Reflective Full Subcategories of the Category of L-Posets Hongping

and Applied Analysis 3 In this paper, L always denotes a complete residuated lattice unless otherwise stated, and L denotes the set of all L-subsets of a nonempty set X. For all A,B ∈ L , we define A ∩ B x A x ∧ B x , A ∪ B x A x ∨ B x , A ∗ B x A x ∗ B x , A −→ B x A x −→ B x . 2.1 Then L, ∗, → ,∨,∧, 0, 1 is also a complete residuated lattice. If no confusion arises, we always do not discriminate the constant value function a with a, for example, a ∗ A x a ∗A x and a → A x a → A x for every x ∈ X. Fuzzy order was first introduced by Zadeh 1 , from then on, different kinds of fuzzy order have been introduced and studied by different authors the reader is referred to 3, 6, 9, 11, 18–20 for details . In this paper, we adopt the definition of fuzzy order introduced by Fan and Zhang 3, 6 . Definition 2.1. An L-partial order e also called an L-order on X is an L-relation satisfying: 1 for all x ∈ X, e x, x 1; 2 for all x, y, z ∈ X, e x, y ∗ e y, z ≤ e x, z ; 3 for all x, y ∈ X, e x, y e y, x 1⇒ x y. Then X, e is called an L-partially ordered set or an L-poset for simplicity. It is worth noting that Bělohlávek defined another fuzzy order in 12, 16 , and it was shown to be equivalent to the above definition by Yao 5 . There are some important L-posets which are mentioned in many papers, such as 3, 4, 19, 21 . For example, in an L-poset X, e , define e x, y e y, x for all x, y ∈ X, then X, e is also an L-poset. In a complete residuated lattice L, define eL : L × L → L by eL x, y x → y for all x, y ∈ L, then L, eL is an L-poset. For all A,B ∈ L , define sub A,B ∧ x∈X A x → B x , then L, sub is an L-poset. In the following, some basic and very important definitions and results related to the theory of L-poset are listed. The reader is referred to 2–9, 11–13, 20 for details. Definition 2.2. Let X, e be an L-poset, A ∈ L , x0 ∈ X. x0 is called a fuzzy join resp., fuzzy meet of A denoted by x0 A resp., x0 A if 1 for all x ∈ X, A x ≤ e x, x0 resp., A x ≤ e x0, x ; 2 for all y ∈ X, ∧x∈X A x → e x, y ≤ e x0, y resp., ∧ x∈X A x → e y, x ≤ e y, x0 . It is easy to see that the fuzzy join or the fuzzy meet is unique in an L-poset X, e if it exists, and x0 A resp., x0 A if and only if e x0, y ∧ x∈X A x → e x, y resp., e y, x0 ∧ x∈X A x → e y, x for all y ∈ X. If A and A exist for all A ∈ L , then X, e is said to be a complete L-lattice. In an L-poset X, e , for any x ∈ X, we usually define ↓x ∈ L as ↓x y e y, x for each y ∈ X and dually define ↑x y e x, y . For any A ∈ L , it is called an L-lower set if A x ∗ e y, x ≤ A y for all x, y ∈ X, or an L-upper set if A x ∗ e x, y ≤ A y for all x, y ∈ X. When A is an L-lower set, we have sub ↓x,A A x , and sub ↑x,A A x when A is an L-upper set. 4 Abstract and Applied Analysis For each map f : X → Y from a set X to an L-poset Y, e , there exist L-forward powerset operators f → L : L X → L defined as f → L A y ∨ f x yA x for all y ∈ Y,A ∈ L , and an L-backward powerset operator f← L : L Y → L defined as f← L B B ◦ f for each B ∈ L . Definition 2.3. Let X, eX , Y, eY be L-posets and f : X → Y be a map. Then 1 f is said to be L-order-preserving or L-monotone if eX x, y ≤ eY f x , f y for all x, y ∈ X; 2 f is an L-order-embedding if eX x, y eY f x , f y for all x, y ∈ X; 3 f is said to be fuzzy-join-preserving if for anyA ∈ L such that A exists, it implies that f → L A exists and f A f → L A ; 4 f is said to be fuzzy-meet-preserving if for any A ∈ L such that A exists, it implies that f → L A exists and f A f → L A . Definition 2.4. Let X, eX , Y, eY be L-posets, f : X → Y , g : Y → X be L-order-preserving maps. If eY f x , y eX x, g y for all x ∈ X, y ∈ Y , then f, g is called a fuzzy Galois connection between X and Y . f is called the left adjoint of g, and dually g is the right adjoint of f . It is worth noting that for anymap f : X → Y , there is a useful fuzzyGalois connection f → L , f ← L between L , sub and L , sub . Theorem 2.5. Let X, eX , Y, eY be L-posets, f : X, eX → Y, eY , and let g : Y, eY → X, eX be maps. 1 If X, eX is complete, then f is L-order-preserving and has a right adjoint if and only if f A f → L A for each A ∈ L ; 2 If Y, eY is complete, then g is L-order-preserving and has a left adjoint if and only if g B g← L B for each B ∈ L . Definition 2.6. Let X, eX be an L-poset, Y, eY a complete L-lattice. If there exists an L-orderembedding φ : X → Y , then Y, eY is said to be a completion of X via φ. Besides, if φ X is join-dense in Y refer to 13 , then we say that Y, eY is a join-completion of X via φ. For an L-poset X, e and A ∈ L , then we have A,A ∈ L which is called upper bound and lower bound of A, respectively.


