In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces based on the concepts of Alexandrov L-filters on X × X. We investigate the categorical relations among Alexandrov L-preuniform convergence spaces, Alexandrov L-preconvergence spaces and reflexive L-fuzzy relations. Moreover, we investigate their topological properties and give their examples as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets.
Höhle [9] introduced an L-filter which is an important mathematical tool as fuzzy topological structures for many valued logics [1–13, 23–28]. Jäger [10–13] introduced stratified L-convergence structures based on the concepts of L-filters where L is a complete Heyting algebra. Yao [28] extended stratified L-convergence structures to complete residuated lattices and investigated the relations between stratified L-convergence structures and L-fuzzy topological spaces. Zhang [27, 32] defined a strong L-topology on the concepts of fuzzy complete lattices. As an extension of Yao [28], Fang [5–8] introduced L-ordered convergence structures on L-ordered filters and investigated the relations between L-ordered convergence structures and strong L-topological spaces. Many researchers [2–8, 23–26] developed the properties of L-convergence structures.
Jäger and Burton [13] introduced stratified L-uniform convergence spaces in a complete Heyting algebra. Fang [6–8] generalized stratified L-semiuniform (L-ordered semiuniform, L-ordered quasiuniform, preuniform) convergence spaces in commutative unital quantales [9].
For an extension of Pawlak’s rough set [22], many researchers [1, 19] developed L-lower and L-upper approximation operators. By using these concepts, information systems and decision makings were investigated [14, 30]. Kim et al. [3, 17] introduced the notion of Alexandrov L-(neighborhood) filters defined as a meet preserving map in a Zhang’s fuzzy complete lattice sense [27, 32]. These filters have stronger conditions than L-filters and are suitable to fuzzy rough sets, Alexandrov L-topologies and Alexandrov L-preconvergence structures.
In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces based on the concepts of Alexandrov L-filters on X × X. In Theorems 3.3 and 3.4, we show that the set of all Alexandrov L-preuniform convergence structures on X × X is a meet-complete lattice in a Zhang’s fuzzy complete lattice sense [27, 32] as a generalization of a complete lattice. In Theorems 3.6, 3.14 and 3.17, there exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of reflexive L-relations. In Theorems 3.9, 3.11, 3.13 and 3.16, there exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of Alexandrov L-convergence spaces. We investigate their topological properties and give their examples as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.
Preliminaries
Definition 2.1. [1, 9] 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 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 and [x] (A) = A (x).
Lemma 2.2. [1, 9] For each x, 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, 9] 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) anti-symmetric 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.
Definition 2.4. [27, 32] 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 2.5. [27, 32] Let (X, eX) be an L-fuzzy partially ordered set and Φ ∈ LX. (1) A point x0 is a join of Φ iff ⋀x∈X (Φ (x) → eX (x, y)) = eX (x0, y). (2) A point x1 is a meet of Φ iff ⋀x∈X (Φ (x) → eX (y, x)) = eX (y, x1).
Definition 2.6. [27, 32] 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 2.7. [17, 32] 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.
Definition 2.8. [3, 17] Let (LX, eLX) and (L, eL) be L-fuzzy partially ordered sets. A map eLX) → (L, eL) 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 2.9. [3, 17] 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.
Definition 2.10. [3, 17] A family is called an Alexandrov L-neighborhood system on X if, for x ∈ X, 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. A map is called N-continuous if, for each B ∈ LY and x ∈ X,
Definition 2.11. [3, 17] A map lim : AF (X) → LX is called an Alexandrov L-preconvergence structure on X iff it satisfies the following conditions: (AC1) lim [x] (x) =⊤. (AC2) for all . An Alexandrov L-preconvergence structure lim is called Alexandrov L-convergence structure on X if it satisfies: (A) ⋁y∈X (lim [y] (x) ⊙ lim [z] (y)) = lim [z] (x) for all x, y ∈ X. Let AC (X) denote the set of all Alexandrov L-preconvergence structures on X. A map is called continuous if for all and x ∈ X where for all B ∈ LY.
Definition 2.12. [3, 17] An Alexandrov L-preconvergence structure is called principal if there exists an Alexandrov L-neighborhood system such that (P) for all .
