Free join- and meet-completions of L -ordered sets

Abstract

In this paper, the free join- and meet-completions of an L-ordered set are built and characterized. Then we give the definitions of Δ₁-directed objects and intermediate structures, and show that the Δ₁-directed object precisely is the Dedekind-MacNeille completion of the intermediate structure in the bounded L-lattice case.

Keywords

Fuzzy order Free join-completion Intermediate structure

1 Introduction

Completions play an important role in the theory of lattices and theoretical computer science. Given a poset P, in order to solve the lack of some completeness of P, we usually extend P to a corresponding completion preserving certain properties. Perhaps the best known completion is the Dedekind-MacNeille completion by cuts [19]. Meanwhile, many kinds of completions have been built and characterized, such as the free join-completion of posets [7]. A pair (Y, α) is called a free join-completion of the poset X if Y is a dcpo and α is an order-embedding from X into Y such that every element of Y is a directed join of elements from α (X) and every element of α (X) is a compact element. In [7], Gehrke and Priestley showed that the free join-completion of posets are useful in the construction of canonical extensions [5, 14] which play a functional role in completeness theorems for various extensions of classical logic such as modal logics [10].

In order to further enrich the theory of fuzzy logic, in this paper, we shall generalize the free join-completions of ordinary posets to L-ordered sets which may lay a foundation for the extension of fuzzy logic. Concretely, we shall prove that the free join-completion of an L-ordered set may be obtained as the L-ordered set of all L-ideals of the L-ordered set. Furthermore, we show that for an L-ordered set X the L-order induced on the union of its free join- and meet-completions in $𝔽_{⊔} (𝔽_{⊓} (X))$ is the same as the oneinduced in $𝔽_{⊓} (𝔽_{⊔} (X))$ . Finally, we give the definitionsof Δ₁-directed objects and intermediate structures, and prove that the Δ₁-directed object precisely is the Dedekind-MacNeille completion of the intermediate structure in the bounded L-lattice case.

2 Preliminaries

In this part, we recall some concepts and results employed in the rest of the paper. We refer to [9 , 13] for fuzzy set theory and to [4, 8] for lattice theory.

A frame is a complete lattice L satisfying the following infinite distributive law:

a ∧ (⋁ B) = ⋁ {a ∧ b ∣ b ∈ B} , ∀ a ∈ L, B ⊆ L.

In fact, a frame is exactly a complete Heyting algebra from the point of view of logic. Moreover, from the definition one can easily see that a frame is a strictly two-sided, unital, commutative quantale with ∧ as the tensor, and the top element ⊤ as the unit. In this paper, unless otherwise stated, L always denotes a frame. Let X be a set. L^X denotes the set of all L-subsets of X, that is, the set of all mappings from X to L.

Proposition 2.1. [2, 11] Suppose L is a frame. Then for all a, b, c ∈ L, {a_j} _j∈J ⊆ L, the following conditions hold:

(1) ⊤ → a = a;

(2) (a → b) ∧ (b → c) ≤ a → c;

(3) (⋁ _j∈Ja_j) → b = ⋀ _j∈J (a_j → b);

(4) a → (⋀ _j∈Ja_j) = ⋀ _j∈J (a → a_j);

(5) a → (b → c) = (a ∧ b) → c;

(6) a ≤ (a → b) → b.

Definition 2.2. [1, 6] A fuzzy order (or an L-order) on a set X is a mapping e : X × X ⟶ L such that for all x, y, z ∈ X,

(E1) e (x, x) =⊤;

(E2) e (x, y) ∧ e (y, z) ≤ e (x, z);

(E3) e (x, y) = e (y, x) =⊤ implies x = y.

The pair (X, e) is called an L-ordered set. We often write simply X for an L-ordered set (X, e) if there would be no confusion about the L-order e.

Example 2.3. (1) Let (X, e) be an L-ordered set. For all x, y ∈ X, let e^op (x, y) = e (y, x). Then (X, e^op) is also an L-ordered set, called the dual of X.

(2) Let (X, e) be an L-ordered set and Y ⊆ X. Then (Y, e ∣ _Y) is an L-ordered set, where e ∣ _Y is the restriction of e to Y × Y.

(3) Let X be a set. For all A, B ∈ L^X, the subsethood degree of A in B ([9]) is defined by sub_X (A, B) = ⋀ _x∈X (A (x) → B (x)) . Then (L^X, sub_X) is an L-ordered set [1].

Definition 2.4. [23] Let (X, e) be an L-ordered set, x₀ ∈ X and A ∈ L^X. The element x₀ is called a join (meet) of A, in symbols x₀ = ⊔ A (x₀ = ⊓ A), if for all x ∈ X,

(1) A (x) ≤ e (x, x₀) (A (x) ≤ e (x₀, x));

(2) ⋀_y∈X (A (y) → e (y, x)) ≤ e (x₀, x) (⋀ _y∈X (A (y) → e (x, y)) ≤ e (x, x₀)).

It is easy to verify by (E3) that if x₁, x₂ are two joins (or meets) of A, then x₁ = x₂. That is, each A ∈ L^X has at most one join (or meet).

Proposition 2.5. [1, 21] Let (X, e) be an L-ordered set and A ∈ L^X. Then x₀ = ⊔ A (x₀ = ⊓ A) iff for all x ∈ X, e (x₀, x) = ⋀ _y∈X (A (y) → e (y, x)) (e (x, x₀) = ⋀ _y∈X (A (y) → e (x, y))).

Definition 2.6. [1, 24] An L-ordered set (X, e) is called a completely lattice L-ordered set (or simply, complete L-lattice) if ⊔A and ⊓A exist for all A ∈ L^X.

Proposition 2.7. [1, 24] Let (X, e) be an L-ordered set. The following statements are equivalent:

(1) (X, e) is a complete L-lattice;

(2) for each A ∈ L^X, ⊔ A exists;

(3) for each A ∈ L^X, ⊓ A exists.

Proposition 2.8. [1 , 21] Let (X, e_X) , (Y, e_Y) be L-ordered sets. Then a mapping f : X ⟶ Y is said to be

(1) an L-order preserving mapping if e_X (x, y) ≤ e_Y (f (x) , f (y)) for all x, y ∈ X;

(2) an L-order embedding if e_X (x, y) = e_Y (f (x) , linebreak [4] f (y)) for all x, y ∈ X;

(3) an L-order isomorphism if it is an L-orderembedding which maps X onto Y. In this case, we say that X and Y are isomorphic.

Let (X, e) be an L-ordered set and A ∈ L^X. A is called a lower L-subset (upper L-subset) of X if A (x) ∧ e (y, x) ≤ A (y) (A (x) ∧ e (x, y) ≤ A (y)) for all x, y ∈ X. Let Low (X) , Upp (X) denote, respectively, the collection of all lower L-subsets of X, the collection of all upper L-subsets of X. Let (X, e) be an L-ordered set and x ∈ X. Define two mappings ι_x, μ_x : X ⟶ L by for all y ∈ X, ι_x (y) = e (y, x) , μ_x (y) = e (x, y). Clearly, we have ⊔ι_x = ⊓ μ_x = x for all x ∈ X. For A ∈ L^X, define ↓A, ↑ A ∈ L^X asfollows: for each y ∈ X,

\begin{array}{l} ↓ A (y) = \underset{x \in X}{⋁} A (x) \land e (y, x), \\ ↑ A (y) = \underset{x \in X}{⋁} A (x) \land e (x, y) . \end{array}

Proposition 2.9. [23] Let (X, e) be an L-ordered set. Then for all A ∈ L^X,

(1) ↓A ∈ Low (X) and ↑A ∈ Upp (X);

(2) if ⊔A exists, then ⊔ ↓ A exists and ⊔A = ⊔ ↓ A; if ⊓A exists, then ⊓ ↑ A exists and ⊓A = ⊓ ↑ A.

Definition 2.10. [16, 22] Let (X, e) be an L-ordered set and D ∈ L^X. D is called an up directed L-subset (or simply, directed L-subset) of X if

(1) ⋁_x∈XD (x) =⊤;

(2) for all x, y ∈ X, D (x) ∧ D (y) ≤ ⋁ _z∈XD (z) ∧ e (x, z) ∧ e (y, z).

Similarly, we can give the definition of down directed L-subsets of X.

A directed L-subset I is called an L-ideal, if I is also a lower L-subset. A down directed L-subset F is called an L-filter, if F is also an upper L-subset. Let $D_{L} (X), D_{L}^{↓} (X), I_{L} (X), F_{L} (X)$ denote, respectively, the collection of all directed L-subsets of X, downdirected L-subsets of X, L-ideals of X, L-filters of X. Clearly, $↓ D \in I_{L} (X)$ for all $D \in D_{L} (X)$ . For all $D \in D_{L} (X)$ , if the join of D exists, then we write the join by ⊔^↑D. For all $F \in D_{L}^{↓} (X)$ , if the meet of F exists, then we write the meet by ⊓^↓F.

