (eP (x, y) ≤ eQ (f (x) , f (y))) for all x, y ∈ P .
Definition 2.11. (See [23]) Let f : X → Y be a map. The Zadeh forward powerset operator and the Zadeh backward powerset operator are, respectively, defined by for all A ∈ LX, y ∈ Y and for all B ∈ LY.
Definition 2.12. (See [21, 30]) A pair of fuzzy order-preserving maps f : P → Q and g : Q → P is called a fuzzy adjunction, in symbols, f ⊢ g if eQ (f (x) , y) = eP (x, g (y)) for all x ∈ P, y ∈ Q. In this case, f is called a left adjoint of g and g is called a right adjoint of f.
The pair is a fuzzy adjunction between LP and LQ. That is, , i.e., for all φ ∈ LP and ψ ∈ LQ.
Definition 2.13. Let (P, eP) , (Q, eQ) be fuzzy posets. A map f : P → Q is called fuzzy join-preserving if for all φ ∈ LP, and is called fuzzy meet-preserving if for all φ ∈ LP.
Proposition 2.14. (See [21, 30]) Letf : P → Qandg : Q → Pbe maps between fuzzy posetsPandQ. The following are equivalent:
(f, g) is a fuzzy adjunction;
eQ (f (x) , y) = eP (x, g (y)) for all x ∈ P andy ∈ Q;
both f and g are fuzzy order-preserving and (f, g) is a classical adjunction between (P, ≤ eP) and (Q, ≤ eQ);
f ∘ g ≤ 1Q and g ∘ f ≥ 1P.
Proposition 2.15. (See [31]) Letf : P → Qandg : Q → Pbe maps between fuzzy posetsPandQ. Then
ifPis a fuzzy complete lattice, thenfis fuzzy order-preserving and has a right adjoint ifffis join-preserving;
ifQis a fuzzy complete lattice, thengis fuzzy order-preserving and has a left adjoint iffgis meet-preserving.
The category whose objects are fuzzy complete lattices and whose morphisms are fuzzy complete homomorphisms (both fuzzy join-preserving and fuzzy meet-preserving) will be denoted by FCLC.
Definition 2.16. (See [29]) Let (P, e) be a fuzzy poset and φ ∈ LP. We define φl, φu as follows:
Proposition 2.17. (See [4, 5]) (the Dedekind-MacNeille completion) Given a fuzzy poset (P, e), ∀A, B ∈ LP, a fuzzy cut is a pair (A, B) such thatA = Bl, B = Au. The collection of all fuzzy cuts, ordered bye ((A, B) , (C, D)) = sub (A, C) = sub (D, B), is a fuzzy complete lattice, called the Dedekind-MacNeille completion ofP. We sometimes useDML (P) ={(A, B) ∣ ∀ A, B ∈ LP, A = Bl, B = Au} orto denote the Dedekind-MacNeille completion ofP, that is to say,DML (P) oris a fuzzy complete lattice.
Clearly, {(A, B) ∣ ∀ A, B ∈ LP, A = Bl, B = Au} is isomorphic to {A ∣ ∀ A ∈ LP, A = Aul}, thus {A ∣ ∀ A ∈ LP, A = Aul} can also denote the Dedekind-MacNeille completion of P. ∀ A, B ∈ LP, Au and Bl are usually called fuzzy upper cut and fuzzy lower cut, respectively.
Let (X, e) be a fuzzy poset and x ∈ X. Define two maps ιx, μx : X → L by ιx (y) = e (y, x) , μx (y) = e (x, y) for all y ∈ X. Clearly, we have ⊔ ιx = ⊓ μx = x and the pair (ιx, μx) is a fuzzy cut of X. Define by η (x) = (ιx, μx) for all x ∈ P, then η is an order-embedding.
Lemma 2.18. (See [4, 29]) Let (P, e) be a fuzzy poset andφ, ψ ∈ LP. Then
φ ≤ φul and φ ≤ φlu;
sub (φ, ψ) ≤ sub (ψu, φu) and sub (φ, ψ) ≤ sub (ψl, φl);
sub (φ, ψ) ≤ sub (φul, ψul) and sub (φ, ψ) ≤ sub (φlu, ψlu);
φu = φulu and φl = φlul.
Fuzzy cut-stable maps
In this section, we introduce the notions of fuzzy lower (upper) cut-continuous maps, fuzzy lower (upper) cut-stable maps, fuzzy residuated (residual) maps and fuzzy cut-preserving maps. The relationships among these maps are also discussed.
Definition 3.1. Let (P, eP) , (Q, eQ) be fuzzy posets, f : P → Q a map. Then we call f a fuzzy lower (upper) cut-continuous map provided that ∀ B ∈ LQ, if B is a fuzzy lower (upper) cut of Q, then is a fuzzy lower (upper) cut of P. We say f is fuzzy cut-continuous if f is both fuzzy lower and upper cut-continuous.
