The Z L -completions of fuzzy posets

Abstract

For a fuzzy subset system Z_L, the concepts of a Δ_{Γ
_L}-completion and a Z_{Γ
_L}-completion of a given fuzzy poset (X, e) are introduced and their universal properties are investigated. In this paper, we prove that: (1) the Δ_{Γ
_L}-completion Δ_{Γ
_L} (X) is a join-completion with the universal property; (2) the Z_{Γ
_L}-completion Z_{Γ
_L} (X) is the smallest Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X) in the case that Z_L is fuzzy subset-hereditary. The results show that the Dedekind-MacNeille completion is a special case of the Z_{Γ
_L}-completion.

Keywords

Fuzzy subset system Fuzzy subset-hereditary

1 Introduction

One important aspect of domain theory [8] is to extend the poset to a complete one. Since the DM-completion was proposed by Dedekind and MacNeille [15], the idea of completions has aroused the interest of many researchers, and there are many researches on this aspect, such as the Alexandroff completion, the cauchy ideal completion, the Frink ideal completion, the D-completion, and so on (see [5, 7, 12, 27]). In 2015, Zhao [28] extended the D-completion to the more general structure Z-subset system, namely Z-completion. Subsequently, Zhang and Li (see [24 –26]) did a sequence of work on completions by Z-subset system: (1) constructed the Z_δ-completions, which generalized the DM-completions of posets; (2) constructed the Z-convergence space completion embedding each T₀-closure space into a Z-convergence one; (3) provided a general framework for poset completions, i.e., the Z_Γ-completion, which includes numerous special cases.

With the development of fuzzy set theory, fuzzy orders have penetrated into many fields of mathematics, such as domain, topology, rough set and so on (see [2, 4, 6, 11, 13, 14, 16 –23, 29, 30]). In the framework of fuzzy domain, Wagner [18] constructed an enriched version of the DM-completion for an Ω-category. Bĕlohlávek [2, 4] used fuzzy concept lattices to describe the DM-completion of a fuzzy poset. And Xie et al. [20] used cuts to construct the DM-completions of fuzzy posets. When we take Ω = L to be a frame, it is easily seen that the three versions of the DM-completion for a fuzzy poset agree with each other. Later, Wang and Zhao [19] provided the join-completion of a fuzzy poset and Su and Li [17] discussed the Δ₁-completion of a fuzzy poset. On the other hand, Ma, Rao and Zhao et al. [14, 16, 30] studied Z_L-subset systems in the fuzzy setting, called fuzzy Z_L-subset systems. The details are as follows: Zhao [30] introduced the definition of Z_L-continuous fuzzy posets and discussed some simple properties; Rao [16] studied section-retraction pair of Z_L-continuous fuzzy posets and obtained an extension theorem of Z_L-algebraic fuzzy posets; Ma and Zhao [14] further investigated the properties of Z_L-continuous fuzzy posets and Z_L-algebraic fuzzy posets. Then, how can we describe the Z_L-completion for fuzzy posets? Can we get the same results as in the classical domain theory? And what is the role of it in fuzzy domain theory? For this purpose, in this paper we first define a Δ_{Γ
_L}-completion of a fuzzy poset, and discuss its universal property. Then we build a Z_{Γ
_L}-completion with the universal property for a fuzzy poset and discuss the relationship with other completions.

The contents are arranged as follows: In Sect. 2, some basic notions and conclusions that will be used throughout this paper are listed. In Sect. 3, for a fuzzy subset system Z_L and a fuzzy subset selection Γ_L of fuzzy Z_L-subsets, the Δ_L-closed fuzzy subsets and Δ_{Γ
_L}-completion for a fuzzy poset are introduced and investigated. Moreover, it is shown that the Δ_{Γ
_L}-completion is a join-completion with the university property. In Sect. 4, the Δ_{Γ
_L}-continuously ⊔-existing fuzzy subsets and Δ_{Γ
_L}-continuous mapping are introduced. Then the concept of a Δ_{Γ
_L}-completion for a fuzzy poset is proposed and discussed. In particular, it is shown that the Z_{Γ
_L}-completion Z_{Γ
_L} (X) is the smallest Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X) and any Δ_{Γ
_L}-continuous mapping f : X ⟶ Y into a Z_L-complete fuzzy poset (Y, e_Y) extends uniquely to a Z_⊔-continuous mapping from the Z_{Γ
_L}-completion Z_{Γ
_L} (X) to Y.

2 Preliminaries

For the convenience of the reader, in this section, some basic concepts are reviewed. In this paper, we will use a complete residuated lattice as the structures of truth values. Such an algebraic structure is significant in fuzzy logic in a narrow sense (see [2, 3, 11]). If no other conditions are imposed, in the sequel, L always denotes a complete residuated lattice. A complete residuated lattice is an algebraic structure (L, ∧ , ∨ , ∗ , → , 0, 1) such that (1) (L, ∧ , ∨ , 0, 1) is a complete lattice with the least element 0 and the greatest element 1; (2) (L, ∗ , 1) is a commutative monoid, i.e., ∗ is commutative, associative, and a ∗ 1 = a holds for all a ∈ L; (3) ∗ and → form an adjoint pair, i.e., a ∗ b ≤ c ⇔ a ≤ b → c for all a, b, c ∈ L.

Proposition 2.1. (Bĕlohlávek [3] and Hájek [10]). Let L be a complete residuated lattice, a, b, c ∈ L, and {a_k} _k∈K, {b_k} _k∈K ⊆ L, then

(1) 0 ∗ a = 0, 1 → a = a, 0 → a = 1;

(2) 1 ≤ a → b ⇔ a ≤ b;

(3) (a → b) ∗ (b → c) ≤ (a → c);

(4) a → (b → c) = (a ∗ b) → c = b → (a → c);

(5) a ∗ ⋁ _k∈Kb_k = ⋁ _k∈Ka ∗ b_k;

(6) (⋁ _k∈Ka_k) → b = ⋀ _k∈K (a_k → b);

(7) a → (⋀ _k∈Kb_k) = ⋀ _k∈K (a → b_k);

(8) (a → b) → (a → c) ≥ b → c, (a → b) → (c → b) ≥ c → a;

(9) a ∗ (a → b) ≤ b, a ≤ (a → b) → b;

(10) a ∗ (b ∧ c) ≤ (a ∗ b) ∧ (a ∗ c);

(11) a → (b ∗ c) ≥ (a → b) ∗ c.

More properties about complete residuated lattices can be found in [3, 10].

L^X denotes the set of all fuzzy subsets of X. For A ⊆ X, χ_A denotes the characteristic function of A. In particular, when A = {x}, χ_A is denoted by χ_x. For a ∈ L and A, B ∈ L^X, write fuzzy subsets a ∗ A, a → A and A ⊆ B by (a ∗ A) (x) = a ∗ A (x), (a → A) (x) = a → A (x) and A (x) ≤ B (x) for all x ∈ X.

Definition 2.2. (Bĕlohlávek [4] and Fan [6]) A fuzzy poset is a pair (X, e) such that X is a set and e : X × X ⟶ L is a mapping, called a fuzzy order, that satisfies for every x, y, z ∈ X,

(1) e (x, x) =1;

(2) e (x, y) ∗ e (y, z) ≤ e (x, z);

(3) e (x, y) = e (y, x) =1 implies x = y.

Let X be a nonempty set and A, B ∈ L^X, the subsethood degree of A in B [9] is defined by sub (A, B) = ⋀ _x∈XA (x) → B (x) and sub is also called the fuzzy inclusion order on L^X. Then (L^X, sub) is a fuzzy poset. Some basic properties of sub are collected in the following proposition, which can be found in [2, 9].

Proposition 2.3. Let sub : L^X × L^X ⟶ L on L^X, A, B, C, D ∈ L^X, and {B_k} _k∈K ⊆ L^X. Then

(1) sub (A, B) =1 iff A ⊆ B;

(2) sub (A, B) ∗ sub (C, D) ≤ sub (A ∗ C, B ∗ D);

(3) sub (A, ⋀ _k∈KB_k) = ⋀ _k∈Ksub (A, B_k);

(4) sub (⋁ _k∈KB_k, A) = ⋀ _k∈Ksub (B_k, A).

Definition 2.4. (Yao [21], Zhang and Fan [23]) Let (X, e) be a fuzzy poset. An element x₀ ∈ X is called a join (resp . , meet) of A, in symbols x₀ = ⊔ A (resp . , x₀ = ⊓ A), iff

(1) for any x ∈ X, A (x) ≤ e (x, x₀) (resp . , A (x) ≤ e (x₀, x));

(2) for any $y \in X, \underset{x \in X}{⋀} (A (x) \to e (x, y)) \leq e (x_{0}, y) (resp ., \underset{x \in X}{⋀} (A (x) \to e (y, x)) \leq e (y, x_{0}))$ .

Proposition 2.5. (Yao [21]) Let (X, e) be a fuzzy poset. Then

(1) x₀ = ⊔ A iff $e (x_{0}, y) = \underset{x \in X}{⋀} A (x) \to e (x, y)$ for all y ∈ X;

(2) x₀ = ⊓ A iff $e (y, x_{0}) = \underset{x \in X}{⋀} A (x) \to e (y, x)$ for all y ∈ X.

