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

Chun-Yan Liao; Xiu-Yun Wu

doi:10.1515/math-2019-0133

Artikel Open Access

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

Chun-Yan Liao und Xiu-Yun Wu

Veröffentlicht/Copyright: 26. Dezember 2019

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 17 Heft 1

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.

Keywords: L-convex space; L-convex enclose relation space; L-topological-convex space; L-topological-convex enclosed relation space

MSC 2010: 54A40; 52A01

1 Introduction

By convex sets, we traditionally refer to convex sets in Euclidean spaces, where the property ‘convexity’ was originally inspired by some elementary geometric problems such as shapes of circles and characterizations of polytopes [1]. However, with increasing fields that convex sets involved and expanding scopes that convex sets were applied, many complex problems compelled people to engage in an axiomatic research of convex sets. This leads to abstract convex structure which is a set-theoretic structure satisfying several axioms [2]. Now, its theory involves many mathematical structures including graphs [3], posets [4], median algebras [5], metric spaces [6], lattices [7] and vector spaces [2].

Convex structure has been extended into fuzzy settings in several ways. Fuzzy convex structures defined by Rosa [8] was further extended into M-convex structures by Maruyama [9]. Latter, some characterizations of L-convex spaces were discussed [10, 11, 12, 13, 14, 15]. Actually, an M-convex structure is a crisp family of M-fuzzy sets satisfying certain set of axioms similar to these that an abstract convex structure has. However, from a totally different point of view, Shi and Xiu introduced M-fuzzifying convex structures [16]. Now, many subsequent properties in M-fuzzifying convex spaces have been studied [17, 18, 19, 20, 21, 22, 23, 24, 25]. Shi and Xiu also introduced (L, M)- fuzzy convex spaces unifying both L-convex spaces and M-fuzzifying convex spaces [26]. Based on this, many characterizations of L-convex spaces and (L, M)- fuzzy convex spaces have been discussed [27, 28].

In this paper, we introduce axiomatic notions of both L-convex bases and L-convex subbase and discuss their relations with L-convex spaces. Further, based on L-concave bases, we introduce the notion of L-topological-convex spaces and obtain several of its characterizations in a view point of category aspect.

This paper is arranged as follows. In Section 2, we recall some notions and results related to L-convex spaces and L-cotopological spaces. In Section 3, we introduce L-convex bases and study its relations with L-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.

2 Preliminaries

Throughout this paper, X and Y are nonempty sets. The power set of X is denoted by 2^X. The set of all finite subsets of X is denoted by 2finX .

(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 L₁ ⊆ J(L) such that a = ⋁_b∈L₁b [29]. A binary relation ≺ on L is defined by a ≺ b iff for each L₁ ⊆ L, b ≤ ⋁ L₁ implies the existence of d ∈ L₁ such that a ≤ d. The mapping β : L → 2^L, defined by β(a) = {b : b ≺ a}, satisfies β(⋁_i∈Ia_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) satisfies 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∈ψA and ⋀ ψ = ⋀_A∈ψA. Further, if ψ is up-directed, we also adopt ⋁ ψ = ⋁A∈ψdir A. For A ∈ L^X, we denote

F(A)={F∈LX:∃φ∈2finβ∗(A),F=⋁φ},

where β^*(A) = ⋃{x_λ : λ ∈ β^*(A(x))}. In particular, we write 𝔉(L^X) for 𝔉(⊤) consisting of all L-fuzzy finite sets on X [28]. Clearly, for A, B ∈ L^X, B ≤ A iff 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 fL→ : L^X → L^Y is defined by fL→ (A)(y) = ⋁{A(x) : f(x) = y} for A ∈ L^X and y ∈ Y, and the mapping fL← : L^Y → L^X is defined by fL← (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 definitions and results related to L-convex spaces and L-cotopological spaces.

Definition 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 satisfies

⊥, ⊤ ∈ 𝓒;
⋀ 𝓒₁ for any 𝓒₁ ⊆ 𝓒;
⋁ 𝓒₂ for any 𝓒₂ ⊆dir 𝓒.

Theorem 2.2

[11] Let (X, 𝓒) be an L-convex space. The L-hull operator co_𝓒 : L^X → L^X (briefly, co) of (X, 𝓒), defined by co(A) = ⋀{B ∈ 𝓒 : A ≤ B}, satisfies

co(⊥) = ⊥;
A ≤ co(A) for all A ∈ L^X;
if A ≤ B, then co(A) ≤ co(B);
co(co(A)) = co(A) for all A ∈ L^X;
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 satisfies (LCO1)–(LCO5), then the set 𝓒_co = {A ∈ L^X : co(A) = A} is an L-convex structure satisfying co_{𝓒_co} = 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 iff it satisfies (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 fL← (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].

Definition 2.3

[32] A subset 𝓣 ⊆ L^X is called an L-cotopology and the pair (X, 𝓣) is called an L-cotopological space, if 𝓣 satisfies

⊥, ⊤ ∈ 𝓣;
⋀ 𝓣₁ ∈ 𝓣 for any 𝓣₁ ⊆ 𝓣;
A ∨ B ∈ 𝓣 for all A, B ∈ 𝓣.

Let (X, 𝓣_X) and (Y, 𝓣_Y) be L-cotopological spaces. A mapping f : X → Y is called an L-continuous mapping if fL← (A) ∈ 𝓣_X for any A ∈ 𝓣_Y. The category of L-cotopological spaces and L-continuous mappings is denoted by L-CTS.

Definition 2.4

[12] A subset φ ⊆ L^X is called an L-closure structure and the pair (X, φ) is called an L-closure space, if it satisfies

⊥, ⊤ ∈ φ;
⋀φ₁ ∈ φ for any φ₁ ⊆ φ.

An L-closure structure is an L-cotopology (resp. L-convex structure) iff it satisfies (LT3) (resp. (LC3)). The L-closure operator of an L-closure space (X, φ) is defined by cl_φ(A) = ⋀ {B ∈∈ φ : A ≤ B} for all A ∈ L^X. Then cl_φ satisfies

cl(⊥) = ⊥;
A ≤ cl(A) for all A ∈ L^X;
if A ≤ B, then cl(A) ≤ cl(B);
cl(cl(A)) = cl(A) for all A ∈ L^X.

Conversely, if an operator cl : L^X → L^X satisfies (LCL1)–(LCL4), then the set φ_cl = {A ∈ L^X : cl(A) = A} is an L-closure structure satisfying cl_{φ_cl} = cl.

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 fL← (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.

Definition 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 ⋞ satisfies

⊥ ⋞ ⊥;
A ⋞ B implies A ≤ B;
A ⋞ ⋀_i∈I B_i iff A ⋞ B_i for all i ∈ I;
A ⋞ B implies C ∈ L^X with A ⋞ C ⋞ B;
A ∨ B ⋞ C iff A ⋞ C and B ⋞ C.

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 : X → Y is called an L-topological enclosed relation dual-preserving mapping, if fL← (A) ⋞_X fL← (B) for all A ⋞_Y B [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, defined by

cl⋞(A)=⋀{B∈LX:A⋞B},

is a closure operator of some L-cotopology, denoted by 𝓣_⋞.

(2) For an L-cotopological space (X, 𝓣), the binary operator ⋞_𝓣, defined by

A⋞TBiffclT(A)≤B.

is an L-topological enclosed relation.

(3) L-CTS is isomorphic to L-TERS.

Definition 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 ⪕ satisfies

⊥ ⪕ ⊥;
A ⪕ B implies A ≤ B;
A ⪕ ⋀_i∈I B_i iff A ⪕ B_i for all i ∈ I;
⋁i∈Idir A_i ⪕ C iff A_i ⪕ C for all i ∈ I;
A ⪕ B implies the existence of C ∈ L^X such that A ⪕ C ⪕ B.

Let (X, ⪕_X) and (Y, ⪕_Y) be L-convex enclosed relation spaces. A mapping f : X → Y is called an L-convex enclosed relation dual-preserving mapping, if fL← (A) ⪕_X fL← (B) for all A ⪕_Y B. 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.