Alexandrov L-preuniform convergence spaces
Definition 3.1. A map UX : AF (X × X) → L is called an Alexandrov L-preuniform convergence structure on X × X iff it satisfies the following conditions:
(AU1) UX ([(x, x)]) =⊤.
(AU2) for all .
An Alexandrov L-preuniform convergence structure UX is called an Alexandrov L-quasiuniform convergence structure if UX ([(x, y)]) ⊙ UX ([(y, z)]) ≤ UX ([(x, z)]) for each x, y, z ∈ X.
Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X.
A map ψ : (X, UX) → (Y, UY) is called uniformly continuous if for all where for all w ∈ LY×Y.
Lemma 3.2.Let UX : AF (X × X) → L 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 3.3.Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Then (1) α → U ∈ AU (X) for each U ∈ AU (X)
(2) ⋀i∈ΓUi, ⋁ i∈ΓUi ∈ AU (X) for each Ui ∈ AU (X), i ∈ Γ.
Theorem 3.4.Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Define eAU(X) : AU (X) × AU (X) → L as
Then (AU (X), eAU(X)) is a meet-complete lattice.
Proof. We easily prove that (AU (X), eAU(X)) is an L-fuzzy partially ordered set. For Φ : AU (X) → L, since ⋀U∈AU(X) (Φ (U) → U) ∈ AU (X) from Theorem 3.3, we have
Hence ⊓Φ = ⋀ U∈AU(X) (Φ (U) → U) ∈ AU (X) . □
Remark 3.5. Let AU (X) denote the set of all Alexandrov L-preuniform convergence structures on X × X. Define Φ : AU (X) → L as Φ (U) = α for U ∈ AU (X) and Φ (U1) =⊥, otherwise. Then
Hence (AU (X), eAU(X)) is not a join-complete lattice.
Theorem 3.6. (1) Let eX be a reflexive L-fuzzy relation (resp. L-fuzzy preorder) on X. Define UeX : AF (X × X) → L as
Then UeX is an Alexandrov L-preuniform (resp. quasiuniform) convergence on X × X.
(2) Let (X, U) be an Alexandrov L-preuniform (resp. quasiuniform) convergence space. Define eU : X × X → L as
Then eU is a reflexive L-fuzzy relation (resp. L-fuzzy preorder).
(3) eUeX = eX and UeU ≥ U.
Proof. (1) (AU1) For x ∈ X,
(AU2) from:
for , by Theorem 2.9 and Lemma 2.2(11),
If eX is an L-fuzzy preorder on X, UeX is an Alexandrov L-quasiuniform convergence on X × X from UeX ([(x, y)]) = eX (x, y).
(2) It is easily proved from (1).
(3) eUeX (x, y) = UeX ([(x, y)]) = eX (x, y) . Since . Thus,
□
Theorem 3.7.Let be an Alexandrov L-neighborhood system on X. Define as
Then the following properties hold.
(1) is an Alexandrov L-preconvergence structure with
(2) If is topological, then is an Alexandrov L-convergence structure on X.
Proof. (1) and
For ,
(2) Since ,
Since
□
Lemma 3.8. (1) For and x ∈ X, there exists with
(2) For each x, y ∈ X,
(3) Let ψ : X → Y be a map. For and x ∈ X,
(4) For , . Moreover,
(5) .
(6) . If x ≠ z, .
Proof. (1) Since ,
(2)
(3) Since (ψ × ψ) ← (w) (x, y) = ψ← (w (ψ (x), -)) (y) = w (ψ (x), ψ (y)),
(4) Since and , (ψ × ψ) ⇒. Moreover,
(5) For ,
Hence
Since , put , and for w ≠ x.
(6) . Since for x ≠ z, =⊥. □
Theorem 3.9.Let (X, U) be an Alexandrov L-preuniform convergence space. Define as
Then the following properties hold.
(1) is an Alexandrov L-preconvergence structure.
(2)
(3) If (X, U) is an Alexandrov L-quasiuniform convergence space, then is an Alexandrov L-convergence structure.
(4) If eX is an L-fuzzy preorder on X, then is an Alexandrov L-convergence structure such that
where .