Let (X, e) be a fuzzy poset. A ∈ L^X is called a fuzzy upper subset (or a fuzzy lower subset) iff ∀x, y ∈ X, A (x) ∗ e (x, y) ≤ A (y) (or A (x) ∗ e (y, x) ≤ A (y)). ↓A ∈ L^X is defined by ↓A (x) = ⋁ _x′∈XA (x′) ∗ e (x, x′) for all x ∈ X. Then ↓A is a fuzzy lower subset and A is a fuzzy lower subset iff A = ↓ A. For x ∈ X, define the mappings ι_x, μ_x : X ⟶ L as follows: ∀y ∈ X, ι_x (y) = e (y, x) , μ_x (y) = e (x, y). Obviously, ι_x is a fuzzy lower subset and μ_x is a fuzzy upper subset. Let Ψ (X) = {ι_x | x ∈ X} denote the set of all fuzzy principal ideals of X.

Given a map f : X → Y between two fuzzy posets (X, e) and (Y, e_Y), as usual, define f^→ : L^X → L^Y and f^← : L^Y → L^X by f^→ (A) (y) = ⋁ _x∈XA (x) ∗ e_Y (y, f (x)) , f^← (B) (x) = B (f (x)) , ∀ y ∈ Y, x ∈ X . Then the following results can be found in [2, 13].

Proposition 2.6. Let f : X → Y be a mapping two fuzzy posets (X, e) and (Y, e_Y), A, B ∈ L^X, and C, D ∈ L^Y. Then the following statements hold:

(1) sub (A, B) ≤ sub_Y (f^→ (A) , f^→ (B));

(2) sub_Y (C, D) ≤ sub (f^← (C) , f^← (D));

(3) A ⊆ f^← (f^→ (A));

(4) f^→ (f^← (C)) ⊆ C and f^← (C) is a fuzzy lower subset of X when C = ↓ C.

Definition 2.7. (Yao [21], Zhang and Fan [23]) Let (X, e_X) , (Y, e_Y) be two fuzzy posets, then a mapping f : X → Y is called

(1) a fuzzy order-preserving if e_X (x, y) ≤ e_Y (f (x) , f (y)) for all x, y ∈ X;

(2) a fuzzy order-embedding if e_X (x, y) = e_Y (f (x) , f (y)) for all x, y ∈ X;

(3) a fuzzy order-isomorphism if f is a fuzzy order-embedding which maps X onto Y.

Let (X, e) be a fuzzy poset and A ∈ L^X. The A^u, A^l ∈ L^X is defined as follows: A^u (x) = ⋀ _x′∈XA (x′) → e (x′, x) , A^l (x) = ⋀ _x′∈XA (x′) → e (x, x′) , ∀ x ∈ X . The cut operator (-) ^δ : L^X ⟶ L^X is defined by A^δ = A^ul. A is called a cut if A = A^δ. Clearly, a cut is always a fuzzy lower subset and the cut operator is a fuzzy closure operator on X, that is, (1) A ⊆ A^δ, (2) sub (A, B) ≤ sub (A^δ, B^δ), and (3) A^δ = (A^δ) ^δ for all A, B ∈ L^X. The Dedekind-MacNeille completion DM_L (X) = {A ∈ L^X : A = A^δ}.

Proposition 2.8. Let (X, e) be a fuzzy poset and A ∈ L^X. Then A^δ = ι_x for some x ∈ X iff ⊔A = x.

Proof. By Lemma 2.5 in [20].

Definition 2.9. (Bĕlohlávek [2], Wang [19], and Xie [20]) Let (X, e) be a fuzzy poset, P ⊆ X, and i : P ⟶ X an inclusion mapping.

(1) If for every x ∈ X there is a fuzzy subset A of P such that x = ⊔ i^→ (A), then P is said to be join-dense in X;

(2) If (Y, e_Y) is a fuzzy complete lattice and φ : X ⟶ Y is a fuzzy order-embedding, then (Y, e_Y) is called a completion of X; For convenience, a completion (Y, e_Y) of X via the fuzzy order-embedding φ is denoted by the pair (Y, φ) (or just Y if there is no ambiguity);

(3) If (Y, φ) is a completion of X and φ (X) is join-dense in Y, then (Y, φ) is called a join-completion of X.

Proposition 2.10. (Wang [19]) Let (X, e) be a fuzzy poset and C a fuzzy closure operator on X. If C is consistent, i.e., C (χ_x) = ι_x for all x ∈ X, then $(T_{C}, φ_{C})$ is a join-completion of X, where $T_{C} = {A \in L^{X} | C (A) = A}$ and $φ_{C} : X ⟶ T_{C}$ is defined by φ_C (x) = ι_x for all x ∈ X.

Let FPos be the category of fuzzy posets with fuzzy order-preserving mappings as morphisms. A fuzzy subset system [14] on FPos is a functor Z_L : FPos ⟶ Set satisfying the following conditions:

(1) $\forall (X, e) \in O b (FPos)$ , Z_L (X, e) ⊆ L^X, for convenience Z_L (X, e) is always denoted by Z_L (X);

(2) $\forall f : (X, e_{X}) ⟶ (Y, e_{Y}) \in M or (FPos)$ , if A ∈ Z_L (X), then Z_L (f) (A) = f^→ (A) ∈ Z_L (Y);

(3) $\exists (X, e) \in O b (FPos)$ such that ∃ A ∈ Z_L (X) satisfies ⋁_x∈XA (x) =1.

For all A ∈ Z_L (X), A is called a fuzzy Z_L-subset of X. By (2), for any P ⊆ X, A ∈ Z_L (P) implies i^→ (A) ∈ Z_L (X), where i : P ⟶ X is an inclusion mapping. By (2) and (3), ι_x ∈ Z_L (X) for all x ∈ X. In fact, by (3), $\exists (Y, e_{Y}) \in O b (FPos)$ such that ∃ B ∈ Z_L (Y) satisfies ⋁_y∈YB (y) =1. We can define a mapping f : Y ⟶ X as f (y) = x for all y ∈ Y. Then f is a fuzzy order-preserving mapping and f^→ (B) (x′) = ⋁ _y∈YB (y) ∗ e (x′, f (y)) = ⋁ _y∈YB (y) ∗ e (x′, x) = e (x′, x) = ι_x (x′) for all x′ ∈ X. So ι_x = f^→ (B) ∈ Z_L (X) by (2). Moreover, if f is a fuzzy order-isomorphism, then A ∈ Z_L (X) ⇔ f^→ (A) ∈ Z_L (Y).

Example 2.11. (Ma [14], Rao [16] and Zhao [30])Let (X, e) be a fuzzy poset, $D_{L} (X) = {A \in L^{X} | A is a fuzzy directed subset of X}$ , $L_{L} (X) = {A \in L^{X} | A is a fuzzy lower subset of X}$ , and $P_{L} (X) = L^{X}$ , then $D_{L}$ , $L_{L}$ and $P_{L}$ are a fuzzy subset system.

(X, e) is called Z_L-complete if ⊔D exists for all D ∈ Z_L (X). P ⊆ X is called a Z_L-complete fuzzy subposet if ∀D ∈ Z_L (P), ⊔i^→ (D) ∈ P. A ∈ L^X is called Z_L-closed if ∀D ∈ Z_L (X) with existing ⊔D implies sub (D, A) ≤ A (⊔ D). Clearly, $Z_{L}^{c} (X) = {A \in L^{X} | A is Z_{L} - closed}$ is a fuzzy closure space defined in [1], then the Z_L-closure operator C_{Z
_L} is defined as follows: for any B ∈ L^X, $C_{Z_{L}} (B) (x) = ⋀_{A \in Z_{L}^{c} (X)} sub (B, A) \to A (x) .$ A mapping f : X ⟶ Y between two fuzzy posets is fuzzy Z_⊔-continuous iff f is a fuzzy order-preserving and f (⊔ D) = ⊔ f^→ (D) for all D ∈ Z_L (X) with existing ⊔D.

Definition 2.12. A fuzzy subset system Z_L is called fuzzy subset-hereditary if for any fuzzy posets (X, e_X), (Y, e_Y) and a fuzzy order-embedding f : X ⟶ Y, D ∈ Z_L (X) iff f^→ (D) ∈ Z_L (Y).

Proposition 2.13. Let (X, e) be a fuzzy poset, P ⊆ X, and D ∈ L^P. If Z_L is fuzzy subset-hereditary, then D ∈ Z_L (P) iff i^→ (D) ∈ Z_L (X).

3 The Δ_{Γ
_L}-completion of fuzzy posets

Let Γ_L (X) ⊆ Z_L (X). If ⊔D exists in X for any D ∈ Γ_L (X), then we say (X, e) is Γ_L-complete. In particular, if (X, e) is Z_L-complete, then (X, e) is Γ_L-complete for any Γ_L. In this section, We define a special type of fuzzy subsets by the fuzzy subset selection Γ_L (X) and construct a Δ_{Γ
_L}-completion of a fuzzy poset based on it.