Definition 2.11. [16, 22] Let (X, e) be an L-ordered set. (X, e) is called an L-dcpo if ⊔^↑D exists for all $D \in D_{L} (X)$ . (X, e) is called a dual L-dcpo if ⊓^↓F exists for all $F \in D_{L}^{↓} (X)$ .

Let (X, e) be an L-ordered set. Then $(I_{L} (X), {sub}_{X})$ is an L-dcpo.

Definition 2.12. [16, 22] Let (X, e_X) be an L-dcpo. For each x ∈ X, we define ⇓x ∈ L^X as follows:

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

For each x ∈ X, x is called an L-compact element of X if ⇓x (x) =⊤.

Let f : X ⟶ Y be a mapping from a set X to an L-ordered set (Y, e_Y). Then we define the forward fuzzy powerset operators $f^{\to}, f_{*}^{\to} : L^{X} ⟶ L^{Y}$ by for all A ∈ L^X, y ∈ Y, f^→ (A) (y) = ⋁_x∈X A (x) ∧ e_Y (y, f (x)), $f_{*}^{\to}$ (A) (y) = ⋁_x∈X A (x) ∧ e_Y (f (x) , y).

Proposition 2.13. [22] Let (X, e_X) , (Y, e_Y) be L-ordered sets. If f : X ⟶ Y is an L-order preserving mapping, then $f^{\to} (D) \in I_{L} (Y)$ for all $D \in D_{L} (X)$ .

Let (X, e_X) , (Y, e_Y) be L-ordered sets. A mapping f : X ⟶ Y is said to be join-preserving if itsatisfies f (⊔ A) = ⊔ f^→ (A) for all A ∈ L^X, and said to be meet-preserving if it satisfies $f (⊓ A) = ⊓ f_{*}^{\to} (A)$ for all A ∈ L^X. Similarly, f is called directed-join-preserving if it satisfies f (⊔ ^↑D) = ⊔ ^↑f^→ (D) for all $D \in D_{L} (X)$ , and called down-directed-meet-preserving if it satisfies $f (⊓^{↓} D) = ⊓^{↓} f_{*}^{\to} (D)$ for all $D \in D_{L}^{↓} (X)$ .

Remark 2.14. Every directed-join-preserving mapping is L-order preserving.

Let A ∈ L^X. The support set of A is the subset {x ∈ X|A (x) ≠0}. A is said to be of finite support if its support set is finite, in this case we say A is a finite L-subset. If the support set of A is the empty set then A is denoted by $\bar{0}$ .

Proposition 2.15. [25] Let (X, e) be an L-ordered set. Then (X, e) is a join L-semilattice iff each finite L-subset has a join; (X, e) is a meet L-semilattice iff each finite L-subset has a meet.

Definition 2.16. [25] An L-ordered set is called an L-lattice if it is simultaneously a join L-semilattice and a meet L-semilattice.

If X is an L-ordered set such that $⊔ \bar{0}$ exists, denote $0 = ⊔ \bar{0}$ , then e (0, x) =⊤ for all x ∈ X and I (0) =⊤ for all $I \in I_{L} (X)$ . Similarly, if X is an L-ordered set such that $⊓ \bar{0}$ exists, denote $1 = ⊓ \bar{0}$ , then e (x, 1) =⊤ for all x ∈ X and F (1) =⊤ for all $F \in F_{L} (X)$ .

Definition 2.17. [25] An L-lattice is said to be bounded if both $⊔ \bar{0}$ and $⊓ \bar{0}$ exist.

Let (X, e) be an L-ordered set. Define a binary relation ≤ on X as follows: ∀ x, y ∈ X, x ≤ y ifandonlyif e (x, y) = ⊤ . Then (X, ≤) is a poset.

Proposition 2.18. [18] Let (X, e) be an L-ordered set, A ⊆ X. $X_{A}$ is the characteristic function of A, that is $X_{A} (x) = ⊤$ if x ∈ A, otherwise $X_{A} (x) = ⊥$ . If $\sup X_{A}$ exists, then ⋁A exists in (X, ≤) and $⋁ A = \sup X_{A}$ . If $\inf X_{A}$ exists, then ⋀A exists in (X, ≤) and $⋀ A = \inf X_{A}$ .

Proposition 2.19. [18] If (X, e) is a join L-semilattice and $⊔ \bar{0}$ exists, then an L-lower set I ∈ L^X is an L-ideal iff I (0) =⊤ and I (x) ∧ I (y) ≤ I (x ∨ y) for all x, y ∈ X, where the join ∨ is computed in (X, ≤).

3 Free join-completions

Definition 3.1. Let (Y, e_Y) be an L-dcpo and X ⊆ Y. We say that Y is ⊔^↑-generated by X if for all y ∈ Y, there exists $A \in I_{L} (X)$ such that y = ⊔ ^↑i^→ (A) where i : X ⟶ Y is an inclusion mapping.

Example 3.2. Let Y = {y_i|i ∈ [0.4, 0.5]} ∪ {y₀}. Define e_Y : Y × Y ⟶ [0, 1] by e_Y (y₀, y₀) =1, e_Y (y₀, y_i) = i, e_Y (y_i, y₀) =0.5 and $e_{Y} (y_{i}, y_{j}) = {\begin{matrix} 1, & if i \leq j, \\ j, & if i > j, \end{matrix}$ for all i, j ∈ [0.4, 0.5]. Then (Y, e_Y) is an L-dcpo and Y is ⊔^↑-generated by {y₀, y_0.4}. In fact, y_i = ⊔ ^↑i^→ (A_i) for all i ∈ [0.4, 0.5] where A_i (y₀) = i, A_i (y_0.4) =1 and $A_{i} \in I_{L} ({y_{0}, y_{0.4}})$ , y₀ = ⊔ ^↑i^→ (A₀) where A₀ (y₀) =1, A₀ (y_0.4) =0.5 and $A_{0} \in I_{L} ({y_{0}, y_{0.4}})$ .

Definition 3.3. Let (X, e_X) be an L-ordered set. If (Y, e_Y) is an L-ordered set and α : X ⟶ Y is an L-order embedding, then we say that (Y, e_Y) is an extension of (X, e_X), denoted by ((Y, e_Y) , α).Such an extension is said a directed join-extension of (X, e_X) if Y is ⊔^↑-generated by α (X).

Consider Example 3.2, we have that ((Y, e_Y) , i) is a directed join-extension of ({y₀, y_0.4} , e_Y|_{{y₀,y_0.4}}) where i : {y₀, y_0.4} ⟶ Y is an inclusion mapping.

Definition 3.4. Let (X, e_X) be an L-ordered set. If ((Y, e_Y) , α) is a directed join-extension of (X, e_X) and (Y, e_Y) is an L-dcpo, then we say that ((Y, e_Y) , α) is a directed join-completion of (X, e_X). Such a completion is said to be compact if ⇓α (x) (α (x)) =⊤ for all x ∈ X, we shall refer to this compact directed join-completion as a free join-completion of X.

Example 3.5. Let X = {x₁, x₂}. Define e_X : X × X ⟶ [0, 1] by e_X (x₁, x₁) = e_X (x₂, x₂) =1, e_X (x₁, x₂) =0.5 and e_X (x₂, x₁) =0.4, then (X, e_X) is an L-ordered set. Let (Y, e_Y) be the L-ordered set given in Example 3.2. Define α : X ⟶ Y by α (x₁) = y_0.4 and α (x₂) = y₀. Then ((Y, e_Y) , α) is a free join-completion of (X, e_X).

Lemma 3.6. Let (X, e_X) be an L-ordered set. Then ι_x is an L-compact element of $I_{L} (X)$ for all x ∈ X.

Theorem 3.7. Let (X, e_X) be an L-ordered set and let ((Y, e_Y) , α) be a directed join-completion of (X, e_X). Then the following statements are equivalent:

(1) ((Y, e_Y) , α) is a free join-completion of (X, e_X);

(2) whenever (Y′, e_Y′) is an L-dcpo and f : X ⟶ Y′ is an L-order preserving mapping, then there exists a unique directed-join-preserving mapping $\bar{f} : Y ⟶ Y^{'}$ such that $\bar{f} \circ α = f$ ;

(3) there exists an L-order isomorphism $η : Y ⟶ I_{L} (X)$ with η (α (x)) = ι_x for all x ∈ X.

Proof. (1) ⇒ (2): Suppose ((Y, e_Y) , α) is a free join-completion of (X, e_X). Define $\bar{f} : Y ⟶ Y^{'}$ by for all y ∈ Y,

$\bar{f} (y) = ⊔^{↑} f^{\to} (A_{y}),$

where A_y ∈ L^X and A_y (a) = e_Y (α (a) , y) for all a ∈ X. Define $A_{y}^{'} \in L^{α (X)}$ by $A_{y}^{'} (b) = e_{Y} (b, y)$ for all b ∈ α (X). Then $y = ⊔ i^{\to} (A_{y}^{'})$ , where i : α (X) ⟶ Y is an inclusion mapping. Clearly, $A_{y} (x) = A_{y}^{'} (α (x))$ for all x ∈ X. Since ((Y, e_Y) , α) is a directed join-completion of (X, e_X), there exists $I_{y} \in I_{L} (α (X))$ such y = ⊔ ^↑i^→ (I_y).