Proof. (1) Since ,
For each ,
(2) It follows from Lemma 3.8(2) and Theorem 3.6(2).
(3) For each x, z ∈ X,
(4) Since , put , and for w ≠ x. Thus,
Example 3.10. Let ([0, 1], ⊙, →, *, 0, 1) be a complete residuated lattice (ref.[1, 9]) as
Let X = {x, y, z} and UeX : AF (X × X) → [0, 1] with for with [0, 1]-fuzzy relations as
(1) Since is not an Alexandrov [0, 1]-preuniform convergence structure. Since is a reflexive [0, 1]-fuzzy relation, by Theorem 3.6, is an Alexandrov [0, 1]-preuniform convergence structure such that
But is not an Alexandrov [0, 1]-quasiuniform convergence structure from:
is an Alexandrov [0, 1]-neighborhood filter with
Then
From Theorem 3.9, we obtain an Alexandrov [0, 1]-preconvergence structure as with
But is not an Alexandrov [0, 1]-convergence structure because
(2) In (1), if eX = ▵ X×X with
Since ▵X×X is a [0, 1]-fuzzy preorder, U▵X×X is an Alexandrov [0, 1]-quasiuniform convergence structure on X × X. Moreover,
Since , by Theorem 3.9(4), is an Alexandrov L-convergence structure.
(3) If eX = 1X×X, then Then U1X×X is an Alexandrov L-quasiuniform convergence structure on X × X. Moreover,
Since , by Theorem 3.9(4), is an Alexandrov L-convergence structure.
(4) Define eA : X × X → [0, 1] as eA (x, y) = A (x) → A (y). Then eA is an [0, 1]-fuzzy preorder. By Theorem 3.6, we obtain an Alexandrov L-quasiuniform convergence UeA : AF (X × X) → [0, 1] as
Theorem 3.11.Let (X, lim) be an Alexandrov L-pre-convergence space. Define Ulim : AF (X × X) → L as
Then the following properties hold.
(1) Ulim is an Alexandrov L-preuniform convergence structure. Moreover, and Ulim ([(x, y)]) ≥ lim([y]) (x) .
(2) If (X, U) be an Alexandrov L-preuniform convergence space, then .
(3) If (X, lim) is a principle Alexandrov L-preconvergence space, then there exists an Alexandrov L-neighborhood system such that
Moreover, and Ulim ([(x, y)]) = lim([y]) (x) .
(4) If (X, lim) is a principle Alexandrov L-convergence space, then Ulim is an Alexandrov L-quasiuniform convergence structure.
Proof. (1) Since from Lemma 3.8(2),
Moreover,
By Lemma 3.8 (6), since and ,
(2) For each ,
(3) Since (X, lim) is a principle Alexandrov L-preconvergence space, there exists an Alexandrov L-neighborhood system such that By Lemma 3.8(5),
Since , if x ≠ w, from Lemma 3.8(5),
Moreover, .
By Lemma 3.8 (6), since ,
By (1), Ulim ([(x, y)]) = lim([y]) (x) .
(4) It follows from Ulim ([(x, y)]) = lim([y]) (x) . □
Example 3.12. (1) Let be an Alexandrov L-neighborhood system on X. Define as
Hence . Since [x] is a topological L-neighborhood filter on X and , is a principle Alexandrov L-convergence structure. We obtain an Alexandrov L-quasiuniform convergence structure as
Then Since for w ≠ x and ,
(2) Let be an Alexandrov L-neighborhood system on X. Define as
Moreover, Hence is a principle Alexandrov L-convergence structure. We obtain an Alexandrov L-quasiuniform convergence structure as
Theorem 3.13. (1) Let (X, UX) and (Y, UY) be Alexandrov L-preuniform convergence spaces. If ψ : (X, UX) → (Y, UY) is a uniformly continuous map, then is a continuous map.
(2) Let and be Alexandrov L-preconvergence spaces. If is a continuous map, then is a uniformly continuous map.
Proof. (1) Since for each ,
(2) For each ,
Theorem 3.14. (1) Let (X, UX) and (Y, UY) be an Alexandrov L-preuniform convergence spaces. If ψ : (X, UX) → (Y, UY) is a uniform continuous map, then ψ : (X, eUX) → (Y, eUY) 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 ψ : (X, UeX) → (Y, UeY) is a uniform continuous map.