Theorem 2.8

For an L-convex enclosed relation space (X, ⪕), the operator co_⪕ : L^X → L^X, defined by

co⪕(A)=⋀{B∈LX:A⪕B},

is an L-hull operator of some L-convex structure, denoted by 𝓒_⪕.
For an L-convex space (X, 𝓒), the binary operator ⪕_𝓒, defined by

A⪕CBiffcoC(A)≤B.

is an L-convex enclosed relation.
The category L-CS is isomorphic to the category L-CERS.

Remark 2.9

(1) L-cotopologies and L-convex structures have a uniform origination, i.e., L-closure structure. L-topological enclosed relations and L-convex enclosed relations also have a uniform origination defined by:

A binary operation ⊏ on X is called an L-enclosed relation and the pair (X, ⊏) is called an L-enclosed relation space, if ⊏ satisfies the following conditions

⊥ ⊏ ⊥;
A ⊏ B implies A ≤ B;
A ⊏ ⋀_i∈I B_i iff A ⊏ B_i for all i ∈ I;
A ⊏ B implies the existence of C ∈ L^X such that A ⊏ C ⊏ B;
C ≤ A ⊏ B implies C ⊏ B.

Clearly, a binary relation satisfying (LER1)–(LER4) is an L-topological (resp. L-convex) enclosed relation if it satisfies (LTER5) (resp. (LCER5)).

For two L-enclosed relations ⊏₁, ⊏₂ on X, we say ⊏₁ is coarse than ⊏₂, denoted by ⊏₁ ≤ ⊏₂, if A ⊏₁ B implies A ⊏₂ B for all A, B ∈ L^X.

3 L-convex bases

In this section, we introduce L-convex bases and discuss its relations with L-convex spaces.

Definition 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 A ∈ 𝓒, there is 𝓑₁ ⊆dir 𝓑 such that A = ⋁ 𝓑₁.

Proposition 3.2

Let (X, 𝓒) be an L-convex space and let 𝓑 ⊆ 𝓒. If co(F) ∈ 𝓑 for any F ∈ 𝔉 (L^X), then 𝓑 is an L-convex base of 𝓒.

Proof

Let A ∈ 𝓒. Since {co(F) : F ∈ 𝔉(A)} ⊆dir 𝓑, we have A = co(A) = co( ⋁F∈F(A)dir F) = ⋁F∈F(A)dir co(F) by (LCO5). Thus 𝓑 is an L-convex base of 𝓒.□

Remark 3.3

A subset 𝓑 ⊆ 𝓒 of a convex space (X, 𝓒) is a base iff it contains all polytopes (i.e., co(F) ∈ 𝓑 for all F ∈ 2finX ) [2]. However, if L ≠ {⊥, ⊤}, the inverse result in Proposition 3.2 fails. For example, let X = {x} and L = [0, 1]. Define B=[0,13)X∪[12,1]X and C=B∪{0,13}X, where φ^X = {z_r ∈ L^X : z ∈ X, r ∈ φ} for any φ ⊆ L. Then 𝓑 is an L-convex base of 𝓒. But x13∈F(LX) and co(x13)=x13∉B. Thus Proposition 3.2 just gives a sufficient condition of L-convex bases. To obtain a necessary and sufficient condition for L-convex bases, we present the following results.

Theorem 3.4

If (X, 𝓒) is an L-convex space and 𝓑 ⊆ 𝓒 is an L-convex base, then

⊤ = ⋁ 𝓑₁ for some 𝓑₁ ⊆dir 𝓑;
for {B_i}_i∈I ⊆ 𝓑, there is 𝓑₁ ⊆dir 𝓑 such that ⋀_i∈I B_i = ⋁ 𝓑₁;
if {A_i}_i∈I ⊆dir L^X, and {B_ij}_{j∈J_i} ⊆dir 𝓑 such that A_i = ⋁_{j∈J_i} B_ij for each i ∈ I, then there is {B_k}_k∈K ⊆dir 𝓑 such that ⋁_i∈I A_i = ⋁_k∈K B_k.

Proof

Since ⊤ ∈ 𝓒, there is 𝓑₁ ⊆dir 𝓑 with ⊤ = ⋁ 𝓑₁.
Since {B_i}_i∈I ⊆ 𝓒, we have ⋀_i∈I B_i ∈ 𝓒 by (LC2). Thus, by Definition 3.1, there is 𝓑₁ ⊆dir 𝓑 such that ⋀_i∈I B_i = ⋁ 𝓑₁.
Let {A_i}_i∈I ⊆dir L^X, and let {B_ij}_{j∈J_i} ⊆dir 𝓑 with A_i = ⋁_{j∈J_i} B_ij for each i ∈ I. For each i ∈ I, A_i = ⋁_{j∈J_i} B_ij ∈ 𝓒 by (LC3) and 𝓑 ⊆ 𝓒. Thus ⋁_i∈I A_i ∈ 𝓒 since {A_i}_i∈I ⊆dir 𝓒. Hence there is 𝓑₁ ⊆dir 𝓑 such that ⋁_i∈I A_i = ⋁ 𝓑₁.□

Theorem 3.5

Let 𝓑 ⊆ L^X be a set satisfying (LCA1)–(LCA3) in Theorem 3.4. Then there is a unique L-concave structure with 𝓑 as an L-convex base.

Proof

We prove that 𝓒_𝓑 is an L-convex structure, where

CB={A∈LX:∃B1⊆dirB,A=⋁B1}.

By (LCB1), we have ⊤ ∈ 𝓒_𝓑. In addition, let ∅ ⊆ 𝓑, then ∅ is up-directed and ⊥ = ⋁ ∅ ∈ 𝓒_𝓑.
If {A_i}_i∈I ∈ 𝓒_𝓑 and 𝓑_i = {B_ij}_{j∈J_i} ⊆dir 𝓑 with A_i = ⋁_{j∈J_i} B_ij, then

⋀i∈IAi=⋀i∈I⋁j∈JiBij=⋁f∈Πi∈IJi⋀i∈IBif(i).

By (LCB2), for each f ∈ Π_i∈I J_i, there is {B_ik}_{k∈K_i} ⊆dir 𝓑 such that ⋀_i∈I B_if(i) = ⋁_{k∈K_i} B_ik. Next, we prove that {⋀_i∈I B_if(i) : f ∈ Π_i∈I J_i} ⊆dir L^X.

Let f, g ∈ Π_i∈I J_i. Then f(i), g(i) ∈ J_i and B_if(i), B_ig(i) ∈ 𝓑_i for each i ∈ I. Since 𝓑_i is up-directed, there is B_{ij_i} ∈ 𝓑_i such that B_if(i), B_ig(i) ≤ B_{ij_i}. Define h : I → Π_i∈I J_i by: h(i) = B_{ij_i} for all i ∈ I. Thus h ∈ Π_i∈I J_i and

⋁i∈IBif(i),⋁i,g(i)Big(i)≤⋁i,h(i)Bih(i).

Hence the aimed set is up-directed. By (LCB3), there is 𝓓 ⊆dir 𝓑 such that

⋀i∈IAi=⋁f∈Πi∈IJi⋀i∈IBif(i)=⋁D∈CB.

(LC3) : Let {A_i}_i∈I ⊆dir 𝓒_𝓑. For each i ∈ I, there is {B_ij}_{j∈J_i} ⊆dir 𝓑 such that A_i = ⋁_{j∈J_i} B_ij. Thus there is {B_k}_k∈K ⊆dir 𝓑 such that ⋁_i∈I A_i = ⋁_k∈K B_k by (LCB3). Therefore ⋁_i∈I A_i ∈ 𝓒_𝓑.

Finally, since 𝓑 ⊆ 𝓒_𝓑 is an L-convex base, we know that 𝓒_𝓑 is unique.□

By Theorems 3.4 and 3.5, we see that(LCB1)–(LCB3) is a necessity and sufficient condition for L-convex bases. Thus we present the axiomatic definition of L-convex bases as follows.

Definition 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 satisfies (LCB1)–(LCB3).

Let (X, 𝓑_X) and (Y, 𝓑_Y) be L-convex bases. A mapping f : X → Y is called an L-convex base preserving mapping if fL← (B) ∈ 𝓑_X for all B ∈ 𝓑_Y. 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.

Theorem 3.8

Let (X, 𝓑_X) and (X, 𝓑_Y) be L-convex base spaces. If f : X → Y is an L-convex base preserving mapping, then f : (X, 𝓒_{𝓑_X}) → (Y, 𝓒_{𝓑_Y}) is an L-convex structure preserving mapping.

