On the category of enriched ( L,M )-convex spaces

Abstract

The theory of abstract convexity exists in many mathematical branches such as algebra, topology and order. The fuzzification is one of important directions towards the discussion of abstract convexity. The theory of (L, M)-convex spaces is one very general fuzzy convexity since it includes many other fuzzy convexities as its special case. In this paper, we shall introduce a subcategory of (L, M)-convex spaces and study its property. This subcategory is called enriched (L, M)-convex spaces and denoted by ELMCS. The main results are: (1) ELMCS is a topological category; (2) ELMCS is a coreflective subcategory of the category of (L, M)-convex spaces; (3) the category of M-fuzzifying convex spaces (denoted by MCS) can be embedded in the category ELMCS as a reflective subcategory.

Keywords

fuzzy set convex space (co)reflective subcategory

1 Introduction

Abstract convexity theory is an important branch of mathematics [28]. These theories exist in many different mathematical research areas, such as convexities in lattices [27, 29], convexities in metric spaces and graphs [9 , 26] and convexities in topology [2, 3, 22].

With the development of fuzzy mathematics, convexity has been interrelated to fuzzy set theory. Many authors discussed many fuzzy convexities and more general lattice-valued convexities [14 , 32]. Recently, consider L and M being completely distributive lattices, three kinds of lattice-valued convexities, namely, L-convexity [7, 14 , 17], M-fuzzifying convexity [19, 20] and (L, M)-convexity [32], were discussed frequently. The relationship between convexity and latticed-valued convexities is an important direction of research. In [16], Pang and Shi introduced some subcategories of L-convex spaces, in particular, they investigated a coreflectively embedding functor from the category of convex spaces to the category of stratified L-convex spaces. In [21, 32], Xiu and Shi discussed the relationship between M-fuzzifying convex spaces and (L, M)-convex spaces. In particular, he constructed a coreflectively embedding functor from the category of M-fuzzifying convex spaces to the category of (L, M)-convex spaces. Pang-Shi’s functor [16] can be regarded as an analogizing of the Lowen functor between the category of topological spaces and the category of stratified fuzzy topological spaces [13]. Shi and Xiu’s functor [21, 32] can be regarded as an analogizing of Yue-Fang’s extended Lowen functor between the category of fuzzifying topological spaces and the category of stratified I-fuzzy topological spaces [33]. The (extended) Lowen functor plays an important role in the study of fuzzy topological spaces [6 , 31].

In this paper, consider L being a continuous lattice, M being a completely distributive lattice, a category of enriched (L, M)-convex spaces (denoted by ELMCS) is introduced. At first, it is proved that ELMCS is a topological category. At second, it is showed that ELMCS is a coreflective subcategory of the category of (L, M)-convex spaces (denoted by LMCS). At last, a functor from the category of M-fuzzifying convex spaces (denoted by MCS) to the category ELMCS is constructed. Then by using this functor it is verified that the category MCS can be embedded in the category ELMCS as a reflective subcategory.

The contents are arranged as follows. In Section 2, we recall some basic notions as preliminary. In Section 3, we present the category ELMCS, then discuss its properties. In Section 4, we focus on the relationship between ELMCS, LMCS and MCS. Finally, we end this paper with a summary of conclusion.

2 Preliminaries

Let L = (L, ≤ , ∨ , ∧ , 0_L, 1_L) be a complete lattice, where 0_L is the smallest element, 1_L is the largest element.

For a, b ∈ L, we say that a is way below b (in symbol, a ⪡ b) if for all directed subsets D ⊆ L, y ≤ ∨ D always implies that x ≤ d for some d ∈ D. A complete lattice L is said to be continuous if ∀x ∈ L, x = ∨ ↓ x, where ↓x = {y ∈ L|y ⪡ x} [5]. For a directed subset D ⊆ L, we use ∨^↑D to denote its union.

For a, b ∈ L, we say that a is wedge below b (in symbol, a ⊲ b) if for all subsets D ⊆ L, b ≤ ∨ D always implies that a ≤ d for some d ∈ D. A complete lattice L is said to be completely distributive if ∀x ∈ L, x = ∨ ⇓ x, where ⇓x = {y ∈ L|y ⊲ x} [5].

Obviously, a completely distributive lattice is a continuous lattice. The continuous lattice has a strong flavor of theoretical computer science [5]. Throughout this paper, let L denote a continuous lattice and let M denote a completely distributive lattice, unless otherwise stated. The following lemmas collect some properties of way below (wedge below) relation on a continuous (completely distributive) lattice.

