Fuzzy cut-stable map and its extension property 1

Abstract

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.

Keywords

Fuzzy poset fuzzy cut-stable map Dedekind-MacNeille completion reflective subcategory

1 Introduction

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. B $\overset{˘}{e}$ lohl $\overset{´}{a}$ vek [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.

2 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 $L =$ (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;

$L$ 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* (⋁ _ib_i) = ⋁ _i (a * b_i) ;

a→ (⋀ _ib_i) = ⋀ _i (a → b_i) ;

(⋁ _ib_i) → a = ⋀ _i (b_i → 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, L^X 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 λ_X.

Example 2.4. (1) (The canonical fuzzy partial order on L) Define e_L : L × L → L by e_L (x, y) = x → y for all x, y ∈ L . Then (L, e_L) is a fuzzy poset.

(2) Let X be a set. For A, B ∈ L^X, 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 (L^X, 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 ∈ L^X. 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 ∈ L^X. 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 ∈ L^X, ⊔ 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 ∈ L^X, ⊔ A exists;

for any A ∈ L^X, ⊓ A exists.

Definition 2.9. (See [30]) Let (P, e) be a fuzzy complete lattice. ∀ x ∈ P, define ⇓x ∈ L^P as follows

$\forall y \in P, ⇓ x (y) = ⋀_{I \in I (P)} (e (x, ⊔ I) \to I (y)) .$

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 k_x : P → L by ∀ y ∈ P, $k_{x} (y) = {\begin{matrix} e (y, x), & y \in K (P), \\ 0, & otherwises. \end{matrix}$

If ⨆k_x = x, then we call X an algebraic lattice. Suppose f is a map between fuzzy posets (P, e_P) and (Q, e_Q). Then f is said to be a fuzzy order-preserving (order-embedding) map if e_P (x, y)≤e_Q (f (x) , f (y)) (e_P (x, y) ≤ e_Q (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 $f_{L}^{\to} : L^{X} \to L^{Y}$ and the Zadeh backward powerset operator $f_{L}^{\leftarrow} : L^{Y} \to L^{X}$ are, respectively, defined by $f_{L}^{\to} (A) (y) = ⋁_{f (x) = y} A (x)$ for all A ∈ L^X, y ∈ Y and $f_{L}^{\leftarrow} (B) = B \circ f$ for all B ∈ L^Y.

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 e_Q (f (x) , y) = e_P (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 $(f_{L}^{\to}, f_{L}^{\leftarrow})$ is a fuzzy adjunction between L^P and L^Q. That is, $f_{L}^{\to} ⊢ f_{L}^{\leftarrow}$ , i.e., $sub (f_{L}^{\to} (φ), ψ) = sub (φ, f_{L}^{\leftarrow} (ψ))$ for all φ ∈ L^P and ψ ∈ L^Q.

Definition 2.13. Let (P, e_P) , (Q, e_Q) be fuzzy posets. A map f : P → Q is called fuzzy join-preserving if $f (⊔ φ) = ⊔ f_{L}^{\to} (φ)$ for all φ ∈ L^P, and is called fuzzy meet-preserving if $f (⊓ φ) = ⊓ f_{L}^{\to} (φ)$ for all φ ∈ L^P.

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;

e_Q (f (x) , y) = e_P (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, ≤ _{e
_P}) and (Q, ≤ _{e
_Q});

f ∘ g ≤ 1_Q and g ∘ f ≥ 1_P.

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 ∈ L^P. We define φ^l, φ^u as follows:

$\begin{matrix} \forall x \in P, φ^{l} (x) & = ⋀_{x^{'} \in P} (φ (x^{'}) \to e (x, x^{'})), φ^{u} (x) \\ = ⋀_{x^{'} \in P} (φ (x^{'}) \to e (x^{'}, x)) . \end{matrix}$

Proposition 2.17. (See [4, 5]) (the Dedekind-MacNeille completion) Given a fuzzy poset (P, e), ∀A, B ∈ L^P, a fuzzy cut is a pair (A, B) such thatA = B^l, B = A^u. 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 useDM_L (P) ={(A, B) ∣ ∀ A, B ∈ L^P, A = B^l, B = A^u} or $N (P)$ to denote the Dedekind-MacNeille completion ofP, that is to say,DM_L (P) or $N (P)$ is a fuzzy complete lattice.

