In this paper, we introduce the notion of Alexandrov L-(neighborhood) filters and Alexandrov L-convergence structures as a topological viewpoint of fuzzy rough sets. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures. Moreover, we investigate their topological properties and give examples.
Ward et al. [22] introduced a complete residuated lattice which is an algebraic structure for many valued logic. It is an important mathematical tool as algebraic structures for many valued logics [1, 10–13].
Pawlak [16, 17] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. For an extension of classical rough sets, many researchers [10, 18–21] developed L-lower and L-upper approximation operators in complete residuated lattices. By using this concepts, information systems and decision rules were investigated in complete residuated lattices [1].
An interesting and natural research topic in rough set theory is the study of rough set theory and topological structures. Lai [13] and Ma [14] investigated the Alexandrov L-topology and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Kim [10, 11] studied the relations between L-fuzzy upper and lower approximation spaces and Alexandrov L-topologies in complete residuated lattices. Moreover, categories of fuzzy preorders, approximating operators and Alexandrov topologies are isomorphic [11].
Jäger [7–9, 15] developed stratified L-convergence structures based on the concepts of L-filters where L is a complete Heyting algebra. Yao [24] extended stratified L-convergence structures to complete residuated lattices and investigated between stratified L-convergence structures and L-fuzzy topological spaces.
Zhang [23] defined a strong L-topology on the concepts of fuzzy complete lattices. As an extension of Yao [24], Fang [2, 3] introduced L-ordered convergence structures on L-ordered filters and investigated between L-ordered convergence structures and strong L-topological spaces.
In this paper, we introduce the notion of Alexandrov L-(neighborhood) filters as an extension of Fang’s L-ordered filters [3]. It is motivated by the topological structure of fuzzy information and fuzzy rough sets in a complete residuated lattice. In section 2, some basic concepts and notions used in this paper are listed. In section 3, we define Alexandrov L-(neighborhood) filters. We investigate the categorical relations between topologicla Alexandrov L-neighborhood filters and L-fuzzy preorders. In section 3, we define Alexandrov L-convergence structures. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures. Moreover, we investigate their topological properties and give examples.
Preliminaries
Definition 2.1. [1, 22] An algebra (L, ∧ , ∨ , ⊙ , → , ⊥ , ⊤) is called a complete residuated lattice if it satisfies the following conditions:
(L1) (L, ≤ , ∨ , ∧ , ⊥ , ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;
(L2) (L, ⊙ , ⊤) is a commutative monoid;
(L3) x ⊙ y ≤ z iff x ≤ y → z for x, y, z ∈ L.
In this paper, we always assume that (L, ≤ , ⊙ , → , ∗) is complete residuated lattice with x* = x→ ⊥ and (x*) * = x for each x ∈ L.
For α ∈ L, A ∈ LX, we denote (α → A) , (α ⊙ A) , αX ∈ LX as (α → A) (x) = α → A (x) , (α ⊙ A) (x) = α ⊙ A (x) , αX (x) = α.
Lemma 2.2. [1, 22] For eachx, y, z, xi, yi, w ∈ L, we have the following properties.
(1) ⊤ → x = x, ⊥ ⊙ x = ⊥ ,
(2) If y ≤ z, then x ⊙ y ≤ x ⊙ z, x → y ≤ x → z and z → x ≤ y → x,
(3) x ≤ y iff x→ y = ⊤.
(4) x → (⋀ iyi) = ⋀ i (x → yi),
(5) (⋁ixi) → y = ⋀ i (xi → y),
(6) x ⊙ (⋁iyi) = ⋁ i (x ⊙ yi),
(7) (x ⊙ y) → z = x → (y → z) = y → (x → z),
(8) (x → y) ⊙ (z → w) ≤ (x ⊙ z) → (y ⊙ w) and x → y ≤ (x ⊙ z) → (y ⊙ z),
(11) x → y ≤ (y → z) → (x → z) and x → y ≤ (z → x) → (z → y).
(12) (x ⊙ y*) * = x → y and x → y = y* → x*.