Lemma 2.1. [5] Let L be a continuous lattice. Then (1) a ⪡ b ⇒ a ≤ b, (2) a ≤ b ⪡ c ≤ d ⇒ a ⪡ d, (3) a ⪡ b ⇒ ∃ c such that a ⪡ c ⪡ b, (4) a ⪡ ∨ ^↑D ⇒ a ⪡ d for some d ∈ D, (5) a ≤ b ⇔ ∀ c ⪡ a, c ≤ b.

Lemma 2.2. [5] Let L be a continuous lattice and let {a_j,k|j ∈ J, k ∈ K (j)} be a nonempty family of element in L such that {a_j,k|k ∈ K (j)} is directed for all j ∈ J. Then the following identity holds. $(DD) ⋀_{j \in J} {\underset{k \in K (j)}{⋁}}^{↑} a_{j, k} = {\underset{h \in N}{⋁}}^{↑} ⋀_{j \in J} a_{j, h (j)},$

where N is the set of all choice functions h : J ⟶ ⋃ _j∈JK (j) with h (j) ∈ K (j) for all j ∈ J.

Lemma 2.3. [5] Let M be a completely distributive lattice. Then (1) a ⊲ b ⇒ a ≤ b, (2) a ≤ b ⊲ c ≤ d ⇒ a ⊲ d, (3) a ⊲ b ⇒ ∃ c such that a ⊲ c ⊲ b, (4) a ⊲ ∨ D ⇒ a ⊲ d for some d ∈ D, (5) a ≤ b ⇔ ∀ c ⊲ a, c ≤ b.

Let X be a nonempty set, the functions X ⟶ L, denoted as L^X, are called the L-subsets on X. The operators on L can be translated onto L^X in a pointwise way. We make no difference between a constant function and its value since no confusion will arise. For a crisp subset A ⊆ X, we also make no difference between A and its characteristic function χ_A. Clearly, χ_A can be regarded as an L-subset on X. Let f : X ⟶ Y be a function. Then define $f_{L}^{\leftarrow} : L^{Y} ⟶ L^{X}$ by $f_{L}^{\leftarrow} (λ) = λ \circ f$ for λ ∈ L^Y. For a nonempty set X, let 2^X denote the powerset of X.

Definition 2.4. [19] A function $C : 2^{X} ⟶ M$ is called an M-fuzzifying convex structure on X if it satisfies:

$C (\emptyset) = C (X) = 1_{M}$ ;

if {A_j} _j∈J ⊆ 2^X is nonempty, then $C (⋂_{j \in J} A_{j})$ $\geq ⋀_{j \in J} C (A_{j})$ ;

if {A_j} _j∈J ⊆ 2^X is directed, then $C (⋃_{j \in J} A_{j})$ $\geq ⋀_{j \in J} C (A_{j})$ .

The pair $(X, C)$ is called an M-fuzzifying convex space. A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is called M-fuzzifying convexity-preserving (M-CP, in short) provided that for each B ∈ 2^X, $C_{Y} (B) \leq C_{X} (f^{- 1} (B))$ . The category whose objects are convex spaces and whose morphisms are M-CP mappings will be denoted by MCS.

Definition 2.5. [32] A function $C : L^{X} ⟶ M$ is called an (L, M)-convex structure on X if it satisfies:

$C (0_{L}) = C (1_{L}) = 1_{M}$ ;

if {λ_j} _j∈J ⊆ L^X is nonempty, then $C (⋀_{j \in J} λ_{j})$ $\geq ⋀_{j \in J} C (λ_{j})$ ;

if {λ_j} _j∈J ⊆ L^X is directed, then $C (⋁_{j \in J}^{↑} λ_{j})$ $\geq ⋀_{j \in J} C (λ_{j})$ .

The pair $(X, C)$ is called an (L, M)-convex space. A mapping $f : (X, C_{X}) ⟶ (Y, C_{Y})$ between (L, M)-convex spaces is called (L, M)-convexity-preserving ((L, M)-CP, in short) provided that μ ∈ L^Y, implies $C_{Y} (μ) \leq C_{X} (f^{\leftarrow} (μ))$ . The category whose objects are (L, M)-convex spaces and whose morphisms are (L, M)-CP mappings will be denoted by LMCS.

Finally,we recall some categoric notions from [1].

Definition 2.6. [1] Suppose that A and B are concrete categories; F : A ⟶ B and G : B ⟶ A are concrete functors. The pair (F, G) is called a Galois correspondence if F ∘ G ≤ id in the sense that for each Y ∈ B, id_Y : F ∘ G (Y) ⟶ Y is a B-morphism; and id ≤ G ∘ F in the sense that for each X ∈ A, id_X : X ⟶ G ∘ F (X) is an A-morphism.