Clearly, {(A, B) ∣ ∀ A, B ∈ L^P, A = B^l, B = A^u} is isomorphic to {A ∣ ∀ A ∈ L^P, A = A^ul}, thus {A ∣ ∀ A ∈ L^P, A = A^ul} can also denote the Dedekind-MacNeille completion of P. ∀ A, B ∈ L^P, A^u and B^l are usually called fuzzy upper cut and fuzzy lower cut, respectively.

Let (X, e) be a fuzzy poset and x ∈ X. Define two maps ι_x, μ_x : X → L by ι_x (y) = e (y, x) , μ_x (y) = e (x, y) for all y ∈ X. Clearly, we have ⊔ ι_x = ⊓ μ_x = x and the pair (ι_x, μ_x) is a fuzzy cut of X. Define $η : P \to N (P)$ by η (x) = (ι_x, μ_x) 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 ∈ L^P. Then

φ ≤ φ^ul and φ ≤ φ^lu;

sub (φ, ψ) ≤ sub (ψ^u, φ^u) and sub (φ, ψ) ≤ sub (ψ^l, φ^l);

sub (φ, ψ) ≤ sub (φ^ul, ψ^ul) and sub (φ, ψ) ≤ sub (φ^lu, ψ^lu);

φ^u = φ^ulu and φ^l = φ^lul.

3 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, e_P) , (Q, e_Q) be fuzzy posets, f : P → Q a map. Then we call f a fuzzy lower (upper) cut-continuous map provided that ∀ B ∈ L^Q, if B is a fuzzy lower (upper) cut of Q, then $f_{L}^{\leftarrow} (B)$ 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, e_P) is a fuzzy poset, then $(N (P), {sub}_{P})$ is also a fuzzy poset. Define $f : (P, e) \to (N (P), sub), x \mapsto ι_{x}$ , $g : (P, e) \to (N (P)^{op},$ sub^op) , x ↦ μ_x 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, e_P) , (Q, e_Q) be fuzzy posets, f : P → Q a map. Then we call f fuzzy lower (upper) cut-stable if ∀ A ∈ L^P, $(f_{L}^{\to} (A^{u}))^{l} =$ $(f_{L}^{\to} (A))^{ul} ((f_{L}^{\to} (A^{l}))^{u} = (f_{L}^{\to} (A))^{lu})$ . We say that f is fuzzy cut-stable if f is both fuzzy lower and upper cut-stable.

Example 3.4. Let (P, e_P) , (Q, e_Q) be fuzzy posets, f : P → Q a map. Define g : L^P → L^Q, h : L^Q → L^P by ∀ A ∈ L^P, B ∈ L^Q, 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 $f_{L}^{\leftarrow}$ is fuzzy cut-stable.

In the following we will use $f_{L}^{\to} (A)^{-}$ to denote $(f_{L}^{\to} (A))^{-}$ for brevity.

Definition 3.5. Let (P, e_P) , (Q, e_Q) 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 $(f_{L}^{\to} (B)^{l}, f_{L}^{\to} (A)^{u})$ is a fuzzy cut of Q.

Example 3.6. By example 2.4(1), we have (L, e_L) is a fuzzy poset. In fact, (L, e_L) is also a fuzzy complete lattice. Define f : L^L → L as follows ∀ A ∈ L^L, 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 $f_{L}^{\to} (A^{ul}) \leq f_{L}^{\to} (A)^{ul}$ $(f_{L}^{\to} (A^{lu}) \leq f_{L}^{\to} (A)^{lu})$ for all A ∈ L^P;

f is fuzzy order-preserving iff $f_{L}^{\to} (A)^{ul} \leq f_{L}^{\to} (A^{u})^{l}$ for all A ∈ L^P;

f is fuzzy cut-preserving iff $f_{L}^{\to} (A^{u})^{lu} = f_{L}^{\to} (A^{ul})^{u}$ iff $f_{L}^{\to} (A^{l})^{ul} = f_{L}^{\to} (A^{lu})^{l}$ for all A ∈ L^P.