Definition 3.1. Let (X, e) be a fuzzy poset. A fuzzy subset A ∈ L^X is called Δ_{Γ
_L}-closed if

(1) A = ↓ A;

(2) ∀D ∈ Γ_L (X), sub (D, A) ≤ sub (D^δ, A).

Proposition 3.2. Let (X, e) be a fuzzy poset and Δ_{Γ
_L} (X) = {A ∈ L^X | A is Δ_{Γ
_L} - closed}. Then (Δ_{Γ
_L} (X) , sub) is a fuzzy complete lattice.

Proof. We only need to show that for any $A \in L^{Δ_{Γ_{L}} (X)}$ , $⊓ A$ exists and $⊓ A \in Δ_{Γ_{L}} (X)$ . Let $B (x) = ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A (x)$ for any x ∈ X, then B ∈ Δ_{Γ
_L} (X). In fact, for any x, y ∈ X, $\begin{matrix} B (x) * e (y, x) & = (⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A (x)) * e (y, x) \\ \leq ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A (x) * e (y, x) \\ \leq ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A (y) = B (y), \end{matrix}$

which implies that B is a fuzzy lower subset of X. Moreover, for any D ∈ Γ_L (X), $\begin{matrix} sub (D, B) \\ = ⋀_{x \in X} D (x) \to (⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A (x)) \\ = ⋀_{x \in X} ⋀_{A \in Δ_{Γ_{L}} (X)} D (x) \to (A (A) \to A (x)) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to sub (D, A) \\ \leq ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to sub (D^{δ}, A) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} ⋀_{x \in X} D^{δ} (x) \to (A (A) \to A (x)) \\ = ⋀_{x \in X} D^{δ} (x) \to B (x) = sub (D^{δ}, B) . \end{matrix}$ Then we can easy check that B is the meet of $A$ , i.e., $⊓ A = ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to A$ .

Define a fuzzy Δ_{Γ
_L}-closure operator C_{Δ
_{Γ
_L}} on X by C_{Δ
_{Γ
_L}} (B) (x) = ⋀ _{A∈Δ_{Γ
_L}(X)}sub (B, A) → A (x) for any B ∈ L^X and x ∈ X. Let $T_{C_{Δ_{Γ_{L}}}} = {A \in L^{X} | C_{Δ_{Γ_{L}}} (A) = A}$ , then $T_{C_{Δ_{Γ_{L}}}} = Δ_{Γ_{L}} (X)$ by Theorem 3.6 in [1].

Proposition 3.3. Let (X, e) be a fuzzy poset, A ∈ L^X and $A \in L^{Δ_{Γ_{L}} (X)}$ .

(1) If A = A^δ, then A ∈ Δ_{Γ
_L} (X). In particular, Ψ (X) ⊆ Δ_{Γ
_L} (X);

(2) $⊔_{Δ_{Γ_{L}} (X)} A = C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A)$ .

Proof. (1) Let A = A^δ, then A is a fuzzy lower subset, i.e., A = ↓ A. For any D ∈ Γ_L (X), sub (D, A) ≤ sub (D^δ, A^δ) = sub (D^δ, A). Therefore, A ∈ Δ_{Γ
_L} (X). Moreover, Ψ (X) ⊆ Δ_{Γ
_L} (X) since ι_x^δ = ι_x.

(2) For any B ∈ Δ_{Γ
_L} (X), $\begin{matrix} sub (B, C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A)) \\ = ⋀_{x \in X} B (x) \to (⋀_{C \in Δ_{Γ_{L}} (X)} sub (⋁_{A \in Δ_{Γ_{L}} (X)} \\ A (A) * A, C) \to C (x)) \\ = ⋀_{C \in Δ_{Γ_{L}} (X)} sub (⋁_{A \in Δ_{Γ_{L}} (X)} \\ A (A) * A, C) \to sub (B, C) \\ \geq sub (B, ⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A) \\ = ⋀_{x \in X} B (x) \to (⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A (x)) \\ \geq ⋀_{x \in X} B (x) \to (A (B) * B (x)) \geq A (B) . \end{matrix}$ Moreover, $\begin{matrix} sub (C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A), B) \\ = ⋀_{x \in X} (⋀_{C \in Δ_{Γ_{L}} (X)} sub (⋁_{A \in Δ_{Γ_{L}} (X)} \\ A (A) * A, C) \to C (x)) \to B (x) \\ \geq ⋀_{x \in X} (sub (⋁_{A \in Δ_{Γ_{L}} (X)} \\ A (A) * A, B) \to B (x)) \to B (x) \\ \geq sub (⋁_{A \in Δ_{Γ_{L}} (X)} A (A) * A, B) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} A (A) \to sub (A, B) . \end{matrix}$

Proposition 3.4. Let (X, e) be a fuzzy poset. Then

(1) C_{Δ
_{Γ
_L}} is a consistent fuzzy closure operator on X, i.e., C_{Δ
_{Γ
_L}} (χ_x) = ι_x for all x ∈ X;

(2) (Δ_{Γ
_L} (X) , η_X) is a join-completion of (X, e), where η_X : X ⟶ Δ_{Γ
_L} (X) defined by η_X (x) = ι_x for all x ∈ X. In this case, we always call (Δ_{Γ
_L} (X) , η_X) or simply Δ_{Γ
_L} (X) the Δ_{Γ
_L}-completion of the fuzzy poset (X, e).

Proof. (1) For any x, x′ ∈ X, it is obvious that sub (ι_x, C_{Δ
_{Γ
_L}} (χ_x)) = C_{Δ
_{Γ
_L}} (χ_x) (x) =1 and $\begin{matrix} C_{Δ_{Γ_{L}}} (χ_{x}) (x^{'}) & = ⋀_{A \in Δ_{Γ_{L}} (X)} sub (χ_{x}, A) \to A (x^{'}) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} A (x) \to A (x^{'}) \\ \leq ι_{x} (x) \to ι_{x} (x^{'}) = ι_{x} (x^{'}) . \end{matrix}$ Therefore C_{Δ
_{Γ
_L}} (χ_x) = ι_x.

(2) It is a direct consequence of (1), Proposition 3.2 and Proposition 3.16 in [19].

Define a mapping $F_{Δ Γ_{L} (X)} : L^{X} ⟶ L^{X}$ by $F_{Δ_{Γ_{L}} (X)} (A) (x) = sub (η_{X} (x), ⊔ η X \to (A))$ for all A ∈ L^X and x ∈ X. Then we have the following result.

Lemma 3.5. For all A ∈ L^X, $F_{Δ_{Γ_{L}} (X)} (A) = C_{Δ_{Γ_{L}}} (A)$ .

Proof. Let A ∈ L^X and x ∈ X, then $F_{Δ_{Γ_{L}} (X)} (A) (x) = sub (η_{X} (x), ⊔ η X \to (A)) = ⊔ η X \to (A) (x)$ since $⊔ η_{X}^{\to} (A) \in Δ_{Γ_{L}} (X)$ is a fuzzy lower subset. By Proposition 3.3, we have that

$\begin{matrix} F_{Δ_{Γ_{L}} (X)} (A) (x) = ⊔ η_{X}^{\to} (A) (x) \\ = C_{Δ_{Γ_{L}}} (⋁_{B \in Δ_{Γ_{L}} (X)} η_{X}^{\to} (A) (B) * B) (x) \\ = C_{Δ_{Γ_{L}}} (⋁_{B \in Δ_{Γ_{L}} (X)} ⋁_{x^{'} \in X} A (x^{'}) * sub (B, η_{X} (x^{'}))) * B) (x) \\ = C_{Δ_{Γ_{L}}} (⋁_{x^{'} \in X} A (x^{'}) * η_{X} (x^{'})) (x) \\ = C_{Δ_{Γ_{L}}} (↓ A) (x) \supseteq C_{Δ_{Γ_{L}}} (A) (x) . \end{matrix}$ On the other hand, $\begin{matrix} sub (C_{Δ_{Γ_{L}}} (⋁_{B \in Δ_{Γ_{L}} (X)} η_{X}^{\to} (A) (B) * B), C_{Δ_{Γ_{L}}} (A)) \\ \geq sub (⋁_{B \in Δ_{Γ_{L}} (X)} η_{X}^{\to} (A) (B) * B, A) \\ = ⋀_{x \in X} ⋀_{B \in Δ_{Γ_{L}} (X)} (η_{X}^{\to} (A) (B) * B (x)) \to A (x) \\ = ⋀_{B \in Δ_{Γ_{L}} (X)} η_{X}^{\to} (A) (B) \to sub (B, A) \end{matrix}$ $\begin{matrix} = ⋀_{B \in Δ_{Γ_{L}} (X)} ⋀_{x \in X} (A (x) * sub (B, η_{X} (x))) \to sub (B, A) \\ \geq ⋀_{x \in X} A (x) \to A (x) = 1, \end{matrix}$ which implies that $F_{Δ_{Γ_{L}} (X)} (A) \subseteq C_{Δ_{Γ_{L}}} (A)$ . Therefore $F_{Δ_{Γ_{L}} (X)} (A) = C_{Δ_{Γ_{L}}} (A)$ .