If (F, G) is a Galois correspondence, then F is a left adjoint of G (equivalently, G is a right adjoint of F), hence F and G form an adjunction F ⊢ G : A ⇀ B.

3 Enriched (L, M)-convex spaces

In this section, we will introduce the category of enriched (L, M)-convex spaces and prove that category is a topological category.

Definition 3.1. An (L, M)-convex space $(X, C)$ is called enriched if it satisfies moreover:

$\forall α \in L, C (α) = 1_{M}$ .

The subcategory of LMCS, consisted of enriched (L, M)-convex spaces, is denoted by ELMCS.

For a set X, let (F_e (X)) F (X) denote its fibre $\begin{matrix} {(X, C) | C is an (enriched) (L, M) - convex \\ structure on X} . \end{matrix}$

For (enriched) (L, M)-convex spaces $(X, C_{1})$ and $(X, C_{2})$ , we say $(X, C_{1})$ is finer than $(X, C_{2})$ , or $(X, C_{2})$ is coarser than $(X, C_{1})$ , if the identity mapping ${id}_{X} : (X, C_{1}) ⟶ (X, C_{2})$ is (L, M)-CP.

Example 3.2. Let X be a nonempty set.

Define $C_{dis} : L^{X} ⟶ M$ as $C_{dis} \equiv 1_{M}$ . Then $C_{dis}$ is the finest (enriched) (L, M)-convex structure on X, called the discrete (enriched) (L, M)-convex structure on X [32].

Define $C_{idis} : L^{X} ⟶ M$ as $C_{idis} (λ) = {\begin{matrix} 1_{M}, & λ = α, \forall α \in L; \\ 0_{M}, & otherwise . \end{matrix}$

Then $C_{idis}$ is the coarsest enriched (L, M)-convex structure on X, called the indiscrete enriched (L, M)-convex structure on X.

Let L = M = [0, 1] and (X, R) be an L-preordered set. For any a, b ∈ L, define a → b = ∨ {c ∈ L|a ∧ c ≤ b}. Then the function $C_{R} : L^{X} ⟶ M$ defined by ∀λ ∈ L^X, $C_{R} (λ) = ⋀_{(x, y) \in X \times X} (R (x, y) \to (λ (x) \to λ (y))$ is an (enriched) (L, M)-convex structure on X [4, 32].

When L being a completely distributive lattice, Xiu [32] proved that for any function φ : L^X ⟶ M, there exists the finest (L, M)-convex space $(X, C_{φ})$ with $C_{φ} \geq φ$ . Next we prove a similar result on enriched (L, M)-convex space.

Lemma 3.3. (Subbase Lemma) Let φ : L^X ⟶ M be a function. Take $B_{φ} : L^{X} ⟶ M$ as ∀μ ∈ L^X, $\begin{matrix} B_{φ} (μ) & = & ⋁ {⋀_{k \in K} φ (μ_{k}) | μ = a_{K} \land ⋀_{k \in K} μ_{k}, \\ {μ_{k}}_{k \in K} \in 2^{(L^{X})}, a_{K} \in L} \end{matrix}$ and take $C_{φ} : L^{X} ⟶ M$ as ∀λ ∈ L^X, $\begin{matrix} C_{φ} (λ) & = & ⋁ {⋀_{j \in J} B_{φ} (λ_{j}) | λ = {⋁_{j \in J}}^{↑} λ_{j}, \\ {λ_{j}}_{j \in J} \in 2^{(L^{X})} is directed} . \end{matrix}$

Then $(X, C_{φ})$ is the coarsest enriched (L, M)-convex space with $C_{φ} \geq φ$ .

Proof. (1) $B_{φ}$ satisfies (LMCS). Then we immediately get that $C_{φ}$ satisfies (LMCS).

For any a ∈ L, take K =∅. Then a ∧ ⋀ _k∈Kμ_k = a ∧ 1_L = a and ⋀_k∈Kφ (μ_k) =1_M. Thus $B_{φ} (a) = 1_{M}$ .

(2) $B_{φ}$ satisfies (LMC2).