Step 1. $\bar{f}$ is well defined. On the one hand, since ${sub}_{α (x)} (I_{y}, A_{y}^{'}) = ⊤$ , then $⋁_{b \in α (X)} A_{y}^{'} (b) \geq ⋁_{b \in α (X)} I_{y} (b) = ⊤$ . On the other hand, since ((Y, e_Y) , α) is a free join-completion of (X, e_X), then e_Y (b, ⊔ i^→ (I_y)) = i^→ (I_y) (b) for all b ∈ α (X). For all b₁, b₂ ∈ α (X), we have

$\begin{matrix} A_{y}^{'} (b_{1}) \land A_{y}^{'} (b_{2}) \\ = e_{Y} (b_{1}, y) \land e_{Y} (b_{2}, y) \\ = e_{Y} (b_{1}, ⊔ i^{\to} (I_{y})) \land e_{Y} (b_{2}, ⊔ i^{\to} (I_{y})) \\ = i^{\to} (I_{y}) (b_{1}) \land i^{\to} (I_{y}) (b_{2}) \\ = (⋁_{b_{3} \in α (X)} I_{y} (b_{3}) \land e_{Y} (b_{1}, b_{3})) \\ \land (⋁_{b_{4} \in α (X)} I_{y} (b_{4}) \land e_{Y} (b_{2}, b_{4})) \\ = ⋁_{b_{3} \in α (X)} ⋁_{b_{4} \in α (X)} (I_{y} (b_{3}) \land I_{y} (b_{4}) \\ \land e_{Y} (b_{1}, b_{3}) \land e_{Y} (b_{2}, b_{4})) \\ \leq ⋁_{b_{3} \in α (X)} ⋁_{b_{4} \in α (X)} ⋁_{b \in α (X)} (I_{y} (b) \land e_{Y} (b_{3}, b) \\ \land e_{Y} (b_{4}, b) \land e_{Y} (b_{1}, b_{3}) \land e_{Y} (b_{2}, b_{4})) \\ \leq ⋁_{b \in α (X)} (I_{y} (b) \land e_{Y} (b_{1}, b) \land e_{Y} (b_{2}, b)) \\ \leq ⋁_{b \in α (X)} (A_{y}^{'} (b) \land e_{Y} (b_{1}, b) \land e_{Y} (b_{2}, b)) . \end{matrix}$

These show $A_{y}^{'} \in D_{L} (α (X))$ . Clearly, $A_{y}^{'}$ is a lower L-subset, then $A_{y}^{'} \in I_{L} (α (X))$ . Since X ≅ α (X) and $A_{y} (x) = A_{y}^{'} (α (x))$ for all x ∈ X, then $A_{y} \in I_{L} (X)$ . Thus $f^{\to} (A_{y}) \in I_{L} (Y)$ and $\bar{f}$ is well defined.

Step 2. For all x ∈ X, y′ ∈ Y′,

$\begin{matrix} e_{Y^{'}} (⊔ f^{\to} (A_{α (x)}), y^{'}) \\ = ⋀_{y \in Y^{'}} f^{\to} (A_{α (x)}) (y) \to e_{Y^{'}} (y, y^{'}) \\ = ⋀_{y \in Y^{'}} (⋁_{a \in X} A_{α (x)} (a) \land e_{Y^{'}} (y, f (a))) \to e_{Y^{'}} (y, y^{'}) \\ = ⋀_{a \in X} A_{α (x)} (a) \to e_{Y^{'}} (f (a), y^{'}) \\ = ⋀_{a \in X} e_{Y} (α (a), α (x)) \to e_{Y^{'}} (f (a), y^{'}) \\ = ⋀_{a \in X} e_{X} (a, x) \to e_{Y^{'}} (f (a), y^{'}) \\ = e_{Y^{'}} (f (x), y^{'}) . \end{matrix}$

Then $(\bar{f} \circ α) (x) = \bar{f} (α (x)) = ⊔ f^{\to} (A_{α (x)}) = f (x)$ for all x ∈ X, hence $\bar{f} \circ α = f$ .

Step 3. We prove that $\bar{f} : Y ⟶ Y^{'}$ is a directed-join-preserving mapping. Since ((Y, e_Y) , α) is a free join-completion of (X, e_X), we have for all $D \in D_{L} (Y), y_{0} \in Y^{'}$ ,

$\begin{matrix} e_{Y^{'}} (\bar{f} (⊔ D), y_{0}) \\ = e_{Y^{'}} (⊔ f^{\to} (A_{⊔ D}), y_{0}) \\ = ⋀_{y^{'} \in Y^{'}} f^{\to} (A_{⊔ D}) (y^{'}) \to e_{Y^{'}} (y^{'}, y_{0}) \\ = ⋀_{x \in X} A_{⊔ D} (x) \to ⋀_{y^{'} \in Y^{'}} e_{Y^{'}} (y^{'}, f (x)) \to e_{Y^{'}} (y^{'}, y_{0}) \\ = ⋀_{x \in X} e_{Y} (α (x), ⊔ D) \to e_{Y^{'}} (f (x), y_{0}) \\ = ⋀_{x \in X} e_{Y} (α (x), ⊔ ↓ D) \to e_{Y^{'}} (f (x), y_{0}) \\ = ⋀_{x \in X} (↓ D) (α (x)) \to e_{Y^{'}} (f (x), y_{0}) \\ = ⋀_{x \in X} (⋁_{y \in Y} D (y) \land e_{Y} (α (x), y)) \to e_{Y^{'}} (f (x), y_{0}) \\ = ⋀_{y \in Y} D (y) \to (⋀_{x \in X} e_{Y} (α (x), y) \to e_{Y^{'}} (f (x), y_{0})) \\ = ⋀_{y \in Y} D (y) \to (⋀_{x \in X} A_{y} (x) \to ⋀_{y_{1} \in Y^{'}} (e_{Y^{'}} (y_{1}, f (x)) \\ \to e_{Y^{'}} (y_{1}, y_{0})) \\ = ⋀_{y \in Y} D (y) \to ⋀_{y_{1} \in Y^{'}} ((⋁_{x \in X} A_{y} (x) \land e_{Y^{'}} (y_{1}, f (x))) \\ \to e_{Y^{'}} (y_{1}, y_{0})) \\ = ⋀_{y \in Y} D (y) \to (⋀_{y_{1} \in Y^{'}} f^{\to} (A_{y}) (y_{1}) \to e_{Y^{'}} (y_{1}, y_{0})) \\ = ⋀_{y \in Y} D (y) \to e_{Y^{'}} (⊔ f^{\to} (A_{y}), y_{0}) \\ = ⋀_{y \in Y} D (y) \to e_{Y^{'}} (\bar{f} (y), y_{0}) \\ = ⋀_{y_{2} \in Y^{'}} {\bar{f}}^{\to} (D) (y_{2}) \to e_{Y^{'}} (y_{2}, y_{0}) \\ = e_{Y^{'}} (⊔ {\bar{f}}^{\to} (D), y_{0}) . \end{matrix}$

Hence $\bar{f} (⊔ D) = ⊔ {\bar{f}}^{\to} (D)$ .

Step 4. Suppose g : Y ⟶ Y′ is a directed-join-preserving mapping and g ∘ α = f. Then $g (y) = g (⊔ i^{\to} (A_{y}^{'})) = ⊔ (g \circ i)^{\to} (A_{y}^{'}) = ⊔ (g \circ α)^{\to} (A_{y}) = ⊔ f^{\to} (A_{y}) = \bar{f} (y)$ for all y ∈ Y. Thus the uniqueness of $\bar{f}$ is proved. (2) ⇒ (3): Define a mapping $f : X ⟶ I_{L} (X)$ by for all x ∈ X, f (x) = ι_x. Clearly, f is an L-order preserving mapping. Since $I_{L} (X)$ is an L-dcpo, then there exists a unique directed-join-preserving mapping $η : Y ⟶ I_{L} (X)$ such that η ∘ α = f. Define a mapping $σ : I_{L} (X) ⟶ Y$ by for all $I \in I_{L} (X)$ , σ (I) = ⊔ α^→ (I). Clearly, σ is well defined. For all $I, I_{1} \in I_{L} (X)$ , we have

$\begin{matrix} ⋀_{I_{2} \in I_{L} (X)} f^{\to} (I) (I_{2}) \to {sub}_{X} (I_{2}, I_{1}) \\ = ⋀_{x \in X} I (x) \to (⋀_{I_{2} \in I_{L} (X)} {sub}_{X} (I_{2}, f (x)) \\ \to {sub}_{X} (I_{2}, I_{1})) \\ = ⋀_{x \in X} I (x) \to {sub}_{X} (ι_{x}, I_{1}) \\ = ⋀_{x \in X} ⋀_{x_{0} \in X} (I (x) \land e_{X} (x_{0}, x) \to I_{1} (x_{0})) \\ = ⋀_{x_{0} \in X} (⋁_{x \in X} I (x) \land e_{X} (x_{0}, x)) \to I_{1} (x_{0}) \end{matrix}$