Introduction
Fuzzy order was originated from the literature of Zadeh 1 and later was broadly studied. A fuzzy ordered set an L-poset is the fundamental idea of fuzzy ordered structures. Some people studied it from the fuzzy set theory such as 2-8 , and others considered it as a category see 9-11 . All these works enriched the theories of fuzzy orders and fuzzy ordered structures. At present, we are about to consider fuzzy ordered sets as objects in the framework of category. As we are all know that the researches on category D can be reflected on its reflective full subcategories whose structures are better than D. Searching for such subcategories provides a new approach to study primary category.
In recent years, the completion theory for fuzzy ordered sets attracts much attention. Wagner 9 introduced the enriched Dedekind-MacNeille completion for a category enriched over a commutative unital quantale. Bělohlávek 12 described the Dedekind-MacNeille completion for an L-ordered set as an application of the theory of concept lattices in the fuzzy setting. Xie et al. 13 built and characterized the Dedekind-MacNeille completions for L-posets. In 14 , Wang and Zhao proposed join-completions for L-ordered sets and investigated their characterizations. It was shown that each join-completion is in bijective correspondence with a consistent fuzzy closure operator C, and it has a universal property 2 Abstract and Applied Analysis with respect to C-homomorphisms. It also mentioned that the relationship between the categories of L-posets and complete L-lattices will be of concern in the future. This problem is closely related to the appropriate morphisms chosen, respectively. How about when Chomomorphisms endowed on L-posets and fuzzy-join-preserving mappings are chosen for complete L-lattices? This is our first motivation for this paper. After analysis, we discover that the latter category is a reflective full subcategory of the former with respect to a special class of fuzzy closure operators, but not for any one a counterexample is given . The special class of fuzzy closure operators is singled out in this paper. Moreover, when a completion has a universal property, we can declare that it is a join-completion. But there are so many join-completions, thus which type is it? This is the other motivation of this paper. We prove that it is exactly the Dedekind-MacNeille completion up to isomorphism. This paper is organized as follows. In Section 2, we list some basic definitions and well-known results for fuzzy order theory. In Section 3, we discuss the relationship between C-homomorphisms and fuzzy-join-preserving mappings then single out a special class of fuzzy closure operators such that the category of complete L-lattices is a reflective full subcategory of the category of L-posets. In Section 4, we characterize the Dedekind-MacNeille completions for L-posets and give an equivalent description for them. In Section 5, we summarize all the content and reach a conclusion.
Abstract and Applied Analysis 3 In this paper, L always denotes a complete residuated lattice unless otherwise stated, and L X denotes the set of all L-subsets of a nonempty set X. For all A, B ∈ L X , we define Then L X , * , → , ∨, ∧, 0, 1 is also a complete residuated lattice. If no confusion arises, we always do not discriminate the constant value function a with a, for example, a * A x a * A x and a → A x a → A x for every x ∈ X. Fuzzy order was first introduced by Zadeh 1 , from then on, different kinds of fuzzy order have been introduced and studied by different authors the reader is referred to 3, 6, 9, 11, 18-20 for details . In this paper, we adopt the definition of fuzzy order introduced by Fan and Zhang 3, 6 . 2 for all x, y, z ∈ X, e x, y * e y, z ≤ e x, z ; 3 for all x, y ∈ X, e x, y e y, x 1 ⇒ x y.
Then X, e is called an L-partially ordered set or an L-poset for simplicity. It is worth noting that Bělohlávek defined another fuzzy order in 12, 16 , and it was shown to be equivalent to the above definition by Yao 5 . There are some important L-posets which are mentioned in many papers, such as 3, 4, 19, 21 . For example, in an L-poset X, e , define e op x, y e y, x for all x, y ∈ X, then X, e op is also an L-poset. In a complete residuated lattice L, define e L : L × L → L by e L x, y x → y for all x, y ∈ L, then L, e L is an L-poset. For all A, B ∈ L X , define sub A, B x∈X A x → B x , then L X , sub is an L-poset. In the following, some basic and very important definitions and results related to the theory of L-poset are listed. The reader is referred to 2-9, 11-13, 20 for details. Definition 2.2. Let X, e be an L-poset, A ∈ L X , x 0 ∈ X. x 0 is called a fuzzy join resp., fuzzy meet of A denoted by x 0 A resp., It is easy to see that the fuzzy join or the fuzzy meet is unique in an L-poset X, e if it exists, and x 0 A resp., x 0 A if and only if e x 0 , y x∈X A x → e x, y resp., e y, x 0 x∈X A x → e y, x for all y ∈ X. If A and A exist for all A ∈ L X , then X, e is said to be a complete L-lattice.
In an L-poset X, e , for any x ∈ X, we usually define ↓ x ∈ L X as ↓ x y e y, x for each y ∈ X and dually define ↑ x y e x, y . For any A ∈ L X , it is called an L-lower set if A x * e y, x ≤ A y for all x, y ∈ X, or an L-upper set if A x * e x, y ≤ A y for all x, y ∈ X. When A is an L-lower set, we have sub ↓ x, A A x , and sub ↑ x, A A x when A is an L-upper set.