Let {μ_j} _j∈J ⊆ L^X be nonempty and take any $\begin{matrix} a ⊲ ⋀_{j \in J} B_{φ} (μ_{j}) \\ = ⋀_{j \in J} \underset{μ_{j} = a_{K (j)} \land \underset{k \in K (j)}{⋀} μ_{j, k}}{⋁} ⋀_{k \in K (j)} φ (μ_{j, k}) . \end{matrix}$

Then by Lemma 2.3 (1), (2), (4), for any j ∈ J there exists a_K(j) ∈ L, {μ_j,k} _k∈K(j) ∈ 2^{(L^X)} such that $\begin{matrix} μ_{j} & = & a_{K (j)} \land \underset{k \in K (j)}{⋀} μ_{j, k} and \forall k \in K (j), \\ a \leq φ (μ_{j, k}) . \end{matrix}$

It follows that $⋀_{j \in J} μ_{j} = (⋀_{j \in J} a_{K (j)}) \land \underset{k \in \cup_{j \in J} K (j)}{⋀} μ_{j, k}$ and $a \leq ⋀_{k \in \cup_{j \in J} K (j)} φ (μ_{j, k}) .$

Thus $a \leq B_{φ} (⋀_{j \in J} μ_{j})$ and so $⋀_{j \in J} B_{φ} (μ_{j}) \leq B_{φ} (⋀_{j \in J} μ_{j})$ by Lemma 2.3 (5).

(3) $C_{φ}$ satisfies (LMC2).

For any {λ_j} _j∈J ⊆ L^X (J≠ ∅), take any $a ⊲ ⋀_{j \in J} C_{φ} (λ_{j}) = ⋀_{j \in J} \underset{λ_{j} = {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k}}{⋁} ⋀_{k \in K (j)} B_{φ} (λ_{j, k}) .$

Then for any j ∈ J there exists directed {λ_j,k} _k∈K(j) ∈ 2^{(L^X)} such that $λ_{j} = {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k} {and \forall k \in K (j), a \leq B_{φ} (λ_{j, k}) .$

It follows that $⋀_{j \in J} λ_{j} = ⋀_{j \in J} {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k} \overset{(DD)}{=} {\underset{h \in N}{⋁}}^{↑} ⋀_{j \in J} λ_{j, h (j)},$

where N is the set of all choice functions h : J ⟶ ⋃ _j∈JK (j) with h (j) ∈ K (j) for all j ∈ J. For any h ∈ N, denote λ_h = ⋀ _j∈Jλ_j,h(j). Because $B_{φ}$ satisfies (LMC2) we have that $a \leq ⋀_{j \in J} B_{φ} (λ_{j, h (j)}) \leq B_{φ} (⋀_{j \in J} λ_{j, h (j)}) = B_{φ} (λ_{h}) .$

It follows that $a \leq ⋀_{h \in N} B_{φ} (λ_{h}) \leq C_{φ} (⋀_{j \in J} λ_{j})$ and so $⋀_{j \in J} C_{φ} (λ_{j}) \leq C_{φ} (⋀_{j \in J} λ_{j})$ .

(4) $C_{φ}$ satisfies (LMC3).

Let {λ_j} _j∈J ⊆ L^X be directed and denote $λ = ⋁_{j \in J}^{↑} λ_{j}$ . Take any $a ⊲ ⋀_{j \in J} C_{φ} (λ_{j}) = ⋀_{j \in J} \underset{λ_{j} = {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k}}{⋁} ⋀_{k \in K (j)} B_{φ} (λ_{j, k}) .$

Then for any j ∈ J there exists directed {λ_j,k} _k∈K(j) ∈ 2^{(L^X)} such that $λ_{j} = {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k} {and \forall k \in K (j), a \leq B_{φ} (λ_{j, k}) .$

It follows that $λ = {⋁_{j \in J}}^{↑} λ_{j} = {⋁_{j \in J}}^{↑} {\underset{k \in K (j)}{⋁}}^{↑} λ_{j, k} .$

For b ∈ L, x ∈ X, we denote x_b as the L-subset values b at x and values 0_L otherwise. For λ ∈ L^X, let pt (λ) = {x_b|b ⪡ λ (x)} and let Fin (λ) denote the set of finite subset of pt (λ). Obviously, λ = ∨ pt (λ) = ∨ {∨ F|F ∈ Fin (λ)}.

Let σ : Fin (λ) ⟶ L^X be a function defined by $σ (F) = \land {λ_{j, k} | \lor F \leq λ_{j, k}} .$

We check below that σ is definable. That is, for any F ∈ Fin (λ), there exists a λ_j,k such that ∨F ≤ λ_j,k. Let F ∈ Fin (λ). Then for each x_b ∈ F, we have $b ⪡ λ (x) = {\underset{j \in J}{⋁}}^{↑} λ_{j} (x)$ . It follows by Lemma 2.1 (4) that b ⪡ λ_{j
_{x
_b}} (x) for some j_{x
_α} ∈ J. Since {λ_j} _j∈J is directed then there exists a j ∈ J, denote as j_F, such that λ_{j
_{x
_b}} ≤ λ_{j
_F} for all j_{x
_b}. By Lemma 2.1 (2) we get b ⪡ λ_{j
_F} (x). This shows that F ∈ Fin (λ_{j
_F}). By a similar discussion on λ_{j
_F} we have that F ∈ Fin (λ_{j_F,k_F}) for some k_F ∈ K (j_F). It follows that ∨F ≤ λ_{j_F,k_F}.