Definition 3.15. [1, 9] 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 AUC be denote the category of Alexandrov L-preuniform convergence spaces and uniform continuous mappings for morphisms.
Let AC be denote the category of Alexandrov L-preconvergence spaces and continuous mappings for morphisms.
Theorem 3.16. (1) F : AC → AUC defined as is a functor.
(2) G : AUC → AC defined as is a functor.
(3) The pair (F, G) is a Galois correspondence between AUC and AC.
Proof. (1) and (2) are follows from Theorems 3.9, 3.11 and 3.13.
(3) By Theorem 3.11(2), if (X, UX) is an Alexandrov L-preuniform convergence space, then . Hence, the identity map is a uniformly continuous map. Moreover, if is an Alexandrov L-preconvergence space, by Theorem 3.11(1), . Hence the identity map is a continuous map. Therefore (F, G) is a Galois correspondence.
Let RFR be denote the category of reflexive L-fuzzy relations and order preserving mappings for morphisms. □
Theorem 3.17. (1) P : RFR → AUC defined as P (X, eX) = (X, UeX) is a functor.
(2) Q : AUC → RFR defined as Q (X, UX) = (X, eUX) is a functor.
(3) The pair (Q, P) is a Galois correspondence between AUC and RFR.
Proof. (1) and (2) are follows from Theorems 3.6 and 3.14.
(3) By Theorem 3.6(3), if (X, UX) is an Alexandrov L-preuniform convergence space, then P (Q (X, UX)) = (X, UeUX) ≥ (X, UX). Hence, the identity map idX : (X, UX) → (X, UeUX) = P (Q (X, UX)) is a uniformly continuous map. Moreover, if (X, eX) is a reflexive L-fuzzy relation, by Theorem 3.6(3), Q (P (X, eX)) = (X, eUeX) = (X, eX). Hence the identity map idX : Q (P (X, eX)) = (X, eUeX) → (X, eX) is an oder preserving map. Therefore (Q, P) is a Galois correspondence. □
Example 3.18. 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 as in Example 3.10. Let R ∈ [0, 1] X×Y be a fuzzy information as follows:
Define L-fuzzy preordered relations by
(1) We obtain a topological Alexandrov L-neighborhood system and by Theorem 3.7, a principal Alexandrov L-convergence structure as follows:
By Theorem 3.11(4), is Alexandrov L-quasiuniform convergence structure on X × X such that
Since is an L-fuzzy preorder, by Theorem 3.6, is an Alexandrov L-quasiuniform convergence structure on X × X such that
(1-A) For B ∈ [0, 1] X with B = (0.4, 0.7, 0.1), put . Then
Then . By Theorems 3.6 and 3.11,
(1-B) Let Then .
(1-C) Let and . Then . By Theorems 3.6 and 3.11,
(1-D) For such that with
(2) By a similar way in (1), is a topological Alexandrov L-neighborhood system as
By Theorem 3.7, we obtain a principal Alexandrov L-convergence structure
By Theorem 3.11(4), is Alexandrov L-quasiuniform convergence structure on X × X such that
Since is an L-fuzzy preorder, by Theorem 3.6, is an Alexandrov L-quasiuniform convergence structure on X × X such that
(2-A) By (1-A),
(2-B) For in (1-C),
(2-C) For in (1-D),
Conclusion
In this paper, we introduce the notion of Alexandrov L-preuniform convergence spaces in a complete residuated lattice. We investigate the relations among Alexandrov L-(neighborhood) filters, L-fuzzy preorders and Alexandrov L-(preuniform)convergence structures. There exists a Galois correspondence between the category of Alexandrov L-preuniform convergence spaces and that of Alexandrov L-convergence spaces (resp. reflexive L-relations). Finally, we give the examples of Alexandrov L-preuniform convergence spaces and Alexandrov L-convergence for a fuzzy information.
In the future, by using the concepts of Alexandrov L-preuniform convergence spaces and Alexandrov L-convergence spaces, information systems and decision rules with a view point of applications to mult-attribute decision-making [14, 30] are investigated in residuated lattices.