Proof. (1) (Necessity) Suppose f is fuzzy lower cut-continuous. ∀ A ∈ L^P, $f_{L}^{\to} (A) \leq f_{L}^{\to} (A)^{ul} \Rightarrow A \leq f_{L}^{\leftarrow} (f_{L}^{\to} (A)^{ul}) \Rightarrow A^{ul} \leq f_{L}^{\leftarrow} (f_{L}^{\to} (A)^{ul}) (A^{ul}$ is thefuzzy lower cut generated by A) $\Rightarrow f_{L}^{\to} (A^{ul})$ $\leq f_{L}^{\to} (A)^{ul} .$

(Sufficiency) $\forall B \in L^{Q}, f_{L}^{\to} (f_{L}^{\leftarrow} (B^{l})^{ul}) \leq f_{L}^{\to} (f_{L}^{\leftarrow}$ (B^l)) ^ul ≤ B^lul = B^l. So $f_{L}^{\leftarrow} (B^{l}) \leq f_{L}^{\leftarrow} (B^{l})^{ul} \leq f_{L}^{\leftarrow} (B^{l})$ , that is, $f_{L}^{\leftarrow} (B^{l}) = f_{L}^{\leftarrow} (B^{l})^{ul}$ . Hence, f is fuzzy lower cut-continuous.

Similarly, we have that f is fuzzy upper cut-continuous iff $f_{L}^{\to} (A^{lu}) \leq f_{L}^{\to} (A)^{lu}$ for all A ∈ L^P.

(2) (Necessity) Suppose f is fuzzy order-preserving. We can easily prove $\forall A \in L^{P}, f_{L}^{\to} (A^{u}) \leq f_{L}^{\to} (A)^{u}$ , thus $f_{L}^{\to} (A)^{ul} \leq f_{L}^{\to} (A^{u})^{l}$ .

(Sufficiency) ∀ x ∈ P, put A = ι_x, then $f_{L}^{\to} (ι_{x})^{ul} \leq$ $f_{L}^{\to} (ι_{x}^{u})^{l} \Rightarrow f_{L}^{\to} (ι_{x}^{u}) \leq f_{L}^{\to} (ι_{x}^{u})^{lu} \leq f_{L}^{\to} (ι_{x})^{ulu} = f_{L}^{\to}$ (ι_x) ^u. Therefore, $ι_{x}^{u} \leq f_{L}^{\leftarrow} (f_{L}^{\to} (ι_{x})^{u})$ . ∀ x, y ∈ P, $\begin{matrix} e_{P} (x, y) & = & ⋀_{t \in P} (ι_{x} (t) \to e (t, y)) \\ = & ι_{x}^{u} (y) \\ \leq & f_{L}^{\leftarrow} (f_{L}^{\to} (ι_{x})^{u}) (y) \\ = & f_{L}^{\to} (ι_{x})^{u} (f (y)) \\ = & ⋀_{z \in Q} (f_{L}^{\to} (ι_{x}) (z) \to e_{Q} (z, f (y))) \\ = & ⋀_{z \in Q} ((⋁_{f (x^{'}) = z} e_{P} (x^{'}, x)) \to e_{Q} (z, f (y))) \\ = & ⋀_{x^{'} \in P} (e_{P} (x^{'}, x) \to e_{Q} (f (x^{'}), f (y))) \\ \leq & e_{Q} (f (x), f (y)) . \end{matrix}$

(3) (Necessity) Suppose f is fuzzy cut-preserving. Since ∀ A ∈ L^P, (A^ul, A^u) is a fuzzy cut of P, it follows that $(f_{L}^{\to} (A^{u})^{l}, f_{L}^{\to} (A^{ul})^{u})$ is a fuzzy cut of Q. Hence, $f_{L}^{\to} (A^{u})^{lu} = f_{L}^{\to} (A^{ul})^{u}$ .

