L-topological-convex spaces generated by L-convex bases

Abstract In this paper, axiomatic definitions of both L-convex bases and L-convex subbases are introduced and their relations with L-convex spaces are studied. Based on this, the notion of L-topological-convex space is introduced as a triple (X, 𝓣, 𝓒), where X is a nonempty set, 𝓒 is an L-convex structure on X and 𝓣 is an L-cotopology on X compatible with 𝓒. It can be characterized by many means.

convex spaces. In Section 4, we introduce L-convex subbases and study its relations with L-convex spaces and L-convex bases. In Section 5, we introduce the notion of L-topological-convex spaces and obtain several of its characterizations. In Section 6, we introduce the notion of L-topological-convex enclosed relation spaces and further obtain some other characterizations of L-topological-convex spaces.

Preliminaries
Throughout this paper, X and Y are nonempty sets. The power set of X is denoted by X . The set of all nite subsets of X is denoted by X n . (L, ∨, ∧) is a completely distributive lattice. The least (resp. largest) element in L is denoted by ⊥ (resp. ). An element a ∈ L is called a co-prime, if for all b, c ∈ L, a ≤ b ∧ c implies a ≤ b or a ≤ c. The set of all co-primes in L\{⊥} is denoted by J(L). For any a ∈ L, there is L ⊆ J(L) such that a = b∈L b [29]. A binary relation ≺ on L is de ned by a ≺ b i for each L ⊆ L, b ≤ L implies the existence of d ∈ L such that a ≤ d. The mapping β : L → L , de ned by β(a) = {b : b ≺ a}, satis es β( i∈I a i ) = i∈I β(a i ) for {a i } i∈I ⊆ L. For any a ∈ L, β(a) and β * (a) = β(a) ∩ J(L) satis es a = β(a) = β * (a) [29]. L X is the set of all L-fuzzy sets on X. The lease (resp, largest) element in L X is denoted by ⊥ (resp. ). A subset {A i } i∈I ⊆ L X is called an up-directed set, simply denoted by {A i } i∈I dir ⊆ L X , if for all i, j ∈ I, there is k ∈ I such that A i , A j ≤ A k . For convenience, if ψ ⊆ L X , we adopt ψ = A∈ψ A and ψ = A∈ψ A. Further, if ψ is up-directed, we also adopt ψ = dir A∈ψ A. For A ∈ L X , we denote where β * (A) = {x λ : λ ∈ β * (A(x))}. In particular, we write F(L X ) for F( ) consisting of all L-fuzzy nite sets on X [28]. Clearly, for A, B ∈ L X , B ≤ A i F(B) ⊆ F(A). In addition, it has been proved that β * (A) ⊆ F(A) dir ⊆ L X , F(A) = A and F( i∈I A i ) = i∈I F(A i ) for all A ∈ L X and {A i } i∈I dir ⊆ L X [28]. For a mapping f : X → Y, the L-fuzzy mapping f → L : L X → L Y is de ned by f → L (A)(y) = {A(x) : f (x) = y} for A ∈ L X and y ∈ Y, and the mapping f ← L : L Y → L X is de ned by f ← L (B)(x) = B(f (x)) for B ∈ L Y and x ∈ X [30].
Terminologies of Category Theory (resp. Convex Theory) used this paper can be seen in [31] (resp. [2]). Next, we recall some basic de nitions and results related to L-convex spaces and L-cotopological spaces.
De nition 2.1. [9] A subset C ⊆ L X is called an L-convex structure and the pair (X, C) is called an L-convex space, if C satis es (LC1) ⊥, ∈ C; (LC2) C for any C ⊆ C; (LC3) C for any C dir ⊆ C.
Theorem 2.2. [11] Let (X, C) be an L-convex space. The L-hull operator co C : L X → L X (brie y, co) of (X, C), (LCO5) co( i∈I A i ) = i∈I co(A i ) for any {A i } i∈I dir ⊆ L X . Conversely, if co : L X → L X satis es (LCO1)-(LCO5), then the set Cco = {A ∈ L X : co(A) = A} is an L-convex structure satisfying co Cco = co.
In [28], it showed that an operator co : L X → L X satisfying (LCO1)-(LCO4) is the L-hull operator of some L-convex space i it satis es (LDF).
(LDF) co(A) = F∈F(A) co(F) for A ∈ L X . Let (X, C X ) and (Y , C Y ) be L-convex spaces. A mapping f : X → Y is called an L-convex structure preserving mapping, if f ← L (A) ∈ C X for any A ∈ C Y . The category of L-convex spaces and L-convex structure preserving mappings is denoted by L-CS [28].