References
1.
R.Bělohlavek, Fuzzy Relational Systems, Kluwer Academic Publishers, New York, 2002.
2.
A.Craig and G.Jäger, A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces, Fuzzzy Sets and Systems160 (2009), 1177–1203.
3.
B.Davvaz and Y.C.Kim, Alexandrov L-topologies and Alexandrov L-convergence structures, Journal of Intelligent and Fuzzy Systems.https://doi.org/10.3233/JIFS-181295
J.Fang and Y.Yue, T-diagonal conditions and Contínuos extension theorem, Fuzzy Sets and Systems321 (2017), 73–89.
9.
U.Höhle and S.E.Rodabaugh, Mathematics of Fuzzy Sets, Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Kluwer Academic Publishers, Dordrecht, 1999.
10.
G.Jäger, A category of L-fuzzy convergence spaces, Quaes-tiones Math24 (2001), 501–517.
11.
G.Jäger, Subcategories of lattice-valued convergence spaces, Fuzzy Sets and Systems156 (2005), 1–24.
12.
G.Jäger, Pretopological and topological lattice-valued convergence spaces, Fuzzy Sets and Systems158 (2007), 424–435.
13.
G.Jäger and M.H.Burton, Stratified L-uniform convergence spaces, QuaestMath28 (2005), 11–36.
14.
H.Jiang, J.Zhan and D.Chen, Covering based variable precision (I,T)-fuzzy rough sets with applications to multiattribute decision-making, IEEE Transactions Fuzzy Systems, https://doi.org/10.1109/TFUZZ.2018.2883023
15.
Y.C.Kim, Categories of fuzzy preorders, approximation operators and Alexandrov topologies, Journal of Intelligent and Fuzzy Systems31 (2016), 1787–1793.
16.
Y.C.Kim and J.M.Ko, Images and preimages of L-filter bases, Fuzzy Sets and Systems173 (2005), 93–113.
17.
J.M.Ko and Y.C.Kim, Alexandrov L-filters and Alexandrov L-convergence spaces, Journal of Intelligent and Fuzzy Systems35(2) (2018) 3255–3266.
18.
L.Q.Li, Q.Jin and K.Hu, On stratified L-convergence spaces: Fisher’s diagonal axiom, Fuzzy Sets and Systems267 (2015), 31–40.
19.
Z.M.Ma and B.Q.Hu, Topological and lattice structures of Lfuzzy rough set determined by lower and upper sets, Information Sciences218 (2013), 194–204.
20.
X.Ma, J.Zhan, M.I.Ali and N.Mehmood, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review49(4) (2018), 511–529.
21.
D.Orpen and G.Jäger, Lattice-valued convergence spaces, Fuzzy Sets and Systems162 (2011), 1–24.
B.Pang and Z.Y.Xiu, Stratified-prefilter convergence structures in stratified L-topological spaces, Soft Computing22 (2018), 7539–7551.
26.
Q.Yu and J.Fang, The category of T-convergence spaces and its catersion-closedness, Iranian Journal of Fuzzy Systems14(3) (2017), 121–138.
27.
W.X.Xie, Q.Y.Zhang and L.Fan, Fuzzy complete lattices, Fuzzy Sets and Systems160 (2009), 2275–2291.
28.
W.Yao, On many-valued L-fuzzy convergence spaces, Fuzzy Sets and Systems159 (2008), 2503–2519.
29.
J.Zhan, B.Sun and J.C.R.Alcantud, Covering based multigranulation (I,T)-fuzzy rough set models and applications in multattribute group decision-making, Information Sciences476 (2019), 290–318.
30.
L.Zhang, J.Zhan and Z.X.Xu, Covering based generalized IF rough sets with applications to mult-attribute decision-making, Information Sciences478 (2019), 275–302.
31.
Q.Y.Zhang and L.Fan, Continuity in quantitive domains, Fuzzy Sets and Systems154 (2005), 118–131.
32.
Q.Y.Zhang, W.X.Xie and L.Fan, The Dedekind-MacNeille completion for fuzzy posets, Fuzzy Sets and Systems160 (2009), 2292–2316.