(Sufficiency) Suppose (A, B) is a fuzzy cut of P. It suffices to show that $(f_{L}^{\to} (B)^{l}, f_{L}^{\to} (A)^{u})$ is a fuzzy cut of Q, i.e., $(f_{L}^{\to} (B)^{l}, f_{L}^{\to} (A)^{u}) = (f_{L}^{\to} (A)^{ul}, f_{L}^{\to} (B)^{lu})$ . It is easy to check that $f_{L}^{\to} (A)^{u} = f_{L}^{\to} (A^{ul})^{u} = f_{L}^{\to} (A^{u})^{lu} = f_{L}^{\to} (B)^{lu}$ , $f_{L}^{\to} (A)^{ul} = f_{L}^{\to} (B)^{lul} = f_{L}^{\to}$ (B) ^l. Similarly, f is fuzzy cut-preserving iff $f_{L}^{\to} (A^{l})^{ul} = f_{L}^{\to} (A^{lu})^{l}$ for all A ∈ L^P.

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 $f_{L}^{\leftarrow} (ι_{q}) = ι_{p}$ (∀ p ∈ P, ∃ q ∈ Q such that $f_{L}^{\to} (ι_{p}) = ι_{q}$ ).

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.

$★ {\begin{matrix} fuzzy residuated (fuzzy residual) \\ ⇕ \\ fuzzy lower cut - stable (fuzzy upper cut - stable) \Rightarrow fuzzy cut - preserving \\ ⇕ \\ fuzzy lower cut - continuous (fuzzy upper cut - continuous) \\ ⇕ \\ fuzzy join - preserving (fuzzy meet - preserving) \Rightarrow fuzzy order - preserving \end{matrix}$

Proposition 3.12.Let (P, e) be fuzzy poset, (Q, e) a fuzzy complete lattice,f : P → Qa map. Thenfis fuzzy cut-preserving iff $⊓ f_{L}^{\to} (A^{u}) = ⊔ f_{L}^{\to} (A^{ul})$ for allA ∈ L^P, where asfis fuzzy lower (upper) cut-stable iff $⊓ f_{L}^{\to} (A^{u}) = ⊔ f_{L}^{\to} (A)$ for allA ∈ L^P $(⊓ f_{L}^{\to} (A^{l}) = ⊔ f_{L}^{\to} (A)$ for allA ∈ L^P).

4 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 completion $N (P)$ and $N (Q)$ extendingf, i.e., making the following diagram commutative:

Proof.(Necessity) Assume that f : P → Q is fuzzy cut-stable. Define $F : N (P) \to N (Q)$ by $\forall (A, B) \in N (P), F (A, B) = (f_{L}^{\to} (B)^{l}, f_{L}^{\to} (A)^{u})$ .

Clearly, it is well-defined. In fact, F has a right adjoint $G : N (Q) \to N (P), (C, D) \mapsto (f_{L}^{\leftarrow} (C), f_{L}^{\leftarrow} (C)^{u})$ and a left adjoint $H : N (Q) \to N (P), (C, D) \mapsto (f_{L}^{\leftarrow} (D)^{l}, f_{L}^{\leftarrow} (D))$ . Since $f_{L}^{\leftarrow} (C) = f_{L}^{\leftarrow} (D^{l}) = f_{L}^{\leftarrow} (D^{l})^{ul} = f_{L}^{\leftarrow} (C)^{ul}, f_{L}^{\leftarrow} (D) = f_{L}^{\leftarrow} (C^{u}) = f_{L}^{\leftarrow}$ $(C^{u})^{lu} = f_{L}^{\leftarrow} (D)^{lu}$ , 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)) = sub $(A, f_{L}^{\leftarrow} (C)) = sub (f_{L}^{\to} (A), C) = sub (f_{L}^{\to} (A)^{ul}, C)$ $= sub (C^{u}, f_{L}^{\to} (A)^{u}) = sub (D, f_{L}^{\to} (A)^{u}) = sub (F (A,$ B) , (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 $\forall x \in P, y \in Q, f_{L}^{\to} (μ_{x})^{l} (y) = ⋀_{z \in Q} (f_{L}^{\to} (μ_{x}) (z) \to e (y, z) 1.1) = ⋀_{x^{'} \in P}$ (e (x, x′) → e (y, f (x′))) = ι_f(x) (y) , then F ∘ η_P $(x) = F (ι_{x}, μ_{x}) = (f_{L}^{\to} (μ_{x})^{l}, f_{L}^{\to} (ι_{x})^{u}) = (ι_{f (x)}, μ_{f (x)})$ = η_Q ∘ f (x).