Note that σ (F) = ∧ {λ_j,k| ∨ F ≤ λ_j,k} and $\forall j \in J, \forall k \in K (j), a \leq B_{φ} (λ_{j, k})$ . Then $a \leq ⋀ {B_{φ} (λ_{j, k}) | \lor F \leq λ_{j, k}} \overset{(LMC 2)}{\leq} B_{φ} (σ (F)) .$

It follows that $a \leq ⋀_{F \in Fin (λ)} B_{φ} (σ (F))$ and so $⋀_{j \in J} C_{φ} (λ_{j}) \leq ⋀_{F \in Fin (λ)} B_{φ} (σ (F))$ .

We prove below that λ = ⋁ ^↑ {σ (F) |F ∈ Fin (λ)}. Then it follows that $⋀_{j \in J} C_{φ} (λ_{j})$ $\leq C_{φ} (⋁_{j \in J}^{↑} λ_{j})$ as desired.

For any F ∈ Fin (λ), it is easily observed that ∨F ≤ σ (F) ≤ λ and then $\begin{matrix} λ & = & \lor {\lor F | F \in Fin (λ)} \\ \leq & \lor {σ (F) | F \in Fin (λ)} \leq λ . \end{matrix}$

This means that λ = ⋁ {σ (F) |F ∈ Fin (λ)}.

If F₁, F₂ ∈ Fin (λ) and F₁ ⊆ F₂ then it is easy to check that σ (F₁) ≤ σ (F₂). Thus for any F₁, F₂ ∈ Fin (λ) we have F₁ ∪ F₂ ∈ Fin (λ) and σ (F₁) , σ (F₂) ≤ σ (F₁ ∪ F₂). This shows that {σ (F) |F ∈ Fin (λ)} is directed.

(5) Obviously, $C_{φ} \geq φ$ . Let $(X, D)$ be an enriched (L, M)-convex space and $D \geq φ$ . Then ∀μ ∈ L^X $\begin{matrix} B_{φ} (μ) & = & ⋁_{μ = a_{K} \land ⋀_{k \in K} μ_{k}} ⋀_{k \in K} φ (μ_{k}) \\ \leq & ⋁_{μ = a_{K} \land ⋀_{k \in K} μ_{k}} ⋀_{k \in K} D (μ_{k}) \\ \overset{(LMCS)}{=} & ⋁_{μ = a_{K} \land ⋀_{k \in K} μ_{k}} (D (a_{K}) \land ⋀_{k \in K} D (μ_{k})) \\ \overset{(LMC 2)}{\leq} & ⋁_{μ = a_{K} \land ⋀_{k \in K} μ_{k}} D (a_{K} \land ⋀_{k \in K} μ_{k}) \\ = & D (μ), \end{matrix}$ and so $B_{φ} \leq D$ . Let λ ∈ L^X. Then it follows that $\begin{matrix} C_{φ} (λ) & = & ⋁_{λ = {\underset{j \in J}{⋁}}^{↑} λ_{j}} ⋀_{j \in J} B_{φ} (λ_{j}) \\ \leq & ⋁_{λ = {\underset{j \in J}{⋁}}^{↑} λ_{j}} ⋀_{j \in J} D (λ_{j}) \overset{(LMC 3)}{\leq} D (λ), \end{matrix}$ and so $C_{φ} \leq D$ .

Therefore, $(X, C_{φ})$ is the finest enriched (L, M)-convex space with $C_{φ} \geq φ$ . □

The following corollaries are easily observed from the the proof of the above lemma.

Corollary 3.4. If φ : L^X ⟶ M satisfies (LMCS) then for μ ∈ L^X $\begin{matrix} B_{φ} (μ) & = & ⋁ {⋀_{k \in K} φ (μ_{k}) | μ = ⋀_{k \in K} μ_{k}, \\ {μ_{k}}_{k \in K} \in 2^{(L^{X})}} . \end{matrix}$

Corollary 3.5. If φ : L^X ⟶ M satisfies (LMC2) then for μ ∈ L^X $B_{φ} (μ) = ⋁ {φ (ν) | μ = a \land ν, a \in L, ν \in L^{X}} .$

Theorem 3.6. The category ELMCS is a topological category in the sense of Preuss [18].