De nition 2.3. [32]
A subset T ⊆ L X is called an L-cotopology and the pair (X, T) is called an L-cotopological The category of L-cotopological spaces and L-continuous mappings is denoted by L-CTS.
Let (X, φ X ) and (Y , φ Y ) be L-closure spaces. A mapping f : X → Y is called an L-closure structure preserving mapping if f ← L (A) ∈ φ X for any A ∈ φ Y . The category of L-closure spaces and L-closure structure preserving mappings is denoted by L-CSS.
De nition 2.5. [33] A binary relation on L X is called an L-topological enclosed relation and the pair (X, ) is called an L-topological enclosed relation space, if satis es (LTER1) ⊥ ⊥; In an L-topological enclosed relation space (X, ), A ≤ B C ≤ D implies that A D. Also, A D implies some C ∈ L X such that A ≤ C C ≤ D [33].
Let (X, X ) and (Y , Y ) be L-topological enclosed relation spaces. A mapping f : [33]. The category of L-topological enclosed relation spaces and L-topological enclosed relation dual-preserving mappings is denoted by L-TERS.
Theorem 2.6. [33] (1) For an L-topological enclosed relation space (X, ), the operator cl : L X → L X , de ned by is a closure operator of some L-cotopology, denoted by T .
(2) For an L-cotopological space (X, T), the binary operator T , de ned by is an L-topological enclosed relation.
De nition 2.7. A binary relation on L X is called an L-convex enclosed relation and the pair (X, ) is called an L-convex enclosed relation space, if satis es The category of L-convex enclosed relation spaces and L-convex enclosed relation dual-preserving mappings is denoted by L-CERS.
Similar to Theorem 2.6, the following result is easy to check. is an L-hull operator of some L-convex structure, denoted by C .
(2) For an L-convex space (X, C), the binary operator C , de ned by is an L-convex enclosed relation.
For two L-enclosed relations , on X, we say is coarse than , denoted by

L-convex bases
In this section, we introduce L-convex bases and discuss its relations with L-convex spaces.
De nition 3.1. Let (X, C) be an L-convex space. A subset B ⊆ C is called an L-convex base of C, if for any where φ X = {zr ∈ L X : z ∈ X, r ∈ φ} for any φ ⊆ L. Then B is an L-convex base of C. But x ∈ F(L X ) and co(x ) = x ∉ B. Thus Proposition 3.2 just gives a su cient condition of L-convex bases. To obtain a necessary and su cient condition for L-convex bases, we present the following results.  Proof. We prove that C B is an L-convex structure, where Hence the aimed set is up-directed.
By Theorems 3.4 and 3.5, we see that(LCB1)-(LCB3) is a necessity and su cient condition for L-convex bases. Thus we present the axiomatic de nition of L-convex bases as follows.
De nition 3.6. A subset B ⊆ L X is called an L-convex base and the pair (X, B) is called an L-convex base space, if B satis es (LCB1)-(LCB3).
The category of L-convex base spaces and L-convex base preserving mappings is denoted by L-CBS. Next, we discuss relations between L-CS and L-CBS. Theorem 3.7. An L-convex structure is an L-convex base of itself.
By Theorems 3.7, the category L-CS is a subcategory of L-CBS. Thus we can de ne a factor E b :L-CS→ L-CBS by: By Theorems 3.5 and 3.8, we can de ne a factor F :L-CBS→L-CS by: An L-closure structure can generate an L-convex structure [28]. Actually, an L-closure structure is an L-convex base showed as follows.
Theorem 3.11. An L-closure structure is an L-convex base.
In addition, it is direct to show that the set Proof. To verify this, let (X, φ) be an L-closure space. By Theorems 3.5 and 3.11, Cφ is an L-convex structure. Thus we only need to prove that id X : (X, Cφ) → (X, φ) is a bicore ector. It su cient to show that the following statements hold.

L-topological-convex spaces
In [2], if X is a set equipped with a convex structure C and a cotopology T, then the triple (X, T, C) is called a topological-convex space provided that T is compatible with C, i.e., all polytopes are closed (co C (F) ∈ T for any F ∈ X n ). In this section, we extend this concept into L-fuzzy settings and obtain some of its characterizations. Before this, we give a brief observation of this concept. Remark 5.1. Let X be a set equipped with a cotopology T and a convex structure C. Then T ∩ C is a closure structure whose closure operator is denoted by cl T∩C .
(1) From de nition of a topological-convex described as above, we can conclude that a set X, equipped with a cotopology T and a convex structure C, is a topological-convex space i co C (F) = cl T∩C (F) for all F ∈ X n . Indeed, if (X, T, C) is a topological-convex space, then co C (F) = cl T (co C (F)) for any F ∈ X n . Thus co C (F) = cl T (co C (F)) ∈ T ∩ C which shows Conversely, if co C (F) = cl T∩C (F) for all F ∈ X n , then it is clear that co C (F) ∈ T for all F ∈ X n . That is, T is compatible with C.
(2) In a convex space (X, C), a subset B ⊆ C is called a base if for each A ∈ C, there is an up-directed subset B ⊆ B such that A = B. As described in [2], a subset of a convex structure is a base i it contains all polytopes. Thus, T is compatible with C i C has a closed base (i.e., C has a base B contained in T).
From (2) of Remark 5.1 and De nition 3.1, we extend the notion of topological-convex spaces into L-fuzzy settings as follows.