Definition 2.3. [1, 13] Let X be a set. A function eX : X × X → L is called:
(E1) reflexive if eX (x, x) =⊤ for all x ∈ X,
(E2) transitive if eX (x, y) ⊙ eX (y, z) ≤ eX (x, z), for all x, y, z ∈ X,
(E3) if eX (x, y) = eX (y, x) =⊤, then x = y.
If e satisfies (E1) and (E2), (X, eX) is an L-fuzzy preordered set. If eX satisfies (E1), (E2) and (E3), (X, eX) is an L-fuzzy partially ordered set.
Example 2.4. (1) We define a function eL : L × L → L as eL (x, y) = x → y. Then (L, eL) is an L-fuzzy partially ordered set.
(2) We define a function eLX : LX × LX → L as eLX (A, B) = ⋀ x∈X (A (x) → B (x)) . Then (LX, eLX) is an L-fuzzy partially ordered set from Lemma 2.2 (9).
Definition 2.5. [23, 26] Let (X, eX) be an L-fuzzy partially ordered set and A ∈ LX.
(1) A point x0 is called a join of A, denoted by x0 = ⊔ A, if it satisfies
(J1) A (x) ≤ eX (x, x0),
(J2) ⋀x∈X (A (x) → eX (x, y)) ≤ eX (x0, y).
A point x1 is called a meet of A, denoted by x1 = ⊓ A, if it satisfies
(M1) A (x) ≤ eX (x1, x),
(M2) ⋀x∈X (A (x) → eX (y, x)) ≤ eX (y, x1).
Remark 2.6. Let (X, eX) be an L-fuzzy partially ordered set and Φ ∈ LX.
(1) If x0 is a join of Φ, then it is unique because eX (x0, y) = eX (y0, y) for all y ∈ X, put y = x0 or y = y0, then eX (x0, y) = eX (y0, y) =⊤ implies x0 = y0. Similarly, if a meet of Φ exist, then it is unique.
(2) A point x0 is a join of Φ iff ⋀x∈X (Φ (x) → eX (x, y)) = eX (x0, y).
(3) A point x1 is a meet of Φ iff ⋀x∈X (Φ (x) → eX (y, x)) = eX (y, x1).
Remark 2.7. Let (L, eL) be an L-fuzzy partially ordered set and A ∈ LL.
(1) Since x0 is a join of A iff ⋀x∈L (A (x) → eL (x, y)) = ⋁ x∈L (x ⊙ A (x)) → y = eL (x0, y), x0 = ⊔ A = ⋁ x∈L (x ⊙ A (x)) .
(2) Since x0 is a join of A iff ⋀x∈L (A (x) → eL (x, y) = y → ⋀ x∈L (A (x) → x) = y → ⊓ A, ⊓A = ⋀ x∈L (A (x) → x) .
Remark 2.8. Let (LX, eLX) be an L-fuzzy partially ordered set and Φ ∈ LLX.
(1) ⊔Φ = ⋁ A∈LX (Φ (A) ⊙ A) from:
(2) ⊓Φ = ⋀ A∈LX (Φ (A) → A) from:
Definition 2.9. [23, 26] Let (X, eX) be an L-fuzzy partially ordered set. The pair (X, eX) is called a fuzzy join (resp. meet) complete lattice if ⊔Φ (resp. ⊓Φ) exists for each Φ ∈ LX.
The pair (X, eX) is called a fuzzy complete lattice if ⊔Φ and ⊓Φ exist for each Φ ∈ LX.
Definition 2.10. [23, 26] Let (X, eX) and (Y, eY) be fuzzy complete lattices and ψ : X → Y a map.
(1) ψ is a join preserving map if ψ (⊔ Φ) = ⊔ ψ→ (Φ) for all Φ ∈ LX, where ψ→ (Φ) (y) = ⋁ ψ(x)=yΦ (x).
(2) ψ is a meet preserving map if ψ (⊓ Φ) = ⊓ ψ→ (Φ) for all Φ ∈ LX.
(3) ψ is an order preserving map if eX (x, y) ≤ eY (ψ (x) , ψ (y)) for all x, y ∈ X.
Alexandrov L-filters and Alexandrov L-neighborhood spaces
We introduce the new notion of Alexandrov L-filters as an extension of Fang’s L-ordered filters [3]. We investigate the relations between topological Alexandrov L-neighborhood filters and L-fuzzy preorders.