Proof

If A_Y ∈ 𝓒_{𝓑_Y}, then there is {B_i}_i∈I ⊆dir 𝓑_Y such that A_Y = ⋁_i∈I B_i. Thus { fL← (B_i)}_i∈I ⊆dir 𝓑_X and fL← (A_Y) = ⋁_i∈I fL← (B_i) ∈ 𝓒_{𝓑_X}.□

By Theorems 3.7, the category L-CS is a subcategory of L-CBS. Thus we can define a factor 𝔼_b : L-CS → L-CBS by:

Eb(X,C)=(X,C),Eb(f)=f.

By Theorems 3.5 and 3.8, we can define a factor 𝔽 : L-CBS → L-CS by:

F(X,B)=(X,CB),F(f)=f.

Theorem 3.9

(𝔼_b, 𝔽) is a Galois’s connection and 𝔾 is a left inverse of 𝔼_b.

Proof

By Theorems 3.5, 3.7 and 3.8, 𝕀_L–CS = 𝔽 ∘ 𝔼_b and 𝔼_b ∘ 𝔽 ≤ 𝕀_L–CBS, where 𝕀_L–CS and 𝕀_L–CBS are identity factors of L-CS and L-CBS, respectively.□

Corollary 3.10

L-CS can be embedded as a coreflective subcategory of L-CBS.

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.

Proof

Let (X, φ) be an L-closure space. We verify that φ satisfies (LCB1)–(LCB3).

Since ⊤ ∈ φ, the result is clear.
If {B_i}_i∈I ∈ φ and 𝓑_i = {B_ij}_{j∈J_i} ⊆dir φ with B_i = ⋁_{j∈J_i} B_ij, then

⋀i∈IBi=⋀i∈I⋁j∈JiBij=⋁f∈Πi∈IJi⋀i∈IBif(i).

In addition, it is direct to show that the set 𝓓 = {⋀_i∈I B_if(i) : f ∈ Π_i∈I J_i} ⊆ φ is up-directed. Hence ⋀_i∈I B_i = ⋁ 𝓓 showing that (LCB2) hold for φ.
Let {A_i}_i∈I ⊆dir L^X and let {B_ij}_{j∈J_i} ⊆dir φ such that A_i = ⋁_{j∈J_i} B_ij for each i ∈ I. Further, take A = ⋁_i∈I A_i, φ_* = {B_ij : i ∈ I, j ∈ J_i} and φ_F = {B ∈ φ_* : F ≤ B} for each F ∈ 𝔉(A).

Now, we prove that φ_F is nonempty for each F ∈ 𝔉(A). Since

F∈F(A)=F(⋁i∈I⋁j∈JiBij)=⋃i∈I⋃j∈JiF(Bij),

there are i_F ∈ I and j_F ∈ J_{i_F} such that F ∈ 𝔉(B_{i_Fj_F}). Thus B_{i_Fj_F} ∈ φ_F.

Further, we prove that the set {⋀φ_F : F ∈ 𝔉(A)} is up-directed.

If F, G ∈ 𝔉(A), then F ∨ G ∈ 𝔉(A) and φ_F∨G ⊆ φ_F ∩ φ_G. Thus ⋀ φ_F, ⋀ φ_G ≤ ⋀ φ_F∨G. Hence {⋀ φ_F : F ∈ 𝔉(A)} is up-directed.

Finally, since F ≤ B_{i_Fj_F} ≤ A_{i_F} ≤ A, we have F ≤ ⋀ φ_F ≤ A and

A=⋁F∈F(A)F≤⋁F∈F(A)⋀φF≤A.

Therefore A = ⋁_F∈𝔉(A) ⋀ φ_F showing that (LCB3) holds for φ.□

Remark 3.12

The inverse result of Theorem 3.11 fails. An L-convex base may not be an L-closure structure. To shows this, let X = {x} and let L = [0, 1]. Define 𝓑 = [0, 13 )^X ∪( 13 , 1)^X. Then 𝓑 is an L-convex base. Let 𝓑₁ = ( 13 , 1)^X ⊆ 𝓑. But 1 ∉ 𝓑 and ⋀ 𝓑₁ ∉ 𝓑. Thus both (LC1) and (LC2) fail for 𝓑.

Theorem 3.13

L-CS is a bicoreflective subcategory of L-CSS, where L-CSS is the category of L-closure spaces and L-closure structure preserving mappings.

Proof

To verify this, let (X, φ) be an L-closure space. By Theorems 3.5 and 3.11, 𝓒_φ is an L-convex structure. Thus we only need to prove that id_X : (X, 𝓒_φ) → (X, φ) is a bicoreflector. It sufficient to show that the following statements hold.

id_X : (X, 𝓒_φ) → (X, φ) is an L-closure structure preserving mapping.
for any L-convex space (Y, 𝓒_Y), if f : (Y, 𝓒_Y) → (X, φ) is an L-closure structure preserving mapping, then f : (Y, 𝓒_Y) → (X, 𝓒_φ) is an L-convex structure preserving mapping.

Since φ ⊆ 𝓒_φ, (1) is clear. For (2), if A ∈ 𝓒_φ, then there is an up-directed set {B_i}_i∈I ⊆ φ such that A = ⋁_i∈I B_i. Since f : (Y, 𝓒_Y) → (X, φ) is an L-closure structure preserving mapping, fL← (A) = ⋁_i∈I fL← (B_i) ∈ 𝓒_Y. Thus (2) holds.□

4 L-convex subbases

Definition 4.1

Let (X, 𝓒) be an L-convex space. A subset 𝓓 ⊆ 𝓒 is called an L-convex subbase of 𝓒 if 𝓑_𝓓 = {⋀ 𝓕 : 𝓕 ⊆ 𝓓} is an L-convex base of 𝓒.

Proposition 4.2

Let (X, 𝓒) be an L-convex space and let 𝓓 ⊆ 𝓒.

If there is 𝓓_F ⊆ 𝓓 such that co(F) = ⋀ 𝓓_F for any F ∈ 𝔉(L^X), then 𝓓 is an L-convex subbase of 𝓒.
An L-convex base is an L-convex subbase.
𝓒 is the coarsest L-convex structure containing 𝓓 if 𝓓 is an L-convex subbase.

Proof

(1) and (2) directly follow from Proposition 3.2 and Definition 4.1.

(3) : Let 𝓒₁ be an L-convex structure with 𝓓 ⊆ 𝓒₁. If A ∈ 𝓒, then there is {B_i}_i∈I ⊆dir 𝓑_𝓓 such that A = ⋁_i∈I B_i. Since there is 𝓓_i ⊆ 𝓓 such that B_i = ⋀ 𝓓_i for each i ∈ I, we have B_i ∈ 𝓒₁ by (LC2), and A ∈ 𝓒₁ by (LC3). Thus 𝓒 ⊆ 𝓒₁.□

Remark 4.3

(1) of Proposition 4.2 just gives a sufficient condition of an L-convex subbase. In the example of Remark 3.3, 𝓑 is an L-concave subbase of 𝓒. However, x23 ∈ 𝔉(L^X) and co(x23)=x23≠⋀D for any 𝓓 ⊆ 𝓑. To obtain a necessary and sufficient condition, we present the following results.

Theorem 4.4

If (X, 𝓒) is an L-convex space with an L-convex subbase 𝓓, then

(LCSB) there is an up-directed set {B_i}_i∈I ⊆ L^X such that ⊤ = ⋁_i∈I B_i, and there is subset {D_ij}_{j∈J_i} ⊆ 𝓓 such that B_i = ⋀_{j∈J_i} D_ij for each i ∈ I.

Proof

The result directly follows from (LCB1) of the L-convex base 𝓑_𝓓.□

Theorem 4.5

Let 𝓓 ⊆ L^X be a set satisfying (LCSB), then there is a unique L-convex structure 𝓒_𝓓 with 𝓓 as an L-convex subbase.

Proof

Let 𝓑_𝓓 = {⋀ 𝓕 : 𝓕 ⊆ 𝓓}. We check that 𝓑_𝓓 is an L-convex base.