By Lemma 3.5 and Theorem 4.5 in [19], we obtain that the Δ_{Γ
_L}-completion Δ_{Γ
_L} (X) of a fuzzy poset (X, e) has the universal property, details as follows:

Theorem 3.6. The Δ_{Γ
_L}-completion Δ_{Γ
_L} (X) of a fuzzy poset (X, e) has the universal property in the following sense: for any C_{Δ
_{Γ
_L}}-homomorphism f : X ⟶ Y into a fuzzy complete lattice (Y, e_Y) there is a unique join-preserving mapping $\hat{f} : Δ_{Γ_{L}} (X) ⟶ Y$ such that $f = \hat{f} \circ η_{X}$ , i.e., the following diagram commutes:

Here C_{Δ
_{Γ
_L}}-homomorphism f : X ⟶ Y means f^← (ι_y) ∈ Δ_{Γ
_L} (X) for all y ∈ Y and $\hat{f} : Δ_{Γ_{L}} (X) ⟶ Y$ is defined by $\hat{f} (A) = ⊔ f^{\to} (A)$ for all A ∈ Δ_{Γ
_L} (X).

Example 3.7. Let (X, e) be a fuzzy poset. (1) When $Z_{L} (X) = P_{L} (X)$ and $Γ_{L} (X) = P_{L} (X)$ , a fuzzy subset A ∈ L^X is $Δ_{P_{L}}$ -closed iff A = A^δ by A ⊆ A and Proposition 3.3. Moreover, $Δ_{P_{L}} (X) = {A \in L^{X} | A = A^{δ}} = {DM}_{L} (X)$ is exactly the Dedekind-Macneille completion of the fuzzy poset (X, e).

(2) When Γ_L (X) = Φ_L (X), where Φ_L (X) =∅, we have that A ∈ L^X is Δ_{Φ
_L}-closed iff A = ↓ A. In this case, Δ_{Φ
_L} (X) = {A ∈ L^X | A = ↓ A} is the Δ_{Φ
_L}-completion of (X, e).

4 The Z_{Γ
_L}-completion of fuzzy posets

To go further, the fuzzy subset systems Z_L we consider in the following part of this paper are required to be fuzzy subset-hereditary. In this section, we construct the smallest Z_{Γ
_L}-completion in Δ_{Γ
_L} (X) and discuss its universal property.

Definition 4.1. A mapping f : X ⟶ Y between two fuzzy posets (X, e_X) and (Y, e_Y) is called a Δ_{Γ
_L}-continuous mapping if

(1) f is a fuzzy order-preserving;

(2) ∀D ∈ Γ_L (X), (f^→ (D)) ^δ = (f^→ (D^δ)) ^δ.

Proposition 4.2. Let f : X ⟶ Y be a mapping between two fuzzy posets (X, e_X) and (Y, e_Y). Consider the following conditions:

(1) f is Δ_{Γ
_L}-continuous;

(2) f^← (A) ∈ Δ_{Γ
_L} (X) for all A ∈ Δ_{Γ
_L} (Y);

(3) f is a fuzzy order-preserving and ⊔f^→ (D) = ⊔ f^→ (D^δ) for all D ∈ Γ_L (X);

(4) f is a fuzzy order-preserving and ⊔f^→ (D) = f (⊔ D) for all D ∈ Γ_L (X).

Then (1) ⇔ (2), (1) ⇔ (2) ⇔ (3) when (Y, e_Y) is Z_L-complete, and (1) ⇔ (2) ⇔ (3) ⇔ (4) when (X, e_X) is Γ_L-complete.

Proof. (1)⇒(2): Suppose that f is Δ_{Γ
_L}-continuous, A ∈ Δ_{Γ
_L} (Y), and D ∈ Γ_L (X), then f^← (A) is a fuzzy lower subset of X, f^→ (D) ∈ Δ_{Γ
_L} (Y) and $\begin{matrix} sub (D, f^{\leftarrow} (A)) & \leq {sub}_{Y} (f^{\to} (D), f^{\to} (f^{\leftarrow} (A))) \\ \leq {sub}_{Y} (f^{\to} (D), A) \\ \leq {sub}_{Y} ((f^{\to} (D))^{δ}, A) \\ = {sub}_{Y} ((f^{\to} (D^{δ}))^{δ}, A) \\ \leq {sub}_{Y} (f^{\to} (D^{δ}), A) \\ \leq sub (f^{\leftarrow} (f^{\to} (D^{δ})), f^{\leftarrow} (A)) \\ \leq sub (D^{δ}, f^{\leftarrow} (A)) . \end{matrix}$ Therefore f^← (A) ∈ Δ_{Γ
_L} (X).

(2)⇒(1): Step 1. We show that f is a fuzzy order-preserving. In fact, for any x, x′ ∈ X, (χ_f(x′)) ^δ = ι_f(x′) ∈ Δ_{Γ
_L} (Y). And so f^← (ι_f(x′)) ∈ Δ_{Γ
_L} (X) and f^← (ι_f(x′)) (x′) =1. Therefore e_X (x, x′) = e_X (x, x′) ∗ f^← (ι_f(x′)) (x′) ≤ f^← (ι_f(x′)) (x) = e_Y (f (x) , f (x′)) .

Step 2. We show that ∀D ∈ Γ_L (X), (f^→ (D)) ^δ = (f^→ (D^δ)) ^δ. Obviously, (f^→ (D)) ^δ ⊆ (f^→ (D^δ)) ^δ since D ⊆ D^δ. On the other hand, for any D ∈ Γ_L (X) we get f^→ (D) ∈ Z_L (Y) and (f^→ (D)) ^δ ∈ Δ_{Γ
_L} (Y) by Proposition 3.3. Then f^← ((f^→ (D)) ^δ) ∈ Δ_{Γ
_L} (X). Moreover, D ⊆ f^← (f^→ (D)) ⊆ f^← ((f^→ (D)) ^δ) and so D^δ ⊆ (f^← ((f^→ (D)) ^δ)) ^δ = f^← ((f^→ (D)) ^δ). Therefore f^→ (D^δ) ⊆ f^→ (f^← ((f^→ (D)) ^δ)) ⊆ (f^→ (D)) ^δ and thus (f^→ (D^δ)) ^δ ⊆ (f^→ (D)) ^δ) ^δ ⊆ (f^→ (D)) ^δ.

Suppose that (Y, e_Y) is Z_L-complete. Obviously (3) ⇒ (1) since (f^→ (D)) ^δ = ι_⊔f^→(D) = ι_{⊔f^→(D^δ)} = (f^→ (D^δ)) ^δ for all D ∈ Γ_L (X). Conversely, let f be a Δ_{Γ
_L}-continuous and D ∈ Γ_L (X), then f^→ (D) ∈ Z_L (Y) and ⊔f^→ (D) exists in Y since (Y, e_Y) is Z_L-complete. Hence (f^→ (D^δ)) ^δ = (f^→ (D)) ^δ = ι_⊔f^→(D) and so ⊔f^→ (D^δ) = ⊔ f^→ (D) by Proposition 2.8. This implies that (1) ⇒ (3), and hence (1) ⇔ (2) ⇔ (3).

Let (X, e_X) be a Γ_L-complete fuzzy poset and D ∈ Γ_L (X), then ⊔D exists and D^δ = ι_⊔D by Proposition 2.8. Suppose that (4) is satisfied, then f is a fuzzy order-preserving and ⊔f^→ (D) = f (⊔ D) = ⊔ f^→ (ι_⊔D) = ⊔ f^→ (D^δ). Thus (4) ⇒ (3). As for the proof of (3) ⇒ (1), it is essentially the same as that of the above paragraph. Now suppose that (1) is true, then for any D ∈ Γ_L (X) and y, z ∈ Y, one has $\begin{matrix} f^{\to} (D) (y) & = ⋁_{x \in X} D (x) * e_{Y} (y, f (x)) \\ \leq ⋁_{x \in X} e_{X} (x, ⊔ D) * e_{Y} (y, f (x)) \\ \leq ⋁_{x \in X} e_{Y} (f (x), f (⊔ D)) * e_{Y} (y, f (x)) \\ \leq e_{Y} (y, f (⊔ D)) \end{matrix}$ and $\begin{matrix} ⋀_{y \in Y} f^{\to} (D) (y) \to e_{Y} (y, z) = ⋀_{x \in X} D (x) \to e_{Y} (f (x), z) \\ = ⋀_{x \in X} D (x) \to f^{\leftarrow} (ι_{z}) (x) = sub (D, f^{\leftarrow} (ι_{z})) \\ \leq sub (D^{δ}, f^{\leftarrow} (ι_{z})) (f^{\leftarrow} (ι_{z}) is Δ_{Γ_{L}} - closed by (1) \Leftrightarrow (2)) \\ = sub (ι_{⊔ D}, f^{\leftarrow} (ι_{z})) \leq sub (f^{\to} (ι_{⊔ D}), f^{\to} (f^{\leftarrow} (ι_{z}))) \\ \leq sub (f^{\to} (ι_{⊔ D}), ι_{z}) = ⋀_{x \in X} e (x, ⊔ D) \to e_{Y} (f (x), z) \\ = e_{Y} (f (⊔ D), z) . \end{matrix}$

Thus f (⊔ D) = ⊔ f^→ (D). Therefore (1) ⇔ (4).