Definition 3.1. Let (LX, eLX) and (L, eL) be L-fuzzy partially ordered sets. A map is called an Alexandrov L-filter on X iff for all Φ ∈ LLX. Let AF (X) denote the set of all Alexandrov L-filters on X.
Theorem 3.2.A map is an Alexandrov L-filter on X iff it satisfies the following conditions:
(F1) for all Ai ∈ LX.
(F2) for all A ∈ LX and α ∈ L.
Proof (⇒) For all Φ ∈ LLX, we have from:
(F1) Define Φ : LX → L as Φ (Ai) =⊤ for i ∈ Γ and Φ (B) =⊥, otherwise. Then
So, ⊓Φ = ⋀ i∈ΓAi.
(F2) Define Φ : LX → L as Φ (A) = α for A ∈ LX and Φ (B) =⊥, otherwise. Then
(⇐) For all Φ ∈ LLX, from:
Theorem 3.3.A map is an Alexandrov L-filter on X iff there exists B ∈ LX such that for all A ∈ LX.
Proof (⇒) For all A ∈ LX, since , by Theorem 3.2 (F1) and (F2), we have . Put . Then .
(⇐) From Theorem 3.2, it is easily proved.
Remark 3.4. Let (LX, eLX) and (L, eL) be L-fuzzy partially ordered sets. From Definition 3.1, we can redefine an L-ordered filter as follows. A map is an L-ordered filter on X in a Fang’s sense [3] iff and for a finite family Φ ∈ LLX.
Theorem 3.5.If is an Alexandrov L-filter on X, then, for all A, B ∈ LX, .
Proof. For A ≤ B, . So, It follows . Put α = eLX (A, B). Then implies .
Theorem 3.6.LetAF (X) denote the set of all Alexandrov L-filters on X. Define eAF(X) : AF (X) × AF (X) → L as
Then (AF (X) , eAF(X)) is a meet-complete lattice.
Proof. We easily prove that (AF (X) , eAF(X)) is an L-fuzzy partially ordered set. For each Φ : AF (X) → L, put .
(F1) For all Ai ∈ LX,
(F2) For all A ∈ LX and α ∈ L,
Hence ⊓Φ is an Alexandrov L-filter on X. Moreover, ⊓Φ is a meet of Φ from:
Theorem 3.7.LetAF (X) denote the set of all Alexandrov L-filters on X. Then (AF (X) , eAF(X)) is a meet-complete lattice iff (1) for each
(2) for each , i ∈ Γ.
Proof. (⇒) (1) Define Φ : AF (X) → L as for and , otherwise. Then
(2) Let be given. Define Φ : AF (X) → L as for i ∈ Γ and , otherwise. Then
(⇐) For Φ : AF (X) → L,
Hence (AF (X) , eAF(X)) is a meet-complete lattice.
Remark 3.8. (1) A map [x] : (LX, eLX) → (L, eL) defined by [x] (A) = A (x) is an Alexandrov L-filter on X. Moreover,
By Theorem 3.7, a map (α → [x]) : (LX, eLX) → (L, eL) defined by (α → [x]) (A) = α → A (x) is an Alexandrov L-filter on X. The converse of Theorem 3.5 is not true from:
but α ⊙ [x] is not an Alexandrov L-filter on X.
(2) A map defined by is an Alexandrov L-filter on X. A map is an Alexandrov L-filter on X
(3) Put R ∈ LX×Y, for each x ∈ X, a map defined by is an Alexandrov L-filter on X from Lemma 2.2 (4,7).
Example 3.9. Let X = {hi ∣ i = {1,. . . , 3}} with hi=house and Y = {e, b, w, c, i} with e=expensive, b= beautiful, w=wooden, c= creative, i=in the green surroundings. Let ([0, 1] , ⊙ , → , *, 0, 1) be a complete residuated lattice (ref. [1, 19]) as x* = 1 - x,
Let R ∈ [0, 1] X×Y be a fuzzy information as follows:
A map defined by is an Alexandrov L-filter on X. For A = (0.3, 0.5, 0.6, 0.1, 0.1),
Definition 3.10. A family is called an Alexandrov L-neighborhood system on X if a map satisfies:
(N1) is an Alexandrov L-filter on X.