$\begin{matrix} = ⋀_{x_{0} \in X} I (x_{0}) \to I_{1} (x_{0}) \\ = {sub}_{X} (I, I_{1}) . \end{matrix}$

Then ⊔f^→ (I) = I, and thus (η ∘ σ) (I) = η (⊔ α^→ (I)) = ⊔ (η ∘ α) ^→ (I) = ⊔ f^→ (I) = I. This means that $η \circ σ = {id}_{I_{L} (X)}$ . For each y ∈ Y, there exists

$I_{y} \in I_{L} (α (X))$ such y = ⊔ ^↑i^→ (I_y). We can easily prove $⋁_{x \in X} I_{y} (α (x)) \land ι_{x} \in I_{L} (X)$ . For all $J_{1} \in I_{L} (X)$ ,

$\begin{matrix} ⋀_{J \in I_{L} (X)} (η \circ i)^{\to} (I_{y}) (J) \to {sub}_{X} (J, J_{1}) \\ = ⋀_{J \in I_{L} (X)} (⋁_{x \in X} I_{y} (α (x)) \land {sub}_{X} (J, (η \circ i) (α (x)))) \\ \to {sub}_{X} (J, J_{1}) \\ = ⋀_{x \in X} (I_{y} (α (x)) \to ⋀_{J \in I_{L} (X)} ({sub}_{X} (J, f (x)) \\ \to {sub}_{X} (J, J_{1}))) \\ = ⋀_{x \in X} I_{y} (α (x)) \to (⋀_{a \in X} ι_{x} (a) \to J_{1} (a)) \\ = ⋀_{a \in X} (⋁_{x \in X} I_{y} (α (x)) \land ι_{x} (a)) \to J_{1} (a) \\ = {sub}_{X} (⋁_{x \in X} I_{y} (α (x)) \land ι_{x}, J_{1}) . \end{matrix}$

Then ⊔^↑ (η ∘ i) ^→ (I_y) = ⋁ _x∈XI_y (α (x)) ∧ ι_x. For all d ∈ Y,

$\begin{matrix} ⋀_{a \in Y} (α^{\to} (⋁_{x \in X} I_{y} (α (x)) \land ι_{x}) (a) \to e_{Y} (a, d)) \\ = ⋀_{a \in Y} (⋁_{b \in X} (⋁_{x \in X} I_{y} (α (x)) \land ι_{x} (b)) \land e_{Y} (a, α (b))) \\ \to e_{Y} (a, d) \\ = ⋀_{b \in X} ⋀_{x \in X} (I_{y} (α (x)) \land e_{X} (b, x) \to ⋀_{a \in Y} (e_{Y} (a, α (b) \\ \to e_{Y} (a, d)))) \\ = ⋀_{b \in X} ⋀_{x \in X} (I_{y} (α (x)) \land e_{X} (b, x) \to e_{Y} (α (b), d)) \\ = ⋀_{x \in X} (I_{y} (α (x)) \to ⋀_{b \in X} (e_{Y} (α (b), α (x)) \\ \to e_{Y} (α (b), d))) \\ = ⋀_{x \in X} I_{y} (α (x)) \to e_{Y} (α (x), d) \\ = e_{Y} (⊔^{↑} i^{\to} (I_{y}), d) \\ = e_{Y} (y, d) . \end{matrix}$

Then ⊔^↑α^→ (⋁ _x∈XI_y (α (x)) ∧ ι_x) = y. We conclude that (σ ∘ η) (y) = (σ ∘ η) (⊔ ^↑i^→ (I_y)) = σ (⊔ ^↑ (η ∘ i) ^→ (I_y)) = σ (⋁ _x∈XI_y (α (x)) ∧ ι_x) = ⊔ α^→ (⋁ _x∈XI_y (α (x)) ∧ ι_x) = y. Then σ ∘ η = id_Y, and thus η is an L-order isomorphism and $Y ≅ I_{L} (X)$ . Clearly, η (α (x)) = (η ∘ α) (x) = f (x) = ι_x for all x ∈ X. (3) ⇒ (1): Clearly, $((I_{L} (X), {sub}_{X}), η \circ α)$ is adirected join-completion of (X, e_X). By Lemma3.6 we have $((I_{L} (X), {sub}_{X}), η \circ α)$ is a freejoin-completion of (X, e_X). Hence ((Y, e_Y) , α) is a free join-completion of (X, e_X).□

Example 3.8. Let (X, e_X) be the L-ordered set given in Example 3.5. Then $I_{L} (X) = {I \in [0, 1]^{X} | I (x_{1}) = 1, I (x_{2}) \in [0.4, 0.5] or I (x_{1}) = 0.5, I (x_{2}) = 1}$ and $(I_{L} (X), {sub}_{X})$ is an L-dcpo. Define $f : X ⟶ I_{L} (X)$ by f (x) = ι_x for all x ∈ X. Then $((I_{L} (X), {sub}_{X}), f)$ is a free join-completion of (X, e_X). Let ((Y, e_Y) , α) be the free join-completion of (X, e_X) given in Example 3.5. Define $η : Y ⟶ I_{L} (X)$ by η (y_i) = I_i where I_i (x₁) =1 and I_i (x₂) = i for all i ∈ [0.4, 0.5], η (y₀) = I₀ where I₀ (x₁) =0.5 and I₀ (x₂) =1, then η is an L-order isomorphism and η ∘ α = f.

Remark 3.9. Every L-ordered set has one, and up to isomorphism only one, free join-completion. Wedenote the free join-completion of X by $𝔽_{⊔} (X)$ and the embedding of X into $𝔽_{⊔} (X)$ by α_X in this paper.

Proposition 3.10. Let (X, e_X) be an L-ordered set. Then $α_{X} : X ⟶ 𝔽_{⊔} (X)$ preserves existing meets.

Proposition 3.11. Let (X, e_X) and (X′, e_X′) be L-ordered sets, and let f : X ⟶ X′ be an L-order preserving mapping. Then we have a unique directed-join-preserving mapping $𝔽_{⊔} (f) : 𝔽_{⊔} (X) ⟶ 𝔽_{⊔} (X^{'})$ such that the diagram in Fig. 1 commutes.

Fig.1

The commutative diagram for the free-join completions

We can, and sometimes will, suppress the embedding of an L-ordered set into its free join-completion and regard X as a subposet of $𝔽_{⊔} (X)$ . When this done, the formula for $𝔽_{⊔} (f)$ takes the form

$𝔽_{⊔} (f) (y) = ⊔^{↑} f^{\to} (A_{y})$ for all $y \in 𝔽_{⊔} (X)$ ,

where A_y ∈ L^X and $A_{y} (a) = e_{𝔽_{⊔} (X)} (a, y)$ for all a ∈ X.

Proposition 3.12. Let (X, e_X) and (X′, e_X′) be L-ordered sets, and let f : X ⟶ X′ be an L-order preserving mapping. Then the following statements hold:

(1) $𝔽_{⊔} (f)$ is the unique extension of f that preserves directed-join;

(2) if f is an L-order embedding, then $𝔽_{⊔} (f)$ is an L-order embedding.

Proof. (1) Clearly, $𝔽_{⊔} (f)$ is the unique directed-join-preserving mapping. For all $x \in X, z \in 𝔽_{⊔} (X^{'})$ , since

$\begin{matrix} e_{𝔽_{⊔} (X^{'})} (𝔽_{⊔} (f) (x), z) \\ = e_{𝔽_{⊔} (X^{'})} (⊔ f^{\to} (A_{x}), z) \\ = ⋀_{x^{'} \in X^{'}} f^{\to} (A_{x}) (x^{'}) \to e_{𝔽_{⊔} (X^{'})} (x^{'}, z) \\ = ⋀_{a \in X} (A_{x} (a) \to ⋀_{x^{'} \in X^{'}} (e_{𝔽_{⊔} (X^{'})} (x^{'}, f (a)) \\ \to e_{𝔽_{⊔} (X^{'})} (x^{'}, z))) \\ = ⋀_{a \in X} (A_{x} (a) \to e_{𝔽_{⊔} (X^{'})} (f (a), z)) \\ = ⋀_{a \in X} (e_{𝔽_{⊔} (X)} (a, x) \to e_{𝔽_{⊔} (X^{'})} (f (a), z)) \\ = e_{𝔽_{⊔} (X^{'})} (f (x), z), \end{matrix}$

we have $𝔽_{⊔} (f) (x) = f (x)$ . Hence $𝔽_{⊔} (f)$ is anextension of f.