Proof. We first prove the existence of initial structures. Let ${(X_{t}, C_{X_{t}})}_{t \in T}$ be a family of enriched (L, M)-convex spaces and let X be a nonempty set. Let further ${f_{t} : X ⟶ (X_{t}, C_{X_{t}})}_{t \in T}$ be a source. Define φ : L^X ⟶ M as ∀λ ∈ L^X, $φ (λ) = ⋁ {C_{X_{t}} (λ_{t}) | t \in T, λ_{t} \in L^{X_{t}}, f_{t}^{\leftarrow} (λ_{t}) = λ} .$

Then by Lemma 3.3 we get that $(X, C_{φ}) \in$ ELMCS.

Let further $(Y, C_{Y}) \in$ ELMCS and let g : Y ⟶ X. Assume that f_t ∘ g is (L, M)-CP for every t ∈ T. Obviously, φ satisfies (LMC1). Then by Lemma 3.3 and Corollary 3.4 we have for λ ∈ L^X, $C_{φ} (λ) = ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{\underset{k \in K (j)}{⋀} λ_{j, k} = λ_{j}} ⋀_{k \in K (j)} φ (λ_{j, k}) .$

In the following we prove that for any λ_j,k, $φ (λ_{j, k}) \leq C_{Y} (g^{\leftarrow} (λ_{j, k}))$ . If there is no μ_t ∈ L^{X
_t} such that $λ_{j, k} = f_{t}^{\leftarrow} (μ_{t})$ then $φ (λ_{j, k}) = 0_{M} \leq C_{Y} (g^{\leftarrow} (λ_{j, k}))$ . Otherwise for any μ_t ∈ L^{X
_t} with $λ_{j, k} = f_{t}^{\leftarrow} (μ_{t})$ we get $\begin{matrix} C_{X_{t}} (μ_{t}) & \leq & C_{Y} ((f_{t} \circ g)^{\leftarrow} (μ_{t})) = C_{Y} (g^{\leftarrow} (f_{t}^{\leftarrow} (μ_{t}))) \\ = & C_{Y} (g^{\leftarrow} (λ_{j, k})) \end{matrix}$ since f_t ∘ g is (L, M)-CP. We have proved that $φ (λ_{j, k}) \leq C_{Y} (g^{\leftarrow} (λ_{j, k}))$ . Then $C_{φ} (λ)$ $\begin{matrix} = & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{\underset{k \in K (j)}{⋀} λ_{j, k} = λ_{j}} ⋀_{k \in K (j)} φ (λ_{j, k}) \\ \leq & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{\underset{k \in K (j)}{⋀} λ_{j, k} = λ_{j}} ⋀_{k \in K (j)} C_{Y} (g^{\leftarrow} (λ_{j, k})) \\ \overset{(LMC 2)}{\leq} & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} C_{Y} (g^{\leftarrow} (λ_{j})) \overset{(LMC 3)}{\leq} C_{Y} (g^{\leftarrow} (λ)) . \end{matrix}$

We have proved that g is (L, M)-CP.

Secondly, it is easily seen that the fiber F_e (X) ⊆2^{((L^X)^M)} is a set. This means that SLMCS is fiber small. Finally, there is only one enriched (L, M)-convex structure on a set X with cardinality one (namely $C_{X} \equiv 1_{M}$ ). This shows that ELMCS has terminal structure. □

Remark 3.7. The category LMCS is not topological in the sense of Preuss [18] since there are more than one (L, M)-convex structures on a set X with cardinality one.

4 The categoric relationship between ELMCS, LMCS and MCS

In this section, we shall discuss the relationship between the categories ELMCS, LMCS and MCS. Precisely, we shall prove that the category ELMCS is a coreflective subcategory, and the category MCS can be embedded in the category ELMCS as a reflective subcategory.

Definition 4.1. Let $(X, D)$ be an (L, M)-convex space. Then by Lemma 3.3 that $(X, C_{D})$ is an enriched (L, M)-convex space. We denote $S (D) = C_{D}$ and call $(X, S (D))$ as the enrichment of $(X, D)$ .

Obviously, $D$ satisfies (LMC2). Then by Corollary 3.5 we get the following lemma.