Abstract and Applied Analysis
For each map f : X → Y from a set X to an L-poset Y, e , there exist L-forward If e Y f x , y e X x, g y for all x ∈ X, y ∈ Y , then f, g is called a fuzzy Galois connection between X and Y . f is called the left adjoint of g, and dually g is the right adjoint of f.
It is worth noting that for any map f : X → Y , there is a useful fuzzy Galois connection f → L , f ← L between L X , sub and L Y , sub .
Theorem 2.5. Let X, e X , Y, e Y be L-posets, f : X, e X → Y, e Y , and let g : Y, e Y → X, e X be maps.
1 If X, e X is complete, then f is L-order-preserving and has a right adjoint if and only if

complete, then g is L-order-preserving and has a left adjoint if and only if
Definition 2.6. Let X, e X be an L-poset, Y, e Y a complete L-lattice. If there exists an L-orderembedding ϕ : X → Y , then Y, e Y is said to be a completion of X via ϕ. Besides, if ϕ X is join-dense in Y refer to 13 , then we say that Y, e Y is a join-completion of X via ϕ.
For an L-poset X, e and A ∈ L X , then we have A u , A l ∈ L X which is called upper bound and lower bound of A, respectively.

Reflective Full Subcategories of L-Posets
The aim of this section is to study the relationship between L-posets and complete L-lattices in categorical terms, which is closely related to the universal property of join-completions determined by a fuzzy closure operator. We refer to 22 for general category theory. Assume that the reader is familiar with the concepts of fuzzy closure operators and fuzzy closure systems. If C is a fuzzy closure operator on X, then there is an associated fuzzy closure system L C {A ∈ L X | C A A} and vice versa refer to 12, 23 for details .