For the uniqueness, suppose there exists $\bar{F} : N (P) \to N (Q)$ such that $\bar{F} \circ η_{P} = η_{Q} \circ f$ .

$\forall (A, B) \in N (P),$ $\begin{matrix} \bar{F} (A, B) & = & \bar{F} (A, A^{u}) \\ = & \bar{F} (⊔ ι_{L}^{\to} (A), ⊓ μ_{L}^{\to} (A)) \\ = & \bar{F} (⊔ (η_{P})_{L}^{\to} (A)) \\ = & ⊔ {\bar{F}}_{L}^{\to} \circ (η_{P})_{L}^{\to} (A) \\ = & ⊔ (\bar{F} \circ η_{P})_{L}^{\to} (A) \\ = & ⊔ (η_{Q} \circ f)_{L}^{\to} (A) \\ = & ⊔ (F \circ η_{P})_{L}^{\to} (A) \\ = & ⊔ F_{L}^{\to} \circ (η_{P})_{L}^{\to} (A) \\ = & F (⊔ (η_{P})_{L}^{\to} (A)) \\ = & F (⊔ ι_{L}^{\to} (A), ⊓ μ_{L}^{\to} (A)) \\ = & F (A, A^{u}) \\ = & F (A, B) . \end{matrix}$ (Sufficiency) Suppose there exists a fuzzy complete homomorphism F : P → Q such that F ∘ η_P =η_Q ∘ f. ∀ A ∈ L^P, $\begin{matrix} (f_{L}^{\to} (A^{u})^{l}, f_{L}^{\to} (A^{u})^{lu}) & = & (⊓ ι_{L}^{\to} (f_{L}^{\to} (A^{u})), ⊔ μ_{L}^{\to} (f_{L}^{\to} (A^{u}))) \\ = & ⊓ (η_{Q} \circ f)_{L}^{\to} (A^{u}) \\ = & ⊓ (F \circ η_{P})_{L}^{\to} (A^{u}) \\ = & ⊓ F_{L}^{\to} \circ (η_{P})_{L}^{\to} (A^{u}) \\ = & F (⊓ (η_{P})_{L}^{\to} (A^{u})) \\ = & F (⊔ (η_{P})_{L}^{\to} (A)) \\ = & ⊔ (F \circ η_{P})_{L}^{\to} (A) \\ = & ⊔ (η_{Q} \circ f)_{L}^{\to} (A) \\ = & ⊔ (η_{Q})_{L}^{\to} \circ f_{L}^{\to} (A) \\ = & (⊔ ι_{L}^{\to} (f_{L}^{\to} (A)), ⊓ μ_{L}^{\to} (f_{L}^{\to} (A))) \\ = & (f_{L}^{\to} (A)^{ul}, f_{L}^{\to} (A)^{u}) . \end{matrix}$ Therefore, $f_{L}^{\to} (A^{u})^{l} = f_{L}^{\to} (A)^{ul}$ . 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 homomorphism $g : N (P) \to Q$ such thatf = g ∘ η_P. In particular,η_Pis fuzzy cut-stable.

The category of fuzzy posets with fuzzy cut-stable maps is denoted by FPOS.

Easily, we can obtain:

$N :$ FPOS⟶FCLC

$P ⟼ N (P)$

$P \overset{f |}{\to} Q \to N (P) \overset{N (f) = F}{\to} N (Q)$

is a functor.