(2) For all $y_{1}, y_{2} \in 𝔽_{⊔} (X)$ , since f is an L-orderembedding, we have

$\begin{matrix} e_{𝔽_{⊔} (X^{'})} (𝔽_{⊔} (f) (y_{1}), 𝔽_{⊔} (f) (y_{2})) \\ = e_{𝔽_{⊔} (X^{'})} (⊔ f^{\to} (A_{y_{1}}), ⊔ f^{\to} (A_{y_{2}})) \\ = ⋀_{x^{'} \in X^{'}} f^{\to} (A_{y_{1}}) (x^{'}) \to e_{𝔽_{⊔} (X^{'})} (x^{'}, ⊔ f^{\to} (A_{y_{2}})) \\ = ⋀_{x \in X} A_{y_{1}} (x) \to e_{𝔽_{⊔} (X^{'})} (f (x), ⊔ f^{\to} (A_{y_{2}})) \\ = ⋀_{x \in X} e_{𝔽_{⊔} (X)} (x, y_{1}) \to e_{𝔽_{⊔} (X^{'})} (f (x), ⊔ f^{\to} (A_{y_{2}})) \\ = ⋀_{x \in X} e_{𝔽_{⊔} (X)} (x, y_{1}) \to f^{\to} (A_{y_{2}}) (f (x)) \\ = ⋀_{x \in X} e_{𝔽_{⊔} (X)} (x, y_{1}) \to ⋁_{x_{1} \in X} (A_{y_{2}} (x_{1}) \land e_{X} (x, x_{1})) \\ = ⋀_{x \in X} (e_{𝔽_{⊔} (X)} (x, y_{1}) \to e_{𝔽_{⊔} (X)} (x, y_{2})) \\ = e_{𝔽_{⊔} (X)} (y_{1}, y_{2}) . \end{matrix}$

Hence $𝔽_{⊔} (f)$ is an L-order embedding. □

We have order dual versions of all of the aboveconstructions and properties. We denote the free meet-completion of an L-ordered set X by $𝔽_{⊓} (X)$ , and the embedding of X into $𝔽_{⊓} (X)$ by β_X. We write $𝔽_{⊓} (f)$ for the down-directed-meet-preserving extension of an L-order preserving mapping f : X ⟶ Y. It is given by

$𝔽_{⊓} (f) (y) = ⊓^{↓} f_{*}^{\to} (B_{y})$ for all $y \in 𝔽_{⊓} (X)$ ,

where B_y ∈ L^X and $B_{y} (a) = e_{𝔽_{⊓} (X)} (y, a)$ for alla ∈ X. We iterate the above constructions to form $𝔽_{⊔} (𝔽_{⊓} (X))$ and $𝔽_{⊓} (𝔽_{⊔} (X))$ and so on. This enablesus to form an infinite hierarchy of completions byalternating free join- and free meet-completions. In the next part we investigate this hierarchy.

4 Δ₁-directed objects and intermediate structures

Fig. 2 shows the hierarchy of completions of an L-ordered set (X, e_X). In this section we want to identify the common part of $𝔽_{⊔} (𝔽_{⊓} (X))$ and $𝔽_{⊓} (𝔽_{⊔} (X))$ , and the common part is called the Δ₁-directed object. If $𝔽_{⊔} (𝔽_{⊓} (X))$ and $𝔽_{⊓} (𝔽_{⊔} (X))$ are subsets of some common set Y, then we would simply take thecommon part to be their intersection. In the absence of such an Y, a natural substitute is to ask for a greatest interpolant Q in Fig. 2. Specifically, in the diagram we want all the maps to be embeddings and the compositions along the upward and downward diagonals to be $α_{𝔽_{⊓} (X)}$ and $β_{𝔽_{⊔} (X)}$ , respectively. And we want thediagram to commute. We now embark on showing how to construct the required interpolant Q.

Fig.2

The hierarchy of completions of X

Theorem 4.1. Let (X, e_X) be an L-ordered set. Then the L-order induced on

$Int (X) = 𝔽_{⊔} (X) \cup 𝔽_{⊓} (X)$

in $𝔽_{⊔} (𝔽_{⊓} (X))$ is the same as that of $𝔽_{⊓} (𝔽_{⊔} (X))$ . For z₁, z₂ ∈ Int (X) it is given by

(1) if $z_{1}, z_{2} \in 𝔽_{⊔} (X)$ , then $e_{Int} (z_{1}, z_{2}) = e_{𝔽_{⊔} (X)} (z_{1}, z_{2})$ ;

(2) if $z_{1}, z_{2} \in 𝔽_{⊓} (X)$ , then $e_{Int} (z_{1}, z_{2}) = e_{𝔽_{⊓} (X)} (z_{1}, z_{2})$ ;

(3) if $z_{1} \in 𝔽_{⊔} (X), z_{2} \in 𝔽_{⊓} (X)$ , then $e_{Int} (z_{1}, z_{2}) = ⋀_{p \in X} ⋀_{q \in X} ((e_{𝔽_{⊔} (X)} (α_{X} (p), z_{1}) \land e_{𝔽_{⊓} (X)} (z_{2}, β_{X} (q))) \to e_{X} (p, q))$ ;

(4) if $z_{1} \in 𝔽_{⊓} (X), z_{2} \in 𝔽_{⊔} (X)$ , then $e_{Int} (z_{1}, z_{2}) = ⋁_{p \in X} (e_{𝔽_{⊓} (X)} (z_{1}, β_{X} (p)) \land e_{𝔽_{⊔} (X)} (α_{X} (p), z_{2}))$ .

Proof. We write simply $e_{⊔}, e_{⊓}, e_{⊓ ⊔}, e_{⊔ ⊓}, 𝔽_{⊔ ⊓}$ for $e_{𝔽_{⊔} (X)}, e_{𝔽_{⊓} (X)}, e_{𝔽_{⊓} (𝔽_{⊔} (X))}, e_{𝔽_{⊔} (𝔽_{⊓} (X))}, 𝔽_{⊔} (𝔽_{⊓} (X))$ , respectively.

(1) Let $z_{1}, z_{2} \in 𝔽_{⊔} (X)$ and consider their embeddings in $𝔽_{⊔} (𝔽_{⊓} (X))$ , namely $𝔽_{⊔} (β_{X}) (z_{1})$ and $𝔽_{⊔} (β_{X}) (z_{2})$ . Since $𝔽_{⊔} (β_{X})$ is an embedding, we have

$e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (z_{1}), 𝔽_{⊔} (β_{X}) (z_{2})) = e_{⊔} (z_{1}, z_{2}) .$

Similarly, $e_{⊓ ⊔} (β_{𝔽_{⊔} (X)} (z_{1}), β_{𝔽_{⊔} (X)} (z_{2})) = e_{⊔} (z_{1}, z_{2}) .$

(2) Let $z_{1}, z_{2} \in 𝔽_{⊓} (X)$ . Similar to (1), we can get

$e_{⊔ ⊓} (α_{𝔽_{⊓} (X)} (z_{1}), α_{𝔽_{⊓} (X)} (z_{2})) = e_{⊓} (z_{1}, z_{2}),$

$e_{⊓ ⊔} (𝔽_{⊓} (α_{X}) (z_{1}), 𝔽_{⊓} (α_{X}) (z_{2})) = e_{⊓} (z_{1}, z_{2}) .$

(3) Let $z_{1} \in 𝔽_{⊔} (X)$ and $z_{2} \in 𝔽_{⊓} (X)$ and consider their embeddings in $𝔽_{⊔} (𝔽_{⊓} (X))$ , namely $𝔽_{⊔} (β_{X}) (z_{1})$ and $α_{𝔽_{⊓} (X)} (z_{2})$ . By Proposition 3.10 and 3.12, $α_{𝔽_{⊓} (X)}$ preserves meets and $𝔽_{⊔} (β_{X})$ preserves directed-joins, it follows that $α_{𝔽_{⊓} (X)} (z_{2}) = α_{𝔽_{⊓} (X)} (⊓ i_{*}^{\to} (B_{z_{2}}^{'})) = ⊓ (α_{𝔽_{⊓} (X)} \circ i)_{*}^{\to} (B_{z_{2}}^{'})$ and $𝔽_{⊔} (β_{X}) (z_{1}) = 𝔽_{⊔} (β_{X})$ $(⊔^{↑} i^{\to} (A_{z_{1}}^{'})) = ⊔^{↑} (𝔽_{⊔} (β_{X}) \circ i)^{\to} (A_{z_{1}}^{'})$ where $B_{z_{2}}^{'} \in L^{β_{X} (X)}$ and $B_{z_{2}}^{'} (b) = e_{⊓} (z_{2}, b)$

for all b ∈ β_X (X), $A_{z_{1}}^{'} \in L^{α_{X} (X)}$ and $A_{z_{1}}^{'} (a) = e_{⊔} (a, z_{1})$ for all a ∈ α_X (X). Since $𝔽_{⊔} (β_{X}) \circ α_{X} = α_{𝔽_{⊓} (X)} \circ β_{X}$ , we have

