Reflective Full Subcategories of the Category of L-Posets

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.