Example 3.2. Suppose (P, eP) is a fuzzy poset, then is also a fuzzy poset. Define , subop) , x ↦ μx and id : P → P, x ↦ x, then f is a fuzzy lower cut-continuous map, g is a fuzzy upper cut-continuous map and id is a fuzzy cut-continuous map.
Definition 3.3. Let (P, eP) , (Q, eQ) be fuzzy posets, f : P → Q a map. Then we call f fuzzy lower (upper) cut-stable if ∀ A ∈ LP, . We say that f is fuzzy cut-stable if f is both fuzzy lower and upper cut-stable.
Example 3.4. Let (P, eP) , (Q, eQ) be fuzzy posets, f : P → Q a map. Define g : LP → LQ, h : LQ → LP by ∀ A ∈ LP, B ∈ LQ, x ∈ P, y ∈ Q, g (A) (y) = ⋁ a∈P (A (a) * e (a, y)) , h (B) (x) = ⋀ b∈Q (e (x, b) → B (b)), respectively. Then g is fuzzy lower cut-stable, h is fuzzy upper cut-stable and is fuzzy cut-stable.
In the following we will use to denote for brevity.
Definition 3.5. Let (P, eP) , (Q, eQ) be fuzzy posets, f : P → Q a map. Then we call f fuzzy cut-preserving if for every fuzzy cut (A, B) of P, the pair is a fuzzy cut of Q.
Example 3.6. By example 2.4(1), we have (L, eL) is a fuzzy poset. In fact, (L, eL) is also a fuzzy complete lattice. Define f : LL → L as follows ∀ A ∈ LL, f (A) = ⊔ A, then f is fuzzy cut-preserving.
Proposition 3.7.Supposef : P → Qis a map. Then
f is fuzzy lower (upper) cut-continuous iff for all A ∈ LP;
f is fuzzy order-preserving iff for all A ∈ LP;
f is fuzzy cut-preserving iff iff for all A ∈ LP.
Proof. (1) (Necessity) Suppose f is fuzzy lower cut-continuous. ∀ A ∈ LP, is thefuzzy lower cut generated by A)
(Sufficiency)(Bl)) ul ≤ Blul = Bl. So , that is, . Hence, f is fuzzy lower cut-continuous.
Similarly, we have that f is fuzzy upper cut-continuous iff for all A ∈ LP.
(2) (Necessity) Suppose f is fuzzy order-preserving. We can easily prove , thus .
(Sufficiency) ∀ x ∈ P, put A = ιx, then (ιx) u. Therefore, . ∀ x, y ∈ P,
(3) (Necessity) Suppose f is fuzzy cut-preserving. Since ∀ A ∈ LP, (Aul, Au) is a fuzzy cut of P, it follows that is a fuzzy cut of Q. Hence, .
(Sufficiency) Suppose (A, B) is a fuzzy cut of P. It suffices to show that is a fuzzy cut of Q, i.e., . It is easy to check that , (B) l. Similarly, f is fuzzy cut-preserving iff for all A ∈ LP.
Corollary 3.8.Every fuzzy cut-preserving map is fuzzy order-preserving. For maps between fuzzy complete lattices the converse implication is also true.
Proposition 3.9.f : P → Qis fuzzy (lower,upper) cut-stable iff it is fuzzy (lower, upper) cut-continuous and fuzzy cut-preserving.
A map f between fuzzy posets P, Q is said to be fuzzy residuated (residual) if ∀ q ∈ Q, ∃ p ∈ P such that (∀ p ∈ P, ∃ q ∈ Q such that ).
Proposition 3.10.A mapf : P → Qis fuzzy residuated (residual) iff it has a right (left) adjointg : Q → P.
Corollary 3.11.LetP, Qbe fuzzy complete lattices,f : P → Qa map. Then the following four properties are equivalent: fuzzy residuated, fuzzy lower cut-stable, fuzzy lower cut-continuous, fuzzy join-preserving.
Putting all the pieces together, we have the following diagram.
Proposition 3.12.Let (P, e) be fuzzy poset, (Q, e) a fuzzy complete lattice,f : P → Qa map. Thenfis fuzzy cut-preserving ifffor allA ∈ LP, where asfis fuzzy lower (upper) cut-stable ifffor allA ∈ LPfor allA ∈ LP).
Extension theorem and its applications
In this section, we show that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps.
Firstly, let’s give the extension property of fuzzy cut-stable maps on fuzzy posets.