It directly follows from (LCSB).
Let {B_i}_i∈I ⊆ 𝓑_𝓓 and let {B_ij}_{j∈J_i} ⊆ 𝓓 with B_i = ⋀_{j∈J_i} B_ij. Then ⋀_i∈I B_i = ⋀_i∈I ⋀_{j∈J_i} B_ij ∈ 𝓑_𝓓. Since {⋀_i∈I B_i} ⊆dir 𝓑_𝓓, (LCB2) holds trivially.
Let {A_i}_i∈I ⊆ L^X be up-directed, and let {B_ij}_{j∈J_i} ⊆dir 𝓑_𝓓 such that A_i = ⋁_{j∈J_i} B_ij for each i ∈ I. Thus, for each i ∈ I and each j ∈ J_i, there is a subfamily {D_ijk}_{k∈K_{J_i}} ⊆ 𝓓 with B_ij = ⋀_{k∈K_{J_i}} D_ijk. Take A = ⋁_i∈I A_i. We have A = ⋁_i∈I A_i = ⋁_i∈I ⋁_{j∈J_i} B_ij. In addition, for each F ∈ 𝔉(A), we have

F∈F(A)=F(⋁i∈I⋁j∈JiBij)=⋃i∈I⋃j∈JjF(Bij).

Thus F ∈ 𝔉(B_ij) for some i ∈ I and j ∈ J_i. Let B_F = ⋀{B_ij : F ∈ 𝔉(B_ij)}. Then

BF=⋀F∈F(Bij)Bij=⋀F∈F(Bij)⋀k∈KJiDijk∈BD.

Further, since {F : F ∈ 𝔉(A)} is up-directed, {B_F : F ∈ 𝔉(A)} is up-directed. So

F≤BF=⋀{Bij:F∈F(Bij)}≤A.

This implies that

A=⋁F∈F(A)F≤⋁F∈F(A)BF≤A.

Hence A = ⋁_F∈𝔉(A) B_F. Therefore (LCB3) holds for 𝓑_𝓓.

By Theorem 3.5, there is a unique L-convex structure with 𝓑_𝓓 as an L-convex base. Thus it is the unique L-convex structure with 𝓓 as an L-convex subbase.□

By Theorems 4.4 and 4.5, (LCSB) is a necessity and sufficient condition for L-convex subbases. Thus we present the following axiomatic definition.

Definition 4.6

A set 𝓓 ⊆ L^X is called an L-convex subbase and the pair (X, 𝓓) is called an L-convex subbase space provided that 𝓓 satisfies (LCSB).

Let (X, 𝓓_X) and (Y, 𝓓_Y) be L-convex subbase spaces. A mapping f : X → Y is called an L-convex subbase preserving mapping if fL← (D) ∈ 𝓓_X for all D ∈ 𝓓_Y.

The category of L-convex subbase spaces and L-convex subbase preserving mappings is denoted by L-CSBS. Next, we study relations between L-CS and L-CSBS.

Theorem 4.7

An L-convex structure is an L-convex subbase of itself.

Theorem 4.8

Let (X, 𝓓_X) and (Y, 𝓓_Y) be L-convex subbase spaces. If f : X → Y is an L-convex subbase preserving mapping, then f : (X, 𝓒_{𝓓_X}) → (Y, 𝓒_{𝓓_Y}) is an L-convex structure preserving mapping.

Proof

If A ∈ 𝓒_{𝓓_Y}, then there is {B_i}_i∈I ⊆dir L^Y with A = ⋁_i∈I B_i where there is {D_ij}_{j∈J_i} ⊆ 𝓓_Y such that B_i = ⋀_{j∈J_i} D_ij for each i ∈ I. Thus

fL←(Bi)=fL←(⋀j∈JiDij)=⋀j∈JifL←(Dij)

Since {B_i}_i∈I is up-directed and f : X → Y is an L-concave subbase preserving mapping, { fL← (B_i)}_i∈I ⊆ 𝓓_X is up-directed. Hence { fL← (B_i)}_i∈I ∈ 𝓑_{𝓓_X} and

fL←(A)=fL←(⋁i∈I⋀j∈JiDij)=⋁i∈IfL←(Bi)∈CDX.

Therefore f : (X, 𝓒_{𝓓_X}) → (Y, 𝓒_{𝓓_Y}) is an L-convex preserving mapping.□

By Theorem 4.7, the category L-CS is a subcategory of the category L-CSBS. Thus we can define a factor 𝔼_s : L-CS → L-CSBS by:

Es(X,C)=(X,C),Es(f)=f.

By Theorems 4.5 and 4.8, we can define a factor 𝔾 : L-CSBS → L-CS by:

G(X,D)=(X,CD),G(f)=f.

Theorem 4.9

(𝔼_s, 𝔾) is a Galois’s connection and 𝔾 is a left inverse of 𝔼_s.

Proof

By Theorems 4.7, 4.5 and 4.8, 𝕀_L–CS = 𝔾 ∘ 𝔼_s and 𝔼_s ∘ 𝔾 ≤ 𝕀_L–CSBS, where 𝕀_L–CS and 𝕀_L–CSBS are identities factors in L-CS and L-CSBS, respectively.□

Corollary 4.10

L-CS can be embedded as a coreflective subcategory of L-CSBS.

From the proofs of Theorems 4.5 and 4.8, we have the following conclusion.

Corollary 4.11

If (X, 𝓓) is an L-convex subbase space, then the set 𝓑_𝓓 = {⋀ 𝓕 : 𝓕 ⊆ 𝓓} is an L-convex base. In addition, if (X, 𝓓_X) and (Y, 𝓓_Y) are L-convex subbase spaces and f : X → Y is an L-convex subbase preserving mapping, then f : (X, 𝓑_{𝓓_X}) → (Y, 𝓑_{𝓓_Y}) is an L-convex base preserving mapping.

Now, we consider relations between L-CBS and L-CSBS.

By (2) of Proposition 4.2, the category L-CBS is a subcategory of the category L-CSBS. Thus we can define a factor 𝔼:L-CBS → L-CSBS by:

E((X,B))=(X,B),E(f)=f.

Conversely, by Corollary 4.11, we can define a factor ℍ : L-CSBS → L-CBS by:

H((X,BD))=(X,D),H(f)=f.

Theorem 4.12

(𝔼, ℍ) is a Galois’s connection and ℍ is a left inverse of 𝔼.

Proof

By Proposition 4.2 and Corollary 4.11, 𝕀_L–CBS = ℍ ∘ 𝔼 and 𝔼 ∘ ℍ ≤ 𝕀_L–CSBS.□

Corollary 4.13

L-CBS can be embedded as a coreflective subcategory of L-CSBS.

5 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 ∈ 2finX ). 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.

From definition 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 iff co_𝓒(F) = cl_𝓣∩𝓒(F) for all F ∈ 2finX . Indeed, if (X, 𝓣, 𝓒) is a topological-convex space, then co_𝓒(F) = cl_𝓣(co_𝓒(F)) for any F ∈ 2finX . Thus co_𝓒(F) = cl_𝓣(co_𝓒(F)) ∈ 𝓣 ∩ 𝓒 which shows

coC(F)≤clT∩C(F)≤clT(coC(F))=coC(F).

Conversely, if co_𝓒(F) = cl_𝓣∩𝓒(F) for all F ∈ 2finX , then it is clear that co_𝓒(F) ∈ 𝓣 for all F ∈ 2finX . That is, 𝓣 is compatible with 𝓒.
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 𝓑₁ ⊆ 𝓑 such that A = ⋃ 𝓑. As described in [2], a subset of a convex structure is a base iff it contains all polytopes. Thus, 𝓣 is compatible with 𝓒 iff 𝓒 has a closed base (i.e., 𝓒 has a base 𝓑 contained in 𝓣).

From (2) of Remark 5.1 and Definition 3.1, we extend the notion of topological-convex spaces into L-fuzzy settings as follows.

Definition 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.

Theorem 5.3

Let X be a set equipped with an L-cotopology 𝓣 and an L-convex structure 𝓒. Let cl_𝓣∩𝓒 is the closure operator of the L-closure structure 𝓣 ∩ 𝓒. Then the following conditions are equivalent.