Theorem 3.4 see 14 .
Let X, e X be a L-poset, C a consistent fuzzy closure operator on X, and Y, e Y a completion of X via ϕ. Then, the following are equivalent: 1 Y, e Y is a join-completion of X via ϕ; 2 ϕ is a C-homomorphism. Moreover, for each complete L-lattice Z, e Z and each Chomomorphism f : X → Z, there exists a unique fuzzy-join-preserving mapping f : Considering Theorem 3.4, we do want to know that whether the category of complete L-lattices with fuzzy-join-preserving mappings, in symbol L-Sup, is a reflective full subcategories of the category whose objects are L-posets and arrows are C-homomorphisms associated with a consistent fuzzy closure operator C, which is denoted by L-POS C . In fact, the answer is positive for some special consistent fuzzy closure operators. But as in Example 3.5, when C ↓, there exists some C-homomorphisms on complete L-lattices but not fuzzy-join-preserving. So L-Sup is not a full subcategory of L-POS ↓ . Sequentially, it is not a reflective full subcategory of L-POS ↓ neither. It easily checked that f is a ↓-homomorphism, but it is not fuzzy-join-preserving.
In the sequel, we try to seek for those special fuzzy closure operators C in order to give some reflective full subcategories for the category of L-posets. Definition 3.6. Let X, e be an L-poset and C a fuzzy closure operator on X. Then, C is called an essential fuzzy closure operator if C A ↓ A for all A ∈ L X with A exists.
Note that an essential fuzzy closure operator is also a consistent fuzzy closure operator, not vice versa. The following proposition shows that "essential" is a sufficient and necessary condition for the equivalence of C-homomorphisms and fuzzy-join-preserving mappings on complete L-lattices. Proposition 3.7. Let X, e X , Y, e Y be complete L-lattices, and let f a mapping from X to Y , C be a consistent fuzzy closure operator on X. Then

Assume that C is essential, we have f is a C-homomorphism if and only if it is fuzzy-joinpreserving;
2 Assume that f is a C-homomorphism if and only if it is fuzzy-join-preserving, then C is essential.

Proof. 1 Suppose that C is essential. If f is an C-homomorphism, then f is L-orderpreserving by Lemma 4.3 in 14 . For any
Conversely, if f is fuzzy-joinpreserving. Then for any y ∈ Y , it is easy to check that f ← L ↓ y ≤ ↓ f ← L ↓ y at first. Furthermore, As shown in the proof of 1 , we can see that ↓ B A 1.
Then f cannot be fuzzyjoin-preserving, and it contradicts with premise. Thus C is essential.

Corollary 3.8. (1) If C is an essential fuzzy closure operator, then L-Sup is a full subcategory of L-POS(C).
(2) If L-Sup is a full subcategory of POS(C) associated with a consistent fuzzy closure operator, then C is essential. Proposition 3.9. Let X, e be an L-poset and C an essential fuzzy closure operator on X. Then there exists a join-completion Y, e of X via an C-homomorphism ϕ. Moreover, for each complete Llattice Z, e and C-homomorphism f : X, e → Z, e , there exists an unique fuzzy-join-preserving mapping f : Y, e → Z, e such that the following diagram commutes: Proof. Since C is an essential fuzzy closure operator, then C is a consistent fuzzy closure operator. By the Literature 14 , we can see that L C , sub is a join-completion of X via Chomomorphism ι. Thus the result follows immediately from Theorem 3.4.