Lemma 4.2. Let $(X, D)$ be an (L, M)-convex space. Then for λ ∈ L^X, $S (D) (λ) = ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land μ_{j} = λ_{j}} D (μ_{j}) .$

Proposition 4.3. If $f : (X, D_{X}) ⟶ (Y, D_{Y})$ is (L, M)-CP then so is $f : (X, S (D_{X}))$ $⟶ (Y, S (D_{Y}))$ .

Proof. Let λ ∈ L^Y. Then by $f : (X, D_{X}) ⟶ (Y, D_{Y})$ is (L, M)-CP $\begin{matrix} S (D_{Y}) (λ) & = & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land μ_{j} = λ_{j}} D_{Y} (μ_{j}) \\ \leq & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land μ_{j} = λ_{j}} D_{X} (f^{\leftarrow} (μ_{j})) \\ \leq & S (D_{X}) (f^{\leftarrow} (λ)) . \end{matrix}$

The last inequality holds because that ${\underset{j \in J}{⋁}}^{↑} λ_{j} = λ$ and a_j ∧ μ_j = λ_j implies that ${\underset{j \in J}{⋁}}^{↑} f^{\leftarrow} (λ_{j}) = f^{\leftarrow} (λ)$ and a_j ∧ f^← (μ_j) = f^← (λ_j), respectively. Thus $f : (X, S (D_{X}))$ $⟶ (Y, S (D_{Y}))$ is (L, M)-CP. □

This shows that the correspondence $(X, D_{X}) ⊢ (X, S (D_{X}))$ defines a concrete functor S: LMCS⟶ ELMCS. Let E: ELMCS⟶ LMCS denote the inclusion functor. Then

Theorem 4.4. (E, S) is a Galois correspondence. Moreover, S is a left inverse of E.

Proof. It is sufficient to show that $E \circ S (C) \geq C$ for any $(X, C) \in$ LMCS and $S \circ E (C) = C$ for any $(X, C) \in$ ELMCS. Indeed, it follows immediately from Lemma 3.3, that is, $(X, S (C))$ is the finest enriched (L, M)-convex space with $S (C) \geq C$ for $(X, C) \in$ LMCS. □

Corollary 4.5. ELMCSis a coreflective subcategory ofLMCS.

Definition 4.6. Let $(X, C)$ be an M-fuzzifying convex space. Define $D_{C} : L^{X} ⟶ M$ as $\forall λ \in L^{X}, D_{C} (λ) = C (U)$ if λ = χ_U otherwise $D_{C} (λ) = 0_{M}$ . It is easily seen that $(X, D_{C})$ is an (L, M)-convex space. Then the enrichment of $D_{C}$ is denoted as $ω (C)$ , called the M-fuzzifying convex generated enriched (L, M)-convex structure by $(X, C)$ .

By Lemma 4.2 we get the following corollary.

Corollary 4.7. Let $(X, C)$ be an M-fuzzifying convex space. Then for λ ∈ L^X, $ω (C) (λ) = ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} C (U_{j}) .$

Lemma 4.8. Let $(X, C)$ be an M-fuzzifying convex space. Then $ω (C) (χ_{U}) = C (U)$ for U ∈ 2^X.

Proof. Let U ∈ 2^X. Then it is obvious that $ω (C) (χ_{U}) \geq C (U)$ . Conversely, take $a ⊲ ω (C) (χ_{U}) = ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = χ_{U}} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} C (U_{j}),$

Then there exists a directed set {a_j ∧ U_j} _j∈J ⊆ L^X such that $χ_{U} = ⋁_{j \in J}^{↑} (a_{j} \land U_{j}),$ and $a \leq C (U_{j})$ for all j ∈ J. Without loss of generality, we assume that a_j ≠ 0_L for all j ∈ J. It is easily seen that $\begin{matrix} x \in U & \Leftrightarrow & χ_{U} (x) = ⋁_{j \in J}^{↑} (a_{j} \land U_{j}) (x) = 1_{L} \\ \Leftrightarrow & x \in ⋃_{j \in J}^{↑} U_{j} . \end{matrix}$

It follows that $a \leq ⋀_{j \in J} C (U_{j}) \overset{(LMC 3)}{\leq} C ({⋃_{j \in J}}^{↑} U_{j}) = C (U) .$

This means that $ω (C) (χ_{U}) \leq C (U)$ . □

Proposition 4.9. $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is M-CP iff $f : (X, ω (C_{X})) ⟶ (Y, ω (C_{Y}))$ is (L, M)-CP.