(N2) for all A ∈ LX.
The pair is called an Alexandrov L-neighborhood space. An Alexandrov L-neighborhood system on X is topological if (TN) for all such that for all y ∈ X.
Let and be Alexandrov L-neighborhood spaces. is called an N-continuos map if, for each B ∈ LY and x ∈ X,
Theorem 3.11.Letx* = x→ ⊥ and (x*) * = x for each x ∈ L. (1) For each x ∈ X, a map is an Alexandrov L-neighborhood filter on X iff there exists a reflexive relation eN ∈ LX×X such that for all A ∈ LX.
(2) For each x ∈ X, a map is a topological Alexandrov L-neighborhood filter on X iff there exists an L-fuzzy preorder eN ∈ LX×X such that for all A ∈ LX.
Proof. (1) (⇒) For all A ∈ LX, since , by Theorem 3.2 (F1) and (F2), we have
Put . Then So, eN is reflexive. Moreover, .
(⇐) From Theorem 3.2, it satisfies (F1),(F(2) and
(2) (⇒) By (1),
Put . Then ⋁y∈X (eN (x, y) ⊙ eN (y, z) ≤ eN (x, z);i.e. eN is transitive. By (1), eN is an L-fuzzy preorder.
(⇐) For each x ∈ X, is topological Alexandrov L-neighborhood filter from (1) and for each A ∈ LX,
Theorem 3.12.Let (X, eX) be an L-fuzzy preorder set. Define as
Then (1) For all x, y ∈ X, and .
(2) is a topological Alexandrov L-neighborhood system on X.
Proof. (1) For all x, y ∈ X, . For each A ∈ LX,
(2) (N1) For each Φ : LX → L, by Remark 2.6 (3),
So,
(N2) Since , , for all A ∈ LX.
(TN) For all A ∈ LX,
Hence is a topological Alexandrov L-neighborhood filter on X
Theorem 3.13.Let be an Alexandrov L-fuzzy neighborhood space. Define as
(1) , for all x, y ∈ X and .
(2) .
(3) If is topological, then and .
(4) If eX ∈ LX×X is reflexive, then .
(5) If eX ∈ LX×X is an L-fuzzy preorder, then .
Proof. (1) For , we have
Moreover, put . Then Since from Theorem 3.11, we have (x, z)) ≤ eN (x, y) .
(2) By (1),
(3) Let be topological. By Theorem 3.11, eN is an L-fuzzy preorder. By (1), . Hence and .
(4) Let eX ∈ LX×X be reflexive. Then is an Alexandrov L-neighborhood structure. By (1), (x, y).
(5) It follows from (4) and .
Remark 3.14. (1) A family is a topological Alexandrov L-neighborhood system on X because [x] ([-] A) = [-] A (x) = [x] (A). By Theorem 3.13,
(2) A family is a topological Alexandrov L-neighborhood system on X because .
(3) Let eX ∈ LX×X be reflexive. A family is an Alexandrov L-neighborhood system on X.
Example 3.15. Let ([0, 1] , ⊙ , → , *, 0, 1) be a complete residuated lattice as in Example 3.9. Let X = {x, y, z} be a set and a reflexive L-fuzzy relation eX ∈ [0, 1] X×X as
where eX ∘ eX (x, z) = ⋁ y∈X (eX (x, y) ⊙ eX (y, z)). Then eX is not an L-fuzzy preorder. Define as
By Theorem 3.10(1), is an Alexandrov L-neighborhood filter for each x ∈ X, but it is not topological because, for A = (0.7, 1, 0.1),
Since eX is not an L-fuzzy preorder, as follows:
Since is not topological, for B = (1, 0.5, 1),
Theorem 3.16.(1) Let (X, eX) and (Y, eY) be L-fuzzy preordered sets. If ψ : (X, eX) → (Y, eY) is an order preserving map, then is an N-continuous map.
(2) Let and be topological Alexandrov L-neighborhood spaces. If is an N-continuous map, then is an order preserving map.
(3) The converses of (1) and (2) are true.