Remark 4.3. We denote “Δ_{Z
_L}” in the case that Γ_L (X) is chosen to be Z_L (X). Then a Δ_{Z
_L}-continuous mapping f : X ⟶ Y between Z_L-complete fuzzy posets is exactly a Z_⊔-continuous mapping by Proposition 4.2 (4). Moreover, A ∈ Δ_{Z
_L} (X) iff A = ↓ A and for any D ∈ Z_L (X), sub (D, A) ≤ A (⊔ D) iff A = ↓ A and A is Z_L-closed.

Lemma 4.4. Let f : X ⟶ Y be a Δ_{Γ
_L}-continuous mapping between fuzzy posets where (Y, e_Y) is Z_L-complete, and A ∈ L^X, then ⊔f^→ (A) exists iff ⊔f^→ (C_{Δ
_{Γ
_L}} (A)) exists, and both imply ⊔f^→ (A) = ⊔ f^→ (C_{Δ
_{Γ
_L}} (A)).

Proof. It suffices to show that for any y₀ ∈ Y is an upper bound of f^→ (A) iff y₀ is an upper bound of f^→ (C_{Δ
_{Γ
_L}} (A)). It is obvious that every upper bound of f^→ (C_{Δ
_{Γ
_L}} (A)) is also an upper bound of f^→ (A) since f^→ (A) ⊆ f^→ (C_{Δ
_{Γ
_L}} (A)). Now suppose that y₀ ∈ Y is an upper bound of f^→ (A). Let A′ (x) = e_Y (f (x) , y₀) ∧ C_{Δ
_{Γ
_L}} (A) (x) for any x ∈ X. Then y₀ is an upper bound of f^→ (A′) since $\begin{matrix} f^{\to} (A^{'}) (y) = ⋁_{x \in X} A^{'} (x) * e_{Y} (y, f (x)) \\ = ⋁_{x \in X} (e_{Y} (f (x), y_{0}) \land C_{Δ_{Γ_{L}}} (A) (x)) * e_{Y} (y, f (x)) \\ \leq ⋁_{x \in X} (e_{Y} (f (x), y_{0}) * e_{Y} (y, f (x))) \land (C_{Δ_{Γ_{L}}} (A) (x) \\ * e_{Y} (y, f (x))) \\ \leq ⋁_{x \in X} e_{Y} (y, y_{0}) \land (C_{Δ_{Γ_{L}}} (A) (x) * e_{Y} (y, f (x))) \\ \leq e_{Y} (y, y_{0}) . \end{matrix}$ For any x, x′ ∈ X, we have $\begin{matrix} C_{Δ_{Γ_{L}}} (A) (x) \geq A^{'} (x) = e_{Y} (f (x), y_{0}) \land C_{Δ_{Γ_{L}}} (A) (x) \\ \geq f^{\to} (A) (f (x)) \land C_{Δ_{Γ_{L}}} (A) (x) \\ = (⋁_{x^{'} \in X} A (x^{'}) * e_{Y} (f (x), f (x^{'}))) \land C_{Δ_{Γ_{L}}} (A) (x) \\ \geq (⋁_{x^{'} \in X} A (x^{'}) * e_{X} (x, x^{'})) \land C_{Δ_{Γ_{L}}} (A) (x) \\ \geq A (x) \land C_{Δ_{Γ_{L}}} (A) (x) = A (x) \end{matrix}$ and $\begin{matrix} A^{'} (x) * e (x^{'}, x) = (e_{Y} (f (x), y_{0}) \land C_{Δ_{Γ_{L}}} (A) (x)) * e (x^{'}, x) \\ \leq (e_{Y} (f (x), y_{0}) * e (x^{'}, x)) \land (C_{Δ_{Γ_{L}}} (A) (x)) * e (x^{'}, x)) \\ \leq (e_{Y} (f (x), y_{0}) * e_{Y} (f (x^{'}), f (x))) \land C_{Δ_{Γ_{L}}} (A) (x^{'}) \\ \leq e_{Y} (f (x^{'}), y_{0}) \land C_{Δ_{Γ_{L}}} (A) (x^{'}) = A^{'} (x^{'}), \end{matrix}$ which implies A′ = ↓ A′. Moreover, for any D ∈ Γ_L (X), one has $\begin{matrix} sub (D^{δ}, A^{'}) = ⋀_{x \in X} D^{δ} (x) \to A^{'} (x) \\ = ⋀_{x \in X} D^{δ} (x) \to (e_{Y} (f (x), y_{0}) \land C_{Δ_{Γ_{L}}} (A) (x)) \end{matrix}$ $\begin{matrix} = ⋀_{x \in X} (D^{δ} (x) \to e_{Y} (f (x), y_{0})) \land (D^{δ} (x) \\ \to C_{Δ_{Γ_{L}}} (A) (x)) \\ \geq e_{Y} (⊔ f^{\to} (D^{δ}), y_{0}) \land sub (D^{δ}, C_{Δ_{Γ_{L}}} (A)) \\ = e_{Y} (⊔ f^{\to} (D), y_{0}) \land sub (D^{δ}, C_{Δ_{Γ_{L}}} (A)) \\ = (⋀_{x \in X} D (x) \to e_{Y} (f (x), y_{0})) \land sub (D^{δ}, C_{Δ_{Γ_{L}}} (A)) \\ \geq (⋀_{x \in X} D (x) \to A^{'} (x)) \land sub (D^{δ}, C_{Δ_{Γ_{L}}} (A)) \\ \geq sub (D, A^{'}) \land sub (D, C_{Δ_{Γ_{L}}} (A)) = sub (D, A^{'}) . \end{matrix}$ Hence A′ ∈ Δ_{Γ
_L} (X) and thus C_{Δ
_{Γ
_L}} (A) ⊆ A′. Therefore C_{Δ
_{Γ
_L}} (A) = A′ and so f^→ (C_{Δ
_{Γ
_L}} (A)) = f^→ (A′), which completes the proof.

Proposition 4.5. For any fuzzy poset (X, e), $\tilde{Ψ (X)} = ⋂ {B \subseteq Δ_{Γ_{L}} (X) | Ψ (X) \subseteq B and B is a Z_{L} - complete fuzzy subposet of Δ_{Γ_{L}} (X)}$ with the fuzzy inclusion order sub is a Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X).

Proof. Let $D \in Z_{L} (\tilde{Ψ (X)})$ and $B$ be a Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X) with $Ψ (X) \subseteq B \subseteq Δ_{Γ_{L}} (X)$ , then $⊔ i_{1}^{\to} (D) = ⊔ i_{B}^{\to} (i_{1}^{\to} (D))$ , where i₁ is the inclusion mapping from Ψ (X) to $B$ and $i_{B}$ is the inclusion mapping from $B$ to Δ_{Γ
_L} (X). Moreover, for any A ∈ Δ_{Γ
_L} (X), $\begin{matrix} i_{B}^{\to} (i_{1}^{\to} (D)) (A) = ⋁_{B \in B} i_{1}^{\to} (D) (B) * sub (A, B) \\ = ⋁_{B \in B} ⋁_{B^{'} \in Ψ (X)} D (B^{'}) * sub (B, B^{'}) * sub (A, B) \\ = ⋁_{B^{'} \in Ψ (X)} D (B^{'}) * sub (A, B^{'}) = i^{\to} (D)) (A), \end{matrix}$ where i is the inclusion mapping from Ψ (X) to Δ_{Γ
_L} (X). Hence $⊔ i^{\to} (D)) = ⊔ i_{1}^{\to} (D) \in B$ , which implies that $⊔ i^{\to} (D)) \in \tilde{Ψ (X)}$ by the arbitrariness of $B$ . Therefore $(\tilde{Ψ (X)}, sub)$ is a Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X).

By Proposition 4.5, we have that $(\tilde{Ψ (X)}, sub)$ is the smallest Z_L-complete fuzzy subposet containing Ψ (X) in Δ_{Γ
_L} (X). Then a natural question is whether $(\tilde{Ψ (X)}, sub)$ has the corresponding universal property. To solve this problem, we give the following conclusions and definitions.

Proposition 4.6. Let (X, e) be a fuzzy poset and $η_{X} : X ⟶ \tilde{Ψ (X)}$ be defined by η_X (x) = ι_x for any x ∈ X. Then the mapping $η_{X} : X ⟶ \tilde{Ψ (X)}$ is Δ_{Γ
_L}-continuous.