Theorem 4.1.SupposeP, Qare fuzzy posets. Thenf : P → Qis fuzzy cut-stable iff there exists a (unique) fuzzy complete homomorphismFbetween the Dedekind-MacNeille completionandextendingf, i.e., making the following diagram commutative:
Proof.(Necessity) Assume that f : P → Q is fuzzy cut-stable. Define by .
Clearly, it is well-defined. In fact, F has a right adjoint and a left adjoint . Since , we have that H, G are well-defined. Let (A, B) be a fuzzy cut of P, (C, D) a fuzzy cut of Q. Since sub ((A, B) , G (C, D)) = subB) , (C, D)), it follows that (F, G) is a fuzzy adjunction. Similarly, (H, F) is also a fuzzy adjunction. Hence, F is a fuzzy complete homomorphism. As (e (x, x′) → e (y, f (x′))) = ιf(x) (y) , then F ∘ ηP= ηQ ∘ f (x).
For the uniqueness, suppose there exists such that .
(Sufficiency) Suppose there exists a fuzzy complete homomorphism F : P → Q such that F ∘ ηP =ηQ ∘ f. ∀ A ∈ LP,
Therefore, . Similarly, f is also fuzzy upper cut-stable.
By the above theorem, we observe that the composite of two fuzzy cut-stable maps is again fuzzy cut-stable.
Corollary 4.2.Suppose (P, e) is a fuzzy poset, (Q, e) is a fuzzy complete lattice,f : P → Qis a map. Thenfis fuzzy cut-stable iff there exists a (unique) fuzzy complete homomorphismsuch thatf = g ∘ ηP. In particular,ηPis fuzzy cut-stable.
The category of fuzzy posets with fuzzy cut-stable maps is denoted by FPOS.
Easily, we can obtain:
FPOS⟶FCLC
is a functor.
Corollary 4.3.The categoryFCLCis a full reflective subcategory of the categoryFPOS.
Next, we will answer the question under what circumstances will be surjective or injective, respectively. First of all, let’s recall the notions of join-denseness and meet-denseness. Suppose (X, e) is a fuzzy poset and Y ⊆ X. Y is said to be join-dense (meet-dense) in X if for all x ∈ X there is a fuzzy subset φ of Y such that . Here i : Y → X is the inclusion map. As usual, a map f : P → Q is said to be join-dense (meet-dense) iff so is its image in Q.
Proposition 4.4.Letfbe a fuzzy cut-stable map between fuzzy posets (P, e) and (Q, e). Then
is surjective iff f is join-dense iff f is meet-dense.
is injective iff is an embedding iff f is an embedding.
is a bijective iff f is a join-dense (meet-dense) embedding.
Proof. (1) Suppose is surjective. For any , there is such that . Thus . Because , we have .
Conversely, if f is join-dense, we need to prove , that is, for all . Since , thus it suffices to check that = C. Firstly, since , we have .
Secondly, ∀ y ∈ Q,
Similarly, is surjective iff f is meet-dense.
(2) If is an embedding, then is injective. Conversely, if is injective, we shall show that f is an embedding. It suffices to prove that ∀ x, y ∈ P, e (y, x) = e (f (y) , f (x)). Since is a fuzzy adjunction and is one-to-one, it holds that , ∀ x ∈ P, =(ιx, μx). Hence, .
If f is an embedding, we need to show that ,i.e., . sub (A, C) for all .
(3) This is an immediate consequence of (1) and (2).
For convenience, in the following we will use to denote the Dedekind-MacNeille completion of P. We say that a fuzzy subset λ : P → L is finite if the support set {x ∈ P : λ (x) ≠0} of λ is a finite set.
Definition 4.5. Let (P, e) be a fuzzy poset. A fuzzy subset I of P is called a (Frink) fuzzy ideal if for each finite fuzzy subset Z of P, we have sub (Z, I) ≤ sub (Zul, I).
Clearly, ∀ x ∈ P, ιx is a (Frink) fuzzy ideal.
Remark 4.6. (1) If , then φ is a (Frink) fuzzy ideal.
(2) If I is a (Frink) fuzzy ideal, then I is a fuzzy lower set.
Lemma 4.7.LetFl (P) denote the set of all (Frink) fuzzy ideals. Then (Fl (P) , sub) is a fuzzy complete lattice.
Definition 4.8. Let P be a fuzzy poset. ∀ y ∈ P, rotatebox []270twoheadrightarrow y is defined by ∀ y ∈ P, rotatebox []270twoheadrightarrow y(x) = ⋀ I∈Fl(P)1.1 (Iul (y) → I (x) 1.1).
Remark 4.9. ∀ y ∈ P, rotatebox []270twoheadrightarrow y is a (Frink) fuzzy ideal.