Proof. (1) For each B ∈ LY,
(2) From Theorem 3.13(3),
(3) From Theorem 3.13 (3,4), since and , the converses of (1) and (2) are true.
Let POS be denote the category of L-fuzzy preordered sets and order preserving maps for morphisms. Let TANS be denote the category of topological Alexandrov L-neighborhood spaces and N-continuous maps for morphisms.
Theorem 3.17.Two categories POS and TANS are isomorphic.
Proof. Define F:POS→ TANS as . Hence F is a functor from Theorem 3.16. Define a functor G : TANS → POS as . Hence G is a functor.
From Theorem 3.13, since G (F (X, eX)) = G (X, and . Thus, POS and TANS are isomorphic.
Alexandrov L-convergence spaces
In this section, we introduce the new notion of Alexandrov L-convergence structure on Alexandrov L-filters. We investigate the categorical relations among Alexandrov L-neighborhood filters, L-fuzzy preorders and Alexandrov L-convergence structures.
Definition 4.1. A map lim : AF (X) → LX is called an Alexandrov L-convergence structure on X iff it satisfies the following conditions:
(AC1) lim [x] (x) =⊤.
(AC2) for all .
Let AC (X) denote the set of all Alexandrov L-convergence structures on X.
A map is called continuous if for all and x ∈ X where for all B ∈ LY.
Lemma 4.2.Let lim : AF (X) → LX be a map. The following statements are equivalent.
(1) for all .
(2) If , then and for all and α ∈ L.
Proof. (1) ⇒ (2). If , then . Hence . Put . Then Hence
(2) ⇒ (1). Since ,
. Hence
Theorem 4.3.LetAC (X) denote the set of all Alexandrov L-convergence structures on X. Then (1) α → lim ∈ AC (X) for each lim ∈ AC (X)
Theorem 4.4.LetAC (X) denote the set of all Alexandrov L-convergence structures on X. Define eAC(X) : AC (X) × AC (X) → L as
Then (AC (X) , eAC(X)) is a meet-complete lattice.
Proof. We easily prove that (AC (X) , eAC(X)) is an L-fuzzy partially ordered set. For Φ : AC (X) → L, since ⋀lim∈AC(X) (Φ (lim) → lim) ∈ AC (X) from Theorem 4.3, we have
Hence ⊓Φ ∈ AC (X) .
Remark 4.5. Let AC (X) denote the set of all Alexandrov L-convergence structures on X. Define Φ : AC (X) → L as Φ (lim) = α for lim ∈ AC (X) and , otherwise. Then
Hence (AC (X) , eAC(X)) is not a join-complete lattice.
Theorem 4.6.Let be an Alexandrov L-neighborhood system on X. Define as
Then (1) is an Alexandrov L-convergence structure on X.
(2) Define Φx : AF (X) → L as
Then is the system of Alexandrov L-neighborhood filters such that .
Proof. (1) (AC1)
(AC2) For all ,
(2) For Φx : AF (X) → L as ,
Since ,
We have from:
Theorem 4.7.Let be an Alexandrov L-convergence space. Define as
Then the following properties hold.
(1) is an Alexandrov L-neghborhood filter on X.
(2) .
(3) .
Proof. (1) By Theorem 3.2, is an Alexandrov L-neghborhood filter on X from:
(2) For each A ∈ LX, from:
(3) For each ,
Definition 4.8. An Alexandrov L-convergence structure is called principal if
(P) for all .
If is topological, then is called topological.
Remark 4.9. Let be an Alexandrov L-neighborhood system on X. By Theorem 4.6, is a pricipal Alexandrov L-convergence structure on X. If is topological, then is topological.
Theorem 4.10.Let be an Alexandrov L-convergence space. Define
Then (1) is an L-fuzzy preorder on X.
(2) If is principal, then .
(3) If is topological principal, then, for all x, y ∈ X,
(4) If is a topological Alexandrov L-neighborhood filter on X, then for all x, y ∈ X.
(5) If eX is an L-fuzzy preorder on X and define
then is a topological principal Alexandrov L-convergence space such that
(6) If is principal, then . Moreover, if is topological principal, then .
Proof. (1) (E1) (E2)
Hence is an L-fuzzy preorder on X.