Proof. Obviously η_X is a fuzzy order-preserving. By Proposition 4.2 and 4.5, we only need to show that $⊔ η_{X}^{\to} (D) = ⊔ η_{X}^{\to} (D^{δ})$ for any D ∈ Γ_L (X). Since $η_{X}^{\to} (D) \in Z_{L} (\tilde{Ψ (X)})$ , $⊔ η_{X}^{\to} (D)$ exists in $\tilde{Ψ (X)}$ . Moreover, any upper bound of $η_{X}^{\to} (D^{δ})$ is also an upper bound of $η_{X}^{\to} (D)$ by $η_{X}^{\to} (D) \subseteq η_{X}^{\to} (D^{δ})$ . Next, we show $⊔ η_{X}^{\to} (D)$ is an upper bound of $η_{X}^{\to} (D^{δ})$ . By Proposition 3.3 and 4.5, we have $⊔ η_{X}^{\to} (D) = ⊔ i^{\to} (η_{X}^{\to} (D))$ and $\begin{matrix} ⊔ i^{\to} (η_{X}^{\to} (D)) = C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} i^{\to} (η_{X}^{\to} (D)) (A) * A) \\ = C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} ⋁_{B \in \tilde{Ψ (X)}} η_{X}^{\to} (D) (B) * sub (A, B) * A) \\ = C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} ⋁_{B \in \tilde{Ψ (X)}} ⋁_{x \in X} D (x) * sub (B, ι_{x}) \\ * sub (A, B) * A) \\ = C_{Δ_{Γ_{L}}} (⋁_{x \in X} D (x) * ι_{x}) . \end{matrix}$ Thus $\begin{matrix} sub (D^{δ}, ⊔ i^{\to} (η_{X}^{\to} (D))) \geq sub (D, ⊔ i^{\to} (η_{X}^{\to} (D))) \\ = sub (D, C_{Δ_{Γ_{L}}} (⋁_{x \in X} D (x) * ι_{x})) \\ = ⋀_{x^{'} \in X} D (x^{'}) \to (⋀_{A \in Δ_{Γ_{L}} (X)} (sub (⋁_{x \in X} D (x) * ι_{x}, A) \\ \to A (x^{'}))) \\ = ⋀_{x^{'} \in X} D (x^{'}) \to (⋀_{A \in Δ_{Γ_{L}} (X)} (sub (D, A) \to A (x^{'}))) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} sub (D, A) \to (⋀_{x^{'} \in X} D (x^{'}) \to A (x^{'})) \\ = ⋀_{A \in Δ_{Γ_{L}} (X)} sub (D, A) \to sub (D, A) = 1 \end{matrix}$ and $\begin{matrix} sub (⊔ i^{\to} (η_{X}^{\to} (D)), D^{δ}) = ⋀_{A \in \tilde{Ψ (X)}} η_{X}^{\to} (D) (A) \to sub (A, D^{δ}) \\ = ⋀_{A \in \tilde{Ψ (X)}} ⋀_{x^{'} \in X} D (x^{'}) * sub (A, ι_{x^{'}}) \to sub (A, D^{δ}) \end{matrix}$ $\begin{matrix} \geq ⋀_{x^{'} \in X} D (x^{'}) \to sub (ι_{x^{'}}, D^{δ}) \\ = ⋀_{x^{'} \in X} D (x^{'}) \to D^{δ} (x^{'}) = 1 . \end{matrix}$ This means that $D^{δ} = ⊔ i^{\to} (η_{X}^{\to} (D))$ and thus $⊔ η_{X}^{\to} (D) = D^{δ} \in \tilde{Ψ (X)}$ . And because for any $A \in \tilde{Ψ (X)}$ , we have $\begin{matrix} η_{X}^{\to} (D^{δ}) (A) \to sub (A, D^{δ}) \\ = (⋁_{x \in X} D^{δ} (x) * sub (A, η_{X} (x))) \to sub (A, D^{δ}) \\ = ⋀_{x \in X} D^{δ} (x) * sub (A, ι_{x}) \to sub (A, D^{δ}) \\ = ⋀_{x \in X} D^{δ} (x) \to (sub (A, ι_{x}) \to sub (A, D^{δ})) \\ \geq ⋀_{x \in X} D^{δ} (x) \to sub (ι_{x}, D^{δ}) \\ = ⋀_{x \in X} D^{δ} (x) \to D^{δ} (x) = 1, \end{matrix}$ $η_{X}^{\to} (D^{δ}) (A) \leq sub (A, D^{δ})$ , which implies that D^δ is an upper bound of $η_{X}^{\to} (D^{δ})$ . This means that the proof completes.

Definition 4.7. For any fuzzy poset (X, e_X), a fuzzy subset A of X is called Δ_{Γ
_L}-continuously ⊔-existing (with respect to the fuzzy subset system Z_L) if for any Z_L-complete poset (Y, e_Y) and Δ_{Γ
_L}-continuous mapping f : X ⟶ Y, ⊔f^→ (A) always exists in Y. For the sake of future discussion, let Z_{Γ
_L} (X) = {A ∈ L^X | A is Δ_{Γ
_L} - continuously ⊔ - existing and Δ_{Γ
_L} - closed}.

It is obvious that all fuzzy Z_L-subsets are Δ_{Γ
_L}-continuously ⊔-existing. In particular, we have all fuzzy principal ideals are Δ_{Γ
_L}-continuously ⊔-existing and so Ψ (X) ⊆ Z_{Γ
_L} (X) ⊆ Δ_{Γ
_L} (X). However, a Δ_{Γ
_L}-continuously ⊔-existing fuzzy subset may fail to be a fuzzy Z_L-subset in general.

By Lemma 4.4 and Definition 4.7, we have the following result.

Remark 4.8. For any Δ_{Γ
_L}-continuous mapping f : X ⟶ Y between fuzzy posets where (Y, e_Y) is Z_L-complete, A ∈ L^X is Δ_{Γ
_L}-continuously ⊔-existing iff C_{Δ
_{Γ
_L}} (A) is Δ_{Γ
_L}-continuously ⊔-existing.

Proposition 4.9. The fuzzy poset (Z_{Γ
_L} (X) , sub) is a Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X).

Proof. Let $D \in Z_{L} (Z_{Γ_{L}} (X))$ , then we only need show that $⊔ i^{\to} (D)$ is Δ_{Γ
_L}-continuously ⊔-existing since ⊔i^→ (D) ∈ Δ_{Γ
_L} (X) is obvious. Let f : X ⟶ Y be a Δ_{Γ
_L}-continuous mapping into a Z_L-complete fuzzy poset (Y, e_Y), then we define $\hat{f} : Z_{Γ_{L}} (X) ⟶ Y$ by $\hat{f} (A) = ⊔ f^{\to} (A)$ for any A ∈ Z_{Γ
_L} (X) and thus $\hat{f}$ is well defined. By the fuzzy order-preserving of f, it follows that $\hat{f}$ is a fuzzy order-preserving since $\begin{matrix} sub (A, B) & \leq sub (f^{\to} (A), f^{\to} (B)) \\ \leq e_{Y} (⊔ f^{\to} (A), ⊔ f^{\to} (B)) = e_{Y} (\hat{f} (A), \hat{f} (B)) \end{matrix}$ for any A, B ∈ Z_{Γ
_L} (X), and thus ${\hat{f}}^{\to} (D) \in Z_{L} (Y)$ and $⊔ {\hat{f}}^{\to} (D)$ exists in Y. Then for any y ∈ Y, $\begin{matrix} e_{Y} (⊔ {\hat{f}}^{\to} (D), y) = ⋀_{y^{'} \in Y} {\hat{f}}^{\to} (D) (y^{'}) \to e_{Y} (y^{'}, y) \\ = ⋀_{A \in Z_{Γ_{L}} (X)} D (A) \to e_{Y} (⊔ f^{\to} (A), y) \\ = ⋀_{A \in Z_{Γ_{L}} (X)} D (A) \to (⋀_{x \in X} A (x) \to e_{Y} (f (x), y)) \\ = ⋀_{A \in Z_{Γ_{L}} (X)} ⋀_{x \in X} D (A) * A (x) \to e_{Y} (f (x), y) \\ = ⋀_{y^{'} \in Y} f^{\to} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A) (y) \to e_{Y} (f (x), y), \end{matrix}$ which implies that $⊔ f^{\to} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A) = ⊔ {\hat{f}}^{\to} (D) \in Y$ . Thus $⊔ f^{\to} (C_{Δ_{Γ_{L}}} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A))$ exists in Y and $⊔ f^{\to} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A) = ⊔ f^{\to} (C_{Δ_{Γ_{L}}} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A))$ by Lemma 4.4. Notice $⊔ i^{\to} (D) = C_{Δ_{Γ_{L}}} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A)$ . Hence $⊔ {\hat{f}}^{\to} (D) = ⊔ f^{\to} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A) = ⊔ f^{\to} (C_{Δ_{Γ_{L}}} (⋁_{A \in Z_{Γ_{L}} (X)} D (A) * A)) = ⊔ f^{\to} (⊔ i^{\to} (D)),$ which implies that $⊔ f^{\to} (⊔ i^{\to} (D))$ exists in Y. Thus $⊔ i^{\to} (D)$ is Δ_{Γ
_L}-continuously ⊔-existing. Therefore (Z_{Γ
_L} (X) , sub) is a Z_L-complete fuzzy subposet of Δ_{Γ
_L} (X).

Since $\tilde{Ψ (X)}$ is the smallest Z_L-complete fuzzy poset containing Ψ (X) in Δ_{Γ
_L} (X), $Ψ (X) \subseteq \tilde{Ψ (X)} \subseteq Z_{Γ_{L}} (X) \subseteq Δ_{Γ_{L}} (X)$ by Proposition 4.9. This means that every $A \in \tilde{Ψ (X)}$ is Δ_{Γ
_L}-continuously ⊔-existing.