Proposition 4.10. (1) ∀ x, y ∈ P, rotatebox []270twoheadrightarrow y(x) ≤ ↓ y (x).
(2) ∀ x, u, v, y∈ P, e (u, x) *rotatebox []270twoheadrightarrow y(x)* e (y, v) ≤rotatebox []270twoheadrightarrow v(u).
Definition 4.11. We call a fuzzy poset P precontinuous if for each x ∈ P, x = ⊔ rotatebox []270twoheadrightarrow x.
Proposition 4.12.A fuzzy posetPis precontinuous iff for eachx ∈ P, (rotatebox []270twoheadrightarrow x) ul (x) =1.
Define Γ, rotatebox []270twoheadrightarrow : LP → LP as follows: ∀ A ∈ LP, x ∈ P, Γ (A) (x) = ⋁ I∈Fl(P) (Iul (x) * sub (I, ↓ A)), rotatebox []270twoheadrightarrow A (x) = ⋁ y∈P(rotatebox []270twoheadrightarrow y (x) * A (y)), respectively.
Proposition 4.13. (1) Γ (A) , rotatebox []270twoheadrightarrow Aare fuzzy lower subsets of P.
(2) rotatebox []270twoheadrightarrow A = rotatebox []270twoheadrightarrow (↓ A) ≤ rotatebox []270twoheadrightarrow Γ (A) ≤ ↓ A ≤ ↓ ΓA = ΓA.
If FLow (P) denotes all the fuzzy lower subsets of P, then restrictions ▵ = Γ ∣ FLow(P), ∇ = rotatebox []270twoheadrightarrow ∣FLow(P) are self-maps of FLow (P) .
Lemma 4.14.Suppose (P, e) is a fuzzy poset. IfPis precontinuous, then the pair (∇ , ▵) is a fuzzy adjunction of the fuzzy complete latticeFLow (P) .
Proof. ∀A ∈ FLow (P) , rotatebox []270twoheadrightarrow Γ (A) ≤ ↓ A = A. Conversely, ∀x ∈ P, A (x) = (rotatebox []270twoheadrightarrow x) ul (x) * A (x) ≤ ⋁ y∈P((rotatebox []270twoheadrightarrow y) ul (x) * A (y)) ≤ ⋁ y∈P ((rotatebox []270twoheadrightarrow y) ul * sub (rotatebox []270twoheadrightarrow y, rotatebox []270twoheadrightarrow A)) ≤ ⋁ I∈Fl(P) (Iul * sub (I, rotatebox []270twoheadrightarrow A) (x)) = rotatebox []270twoheadrightarrow Γ (A) (x). Hence, ▵ ∘ ∇ ≥ idFLow(P) and ∇ ∘ ▵ ≤ idFLow(P).
By the above lemma, we have that the operator Γ ∣ FLow(P) is a fuzzy meet-preserving map. Define by . We can easily show that is a fuzzy closure operator and . Since for all I ∈ Fl (P), it follows that is a fuzzy meet-preserving map of fuzzy (Frink) ideals and induces a fuzzy join-preserving map from Fl (P) onto its image .
Theorem 4.15.Suppose (P, e) is a fuzzy poset. IfPis precontinuous, then the normal completionis a fuzzy continuous lattice.
Proof. For each I ∈ Fl (P), define kI : Fl (P) → L as follows: ∀ A ∈ Fl (P) ,
It is easy to show that if I = ⨆ kI, then Fl (P) is algebraic. Therefore, the normal completion is a fuzzy continuous lattice.
Let FCONT denote the category of fuzzy continuous lattices with fuzzy complete homomorphisms and PFPOS denote the category of fuzzy precontinuous posets with fuzzy cut-stable maps. Similarly to Corollary 4.3, we have the following Theorem.
Theorem 4.16.The categoryFCONTis a full reflective subcategory of the categoryPFPOS.
Conclusions and future works
In this paper, we introduce the concept of fuzzy cut-stable maps and discuss the relationship between fuzzy cut-stable maps and some special maps. We show an important property of fuzzy cut-stable maps, that is, extension property. Furthermore, the property is applied to obtain that the category of fuzzy complete (continuous) lattices with fuzzy complete homomorphisms is a full reflective subcategory of the category of fuzzy (precontinuous) posets with fuzzy cut-stable maps.
Future work would focus on two parts. The first is to continue studying fuzzy cut-stable maps, we hope to show the relationship between the category of fuzzy compactly generated posets with fuzzy cut-stable maps and the category of fuzzy algebraic lattices with fuzzy complete homomorphisms. The second is to investigate the extension property of fuzzy weakly cut-stable maps, discuss the relationship between fuzzy weakly cut-stable maps and some special maps, and how to characterize fuzzy weakly cut-stable maps.
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