(2)
(3) and (4) follow from Theorem 3.13(3).
(5) is a principal Alexandrov L-convergence structure from:
By Theorem 3.13(5), .
(6) For each x, y ∈ X,
By (5),
Let be topological principal. By (2), and .
Remark 4.11. (1) Define as
Then is a principle Alexandrov L-convergence structure from
Moreover, (A) → [x] (A)) = ▵ X×X (x, y) .
(2) Define as
Then is a principle Alexandrov L-convergence structure from
Moreover, A (z) → ⋀ z∈XA (z)) = ⊤ .
(3) Let eX ∈ LX×X be reflexive. Define as
Then is a principle Alexandrov L-convergence structure from
Moreover, ≤eX (x, y). If eX is transitive, .
If eX is transitive,
Theorem 4.12.(1) Let and be Alexandrov convergence spaces. If is a continuous map, then is an N-continuous map.
(2) Let and be Alexandrov L-neighborhood spaces. If is an N-continuous map, then is a continuous map.
Proof. (1) For each B ∈ LY,
(2) For each ,
Theorem 4.13.(1) Let and be a topological principal Alexandrov convergence spaces. If is a continuous map, then is an order preserving map.
(2) Let (X, eX) and (Y, eY) be L-fuzzy preorder spaces. If ψ : (X, eX) → (Y, eY) is an order preserving map, then is a continuous map.
Proof. (1) Since and are topological, by Theorem 3.13(3),
(2) From Theorem 3.16, for each B ∈ LY, Thus
Definition 4.14. [1] Suppose that are concrete functors. The pair (F, G) is called a Galois correspondence between and if for each idY : F ∘ G (Y) → Y is a -morphism, and for each , idX : X → G ∘ F (X) is a -morphism.
If (F, G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F .
Let ACS (resp.TPACS) denote the category of (resp. topological principal) Alexandrov L-convergence spaces and continuous mappings for morphisms. Let ANS denote the category of Alexandrov L-neighborhood spaces continuous mappings for morphisms.
Theorem 4.15.(1) F : ACS → ANS defined as is a functor.
(2) G : ANS → ACS defined as is a functor.
(3) The pair (F, G) is a Galois correspondence between ANS and ACS.
(4) Two categories TPACS and POS are isomorphic.
Proof. (1) and (2) are follows from Theorems 4.6, 4.7 and 4.12.
(3) By Theorem 4.7(2), if is an Alexandrov L-neighborhood space, then . Hence, the identity map is a continuous map. Moreover, if is an Alexandrov L-convergence space, by Theorem 4.7(3), . Hence the identity map is a continuous map. Therefore (F, G) is a Galois correspondence.
(4) Define H : POS → TPACS as . Define a functor K : TPACS → POS as . By Theorem 4.13, H and K are functors. Since from Theorem 4.10(5) and from Theorem 4.10(6). Thus, POS and TPACS are isomorphic.
Example 4.16. Let ([0, 1] , ⊙ , → , *, 0, 1) be a complete residuated lattice (ref. [1, 19]) as
Let X = {x, y, z} be a set and an L-fuzzy preorder eX ∈ [0, 1] X×X as
Define an Alexandrov L-neighborhood filter as
(TN) Since eX ∘ eX = eX, we have
Hence is a topological Alexandrov L-neighborhood system. We obtain:
Hence is a topological principal Alexandrov convergence structure. By Theorem 4.10(4),
Conclusion
In this paper, as a viewpoint for fuzzy rough sets, we can consider Alexandrov L-(neighborhood) filters, L-fuzzy preorders and Alexandrov L-convergence structures. In section 2, the category of L-fuzzy preordered sets and that of topological Alexandrov L-neighborhood spaces are isomorphic. In section 3, there exists a Galois correspondence between the category of Alexandrov L-convergence spaces and that of Alexandrov L-neighborhood spaces. Moreover, the category of L-fuzzy preordered sets and that of topological principal Alexandrov L-convergence spaces are isomorphic. Moreover, we investigate the their topological properties and give their examples.
In the future, we study Alexandrov L-uniform convergence spaces and Alexandrov L-fuzzy topological spaces.
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