$\begin{matrix} e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (z_{1}), α_{𝔽_{⊓} (X)} (z_{2})) \\ = e_{⊔ ⊓} (⊔^{↑} (𝔽_{⊔} (β_{X}) \circ i)^{\to} (A_{z_{1}}^{'}), α_{𝔽_{⊓} (X)} (z_{2})) \\ = ⋀_{r \in 𝔽_{⊔ ⊓}} (𝔽_{⊔} (β) \circ i)^{\to} (A_{z_{1}}^{'}) (r) \to e_{⊔ ⊓} (r, α_{𝔽_{⊓} (X)} (z_{2})) \\ = ⋀_{p \in X} A_{z_{1}}^{'} (α_{X} (p)) \to ⋀_{r \in 𝔽_{⊔ ⊓}} e_{⊔ ⊓} (r, 𝔽_{⊔} (β_{X}) (α_{X} (p))) \\ \to e_{⊔ ⊓} (r, α_{𝔽_{⊓} (X)} (z_{2})) \\ = ⋀_{p \in X} e_{⊔} (α_{X} (p), z_{1}) \\ \to e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (α_{X} (p)), α_{𝔽_{⊓} (X)} (z_{2})) \\ = ⋀_{p \in X} (e_{⊔} (α_{X} (p), z_{1}) \to ⋀_{n \in 𝔽_{⊔ ⊓}} ((α_{𝔽_{⊓} (X)} \circ i)_{*}^{\to} \\ (B_{z_{2}}^{'}) (n) \to e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (α_{X} (p)), n))) \\ = ⋀_{p \in X} (e_{⊔} (α_{X} (p), z_{1}) \to ⋀_{q \in X} (e_{⊓} (z_{2}, β_{X} (q)) \\ \to e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (α_{X} (p)), α_{𝔽_{⊓} (X)} (β_{X} (q))))) \\ = ⋀_{p \in X} ⋀_{q \in X} (e_{⊔} (α_{X} (p), z_{1}) \land e_{⊓} (z_{2}, β_{X} (q)) \\ \to e_{⊔ ⊓} (𝔽_{⊔} (β_{X}) (α_{X} (p)), α_{𝔽_{⊓} (X)} (β_{X} (q)))) \\ = ⋀_{p \in X} ⋀_{q \in X} e_{⊔} (α_{X} (p), z_{1}) \land e_{⊓} (z_{2}, β_{X} (q)) \\ \to e_{X} (p, q) . \end{matrix}$

Similarly, $e_{⊓ ⊔} (β_{𝔽_{⊔} (X)} (z_{1}), 𝔽_{⊓} (α_{X}) (z_{2})) = ⋀_{p \in X} ⋀_{q \in X} e_{⊔} (α_{X} (p), z_{1}) \land e_{⊓} (z_{2}, β_{X} (q)) \to e_{X} (p, q)$ .

(4) Let $z_{1} \in 𝔽_{⊓} (X)$ and $z_{2} \in 𝔽_{⊔} (X)$ and consider their embeddings in $𝔽_{⊔} (𝔽_{⊓} (X))$ , namely $α_{𝔽_{⊓} (X)} (z_{1})$ and $𝔽_{⊔} (β_{X}) (z_{2})$ . Since $𝔽_{⊔} (β_{X}) \circ α_{X} = α_{𝔽_{⊓} (X)} \circ β_{X}$ , we have

$\begin{matrix} e_{⊔ ⊓} (α_{𝔽_{⊓} (X)} (z_{1}), 𝔽_{⊔} (β_{X}) (z_{2})) \\ = e_{⊔ ⊓} (α_{𝔽_{⊓} (X)} (z_{1}), ⊔ (𝔽_{⊔} (β_{X}) \circ i)^{\to} (A_{z_{2}}^{'})) \\ = (𝔽_{⊔} (β_{X}) \circ i)^{\to} (A_{z_{2}}^{'}) (α_{𝔽_{⊓} (X)} (z_{1})) \\ = ⋁_{p \in X} (e_{⊔} (α_{X} (p), z_{2}) \\ \land e_{⊔ ⊓} (α_{𝔽_{⊓} (X)} (z_{1}), 𝔽_{⊔} (β_{X}) (α_{X} (p)))) \\ = ⋁_{p \in X} (e_{⊔} (α_{X} (p), z_{2}) \\ \land e_{⊔ ⊓} (α_{𝔽_{⊓} (X)} (z_{1}), α_{𝔽_{⊓} (X)} (β_{X} (p)))) \\ = ⋁_{p \in X} e_{⊔} (α_{X} (p), z_{2}) \land e_{⊓} (z_{1}, β_{X} (p)) . \end{matrix}$

Similarly, $e_{⊓ ⊔} (𝔽_{⊓} (α_{X}) (z_{1}), β_{𝔽_{⊔} (X)} (z_{2})) = ⋁_{p \in X} e_{⊔} (α_{X} (p), z_{2}) \land e_{⊓} (z_{1}, β_{X} (p)) .$ □

According to the statement in Theorem 4.1, α_X (x) = β_X (x) in Int (X) for all x ∈ X. Hence we can regard Int (X) as being L-ordered and as containing X as a subset. We refer to Int (X) as the intermediate structure of X.

Let (X, e) be an L-ordered set and P ⊆ X. P is said to be join-dense in X if for every x ∈ X there is an L-subset A of P such that x = ⊔ i^→ (A), and it is said to be meet-dense if for every x ∈ X there is an L-subset B of P such that $x = ⊓ i_{*}^{\to} (B)$ , where i : P ⟶ X is an inclusion mapping.

Proposition 4.2. Let (X, e_X) be an L-ordered set. Then $𝔽_{⊔} (X)$ is meet-dense in Int (X) and $𝔽_{⊓} (X)$ is join-dense in Int (X).

Proposition 4.3. Let (X, e_X) be an L-ordered set. Then for all $D \in D_{L} (𝔽_{⊔} (X))$ , ⊔^↑D exists in Int (X), and may be regard as being calculated either in $𝔽_{⊔} (X)$ or in Int (X). Dual assertion holds for ⊓^↓F where $F \in D_{L}^{↓} (𝔽_{⊓} (X)) .$

Proof. For all $D \in D_{L} (𝔽_{⊔} (X))$ , we have for all z ∈ Int (X),

(1) if $z \in 𝔽_{⊓} (X)$ , then

$\begin{matrix} ⋀_{x \in Int (X)} (D (x) \to e_{Int} (x, z)) \\ = ⋀_{x \in 𝔽_{⊔} (X)} (D (x) \to e_{Int} (x, z)) \\ = ⋀_{x \in 𝔽_{⊔} (X)} (D (x) \to ⋀_{p \in X} ⋀_{q \in X} (e_{𝔽_{⊔} (X)} (α_{X} (p), x) \\ \land e_{𝔽_{⊓} (X)} (z, β_{X} (q)) \to e_{X} (p, q))) \\ = ⋀_{p \in X} ⋀_{q \in X} ⋀_{x \in 𝔽_{⊔} (X)} (D (x) \land e_{𝔽_{⊔} (X)} (α_{X} (p), x) \\ \land e_{𝔽_{⊓} (X)} (z, β_{X} (q)) \to e_{X} (p, q)) \\ = ⋀_{p \in X} ⋀_{q \in X} ((⋁_{x \in 𝔽_{⊔} (X)} D (x) \land e_{𝔽_{⊔} (X)} (α_{X} (p), x)) \\ \land e_{𝔽_{⊓} (X)} (z, β_{X} (q)) \to e_{X} (p, q)) \\ = ⋀_{p \in X} ⋀_{q \in X} (e_{𝔽_{⊔} (X)} (α_{X} (p), ⊔_{𝔽_{⊔} (X)}^{↑} D) \\ \land e_{𝔽_{⊓} (X)} (z, β_{X} (q)) \to e_{X} (p, q)) \\ = e_{Int} (⊔_{𝔽_{⊔} (X)}^{↑} D, z) . \end{matrix}$

(2) if $z \in 𝔽_{⊔} (X)$ , then

$⋀_{x \in Int (X)} (D (x) \to e_{Int} (x, z)) = ⋀_{x \in 𝔽_{⊔} (X)} (D (x) \to e_{Int} (x, z)) = ⋀_{x \in 𝔽_{⊔} (X)} (D (x) \to e_{𝔽_{⊔} (X)} (x, z)) = e_{𝔽_{⊔} (X)} (⊔_{𝔽_{⊔} (X)}^{↑} D, z) = e_{Int} (⊔_{𝔽_{⊔} (X)}^{↑} D, z) .$ Hence $⊔_{Int}^{↑} D = ⊔_{𝔽_{⊔} (X)}^{↑} D$ for all $D \in D_{L} (𝔽_{⊔} (X))$ . □

Now we have established that Int (X) works as an interpolant, but we would like to find the Δ₁-directed object in L-ordered set. In order to light the notations, sometimes we will regard X as a subposet of Y if X can embed into Y. Let DM (X) denote the Dedekind-MacNeille completion of X.