(X, 𝓣, 𝓒) is an L-topological-convex space.
co_𝓒(A) = ⋁_F∈𝔉(A) cl_𝓣∩𝓒(F) for any A ∈ L^X.
co_𝓒(F) = ⋁_G∈𝔉(F) cl_𝓣∩𝓒(G) for any F ∈ 𝔉(L^X).

Proof

(1) ⇒ (2) : Let 𝓑 ⊆ 𝓣 ∩ 𝓒 be an L-convex base of 𝓒. Let A ∈ L^X. By (LDF),

coC(A)=⋁F∈F(A)coC(F)≤⋁F∈F(A)clT∩C(F).

Conversely, since 𝓑 is an L-convex base of 𝓒, there is an up-directed set 𝓑₁ ⊆ 𝓑 such that co_𝓒(A) = ⋁ 𝓑₁. For each F ∈ 𝔉(A), we have

F∈F(A)⊆F(coC(A))=F(⋁B1)=⋃B∈B1F(B).

Thus there is B ∈ 𝓑 such that F ∈ 𝔉(B). Since B ∈ 𝓑₁ ⊆ 𝓑 ⊆ 𝓣 ∩ 𝓒, we have

clT∩C(F)≤clT∩C(B)=B≤coC(A).

Hence ⋁_F∈𝔉(A) cl_𝓣∩𝓒(F) ≤ co_𝓒(A).

(2) ⇒ (3) : Clear.

(3) ⇒ (1) : We have 𝓑 = {cl_𝓣∩𝓒(F) : F ∈ 𝔉(L^X)} ⊆ 𝓣 ∩ 𝓒. Thus for each C ∈ 𝓒,

C=coC(C)=⋁F∈F(C)coC(F)=⋁F∈F(C)⋁G∈F(F)clT∩C(F)=⋁G∈F(C)clT∩C(G).

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

Cφ={A∈LX:∃D⊆dir⁡φ,A=⋁D}

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

coCT(F)=⋁G∈F(F)coCT(G)≤⋁G∈F(F)clT∩CT(G).

Conversely, let G ∈ 𝔉(F). For each B ∈ 𝓒_𝓣 with F ≤ B, there is an up-directed set 𝓓 ⊆ 𝓣 such that B = ⋁ 𝓓. Thus G ∈ 𝔉 (B) = ⋃_D∈𝓓 𝔉(D). Hence there is D_G ∈ 𝓓 such that G ∈ 𝔉(D_G). This shows D_G ∈ 𝓣 ∩ 𝓒_𝓣 and G ≤ D_G ≤ B. So

coCT(F)=⋀{B∈CT:F≤B}≥DG=clT∩CT(DG)≥clT∩CT(G).

Therefore ⋁_G∈𝔉(F) cl_{𝓣∩𝓒_𝓣}(G) ≤ co_{𝓒_𝓣}(F).□

Remark 5.5

Unlike topological-convex spaces, an L-topological-convex space (X, 𝓣, 𝓒) fails to imply co_𝓒(F) ∈ 𝓣 for all F ∈ 𝔉(L^X). For example, let X = {x} and L = [0, 1]. Define 𝓣 = {x_r : r ∈ [0, 12 ) ∪ {1}} and 𝓒 = 𝓣 ∪ { x12 }. Then (X, 𝓣, 𝓒) is an L-topological-convex space. However, x12 ∈ 𝔉(L^X) and co_𝓒( x12 ) = x12 ∉ 𝓣. Hence cl_𝓣∩𝓒( x12 ) = cl_𝓣( x12 ) = x₁ which shows that co_𝓒( x12 ) ≠ cl_𝓣∩𝓒( x12 ).
In the L-convex structure generated by an L-cotopology, each element in 𝓒 is the supermum of an up-directed subset of 𝓣. This property is also preserved by an L-topological-convex space. The L-convex structure generated by an L-cotopology containing this L-cotopology. But an L-topological-convex space (X, 𝓣, 𝓒), 𝓣 ⊆ 𝓒 may fail. For example, let X = {x, y} and let L = [0, 1]. Define C={0_,x12,y12,1_} and T=C∪{12_}. Then (X, 𝓣, 𝓒) is an L-topological-convex space. But 𝓣 ⊈ 𝓒.
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 ∈ 2finD }, where ⊔𝓖 stands for ⋁𝓖 (𝓖 is finite). 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.

Lemma 5.6

Let X be equipped with an L-cotopology 𝓣 and an L-convex structure 𝓒. Let φ = 𝓣 ∩ 𝓒. Define

Tw={A∈LX:∃B⊆φ⊔,A=⋀B},

where φ^⊔ = {⊔ 𝓓 : 𝓓 ∈ 2finφ }. Then 𝓣_w is an L-cotopology satisfying 𝓣_w = ⋀{𝓢 ∈ 𝔗(X) : φ ⊆ 𝓢} and 𝓣_w ⊆ 𝓣, where 𝔗(X) is the set of all L-cotopologies on X.

Proof

Since ⊤, ⊥ ∈ φ ⊆ φ^⊔, we have ⊥, ⊤ ∈ 𝓣_w.
Let {A_i}_i∈I ⊆ 𝓣_w. For each i ∈ I, there is 𝓑_i ⊆ φ^⊔ such that A_i = ⋀𝓑_i. Let 𝓑 = ⋃_i∈I𝓑_i. Then 𝓑 ⊆ φ^⊔ and ⋀_i∈IA_i = ⋀ 𝓑 ∈ 𝓣_w.
We firstly prove that A ∨ B ∈ φ^⊔ for all A, B ∈ φ^⊔.

Since A, B ∈ φ^⊔, there are 𝓓₁, 𝓓₂ ∈ 2finφ such that ⊔𝓓₁ = A and ⊔𝓓₂ = B. Since 𝓑 = 𝓓₁ ∪ 𝓓₂, A ∨ B = ⊔𝓑 ∈ φ^⊔. Next, we prove that 𝓣_w satisfies (LT3).

If A, B ∈ 𝓣_w, 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

A∨B=⋀i∈I,j∈JDi∨Dj=⋀i∈I,j∈JDij∈Tw.

Therefore 𝓣_w is an L-cotopology.

To prove that 𝓣_w ⊆ 𝓣, let A ∈ 𝓣_w. Then there is 𝓑 ⊆ φ^⊔ such that A = ⋀𝓑. Since 𝓑 ⊆ φ^⊔ ⊆ 𝓣, we have 𝓑 ⊆ 𝓣 and A = ⋀ 𝓑 ∈ 𝓣. Therefore 𝓣_w ⊆ 𝓣.

Finally, since 𝓣_w ⊇ φ^⊔ ⊇ φ, we have ⋂{𝓢 ∈ 𝔗(X) : φ ⊆ 𝓢} ⊆ 𝓣_w.

Conversely, let φ ⊆ 𝓢 ∈ 𝔗(X). If A ∈ 𝓣_w, then there is 𝓑 ⊆ φ^⊔ such that A = ⋀ 𝓑. Thus, for each B ∈ 𝓑, there is 𝓓 ∈ 2finφ such that B = ⊔𝓓. Hence B ∈ 𝓢 and A = ⋀ 𝓑 ∈ 𝓢. So 𝓣_w ⊆ ⋂{𝓢 ∈ 𝔗(X) : φ ⊆ 𝓢}.□

With help of 𝓣_w, 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 iff (X, 𝓣_w, 𝓒) is an L-topological-convex space.

Proof

𝓣_w ∩ 𝓒 = 𝓣 ∩ 𝓒 by Lemma 5.6. Thus the result follows from Theorem 5.3.□

6 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 L-topological-convex spaces.

For L-topological-convex spaces (X, 𝓣_X, 𝓒_Y) and (Y, 𝓣_Y, 𝓒_Y), a mapping f : X → Y is called an L-topological-convex structure preserving mapping, if f : (X, 𝓣_X) → (Y, 𝓣_Y) is L-continuous and f : (X, 𝓒_X) → (Y, 𝓒_Y) is L-convex structure preserving.

The category of L-topological-convex spaces and L-topological-convex structure preserving mappings is denoted by L-TCS.

Definition 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,

F⪕Bimplies∀G∈F(F),∃D∈LXsuchthatG≤D⪕D⋞D≤B.