De nition 5.2.
Let X be a set equipped with an L-cotopology T and an L-convex structure C. The triple (X, T, C) is called an L-topological-convex space, if T is compatible with C, that is, C has a closed L-convex base (i.e., there is B ⊆ T such that B is a base of C).
Next, we give some characterizations on L-topological-convex spaces as follows. .
Hence F∈F(A) cl T∩C (F) ≤ co C (A). Since the set {G : G ∈ F(C)} is up-directed, the set {cl T∩C (G) : G ∈ F(C)} ⊆ B is also up-directed. Therefore B is a closed L-convex base of C.
In [28], it has been proved that for an L-closure space (X, φ), the set is an L-convex structure generated by φ. Thus an L-cotopology T naturally induces an L-convex structure denoted by C T . Moreover, we have the following result.
Theorem 5.4. Let (X, T) be an L-cotopological space. Then (X, T, C T ) is an L-topological-convex space.
Proof. Let F ∈ F(L X ). Since T ⊆ C T , it directly follows from (LDF) that (3) For an L-cotopological space (X, T), C T is an L-Alexander topology.
In fact, if C T is the L-convex space generated by an L-topology T and {A i } i∈I ⊆ C T , then there is an up-directed set D i ⊆ T such that A = D i for any i ∈ I. Let D = i∈I D i and let B = { G : G ∈ D n }, where G stands for G (G is nite). Then B ⊆ T is up-directed and i∈I A i = B ∈ C T . So C T is an L-Alexander topology. However, in an L-topological-convex space (X, T, C), C may not be an L-Alexander topology. The example in (1) is of this type.
Next, we get a weaker L-topology by an L-topology and an L-convex structure.
(LT3): We rstly prove that A ∨ B ∈ φ for all A, B ∈ φ . Since A, B ∈ φ , there are D , D ∈ φ n such that D = A and D = B. Since B = D ∪ D , A ∨ B = B ∈ φ . Next, we prove that Tw satis es (LT3).
If A, B ∈ Tw, then there are {D i } i∈I , {D j } j∈J ⊆ φ such that A = i∈I D i and B = j∈J D j . Let D ij = D i ∨ D j for any i ∈ I and any j ∈ J. We have Therefore Tw is an L-cotopology.
With help of Tw, we can characterize L-topological-convex space as following.
Theorem 5.7. Let X be a set equipped with an L-cotopology T and an L-convex structure C. Then (X, T, C) is an L-topological-convex space i (X, Tw , C) is an L-topological-convex space.
Proof. Tw ∩ C = T ∩ C by Lemma 5.6. Thus the result follows from Theorem 5.3.