Let (X, e) be an L-ordered set. Define X^δ by $X^{δ} = {u \in DM (Int (X)) : D_{u} \in D_{L} (𝔽_{⊓} (X)) and F_{u} \in D_{L}^{↓} (𝔽_{⊔} (X))}$ where $D_{u} \in L^{𝔽_{⊓} (X)}$ and D_u (y) = e_DM (y, u) for all $y \in 𝔽_{⊓} (X)$ , $F_{u} \in L^{𝔽_{⊔} (X)}$ and F_u (z) = e_DM (u, z) for all $z \in 𝔽_{⊔} (X)$ .

Proposition 4.4. Let (X, e) be an L-ordered set. Then X^δ is the Δ₁-directed object.

Proof. Suppose Q is an interpolant as in Fig. 2. For each q ∈ Q, since Q embeds in $𝔽_{⊔} (𝔽_{⊓} (X))$ we define $D_{q} \in L^{𝔽_{⊓} (X)}$ by for all $y \in 𝔽_{⊓} (X)$ , $D_{q} (y) = e_{𝔽_{⊔} (𝔽_{⊓} (X))} (y, q)$ , then $D_{q} \in D_{L} (𝔽_{⊓} (X))$ and q = ⊔ i^→ (D_q). Similarly, we define $F_{q} \in L^{𝔽_{⊔} (X)}$ by for all $z \in 𝔽_{⊔} (X)$ , $F_{q} (z) = e_{𝔽_{⊓} (𝔽_{⊔} (X))} (q, z)$ , then $F_{q} \in D_{L}^{↓} (𝔽_{⊔} (X))$ and $q = ⊓ i_{*}^{\to} (F_{q})$ . Then Int (X) is both join-dense and meet-dense in Q, hence Q is a subposet of the Dedekind-MacNeille completion DM (Int (X)). Since $𝔽_{⊔} (𝔽_{⊓} (X))$ embedsinto DM (Int (X)) and $𝔽_{⊓} (𝔽_{⊔} (X))$ embeds into DM (Int (X)), then for each q ∈ Q,

$e_{𝔽_{⊔} (𝔽_{⊓} (X))} (y, q) = e_{DM} (y, q)$ for all $y \in 𝔽_{⊓} (X)$ ,

$e_{𝔽_{⊓} (𝔽_{⊔} (X))} (q, z) = e_{DM} (q, z)$ for all $z \in 𝔽_{⊔} (X)$ .

Then X^δ is the greatest interpolant Q in Fig. 2. Hence X^δ is the Δ₁-directed object. □

Proposition 4.5. [21] Let (X, e) be an L-ordered set. If ((DM (X) , sub) , f) is a Dedekind-MacNeillecompletion of (X, e), then f preserves all meets of L-ordered sets which exist in X.

Proposition 4.6. Let (X, e) be a bounded L-lattice. Then $⊔_{𝔽_{⊔} (X)} X_{{α_{X} (x), α_{X} (y)}}$ exists and $α_{X} (⊔_{X} X_{{x, y}}) = ⊔_{𝔽_{⊔} (X)} X_{{α_{X} (x), α_{X} (y)}}$ for all x, y ∈ X.

Proof. For all x, y ∈ X, since (X, e) is a bounded L-lattice, then $⊔_{X} X_{{x, y}}$ exists and $e_{X} (⊔_{X} X_{{x, y}}, b) = e_{X} (x, b) \land e_{X} (y, b)$ for all b ∈ X. We can easily prove $X_{{α_{X} (x), α_{X} (y)}} (z) \leq e_{⊔} (z, α_{X} (⊔_{X} X_{{x, y}}))$ for all $z \in 𝔽_{⊔} (X)$ . For all $a \in 𝔽_{⊔} (X)$ ,

$\begin{matrix} ⋀_{c \in 𝔽_{⊔} (X)} (X_{{α_{X} (x), α_{X} (y)}} (c) \to e_{⊔} (c, a)) \\ = e_{⊔} (α_{X} (x), a) \land e_{⊔} (α_{X} (y), a) \\ = e_{⊔} (α_{X} (x), ⊔ i^{\to} (A_{a}^{'})) \land e_{⊔} (α_{X} (y), ⊔ i^{\to} (A_{a}^{'})) \\ = i^{\to} (A_{a}^{'}) (α_{X} (x)) \land i^{\to} (A_{a}^{'}) (α_{X} (y)) \\ = (⋁_{b_{1} \in X} (A_{a}^{'} (α_{X} (b_{1})) \land e_{X} (x, b_{1}))) \\ \land (⋁_{b_{2} \in X} (A_{a}^{'} (α_{X} (b_{2})) \land e_{X} (y, b_{2}))) \\ = ⋁_{b_{1}, b_{2} \in X} (A_{a}^{'} (α_{X} (b_{1})) \land A_{a}^{'} (α_{X} (b_{2})) \\ \land e_{X} (x, b_{1}) \land e_{X} (y, b_{2})) \\ \leq ⋁_{b_{1}, b_{2} \in X} (⋁_{b \in X} A_{a}^{'} (α_{X} (b)) \land e_{X} (b_{1}, b) \\ \land e_{X} (b_{2}, b) \land e_{X} (x, b_{1}) \land e_{X} (y, b_{2})) \\ \leq ⋁_{b \in X} A_{a}^{'} (α_{X} (b)) \land e_{X} (x, b) \land e_{X} (y, b) \\ = ⋁_{b \in X} A_{a}^{'} (α_{X} (b)) \land e_{X} (⊔_{X} X_{{x, y}}, b) \\ = ⋁_{b \in X} A_{a}^{'} (α_{X} (b)) \land e_{⊔} (α_{X} (⊔_{X} X_{{x, y}}), α_{X} (b)) \\ = i^{\to} (A_{a}^{'}) (α_{X} (⊔_{X} X_{{x, y}})) \\ = e_{⊔} (α_{X} (⊔_{X} X_{{x, y}}), ⊔ i^{\to} (A_{a}^{'})) \\ = e_{⊔} (α_{X} (⊔_{X} X_{{x, y}}), a) . \end{matrix}$

Hence $⊔_{𝔽_{⊔} (X)} X_{{α_{X} (x), α_{X} (y)}} = α_{X} (⊔_{X} X_{{x, y}})$ . □

Clearly, $I_{L} (X)$ is a poset under the pointwise order. Define an order $\leq_{𝔽_{⊔} (X)}$ on $𝔽_{⊔} (X)$ as follows:

$\forall x, y \in 𝔽_{⊔} (X), x \leq_{𝔽_{⊔} (X)} y$ iff η (x) ≤ η (y),

where $η : 𝔽_{⊔} (X) ⟶ I_{L} (X)$ is an L-order isomorphism.

Proposition 4.7. If (X, e) is a join L-semilattice and $⊔ \bar{0}$ exists, then $(𝔽_{⊔} (X)$ , $\leq_{𝔽_{⊔} (X)})$ is a complete lattice.

Proof. Since $𝔽_{⊔} (X) ≅ I_{L} (X)$ , we only need to prove that $(I_{L} (X), \leq)$ is a complete lattice. Define I₀ ∈ L^X by for all x ∈ X, I₀ (x) =⊤. Clearly, $I_{0} \in I_{L} (X)$ . For all ${I_{k}}_{k \in K} \subseteq I_{L} (X)$ ,

(1) if K =∅, then⋀_k∈KI_k = I₀;

(2) if K≠ ∅, it is easy to see that (⋀ _k∈KI_k) (0) = ⊤. For all x, y ∈ X, (⋀ _k∈KI_k) (x) ∧ e (y, x) ≤ ⋀ _k∈K (I_k (x) ∧ e (y, x)) ≤ (⋀ _k∈KI_k) (y). By Proposition 2.19, we have (⋀ _k∈KI_k) (x) ∧ (⋀ _k∈KI_k) (y) = (⋀ _k∈KI_k (x)) ∧ (⋀ _k∈KI_k (y)) ≤ ⋀ _k∈K (I_k (x) ∧ I_k (y)) ≤ ⋀ _k∈KI_k (x ∨ y) = (⋀ _k∈KI_k) (x ∨ y). Then $⋀_{k \in K} I_{k} \in I_{L} (X)$ . Thus $(I_{L} (X), \leq)$ is a complete lattice.

Proposition 4.8. Let (X, e) be a bounded L-lattice. Then $⊓ X_{A}$ exists and $⊓ X_{A} = ⋀ A$ for all $A \subseteq 𝔽_{⊔} (X)$ , where the meet ⋀ is computed in $(𝔽_{⊔} (X)$ , $\leq_{𝔽_{⊔} (X)})$ .