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 : (X, ⪕_X) → (Y, ⪕_Y) is an L-convex enclosed relation dual-preserving mapping. That is, for all A, B ∈ L^Y, A ⋞_Y B (resp. A ⪕_Y B) implies fL← (A) ⋞_X fL← (B) (resp. fL← (A)⪕_X fL← (B)).

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.

Lemma 6.2

Let (X, ⋞) be an L-topological enclosed relation space. Define a binary operator ⪕_⋞ on X by:

∀A,B∈LX,A⪕⋞Biff∃D⊆dir⁡LX,s.t.A=⋁D;∀D∈D,D⋞B.

Then ⪕_⋞ is an L-convex enclosed relation generated by ⋞. Moreover, for all A, B ∈ L^X, A ⪕_⋞ B iff F ⋞ B for all F ∈ 𝔉(A).

Proof

(LCER1): ⊥ ⪕_⋞ ⊥ since {⊥} ⊆dir L^X and ⊥ ⋞ ⊥.

(LCER2): If A ⪕_⋞ B, then there is 𝓓 ⊆dir L^X such that A = ⋁ 𝓓 and D ⋞ B for all D ∈ 𝓓. Thus A ≤ B by (LTER2).

(LCER3): Let B = ⋀_i∈I B_i. If A ⪕_⋞ B, then there is 𝓓 ⊆dir L^X such that A = ⋁ 𝓓 and D ⋞ B for each D ∈ 𝓓. Since B ≤ B_i for each i ∈ I, we have D ⋞ B_i for each D ∈ 𝓓. Thus A ⪕_⋞ B_i for each i ∈ I.

Conversely, let A ⪕_⋞ B_i for each i ∈ I and let 𝓓_i ⊆dir L^X such that A = ⋁ 𝓓_i and D_i ⋞ B_i for all D_i ∈ 𝓓_i. Let 𝓢 be the set of all choice mappings s : I → ⋃_i∈I 𝓓_i with s(i) ∈ 𝓓_i. Then the set 𝓓 = {⋀_i∈I s(i) : s ∈ 𝓢} has the following properties.

𝓓 is up-directed.

Let B₁, B₂ ∈ 𝓓. Then there are s₁, s₂ ∈ 𝓢 such that B₁ = ⋀_i∈I s₁(i) and B₂ = ⋀_i∈I s₂(i). Thus there is D_i ∈ 𝓓_i such that s₁(i), s₂(i) ≤ D_i for each i ∈ I. Define s₃ : I → ⋃_i∈I 𝓓_i by s₃(i) = D_i for each i ∈ I. Then s₃ ∈ 𝓢 and B₁, B₂ ≤ B₃ = ⋀_i∈I s₃(i) ∈ 𝓓. Therefore 𝓓 is up-directed.
A = ⋁ 𝓓.

Let x_λ ≺ A. Since A ⪕_⋞ B_i for each i ∈ I, there is 𝓓_i ⊆dir L^X such that A = ⋁ 𝓓_i. Thus there is D_i ∈ 𝓓_i such that x_λ ≤ D_i. Define s : I → ⋃_i∈I 𝓓_i by s(i) = D_i for each i ∈ I. Then s ∈ 𝓢 and x_λ ≤ ⋀_i∈I s(i) ∈ 𝓓. Hence A ≤ ⋁ 𝓓. Conversely, if y_μ ≺ ⋁ 𝓓, then there is s ∈ 𝓢 such that y_μ ≤ ⋀_i∈I s(i). Since s(i) ∈ 𝓓_i, we have y_μ ≤ ⋁ 𝓓_i = A. Thus ⋁ 𝓓 ≤ A. Therefore A = ⋁ 𝓓.
D ⋞ B for any D ∈ 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)–(iii), we find that A ⪕_⋞ ⋀_i∈I B_i.

(LCER4): Let {A_i}_i∈I ⊆dir L^X with A = ⋁i∈Idir A_i. If A ⪕_⋞ B, then there is 𝓓 ⊆dir L^X such that A = ⋁ 𝓓 and D ⋞ B for each D ∈ 𝓓. For each i ∈ I, it is clear that 𝓓_i = {A_i ∧ D : D ∈ 𝓓} ⊆dir L^X and A_i = ⋁ 𝓓_i. In addition, A_i ∧ D ≤ D ⋞ B and so A_i ∧ D ⋞ B for each D ∈ 𝓓. Thus A_i ⪕_⋞ B for each i ∈ I.

Conversely, let A_i ⪕_⋞ B for each i ∈ I. Then, for each i ∈ I, there is 𝓓_i ⊆dir L^X such that A_i = ⋁ 𝓓_i and D_i ⋞ B for all D_i ∈ 𝓓_i. Let 𝓓 = ⋃_i∈I 𝓓_i, G = ⋁ 𝓓 and φ_F = {D : F ≤ D ∈ 𝓓} for all F ∈ 𝔉(A). We have the following results.
1. φ_F is nonempty for all F ∈ 𝔉(A).
  
  Since {A_i}_i∈I ⊆dir L^X, there is i_F ∈ I such that F ∈ 𝔉(A_{i_F}) = 𝔉 (⋁ 𝓓_{i_F}) = ⋃_{D_{i_F}} ∈ 𝓓_{i_F} 𝔉(D_{i_F}). So there is D_{i_F} ∈ 𝓓_{i_F} ⊆ 𝓓 with F ∈ 𝔉(D_{i_F}). Hence D_{i_F} ∈ φ_F.
2. A = ⋁_F∈𝔉(A) ⋀ φ_F.
  
  Since F ≤ D_{i_F} ≤ A_{i_F} ≤ A for all F ∈ 𝔉(A), we have F ≤ ⋀ φ_F ≤ A. Thus
  
  A=⋁F∈F(A)F≤⋁F∈F(A)⋀φF≤A.
  
  Hence A = ⋁_F∈𝔉(A) ⋀ φ_F.
3. {⋀ φ_F : F ∈ 𝔉(A)} ⊆ 𝓓 is up-directed.
Let F, G ∈ 𝔉(A). Then there is H ∈ 𝔉(A) such that F, G ≤ H. Since D ∈ φ_F and D ∈ φ_G for each D ∈ φ_H, we have ⋀ φ_F, φ_G ≤ φ_H. Thus (iii) holds.
⋀ φ_F ⋞ B for each F ∈ 𝔉(A).

We have ⋀ φ(F) ≤ D ⋞ B for each D ∈ φ(F). Thus ⋀ φ(F) ⋞ B.

Combining (i)–(iv), we conclude that ⋁i∈Idir A_i ⪕_⋞ B.

(LCER5): If A ⪕_⋞ B, then there is 𝓓 ⊆dir L^X such that A = ⋁ 𝓓 and D ⋞ B for each D ∈ 𝓓. Thus, for each D ∈ 𝓓, there is E_D ∈ L^X such that D ≤ E_D ⋞ E_D ≤ B by (LTER5). This shows D ⋞ E_D ⋞ B.

Let φ_D = {E_D ∈ L^X : D ⋞ E_D ⋞ B} for each D ∈ 𝓓. Then φ_D is nonempty. Further, the set {⋀ φ_D : D ∈ 𝓓} is up-directed and D ⋞ ⋀ φ_D by (LTER3). Also, it is clear that ⋀ φ_D ⋞ B. Let C = ⋁_D∈𝓓 ⋀ φ_D. We have D ⋞ C for each D ∈ 𝓓. Hence A ⪕_⋞ C ⪕_⋞ B as desired.

Therefore ⪕_⋞ is an L-convex enclosed relation.

Finally, let A, B ∈ L^X. We prove that A ⪕_⋞ B iff F ⋞ B for each F ∈ 𝔉(A).

If A ⪕_⋞ B, then there is an up-directed set 𝓓 ⊆ L^X such that A = ⋁ 𝓓 and D ⋞ B for each D ∈ 𝓓. Thus, for each F ∈ 𝔉 (A), we have F ∈ 𝔉(A) = 𝔉(⋁ 𝓓) = ⋃_D∈𝓓 𝔉(D). Hence there is D ∈ 𝓓 such that F ≤ D ⋞ B. Therefore F ⋞ B.

Conversely, let F ⋞ B for all F ∈ 𝔉(A). Since {F : F ∈ 𝔉(A)} is up-directed and A = ⋁_F∈𝔉(A) G, we have A ⪕_⋞ B by definition.□

Theorem 6.3

For an L-topological enclosed relation space (X, ⋞), the triple (X, ⋞, ⪕_⋞) is an L-topological-convex enclosed space.