Theorem 3.10. (1) If C is an essential fuzzy closure operator, then L-Sup is a reflective full subcategory of L-POS(C).
(2) If L-Sup is a reflective full subcategory of L-POS(C) associated with a consistent fuzzy closure operator, then C is essential.
Proof. It follows from Corollary 3.8 and Proposition 3.9.
Remark 3.11. For an L-poset X, e , let C ul . By Lemma 2.5 3 in 13 , we can see that C A ↓ A. So C ul is an essential fuzzy closure operator, and its associated completion is exactly the Dedekind-MacNeille completion DM L X , sub via C-homomorphism ι it is called lowercontinuous mapping in Section 4 . Thus L-Sup is a reflective full subcategory of L-POS ul .
It is easy to see that DM op L X, e can be viewed as DM L X, e op . With the duality of the L-order, when mappings dual to C-homomorphism are chosen to serve as morphisms on L-posets, the corresponding morphisms on complete L-lattices need to be changed to fuzzymeet-preserving mappings. Then the dual result is obtained immediately. We list it in the following and leave the proof to the reader. Let L-Inf denote the category of complete L-lattices with fuzzy-meet-preserving mappings, and the L-Pos denote the category whose objects are all L-posets and arrows are upper-continuous mappings.

The Characterizations of the Dedekind-MacNeille Completions for L-Posets
From previous works, we have known that for an L-poset, each join-completion has a universal property, and any completion with a universal property must be a join-completion.

Abstract and Applied Analysis
In this section, we study the characterizations of one special join-completion-the Dedekind-MacNeille completion and obtain a stronger result that a complete L-lattice with a universal property is exactly the Dedekind-MacNeille completion up to isomorphism.
sub A, G B , it implies that F, G forms a fuzzy Galois connection. By Theorem 2.5, F is fuzzy-join-preserving.
Next, to show that the diagram commutes amount to showing that f → In the other direction, since f → L ↓ X x ul is an L-lower set, so for any y ∈ Y , we as needed. Finally, we show that F is unique. Suppose that there is another fuzzy-join-preserving mapping F * satisfying the diagram commutes; we show that F A F * A for all A ∈ DM L X . Since ι X X is join-dense in DM L X , it suffices to show that F ↓ x F * ↓ x for all x ∈ X. That is obvious since the diagram commutes. Proof. By Theorem 4.2, there exists a commutative diagram as follows, and the mapping ι Y is an L-order-isomorphism since Y is a complete L-lattice. Let f ι −1 Y • F, then f is a unique fuzzy-join-preserving mapping such that the following diagram commutes:  There exists a lower-continuous mapping ϕ : X → Y such that for every lower-continuous mapping f : X → Z of X into a complete L-lattice Z, e Z , there exists a unique fuzzy-joinpreserving mapping f : Y → Z such that the following diagram commutes: Proof. Suppose that Y is isomorphic to DM L X , then there exists an L-order-isomorphism g : DM L X → Y . We define ϕ : X → Y as ϕ g • ι X , then it is easily checked that ϕ is a lower-continuous mapping. If f : X → Z is a lower-continuous mapping of X into complete L-lattice Z, e Z . By Corollary 4.3, there exists a unique fuzzy-join-preserving mapping h : For the second part, assume that the condition holds. By Corollary 4.3, the diagrams below commute:

10
Abstract and Applied Analysis By the uniquess condition, ϕ • ι X id DM L X and ι X • ϕ id Y . Thus Y is isomorphic to DM L X .
Since DM op L X, e can be regarded as DM L X, e op , when we replace the lowercontinuous mapping by upper-continuous mapping and fuzzy-join-preserving mapping by fuzzy-meet-preserving mapping, then the dual results of Theorem 4.2 and Corollary 4.3 are obtained naturally. Sequentially, the following observations are immediate, and the proofs are trivial, so we omit them.

Conclusion
In this paper, we consider the relationship between L-posets and complete L-lattices in the categorical framework. After analysis of the characterizations of the join-completions, we prove that the category of complete L-lattices with fuzzy-join-preserving mappings is a reflective full subcategory of the category of L-posets with C-homomorphisms related to an essential fuzzy closure operator. Furthermore, we characterize the Dedekind-MacNeille completions. It is shown that they are the unique free objects up to isomorphism generated by an L-poset in the category of complete L-lattices. It offers an equivalent description for them. Maybe we will continue to focus on the completion theory for fuzzy ordered sets, we will try to propose and characterize other completions for L-posets in the future.