Proof. Let $f : (X, C_{X}) ⟶ (Y, C_{Y})$ be M-CP. Take $a ⊲ ω (C_{X}) (λ) = ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} C_{X} (U_{j}) .$

Then there exists a directed set {a_j ∧ U_j} _j∈J ⊆ L^X such that $λ_{j} = ⋁_{j \in J}^{↑} (a_{j} \land U_{j}),$ and $a \leq C_{X} (U_{j})$ for all j ∈ J. By $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is M-CP and $\begin{matrix} f^{\leftarrow} (λ) & = & f^{\leftarrow} (⋁_{j \in J}^{↑} (a_{j} \land U_{j})) \\ = & ⋁_{j \in J}^{↑} (a_{j} \land f^{- 1} (U_{j})), \end{matrix}$ we have $\begin{matrix} a \leq ⋀_{j \in J} C_{X} (U_{j}) & \leq & ⋀_{j \in J} C_{Y} (f^{- 1} (U_{j})) \\ \leq & ω (C_{Y}) (f^{\leftarrow} (λ)) . \end{matrix}$

It follows that $ω (C_{X}) (λ) \leq ω (C_{Y}) (f^{\leftarrow} (λ))$ . Thus $f : (X, ω (C_{X})) ⟶ (Y, ω (C_{Y}))$ is (L, M)-CP.

Conversely, let $f : (X, ω (C_{X})) ⟶ (Y, ω (C_{Y}))$ be (L, M)-CP. Then for any U ∈ 2^Y, by Lemma 4.8 we have $\begin{matrix} C_{Y} (U) & = & ω (C_{Y}) (χ_{U}) \leq ω (C_{X}) (f^{\leftarrow} (χ_{U})) \\ = & ω (C_{X}) (χ_{f^{- 1} (U)}) = C_{X} (f^{- 1} (U)) . \end{matrix}$

Thus $f : (X, C_{X}) ⟶ (Y, C_{Y})$ is M-CP. □

It is easily seen that the correspondence $(X, C) \mapsto (X, ω (C))$ defines an embedding functor $ω : MCS ⟶ ELMCS .$

Proposition 4.10. Let $(X, C)$ be an enriched (L, M)-convex space. Then it is easily seen that the function $ρ (C) : 2^{X} ⟶ M$ defined by $\forall U \in 2^{X}, ρ (C) (U) = C (χ_{U})$ is an M-fuzzifying convex structure on X and the correspondence $(X, C) \mapsto (X, ρ (C))$ defines a concrete functor $ρ : ELMCS ⟶ LMCS .$

Theorem 4.11. The pair (ρ, ω) is a Galois correspondence and ρ is a left inverse of ω.

Proof. It is sufficient to show that $ρ_{L} \circ ω (C) = C$ for any $(X, C) \in$ MCS and $ω \circ ρ (C) \leq C$ for any $(X, C) \in$ ELMCS.

$ρ_{L} \circ ω (C) = C$ . It follows immediately by Lemma 4.8.

$ω \circ ρ (C) \leq C$ . For any λ ∈ L^X, by Corollary 4.7, we have $ω \circ ρ (C) (λ)$ $\begin{matrix} = & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} ρ (C) (U_{j}) \\ = & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} C (χ_{U_{j}}) \\ \overset{(LMCS)}{=} & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} (C (a_{j}) \land C (χ_{U_{j}})) \\ \overset{(LMC 2)}{\leq} & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} ⋁_{a_{j} \land U_{j} = λ_{j}} C (a_{j} \land U_{j}) \\ = & ⋁_{{\underset{j \in J}{⋁}}^{↑} λ_{j} = λ} ⋀_{j \in J} C (λ_{j}) \overset{(LMC 3)}{\leq} C (λ) . □ \end{matrix}$

Corollary 4.12. The category MCS can be embedded in the category ELMCS as a reflective subcategory.

Remark 4.13. It seems that we cannot use ω to prove that the category MCS can be embedded in the category LMCS as a reflective subcategory. The reason is that $ω \circ ρ (C)$ is enriched for a non enriched (L, M)-convergence space $(X, C)$ . Thus we cannot get that $ω \circ ρ (C) \leq C$ .

5 Conclusions

Let L be continuous lattice and let M be completely distributive lattice. Then the category ELMCS is proposed and its nice properties are obtained: (1) ELMCS is a topological category; (2) ELMCS is a coreflective subcategory of LMCS; (3) MCS can be embedded in ELMCS as a reflective subcategory. Note that the properties (1) and (3) are not possessed by LMCS.

Footnotes

Acknowledgments

The author thank the reviewers and the associated editor for their valuable comments and suggestions. This work is supported by National Natural Science Foundation of China (11501278) and Shandong Provincial Natural Science Foundation, China (ZR2013AQ011, ZR2014AQ011).

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