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 D_G ∈ L^X such that G ≤ D_G ⋞ D_G ≤ B. Since D_G ⋞ D_G and {D_G} ⊆ L^X is up-directed, we have D_G ⪕_⋞ D_G. Hence

G≤DG⪕⋞DG⋞DG≤B.

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. Define

A⋞UBiff∃U∈UsuchthatA≤⋀yμ≰BU(yμ);A⪕UBiff∃D⊆dir⁡LXsuchthatA=⋁D;∀D∈D,D⋞UB.

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 6.5

Let (X, δ) be a pointwise S-quasi-proximity space. Define

A⋞δBiff∀yμ≰B,δ(yμ,A)=0;A⪕δBiff∃D⊆dir⁡LXsuchthatA=⋁D;∀D∈D,D⋞δB.

Then (X, ⋞_δ, ⪕_δ) is an L-topological-convex enclosed relation space by Theorem 4.7 in [33], Lemma 6.2 and Theorem 6.3.

Example 6.6

Let (X, d) be a pointwise pseudo-metric space. Define

A⋞dBiff∀yμ≰B,⋀xη≤Ad(xη,yμ)>0;A⪕dBiff∃D⊆dir⁡LXsuchthatA=⋁D;∀D∈D,D⋞dB.

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.

Theorem 6.7

If (X, ⋞, ⪕) is an L-topological-convex enclosed relation space, then (X, 𝓣_⋞, 𝓒_⪕) is an L-topological-convex space.

Proof

We firstly verify that co_⪕ : L^X → L^X, defined by co_⪕(A) = ⋀{B ∈ L^X : A ⪕ B} in Theorem 2.8, satisfies co_⪕(A) = ⋁_F∈𝔉(A) co_⪕(F) for all A ∈ L^X.

Clearly, ⋁_F∈𝔉(A) co_⪕(F) ≤ co_⪕(A). Conversely, for each G ∈ 𝔉(A), we have

G⪕co⪕(G)≤⋁F∈F(A)co⪕(F).

Thus

A=⋁G∈F(A)G⪕⋁F∈F(A)co⪕(F).

Hence co_⪕(A) ≤ ⋁_F∈𝔉(A) co_⪕(F).

To prove the desired result, let F ∈ 𝔉(L^X). We need to prove that

coC⪕(F)=⋁G∈F(F)clT⋞∩C⪕(G).

Since 𝓣_⋞ ∩ 𝓒_⪕ ⊆ 𝓒_⪕, we have cl_{𝓒_⪕} ≤ cl_{𝓣_⋞∩𝓒_⪕}. Thus

coC⪕(F)=⋁G∈F(F)coC⪕(G)≤⋁G∈F(F)clT⋞∩C⪕(G).

Conversely, let G ∈ 𝔉(F). To prove that cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ co_{𝓒_⪕}(F), let B ∈ L^X with co_{𝓒_⪕}(F) ≤ B. We next prove that cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ B.

Since co_{𝓒_⪕}(F) ≤ B, we have F ⪕ B. Thus there is D ∈ L^X such that G ≤ D ⪕ D ⋞ D ≤ B and so

G≤D=clT⋞(D)=coC⪕(D).

Hence G ≤ D ∈ 𝓣_⋞ ∩ 𝓒_⪕ which implies cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ D ≤ B. In conclusion, ⋁_G∈𝔉(F) cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ B. Therefore ⋁_G∈𝔉(F) cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ co_{𝓒_⪕}(F).

Thus, by Theorem 5.3, (X, 𝓣_⋞, 𝓒_⪕) is an L-topological-convex space.□

Theorem 6.8

If (X, 𝓣, 𝓒) is an L-topological-convex space, then (X, ⋞_𝓣, ⪕_𝓒) is an L-topological-convex enclosed space.

Proof

Let F ∈ 𝔉(L^X) and B ∈ L^X such that F ⪕_𝓒 B. Then

⋁G∈F(F)clT∩C(G)=coC(F)≤B.

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

G≤DG⪕CDG⋞TDG≤B.

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-topological-convex 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 define a factor 𝕋 : L-TCERS → L-TCS by

T(X,⋞,⪕)=(X,T⋞,C⪕)andT(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 L-concave enclosed relation.

Lemma 6.13

Let X be a set equipped with an L-topological enclosed relation ⋞ and an L-convex enclosed relation ⪕. Define a binary operator ⊏_w on X by

A⊏¯wBiffcl¯(A)≤B,

where the operator cl : L^X → L^X is defined by:

cl¯(A)=⋀{cl~(A):cl⋞∨co⪕≤cl~∈C(X)},

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.

(LCL1)–(LCL3) are easy.

(LCL4): Let A ∈ L^X. Since cl ≤ cl~ for any cl~ ∈ ℭ(X) with cl_⋞ ∨ co_⪕ ≤ cl~ ,

cl¯(cl¯(A))=⋀{cl~(cl¯(A)):cl⋞∨co⪕≤cl~∈C(X)}≤⋀{cl~(cl~(A)):cl⋞∨co⪕≤cl~∈C(X)}≤⋀{cl~(A):cl⋞∨co⪕≤cl~∈C(X)}=cl¯(A).

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

⊏¯=⋞cl⊏¯≤⋞cl¯=⊏¯w.

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.

φ_{⊏_w} = 𝓣_⋞ ∩ 𝓒_⪕, where φ_{⊏_w} is the L-closure structure induced by ⊏_w.
𝓣_{⊏_w} = 𝓣_w ⊆ 𝓣_⋞, where 𝓣_{⊏_w} is the L-cotopology generated by φ_{⊏_w}, and 𝓣_w is the L-cotopology generated by 𝓣_⋞ ∩ 𝓒_⪕.
Define a binary operator ⋞_{⊏_w} on X by:

A⋞⊏¯wBiff∀⊏¯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 L-topological enclosed relation generated by ⊏_w.

Proof

Let A ∈ L^X. We have

A∈φ⊏¯w⇔A⊏¯wA⇔A=cl¯(A)≥cl⋞(A)∨co⪕(A)≥A⇒cl⋞(A)=co⪕(A)=A⇔A∈T⋞∩C⪕.

Thus φ_{⊏_w} ⊆ 𝓣_⋞ ∩ 𝓒_⪕. Conversely, since 𝓣_⋞ ∩ 𝓒_⪕ is an L-closure structure, we have

cl⋞∨co⪕≤clT⋞∩C⪕∈C(X).

Thus cl ≤ cl_{𝓣_⋞∩𝓒_⪕}. Hence, for any A ∈ 𝓣_⋞ ∩ 𝓒_⪕, we have

A≤cl¯(A)≤clT⋞∩C⪕(A)=A.

This implies cl(A) = A and A ∈ φ_{⊏_w}. Therefore 𝓣_⋞ ∩ 𝓒_⪕ ⊆ φ_{⊏_w}.
The result directly follows from (1).
For any ⋞͠ ∈ 𝔈(X) with ⊏_w ≤ ⋞͠, we have φ_{⊏_w} ⊆ 𝓣_⋞͠. Thus it follows that 𝓣_{⊏_w} = ⋂{𝓣 : φ_{⊏_w} ⊆ 𝓣 ∈ 𝔗(X)} ⊆ 𝓣_⋞͠. Hence

A⋞T⊏¯wB⇔clT⊏¯w(A)≤B⇒∀⊏¯w≤⋞~∈E(X),clT⋞~(A)≤clT⊏¯w(A)≤B⇒∀⊏¯w≤⋞~∈E(X),A⋞T⋞~B⇔∀⊏¯w≤⋞~∈E(X),A⋞~B⇔A⋞⊏¯wB.

Therefore ⋞_{𝓣_{⊏_w}} ≤ ⋞_{⊏_w}.

Conversely, if A ⋞_{⊏_w} B, then A ⋞͠ B (i.e., cl_⋞͠(A) ≤ B) for each ⊏_w ≤ ⋞͠ ∈ 𝔈(X). Let D = ⋀_{⊏_w≤⋞͠∈𝔈(X)} cl_⋞͠(A). Then for each ⊏_w ≤ ⋞͠ ∈ 𝔈(X), we have

D≤clT⊏¯w(D)≤clT⋞~(D)≤clT⋞~(cl⋞~(A))=cl⋞~(A)≤B.