L-topological-convex enclosed relation spaces
Except for Theorems 5.3 and 5.7, there are others ways to characterize L-topological-convex spaces. In this section, we introduce the notions of L-topological-convex enclosed relations, by which, we characterize Ltopological-convex spaces.
For L-topological-convex spaces (X, The category of L-topological-convex spaces and L-topological-convex structure preserving mappings is denoted by L-TCS. De nition 6.1. Let X be a set equipped with an L-topological enclosed relation and an L-convex enclosed relation . The triple (X, , ) is called an L-topological-convex enclosed relation spaces provided that is compatible with , that is, for any F ∈ F(L X ) and any B ∈ L X , Let (X, X , X ) and (Y , Y , Y ) be L-topological-convex enclosed relation spaces. A mapping f : X → Y is called an L-topological-convex enclosed relation dual-preserving mapping, if f : (X, X ) → (Y , Y ) is an L-topological enclosed relation dual-preserving mapping, and f : ( ). The category of L-topological-convex enclosed relation spaces and L-topological-convex enclosed relation dual-preserving mappings is denoted by L-TCERS.
Next, we prove that the L-convex enclosed relation generated by an L-topological enclosed relation is compatible with this L-topological enclosed relation. (ii) A = D.
For any D ∈ D, there is s ∈ S such that D = i∈I s(i). Fix any j ∈ I. We have D ≤ s(j) B j . Thus D B. Therefore (iii) holds.
Combining (i) φ F is nonempty for all F ∈ F(A).
Combining (i)-(iv), we conclude that dir i∈I A i B. Proof. Let F ∈ F(L X ) and let B ∈ L X with F B. If G ∈ F(F), then G B by Lemma 6.2. Thus there is Therefore (X, , ) is an L-topological-convex enclosed relation space.
By Theorem 6.3, we list some L-topological-convex enclosed relations as follows.
Example 6.4. Let (X, U ) be a pointwise quasi-uniform space. De ne Then (X, U , U ) is an L-topological-convex enclosed relation space by Theorem 4.6 in [33], Lemma 6.2 and Theorem 6.3.
Example 6.6. Let (X, d) be a pointwise pseudo-metric space. De ne Then (X, d , d ) is an L-topological-convex enclosed relation space by Theorem 4.2 in [33], Lemma 6.2 and Theorem 6.3.
Next, we discuss relationships between L-TCERS and L-TCS. Hence co (A) ≤ F∈F(A) co (F).
To prove the desired result, let F ∈ F(L X ). We need to prove that Since T ∩ C ⊆ C , we have cl C ≤ cl T ∩C . Thus Thus, by Theorem 5.3, (X, T , C ) is an L-topological-convex space. Let D G = cl T∩C (G) for each G ∈ F(F). Then G ≤ D G ∈ T ∩ C. By D G ∈ C, we have co C (D G ) = D G and so D G C D G . Similarly, by D G ∈ T, we have cl T (D G ) = D G and so D G T D G . Hence Therefore (X, T , C ) is an L-topological-convex enclosed space.
The following three results directly follow from Theorems 2.6 and 2.8. Theorem 6.9. Let (X, X , X ) and (Y , Y , Y ) be L-topological-convex enclosed relation spaces. If f : X → Y is an L-topological-convex enclosed relation dual-preserving mapping, then f : (X, T X , C X ) → (Y , T Y , C Y ) is an L-topology-convexity preserving mapping. Theorem 6.10. Let (X, T X , C X ) and (Y , T Y , C Y ) be L-topological-convex spaces. If f : X → Y is L-topologicalconvex structure preserving, then f : (X, T X , C X ) → (Y , T Y , C Y ) is L-topological-convex enclosed relation dual-preserving. Theorem 6.11. If (X, , ) is an L-topological-convex enclosed relation space, then T = and C = . Conversely, if (X, T, C) is an L-topological-convex space, then T T = T and C C = C. From Theorems 6.7 and 6.9, we can de ne a factor T : L-TCERS →L-TCS by T(X, , ) = (X, T , C ) and T(f ) = f . From Theorems 6.7 to 6.11, T is an isomorphic factor. Corollary 6.12. L-TCERS is isomorphic to L-TCS.
Next, we obtain a new L-topological enclosed relation via an L-topological enclosed relation and an Lconcave enclosed relation. Lemma 6.13. Let X be a set equipped with an L-topological enclosed relation and an L-convex enclosed relation . De ne a binary operator w on X by where the operator cl : L X → L X is de ned by: where C(X) is the set of all L-closure operator on X and cl ∨ co ≤ cl means cl (A) ∨ co (A) ≤ cl(A) for any A ∈ L X . Then w is an L-enclosed relation generated by and . In addition, w is the biggest L-enclosed relation with respect to w ≤ and w ≤ .
Proof. Note that the set { cl ∈ C(X) : cl ∨ co ≤ cl} is not empty since it contains the closure operator of the indiscrete L-topology on X. To prove that w is an L-enclosed operator, we only need to verify that cl is an L-closure operator.

Thus cl(cl(A)) = cl(A).
Hence cl is an L-closure operator. Therefore w is an L-enclosed relation. Finally, w ≤ and w ≤ by cl ∨ co ≤ cl. Now, let be any L-enclosed relation on X with ≤ and ≤ . Then cl ∨ co ≤ cl and so cl ≤ cl . Hence So w is the biggest L-enclosed relation with respect to w ≤ and w ≤ . Theorem 6.14. Let X be a set equipped with an L-topological enclosed relation and an L-convex enclosed relation . Let w be the L-enclosed relation generated by and . Then the following results are valid.
(1) φ w = T ∩ C , where φ w is the L-closure structure induced by w. (2) T w = Tw ⊆ T , where T w is the L-cotopology generated by φ w , and Tw is the L-cotopology generated by T ∩ C .
(3) De ne a binary operator w on X by: A w B i ∀ w ≤ ∈ E(X), A B, where E(X) is the set of all L-topological enclosed relation on X. Then w = T w which is called the Ltopological enclosed relation generated by w.