Proof. Since $𝔽_{⊔} (X) ≅ I_{L} (X)$ , we only prove that $⊓ X_{B}$ exists and $⊓ X_{B} = ⋀ B$ for all $B \subseteq I_{L} (X)$ , where the meet ⋀ is computed in $(I_{L} (X), \leq)$ . By Proposition 4.7, we have $⋀ B \in I_{L} (X)$ . Clearly, $X_{B} (I) \leq {sub}_{X} (⋀ B, I)$ for all $I \in I_{L} (X)$ . For all $I^{'} \in I_{L} (X)$ ,

$\begin{matrix} ⋀_{I \in I_{L} (X)} (X_{B} (I) \to {sub}_{X} (I^{'}, I)) \\ = ⋀_{I \in B} {sub}_{X} (I^{'}, I) \\ = ⋀_{I \in B} ⋀_{x_{I} \in X} (I^{'} (x_{I}) \to I (x_{I})) \\ \leq ⋀_{x \in X} ⋀_{I \in B} (I^{'} (x) \to I (x)) \\ = ⋀_{x \in X} (I^{'} (x) \to ⋀_{I \in B} I (x)) \\ = ⋀_{x \in X} (I^{'} (x) \to (⋀ B) (x)) \\ = {sub}_{X} (I^{'}, ⋀ B) . \end{matrix}$

Hence $⊓ X_{B}$ exists and $⊓ X_{B} = ⋀ B$ . □

Lemma 4.9. Let (X, e_X) be a bounded L-lattice. Then $⊓_{DM (Int (X))} X_{{x, y}} = x \land y$ for all $x, y \in 𝔽_{⊔} (X)$ , where the meet ∧ is computed in $(𝔽_{⊔} (X)$ , $\leq_{𝔽_{⊔} (X)})$ .

Proof. By Proposition 4.8, we know that $⊓_{𝔽_{⊔} (X)} X_{{x, y}}$ exists and $⊓_{𝔽_{⊔} (X)} X_{{x, y}} = x \land y$ for all $x, y \in 𝔽_{⊔} (X)$ , where the meet ∧ is computed in $(𝔽_{⊔} (X)$ , $\leq_{𝔽_{⊔} (X)})$ . For all z ∈ Int (X),

(1) if $z \in 𝔽_{⊓} (X)$ , on the one hand, $X_{{x, y}} (m) \leq e_{Int} (⊓_{F_{⊔} (X)} X_{{x, y}}, m)$ for all m ∈ Int (X). One the otherhand, by the order dual version of Proposition 4.6, we have $⊓_{𝔽_{⊓} (X)} X_{{β_{X} (p_{1}), β_{X} (p_{2})}}$ exists and $β_{X} (p_{1} \land_{X} p_{2}) [4] = β_{X} (p_{1}) \land_{𝔽_{⊓} (X)} β_{X} (p_{2})$ for all p₁, p₂ ∈ X. Hence $e_{⊓} (z, β_{X} (p_{1} \land_{X} p_{2})) = e_{⊓} (z, β_{X} (p_{1}) \land_{𝔽_{⊓} (X)} β_{X} (p_{2})) = e_{⊓} (z, ⊓_{𝔽_{⊓} (X)} X_{{β_{X} (p_{1}), β_{X} (p_{2})}}) = e_{⊓} (z, [4] β_{X} (p_{1})) \land e_{⊓} (z, β_{X} (p_{2}))$ . By Proposition 3.10, α_X $(p_{1} \land_{X} p_{2}) = α_{X} (⊓_{X} X_{{p_{1}, p_{2}}}) = ⊓ α_{X}^{\to} (X_{{p_{1}, p_{2}}}) = ⊓_{𝔽_{⊔} (X)} X_{{α_{X} (p_{1}), α_{X} (p_{2})}} = α_{X} (p_{1}) \land α_{X} (p_{2})$ . Then

$\begin{matrix} ⋀_{a \in Int (X)} (X_{{x, y}} (a) \to e_{Int} (z, a)) \\ = e_{Int} (z, x) \land e_{Int} (z, y) \\ = (⋁_{p_{1} \in X} e_{⊓} (z, β_{X} (p_{1})) \land e_{⊔} (α_{X} (p_{1}), x)) \\ \land (⋁_{p_{2} \in X} e_{⊓} (z, β_{X} (p_{2})) \land e_{⊔} (α_{X} (p_{2}), y)) \\ = ⋁_{p_{1}, p_{2} \in X} (e_{⊓} (z, β_{X} (p_{1})) \land e_{⊓} (z, β_{X} (p_{2})) \\ \land e_{⊔} (α_{X} (p_{1}), x) \land e_{⊔} (α_{X} (p_{2}), y)) \\ = ⋁_{p_{1}, p_{2} \in X} (e_{⊓} (z, β_{X} (p_{1} \land_{X} p_{2})) \\ \land e_{⊔} (α_{X} (p_{1}), x) \land e_{⊔} (α_{X} (p_{2}), y)) \\ \leq ⋁_{p_{1}, p_{2} \in X} (e_{⊓} (z, β_{X} (p_{1} \land_{X} p_{2})) \\ \land e_{⊔} (α_{X} (p_{1}) \land α_{X} (p_{2}), x \land y)) \\ = ⋁_{p_{1}, p_{2} \in X} (e_{⊓} (z, β_{X} (p_{1} \land_{X} p_{2})) \\ \land e_{⊔} (α_{X} (p_{1} \land_{X} p_{2}), x \land y)) \\ = ⋁_{p \in X} (e_{⊓} (z, β_{X} (p)) \land e_{⊔} (α_{X} (p), x \land y)) \\ = e_{Int} (z, x \land y) \\ = e_{Int} (z, ⊓_{𝔽_{⊔} (X)} X_{{x, y}}) . \end{matrix}$

(2) if $z \in 𝔽_{⊔} (X)$ , then $⋀_{a \in Int (X)} (X_{{x, y}} (a) \to e_{Int} (z, a)) = e_{Int} (z, x) \land e_{Int} (z, y) = e_{⊔} (z, x) \land e_{⊔} (z, y) = ⋀_{a \in 𝔽_{⊔} (X)} (X_{{x, y}} (a) \to e_{⊔} (z, a)) = e_{⊔} (z, ⊓_{𝔽_{⊔} (X)} X_{{x, y}}) = e_{Int} (z, ⊓_{𝔽_{⊔} (X)} X_{{x, y}})$ . Hence $⊓_{Int} X_{{x, y}}$ exists and $⊓_{Int} X_{{x, y}} = ⊓_{𝔽_{⊔} (X)} X_{{x, y}} = x \land y$ . By Proposition 4.5, we have $⊓_{Int} X_{{x, y}} = ⊓_{DM (Int (X))} X_{{x, y}}$ .

Hence $⊓_{DM (Int (X))} X_{{x, y}} = x \land y$ . □

Theorem 4.10. Let (X, e) be a bounded L-lattice. Then the Δ₁-directed object precisely is DM (Int (X)).

Proof. For each u ∈ DM (Int (X)), we define $F_{u} \in L^{𝔽_{⊔} (X)}$ by for all $y \in 𝔽_{⊔} (X)$ , F_u (y) = e_DM (u, y).

(1) $⋁_{a \in 𝔽_{⊔} (X)} F_{u} (a) \geq F_{u} (1) = e_{DM} (u, 1) = ⊤$ .

(2) For all $x, y \in 𝔽_{⊔} (X)$ , by Lemma 4.9, we have that $⊓_{DM (Int (X))} X_{{x, y}} = x \land y$ , where the meet ∧ is computed in $(𝔽_{⊔} (X)$ , $\leq_{𝔽_{⊔} (X)})$ . Then

$\begin{matrix} F_{u} (x) \land F_{u} (y) \\ = e_{DM} (u, x) \land e_{DM} (u, y) \\ = ⋀_{n \in DM (Int (X))} (X_{{x, y}} (n) \to e_{DM} (u, n)) \\ = e_{DM} (u, ⊓_{DM (Int (X))} X_{{x, y}}) \\ = e_{DM} (u, x \land y) \\ = F_{u} (x \land y) . \end{matrix}$

Hence $F_{u} \in D_{L}^{↓} (𝔽_{⊔} (X))$ . Similarly, we can define $D_{u} \in L^{𝔽_{⊓} (X)}$ by for all $z \in 𝔽_{⊓} (X)$ , D_u (z) = e_DM (z, u). Then $D_{u} \in D_{L} (𝔽_{⊓} (X))$ . Thus X^δ = DM (Int (X)). By Proposition 4.4, we have that the Δ₁-directed object of (X, e) precisely is DM (Int (X)). □

5 Conclusion

In this paper, we build and characterize the free join- and meet-completions of an L-ordered set and prove that the free join-completion of an L-ordered set may be obtained as the L-ordered set of all L-ideals of the L-ordered set. Then we show that for an L-ordered set X the L-order induced on the union of its free join- and meet-completions in $𝔽_{⊔} (𝔽_{⊓} (X))$ is the same as the one induced in $𝔽_{⊓} (𝔽_{⊔} (X))$ . Finally, we give thedefinitions of Δ₁-directed objects and intermediate structures, and prove that the Δ₁-directed object precisely is the Dedekind-MacNeille completion of the intermediate structure in the bounded L-lattice case. In the future, we can try to consider the applications of free join-completions to other fields, such as fuzzy logic [3], many valued topology [15, 17] and theoretical computer science [20].

Footnotes

Acknowledgements

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China(11531009), and the authors also wish to express their sincere thanks to the anonymous referees for their careful reading and helpful comments which have improved the quality of the paper.

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