Corollary 4.3.The categoryFCLCis a full reflective subcategory of the categoryFPOS.

Next, we will answer the question under what circumstances $N (f)$ 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 $x = ⊔ i_{L}^{\to} (φ) (x = ⊓ i_{L}^{\to} (ψ))$ . 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

$N (f)$ is surjective iff f is join-dense iff f is meet-dense.

$N (f)$ is injective iff $N (f)$ is an embedding iff f is an embedding.

$N (f)$ is a bijective iff f is a join-dense (meet-dense) embedding.

Proof. (1) Suppose $N (f)$ is surjective. For any $q \in Q, (ι_{q}, μ_{q}) \in N (Q)$ , there is $(A, B) \in N (P)$ such that $N (f) (A, B) = (f_{L}^{\to} (B)^{l}, f_{L}^{\to} (A)^{u}) = (ι_{q}, μ_{q})$ . Thus $f_{L}^{\to} (A)^{u} = μ_{q}$ . Because $\forall y \in Q, f_{L}^{\to} (A)^{u} (y) = ⋀_{y^{'} \in Q} (f_{L}^{\to} (A) (y^{'}) \to e (y^{'}, y)) = μ_{q} (y) = e (q, y)$ , we have $q = ⨆ i_{L}^{\to} (f_{L}^{\to} (A))$ .

Conversely, if f is join-dense, we need to prove $N (f) \circ G = {id}_{N (P)}$ , that is, $N (f) (G (C, C^{u})) = (C, C^{u})$ for all $(C, C^{u}) \in N (Q)$ . Since $N (f) (G$ $(C, C^{u})) = N (f) (f_{L}^{\leftarrow} (C), f_{L}^{\leftarrow} (C)^{u}) = (f_{L}^{\to} (f_{L}^{\leftarrow} (C)^{u})^{l},$ $f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u}) = (f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{ul}, f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u}) = (C, C^{u})$ , thus it suffices to check that $f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{ul}$ = C. Firstly, since $f_{L}^{\to} (f_{L}^{\leftarrow} (C)) \leq C$ , we have $f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{ul} \leq C^{ul} = C$ .

Secondly, ∀ y ∈ Q, $\begin{matrix} f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{ul} (y) & = ⋀_{a \in Q} (f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u} (a) \to e (y, a)) \\ = ⋀_{a \in Q} (f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u} \\ (a) \to e (⨆ i_{L}^{\to} (ι_{y} ∣_{f (P)}), a)) \\ = ⋀_{a \in Q} (f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u} \\ (a) \to (⋀_{b \in P} ι_{y} (f (b)) \to e (f (b), a))) \\ = ⋀_{a \in Q} (f_{L}^{\to} (f_{L}^{\leftarrow} (C))^{u} \\ v (a) \to (⋀_{b \in P} e (f (b), y) \to e (f (b), a))) \\ = ⋀_{a \in Q} (1.1 (⋀_{b^{'} \in P} C (f (b^{'}) \to e (f (b^{'}), a)) \\ \to (⋀_{b \in P} e (f (b), y) \to e (f (b), a))) \\ \geq ⋀_{a \in Q} ⋀_{b \in P} ((C (f (b)) \to e (f (b), a)) \\ \to (e (f (b), y) \to e (f (b), a))) \\ \geq ⋀_{a \in Q} (⋀_{b \in P} e (f (b), y) \to C (f (b))) \\ \geq C (y) . \end{matrix}$ Similarly, $N (f)$ is surjective iff f is meet-dense.

(2) If $N (f)$ is an embedding, then $N (f)$ is injective. Conversely, if $N (f)$ 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 $(N (f), G)$ is a fuzzy adjunction and $N (f)$ is one-to-one, it holds that $G \circ N (f) = {id}_{N (P)}$ , ∀ x ∈ P, $G \circ N (f) (ι_{x}, μ_{x}) = G (f_{L}^{\to} (μ_{x})^{l}, f_{L}^{\to} (ι_{x})^{u}) = (f_{L}^{\leftarrow} (f_{L}^{\to}$ $(μ_{x})^{l}), f_{L}^{\leftarrow} (f_{L}^{\to} (μ_{x})^{l})^{u}) = (f_{L}^{\to} (ι_{f (x)}), f_{L}^{\to} (ι_{f (x)})^{u})$ =(ι_x, μ_x). Hence, $e (f (y), f (x)) = f_{L}^{\leftarrow} (ι_{f (x)}) (y) = ι_{x} (y) = e (y, x)$ .