Theorem 4.10. The $η_{X} : X ⟶ \tilde{Ψ (X)}$ is universal in the following sense: for any Δ_{Γ
_L}-continuous mapping f : X ⟶ Y into a Z_L-complete fuzzy poset (Y, e_Y) there is a unique Z_⊔-continuous mapping $\hat{f} : \tilde{Ψ (X)} ⟶ Y$ such that $f = \hat{f} \circ η_{X}$ , i.e., the following diagram commutes:

Proof. Let f : X ⟶ Y be a Δ_{Γ
_L}-continuous mapping into a Z_L-complete fuzzy poset (Y, e_Y). We define $\hat{f} : \tilde{Ψ (X)} ⟶ Y$ by $\hat{f} (A) = ⊔ f^{\to} (A)$ for any $A \in \tilde{Ψ (X)}$ . The mapping $\hat{f}$ is well defined since every $A \in \tilde{Ψ (X)}$ is Δ_{Γ
_L}-continuously ⊔-existing. We shall show that $\hat{f} (A)$ is Z_⊔-continuous. Obviously, $\hat{f}$ is a fuzzy order-preserving by the proof Proposition 4.9. Moreover, for any $D \in Z_{L} (\tilde{Ψ (X)})$ , we have $\begin{matrix} \hat{f} (⊔ D) & = \hat{f} (⊔ i^{\to} (D)) \\ = \hat{f} (C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} i^{\to} (D) (A) * A)) \\ = ⊔ f^{\to} (C_{Δ_{Γ_{L}}} (⋁_{A \in Δ_{Γ_{L}} (X)} i^{\to} (D) (A) * A)) \\ = ⊔ f^{\to} (⋁_{A \in Δ_{Γ_{L}} (X)} i^{\to} (D) (A) * A) \\ = ⊔ f^{\to} (⋁_{A \in \tilde{Ψ (X)}} D (A) * A) = ⊔ {\hat{f}}^{\to} (D) . \end{matrix}$

The proof for the last equation above is similar to the proof Proposition 4.9. Hence $\hat{f}$ is a Z_⊔-continuous mapping. Obviously $\hat{f} \circ η_{X} (x) = \hat{f} (ι_{x}) = ⊔ f^{\to} (ι_{x})$ . Moreover ⊔f^→ (ι_x) = f (x). In fact, for any y ∈ Y,

$\begin{matrix} ⋀_{z \in Y} f^{\to} (ι_{x}) (z) \to e_{Y} (z, y) \\ = ⋀_{x^{'} \in X} e (x^{'}, x) \to e_{Y} (f (x^{'}), y) \\ \geq ⋀_{x^{'} \in X} e_{Y} (f (x^{'}), f (x)) \to e_{Y} (f (x^{'}), y) \geq e_{Y} (f (x), y) \end{matrix}$ and $\begin{matrix} ⋀_{z \in Y} f^{\to} (ι_{x}) (z) \to e_{Y} (z, y) \\ = ⋀_{x^{'} \in X} e (x^{'}, x) \to e_{Y} (f (x^{'}), y) \\ \leq e (x, x) \to e_{Y} (f (x), y) = e_{Y} (f (x), y) . \end{matrix}$ Therefore $f = \hat{f} \circ η_{X}$ . Next, we show that the uniqueness of $\hat{f}$ . Let $g : \tilde{Ψ (X)} ⟶ Y$ be a Z_⊔-continuous mapping such that f = g ∘ η_X and $B = {A \in \tilde{Ψ (X)} | g (A) = \hat{f} (A)}$ . Since $g \circ η_{X} (x) = \hat{f} \circ η_{X} (x)$ for all x ∈ X, $η_{X} (X) = Ψ (X) \subseteq B \subseteq \tilde{Ψ (X)}$ . If we show that $B$ is a Z_L-complete fuzzy subposet of $\tilde{Ψ (X)}$ , then $B \supseteq \tilde{Ψ (X)}$ since $\tilde{Ψ (X)}$ is the smallest Z_L-complete fuzzy poset containing Ψ (X) in Δ_{Γ
_L} (X) and so $B = \tilde{Ψ (X)}$ , which implies that $g = \hat{f}$ . In fact, for any $D \in Z_{L} (B)$ , $⊔ i^{\to} (D)$ exists in $\tilde{Ψ (X)}$ , where i is the inclusion mapping from $B$ to $\tilde{Ψ (X)}$ . Moreover, since $\begin{matrix} e_{Y} (g (⊔ i^{\to} (D)), \hat{f} (⊔ i^{\to} (D))) = e_{Y} (⊔ g (i^{\to} (D)), \\ ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ = ⋀_{y \in Y} g (i^{\to} (D)) (y) \to e_{Y} (y, ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ = ⋀_{A \in \tilde{Ψ (X)}} i^{\to} (D) (A) \to e_{Y} (g (A), ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ = ⋀_{A \in \tilde{Ψ (X)}} ⋀_{A^{'} \in B} D (A^{'}) * sub (A, A^{'}) \to e_{Y} (g (A), \\ ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ \geq ⋀_{A \in \tilde{Ψ (X)}} ⋀_{A^{'} \in B} D (A^{'}) * e_{Y} (g (A), g (A^{'})) \to e_{Y} (g (A), \\ ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ = ⋀_{A \in \tilde{Ψ (X)}} ⋀_{A^{'} \in B} D (A^{'}) \to (e_{Y} (g (A), g (A^{'})) \\ \to e_{Y} (g (A), ⊔ {\hat{f}}^{\to} (i^{\to} (D)))) \\ \geq ⋀_{A^{'} \in B} D (A^{'}) \to e_{Y} (g (A^{'}), ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ = ⋀_{A^{'} \in B} D (A^{'}) \to e_{Y} (\hat{f} (A^{'}), ⊔ {\hat{f}}^{\to} (i^{\to} (D))) \\ \geq ⋀_{A^{'} \in B} D (A^{'}) \to {\hat{f}}^{\to} (i^{\to} (D)) (\hat{f} (A^{'})) \\ = ⋀_{A^{'} \in B} D (A^{'}) \to (⋁_{A^{″} \in \tilde{Ψ (X)}} (i^{\to} (D) (A^{″}) \\ * e_{Y} (\hat{f} (A^{'}), \hat{f} (A^{″}))) \end{matrix}$ $\geq ⋀_{A^{'} \in B} D (A^{'}) \to i^{\to} (D) (A^{'}) = 1,$

$g (⊔ i^{\to} (D)) \leq \hat{f} (⊔ i^{\to} (D))$ . By the same way, we can check that $g (⊔ i^{\to} (D)) \supseteq \hat{f} (⊔ i^{\to} (D))$ , and thus $g (⊔ i^{\to} (D)) = \hat{f} (⊔ i^{\to} (D))$ , which implies $⊔ i^{\to} (D) \in B$ . Therefore $B$ is a Z_L-complete fuzzy subposet of $\tilde{Ψ (X)}$ . The proof is complete.

Definition 4.11. For a fuzzy poset (X, e), we call $(\tilde{Ψ (X)}, η_{X})$ is a Z_{Γ
_L}-completion of (X, e).

By Theorem 4.10, we have that the Z_{Γ
_L}-completion $(\tilde{Ψ (X)}, η_{X})$ has the following universal property: for any Δ_{Γ
_L}-continuous mapping f : X ⟶ Y into a Z_L-complete fuzzy poset (Y, e_Y), there is a unique Z_⊔-continuous mapping $\hat{f} : \tilde{Ψ (X)} ⟶ Y$ such that $f = \hat{f} \circ η$ . To better understand the structure of the elements in $\tilde{Ψ (X)}$ , we naturally want to know whether $\tilde{Ψ (X)}$ is exactly Z_{Γ
_L} (X).

Lemma 4.12. A ∈ L^X is Δ_{Γ
_L}-continuously ⊔-existing iff $η_{X}^{\to} (A)$ is Δ_{Z
_L}-continuously ⊔-existing in $\tilde{Ψ (X)}$ .

Proof. Let $η_{X}^{\to} (A)$ be a Δ_{Z
_L}-continuously ⊔-existing fuzzy subset of $\tilde{Ψ (X)}$ and f : X ⟶ Y be a Δ_{Γ
_L}-continuous mapping into a Z_L-complete fuzzy poset (Y, e_Y). By Theorem 4.10 and Remark 4.3, there is a unique Δ_{Z
_L}-continuous mapping $\hat{f} : \tilde{Ψ (X)} ⟶ Y$ such that $f = \hat{f} \circ η_{X}$ . Since $η_{X}^{\to} (A)$ is a Δ_{Z
_L}-continuously ⊔-existing, $⊔ {\hat{f}}^{\to} (η_{X}^{\to} (A))$ exists in Y. Then $⊔ f^{\to} (A) = ⊔ (\hat{f} \circ η_{X})^{\to} (A) = ⊔ {\hat{f}}^{\to} (η_{X}^{\to} (A))$ exists in Y, and thus A is Δ_{Γ
_L}-continuously ⊔-existing.