By arbitrariness of ⊏_w ≤ ⋞͠ ∈ 𝔈(X), we have

D≤clT⊏¯w(D)≤⋀⊏¯w≤⋞~∈E(X)cl⋞~(A)=D.

Thus cl_{𝓣_{⊏_w}}(A) ≤ D ∈ 𝓣_{⊏_w} which shows A ⋞_{𝓣_{⊏_w}} B. Therefore ⋞_{⊏_w} ≤ ⋞_{𝓣_{⊏_w}}.□

Theorem 6.15

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 ⪕, and ⋞_{⊏_w} be the L-topological enclosed relation generated by ⊏_w. For any F ∈ 𝔉(L^X), we denote 𝔉_F(L^X) = {H ∈ 𝔉(L^X) : F ∈ 𝔉(H)} and C_F = ⋀_{H∈𝔉_F(L^X)} co_⪕(H). Then the following conditions are equivalent.

(X, ⋞, ⪕) is an L-topological-convex enclosed relation space;
(X, 𝓣_{⊏_w}, 𝓒_⪕) is an L-topological-convex space;
(X, ⋞_{⊏_w}, ⪕) is an L-topological-convex enclosed relation space;
C_F ∈ 𝓣_{⊏_w} for all F ∈ 𝔉(L^X);
C_F ∈ 𝓣_⋞ for all F ∈ 𝔉(L^X);
C_F ∈ φ_{⊏_w} for all F ∈ 𝔉(L^X).

Proof

(1) ⇒ (2) : By (2) of Theorem 6.14, 𝓣_{⊏_w} = 𝓣_w which is an L-topology with 𝓣_⋞ ∩ 𝓒_⪕ as its closed subbase. Let F ∈ 𝔉(L^X). By (LDF),

co⪕(F)=⋁G∈F(F)co⪕(G)≤⋁G∈F(F)clT⊏¯w∩C⪕(G).

Conversely, since (X, ⋞, ⪕) is an L-topological-convex enclosed relation space,

co⪕(F)=⋁G∈F(F)clT⋞∩C⪕(G).

Thus cl_{𝓣_⋞∩𝓒_⪕}(G) ≤ co_⪕(F) for each G ∈ 𝔉(F). Further, for any G ∈ 𝔉(F),

G∈F(F)=F(⋁R∈F(F)R)=⋃R∈F(F)F(R).

Thus there is R ∈ 𝔉(F) such that G ∈ 𝔉(R). Let B_G = ⋀_{H∈𝔉_G(L^X)} cl_{𝓣_⋞∩𝓒_⪕}(H). We have G ≤ B_G ∈ 𝓣_⋞ ∩ 𝓒_⪕ which implies cl_{𝓣_{⊏_w}∩𝓒_⪕}(G) ≤ B_G.

Since R ∈ 𝔉(F) and G ∈ 𝔉(R), we have B_G ≤ cl_{𝓣_⋞∩𝓒_⪕}(R) ≤ co_⋞(F). Hence cl_{𝓣_{⊏_w}∩𝓒_⪕}(G) ≤ co_⋞(F) for any G ∈ 𝔉(F). So ⋁_G∈𝔉(F) cl_{𝓣_{⊏_w}∩𝓒_⪕}(G) ≤ co_⪕(F). Therefore (X, 𝓣_{⊏_w}, 𝓒_⪕) is an L-topological-convex space.

(2) ⇔ (3) : It directly follows from (3) of Theorem 6.14 and Theorem 6.8.

(2) ⇒ (4) : If x_λ ≰ C_F, then there is H ∈ 𝔉_F(L^X) such that x_λ ≰ co_⪕(H). Since F ∈ 𝔉(H), there is R ∈ 𝔉(H) such that F ∈ 𝔉(R).

Since (X, 𝓣_{⊏_w}, 𝓒_⪕) is an L-topological-convex space, we have

clT⊏¯w∩C⪕(R)≤⋁G∈F(H)clT⊏¯w∩C⪕(G)=co⪕(H).

Thus x_λ ≰ cl_{𝓣_{⊏_w}∩𝓒_⪕}(R). Further, we have cl_{⊏_w}(C_F) ≤ cl_{𝓣_{⊏_w}∩𝓒_⪕}(R) since

CF=⋀H∈FF(LX)co⪕(H)≤co⪕(R)≤clT⊏¯w∩C⪕(R)∈T⊏¯w∩C⪕.

Hence x_λ ≰ cl_{⊏_w}(C_F). Therefore cl_{⊏_w}(C_F) ≤ C_F which shows that C_F ∈ 𝓣_{⊏_w}.

(4) ⇒ (5) : Since 𝓣_{⊏_w} = 𝓣_w ⊆ 𝓣_⋞, the result is clear.

(5) ⇒ (6) : Since C_F ∈ 𝓣_⋞, we have

cl⋞(CF)=CF=⋀H∈FF(LX)co⪕(H)∈C⪕.

Thus C_F ∈ 𝓣_⋞ ∩ 𝓒_⪕ = φ_{⊏_w} by (1) of Theorem 6.14.

(6) ⇒ (1) : Let B ∈ L^X with F ⪕ B. Then co_⪕(F) ≤ B. Let G ∈ 𝔉(F). Then F ∈ 𝔉_G(L^X) and so C_G ≤ co_⪕(F). In addition, by (2) of Theorem 6.14 and (6), G ≤ C_G ∈ φ_{⊏_w} = 𝓣_⋞ ∩ 𝓒_⪕. Thus

G≤CG⪕CG⋞CG≤co⪕(F)≤B.

Therefore (X, ⋞, ⪕) is an L-topological-convex enclosed relation space.□

7 Conclusions

In this paper, axiomatic definitions of both L-convex bases and L-convex subbases are introduced. It is proved that the category of L-convex spaces is a coreflective subcategory of both the category of L-convex base spaces and the category of L-convex subbase spaces. In particular, it is also proved that the category of L-convex spaces is a bireflective subcategory of the category of L-closure spaces. Further, by L-convex bases, the notion of L-topological-convex space is introduced which is a triple consisting of an L-cotopology and a compatible L-convex structure on the same set. L-topological-convex spaces can be characterized by many means including L-topological-convex enclosed relation spaces.

L-convex bases are quite different to convex bases when L ≠ {⊥, ⊤}. Specifically, a subset of a convex structure is a base iff it contains all polytopes [2]. That is, if (X, 𝓒) is a convex space, then 𝓑 ⊆ 𝓒 is a base of 𝓒 iff co(F) ∈ 𝓑 for any F ∈ 2finX . However, as we can see in Remark 3.3, an L-convex base 𝓑 of an L-convex space (X, 𝓒) may fails to imply co(F) ∈ 𝓑 for all F ∈ 𝔉(L^X). Thus we didn’t directly extend the original definition of topological-convex spaces to L-setting.

As we can see, L-convergence spaces are closely related to L-topological spaces [34, 35, 36, 37]. Similar to the compatibility between an L-cotopology and an L-convex structure, it could be possible to discuss the compatibility between an L-convergence structure and an L-convex structure. Further, it could be possible to characterize L-topological-convex spaces by such compatibility.

Topological-convex spaces in Convex Theory is a basic notion in combining Topology Theory and Convex Theory. With such spaces, many combined properties can be investigated including continuities of hull operators, compactness and uniformity of convex spaces. Thus this paper could be helpful in discussing L-topological-concave spaces in the future.

Acknowledgements

Authors would like to express their sincere thanks to the reviewers for their insights, precious comments and suggestions.

This work is supported by the Youth Science Foundation of Hunan province (No. 2018JJ3192), the Key Project of Hunan Educational Commission (No. 18A474), the Project of Hunan Educational Commission (No. 19C0822) and 2018 Hunan Research Study and Innovative Experiment Project for College Students (No. 960).

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Received: 2018-11-09

Accepted: 2019-10-13

Published Online: 2019-12-26

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https://doi.org/10.1515/math-2019-0133

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L-convex space; L-convex enclose relation space; L-topological-convex space; L-topological-convex enclosed relation space

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