If f is an embedding, we need to show that $sub ((A, B), (C, D)) = sub (N (f) (A, B), N (f) (C, D))$ ,i.e., $sub (A, C) = sub (f_{L}^{\to} (C)^{u}, f_{L}^{\to} (A)^{u})$ . sub (A, C) $\leq sub (f_{L}^{\to} (A), f_{L}^{\to} (C))$ $\leq sub (f_{L}^{\to} (C)^{u}, f_{L}^{\to} (A)^{u})$ for all $(A, B), (C, D) \in N (P)$ .

$\begin{matrix} sub (A, C) & = & sub (A, C^{ul}) \\ = & ⋀_{x \in P} (A (x) \to (⋀_{y \in P} C^{u} (y) \to e (x, y))) \\ = & ⋀_{x \in P} (A (x) \to (⋀_{y \in P} f_{L}^{\to} (C)^{u} (f (y)) \to e (x, y))) \\ = & ⋀_{x \in P} ⋀_{y \in P} (A (x) * f_{L}^{\to} (C)^{u} (f (y)) \to e (f (x), f (y))) \\ \geq & f_{L}^{\to} (C)^{u} (f (y)) \to f_{L}^{\to} (A)^{u} (f (y)) \\ \geq & ⋀_{z \in Q} (f_{L}^{\to} (C)^{u} (z) \to f_{L}^{\to} (A)^{u} (z)) \\ = & sub (f_{L}^{\to} (C)^{u}, f_{L}^{\to} (A)^{u}) . \end{matrix}$ (3) This is an immediate consequence of (1) and (2).

For convenience, in the following we will use $N (P) = {φ \in L^{P} ∣ φ = φ^{ul}}$ 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 (Z^ul, I).

Clearly, ∀ x ∈ P, ι_x is a (Frink) fuzzy ideal.

Remark 4.6. (1) If $φ \in N (P)$ , 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 (I^ul (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 : L^P → L^P as follows: ∀ A ∈ L^P, x ∈ P, Γ (A) (x) = ⋁ _I∈Fl(P) (I^ul (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) (I^ul * sub (I, rotatebox []270twoheadrightarrow A) (x)) = rotatebox []270twoheadrightarrow Γ (A) (x). Hence, ▵ ∘ ∇ ≥ id_FLow(P) and ∇ ∘ ▵ ≤ id_FLow(P).

By the above lemma, we have that the operator Γ ∣ _FLow(P) is a fuzzy meet-preserving map. Define $C : Fl (P) \to Fl (P)$ by $\forall φ \in Fl (P), C (φ) = φ^{ul}$ . We can easily show that $C$ is a fuzzy closure operator and $N (P) = C (Fl (P))$ . Since $N (I) = C (I) = I^{ul} = Γ (I)$ for all I ∈ Fl (P), it follows that $N$ is a fuzzy meet-preserving map of fuzzy (Frink) ideals and $N$ induces a fuzzy join-preserving map from Fl (P) onto its image $N (P)$ .

Theorem 4.15.Suppose (P, e) is a fuzzy poset. IfPis precontinuous, then the normal completion $N (P)$ is a fuzzy continuous lattice.

Proof. For each I ∈ Fl (P), define k_I : Fl (P) → L as follows: ∀ A ∈ Fl (P) ,

$k_{I} (A) = {\begin{matrix} sub (A, I), & A \in K (Fl (P)), \\ 0, & otherwises. \end{matrix}$

It is easy to show that if I = ⨆ k_I, then Fl (P) is algebraic. Therefore, the normal completion $N (P)$ 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.

5 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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