Conversely, let A be Δ_{Γ
_L}-continuously ⊔-existing and $g : \tilde{Ψ (X)} ⟶ Y$ be a Δ_{Z
_L}-continuous mapping into a Z_L-complete fuzzy poset (Y, e_Y). We first show that g ∘ η_X is Δ_{Γ
_L}-continuous. In fact, for any B ∈ Δ_{Z
_L} (Y), since g is Δ_{Z
_L}-continuous and η_X is Δ_{Γ
_L}-continuous, we have $g^{\leftarrow} (B) \in Δ_{Z_{L}} (\tilde{Ψ (X)})$ , and so $(g \circ η_{X})^{\leftarrow} (B) = η_{X}^{\leftarrow} (g^{\leftarrow} (B)) \in Δ_{Γ_{L}} (X)$ , which implies g ∘ η_X is Δ_{Γ
_L}-continuous by Proposition 4.2. Then $⊔ (g \circ η_{X})^{\to} (A) = ⊔ g^{\to} (η_{X}^{\to} (A))$ exists in Y and thus $η_{X}^{\to} (A)$ is Δ_{Z
_L}-continuously ⊔-existing.

Theorem 4.13. For any fuzzy poset (X, e), $Z_{Γ_{L}} (X) = \tilde{Ψ (X)}$ .

Proof. We only need to show that $Z_{Γ_{L}} (X) \subseteq \tilde{Ψ (X)}$ , i.e., $\forall A \in Z_{Γ_{L}} (X) \Rightarrow A \in \tilde{Ψ (X)}$ . In fact, for any A ∈ Z_{Γ
_L} (X), we have that $η_{X}^{\to} (A)$ is a Δ_{Z
_L}-continuously ⊔-existing fuzzy subset of $\tilde{Ψ (X)}$ by Lemma 4.12. Then $C_{Δ_{Z_{L}}} (η_{X}^{\to} (A))$ is also Δ_{Z
_L}-continuously ⊔-existing by Remark 4.8. Then $C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) \in Δ_{Γ_{L}} (\tilde{Ψ (X)})$ , and thus $⊔ C_{Δ_{Z_{L}}} (η_{X}^{\to} (A))$ exists. Moreover, for any x ∈ X, $\begin{matrix} ⊔ C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (x) \\ = C_{Δ_{Γ_{L}}} (⋁_{B \in \tilde{Ψ (X)}} C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) * B) (x) \\ \geq ⋁_{B \in \tilde{Ψ (X)}} C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) * B (x) \\ \geq ⋁_{B \in \tilde{Ψ (X)}} η_{X}^{\to} (A) (B) * B (x) \\ = ⋁_{B \in \tilde{Ψ (X)}} ⋁_{x^{'} \in X} A (x^{'}) * sub (B, ι_{x^{'}}) * B (x) \\ = ⋁_{x^{'} \in X} A (x^{'}) * ι_{x^{'}} (x) = A (x) . \end{matrix}$ Let $B (B) = C_{Δ_{Z_{L}}} (η X \to (A)) (B) \land sub (B, A), \forall B \in \tilde{Ψ (X)}$ , then $B \in L^{\tilde{Ψ (X)}}, ⊔ B \subseteq A$ and $B \subseteq C_{Δ_{Z_{L}}} (η X \to (A))$ . Next, we first show that $B \in Δ_{Z_{L}} (\tilde{Ψ (X)})$ . In fact, for any $B, C \in \tilde{Ψ (X)}$ , $\begin{matrix} B (B) * sub (C, B) \\ = (C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) \land sub (B, A)) * sub (C, B) \\ \leq (C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) * sub (C, B)) \land (sub (B, A) \\ * sub (C, B)) \\ \leq C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (C) \land sub (C, A)) = B (C) \end{matrix}$ and for any $D \in Z_{L} (\tilde{Ψ (X)})$ ,

$\begin{matrix} \tilde{e} (D^{δ}, B) = ⋀_{B \in \tilde{Ψ (X)}} D^{δ} (B) \to B (B) \\ = ⋀_{B \in \tilde{Ψ (X)}} D^{δ} (B) \to (C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) \land sub (B, A)) \\ = ⋀_{B \in \tilde{Ψ (X)}} (D^{δ} (B) \to C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B)) \land (D^{δ} (B) \\ \to sub (B, A)) \end{matrix}$ $\begin{matrix} \geq \tilde{e} (D^{δ}, C_{Δ_{Z_{L}}} (η_{X}^{\to} (A))) \land \tilde{e} (D^{δ}, ι_{A}) \\ \geq \tilde{e} (D, C_{Δ_{Z_{L}}} (η_{X}^{\to} (A))) \land \tilde{e} (D, ι_{A}) \\ = \tilde{e} (D, C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) \land ι_{A}) = \tilde{e} (D, B), \end{matrix}$ where $\tilde{e}$ is the fuzzy inclusion order on L^{L
^X}. Then we show that $η_{X}^{\to} (A) \subseteq B$ . In fact, for any $B \in \tilde{Ψ (X)}$ , $\begin{matrix} η_{X}^{\to} (A) (B) \to B (B) \\ = η_{X}^{\to} (A) (B) \to (C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B) \land sub (B, A)) \\ = (η_{X}^{\to} (A) (B) \to C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) (B)) \land \\ (η_{X}^{\to} (A) (B) \to sub (B, A)) \\ = η_{X}^{\to} (A) (B) \to sub (B, A) \\ = ⋀_{x \in X} A (x) \to sub (ι_{x}, A) \\ = ⋀_{x \in X} A (x) \to A (x) = 1 . \end{matrix}$ Hence $⊔ η_{X}^{\to} (A) \subseteq ⊔ B$ and $B \supseteq C_{Δ_{Z_{L}}} (η X \to (A))$ , and thus $B = C_{Δ_{Z_{L}}} (η X \to (A)) .$ To sum up, we have $A \subseteq ⊔ B = ⊔ C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) \subseteq A$ and so $⊔ C_{Δ_{Z_{L}}} (η_{X}^{\to} (A)) = A \in \tilde{Ψ (X)}$ .

By Example 3.7, we have the following examples of Z_{Γ
_L}-completion.

Example 4.14. Let (X, e) be a fuzzy poset.

(1) $Z_{P_{L}} (X) = Δ_{P_{L}} (X) = {A \in L^{X} | A = A^{δ}} = {DM}_{L} (X)$ is exactly the Dedekind-Macneille completion of the fuzzy poset (X, e).

(2) When Γ_L (X) = Φ_L (X), a mapping f : X ⟶ Y between fuzzy posets (X, e) and (Y, e_Y) is Δ_{Φ
_L}-continuous iff f is a fuzzy order-preserving. Let I_{Z
_L} (X) = {↓ D | D ∈ Z_L (X)}, then Ψ (X) ⊆ I_{Z
_L} (X) ⊆ Δ_{Φ
_L} (X). A fuzzy subset system Z_L is union-complete if for any $A \in Z_{L} (I_{Z_{L}} (X))$ , $⋁_{A \in} A (A) * A \in I_{Z_{L}} (X)$ . In this case, (I_{Z
_L} (X) , sub) is Z_L-complete and I_{Z
_L} (X) = Z_{Φ
_L} (X), which implies that I_{Z
_L} (X) is the Z_{Φ
_L}-completion of (X, e).

5 Conclusion

Fuzzy posets as the generalization of posets have received more and more attention. Just as the completion of posets is a very important topic in the classical poset theory, the completion of fuzzy posets deserves discussion in fuzzy poset theory. In this paper, we focused on the completion of fuzzy posets based on a fuzzy subset system Z_L and a fuzzy subset selection Γ_L. Firstly, we constructed the Δ_{Γ
_L}-completions Δ_{Γ
_L} (X) of a fuzzy poset (X, e) and showed that it is a join-completeion with universal property. Then in the case that Z_L is fuzzy subset-hereditary, we defined the Z_{Γ
_L}-completion of a fuzzy poset (X, e) to be the smallest Z_L-complete subposet containing all fuzzy principal ideals in Δ_{Γ
_L} (X). Moreover, in order to discuss the corresponding universal property of the Z_{Γ
_L}-completion, we introduced a special type of fuzzy subsets, called Δ_{Γ
_L}-continuously ⊔-existing subsets. The results show that the Z_{Γ
_L}-completion of a fuzzy poset (X, e) is exactly the set Z_{Γ
_L} (X) of all Δ_{Γ
_L}-continuously ⊔-existing subsets and has the universal property. As a special case, we obtained that the DM_L (X) and I_{Z
_L} (X) of a fuzzy poset (X, e) are the Z_{Γ
_L}-completion.

Footnotes

Acknowledgement

The authors would like to thank very much for the support of the National Natural Science Foundation of China (No. 11861006).

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The Z L -completions of fuzzy posets

Abstract

Keywords

1 Introduction

2 Preliminaries

3 The Δ Γ L -completion of fuzzy posets

4 The Z Γ L -completion of fuzzy posets

5 Conclusion

Footnotes

Acknowledgement

References

3 The Δ_{Γ
_L}-completion of fuzzy posets

4 The Z_{Γ
_L}-completion of fuzzy posets