# On the embedding of convex spaces in stratified *L*-convex spaces

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## Abstract

Consider *L* being a continuous lattice, two functors from the category of convex spaces (denoted by **CS**) to the category of stratified *L*-convex spaces (denoted by *SL*-**CS**) are defined. The first functor enables us to prove that the category **CS** can be embedded in the category *SL*-**CS** as a reflective subcategory. The second functor enables us to prove that the category **CS** can be embedded in the category *SL*-**CS** as a coreflective subcategory when *L* satisfying a multiplicative condition. By comparing the two functors and the well known Lowen functor (between topological spaces and stratified *L*-topological spaces), we exhibit the difference between (stratified *L*-)topological spaces and (stratified *L*-)convex spaces.

### Keywords

Fuzzy set Convex space Stratified*L*-convex space Embedding functor Subcategory

## Background

Abstract convexity theory is an important branch of mathematics (Van De Vel 1993). The notion of convexity considered here is considerably broader the classical one; specially, it is not restricted to the context of vector spaces. Now, a convexity on a set is a family of subsets closed for intersection and directed union. Convexities exist in many different mathematical research areas, such as convexities in lattices (Van De Vel 1984; Varlet 1975), convexities in metric spaces and graphs (Lassak 1977; Menger 1928; Soltan 1983) and convexities in topology (Chepoi 1994; Eckhoff 1968; Sierkama 1975; Van Mill 1977).

With the development of fuzzy mathematics, convexity has been interrelated to fuzzy set theory. Many authors have investigated fuzzy convexity by taking the interval [0, 1] as the truth table (Philip 2010; Rosa 1994a, b). However, as (Goguen 1967) pointed out, in some situations it may be impossible to represent degrees of membership by the linearly ordered set [0, 1]. Thus some lattice structures were proposed to replace the interval [0, 1] as the truth value table for membership degrees.

It is easily seen that there is some similarity between convexity and topology (a family of subsets of a set closed for union and finite intersection). Thus, similar to lattice-valued topologies (Höhle and Rodabaugh 1999; Li and Li 2015; Liu and Luo 1997; Sostak 1985; Wang 1988; Ying 1991), there are at least three kinds of lattice-valued convexities, namely, *L*-convexity (Maruyama 2009; Pang and Shi 2016), *M*-fuzzifying convexity (Shi and Xiu 2014; Shi and Li 2016) and (*L*, *M*)-convexity (Xiu 2015), where *L* and *M* are some complete lattices. When \(L=2\), an (*L*, *M*)-convexity is precisely an *M*-fuzzifying convexity; when \(M=2\), an (*L*, *M*)-convexity is precisely an *L*-convexity; and when \(L=M=2\), an (*L*, *M*)-convexity is precisely a convexity. Similar to (lattice-valued) topology, the categorical relationships between convexity and latticed-valued convexity is an important direction of research. When *L* being a completely distributive complete lattices with some additional conditions, Pang and Shi (2016) proved that the category of convex spaces can be embedded in the category of stratified *L*-convex spaces as a coreflective subcategory.

In this paper, we shall continue to study the categorical relationships between convex spaces and stratified *L*-convex spaces. We shall investigate two embedding functors from the category of convex spaces (denoted by **CS**) to the category of stratified *L*-convex spaces (denoted by *SL*-**CS**). The first functor enables us to prove that the category **CS** can be embedded in the category *SL*-**CS** as a reflective subcategory when *L* being a continuous lattice. The second functor enables us to prove that the category **CS** can be embedded in the category *SL*-**CS** as a coreflective subcategory when the continuous lattice *L* satisfying a multiplicative condition. The second functor is an extension of Pang and Shi’s functor (2016) from the lattice-context. Precisely, from completely distributive complete lattice to continuous lattice. And the second functor can be regarded as an analogizing of the well known (extended) Lowen functor between the category of topological spaces and the category of stratified *L*-topological spaces (Höle and Kubiak 2007; Lai and Zhang 2005; Li and Jin 2011; Lowen 1976; Warner 1990; Yue and Fang 2005). By comparing the two functors and Lowen functor, we exhibit the difference between (stratified *L*-)topological spaces and (stratified *L*-)convex spaces from the categorical sense.

The contents are arranged as follows. In “Preliminaries” section, we recall some basic notions as preliminary. In “ **CS** reflectively embedding in *SL*-**CS** ” section, we present the reflective embedding of the category **CS** in the category *SL*-**CS**. In “ **CS** coreflectively embedding in *SL*-**CS** ” section, we focus on the coreflective embedding of the category **CS** in the category *SL*-**CS**. Finally, we end this paper with a summary of conclusion.

## Preliminaries

Let \(L=(L,\le ,\vee ,\wedge ,0,1)\) be a complete lattice with 0 is the smallest element, 1 is the largest element. For \(a,b\in L\), we say that *a* is way below *b* (in symbol, \(a\ll b\)) if for all directed subsets \(D\subseteq L\), \(b \le \vee D\) always implies that \(a\le d\) for some \(d \in D\). A complete lattice *L* is said to be continuous if \(\forall x\in L\), \(x = \vee \Downarrow x\), where \(\Downarrow x=\{y\in L|\ y\ll x\}\) (Gierz et al. 2003). For a directed subset \(D\subseteq L\), we use \(\vee ^\uparrow D\) to denote its union.

Throughout this paper, *L* denote a continuous lattice, unless otherwise stated. The continuous lattice has a strong flavor of theoretical computer science (Gierz et al. 2003). The following lemmas collect some properties of way below relation on a continuous lattice.

**Lemma 1**

(Gierz et al. 2003) *(1)* \(a\ll b\Rightarrow a\le b\), *(2)* \(a\le b\ll c\le d\) \(\Rightarrow a\ll d\), *(3)* \(a\ll b \Rightarrow \exists c\) *such that* \(a\ll c\ll b\), *(4)* \(a\ll \vee ^\uparrow D\Rightarrow a\ll d\) *for some* \(d\in D\).

**Lemma 2**

*Let*

*L*

*be a continuous lattice and let*\(\{a_{j,k}|\ j\in J, k\in K(j)\}\)

*be a nonempty family of element in*

*L*

*such that*\(\{a_{j,k}|\ k\in K(j)\}\)

*is directed for all*\(j\in J\).

*Then the following identity holds.*

*where*

*N*

*is the set of all choice functions*\(h:J\longrightarrow \bigcup _{j\in J}K(j)\)

*with*\(h(j)\in K(j)\)

*for all*\(j\in J\).

Let *X* be a nonempty set, the functions \(X\longrightarrow L\), 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 \(A\subseteq X\), we also make no difference between *A* and its characteristic function \(\chi _A\). Clearly, \(\chi _A\) can be regarded as an *L*-subset on *X*. Let \(f:X\longrightarrow Y\) be a function. Then define \(f^{\leftarrow }_L: L^Y\longrightarrow L^X\) by \(f^{\leftarrow }_L(\lambda )=\lambda \circ f\) for \(\lambda \in L^Y\). For a nonempty set *X*, let \(2^X\) denotes the powerset of *X*.

**Definition 1**

*X*if it satisfies:

- (C1)
\(\emptyset ,X\in {\mathcal {C}}\);

- (C2)
if \(\{A_j\}_{j\in J}\subseteq {\mathcal {C}}\), then \(\bigcap _{j\in J}A_j\in {\mathcal {C}}\), where \(J\ne \emptyset\);

- (C3)
if \(\{A_j\}_{j\in J}\subseteq {\mathcal {C}}\) is directed, then its union, denoted as \(\bigcup _{j\in J}^\uparrow A_j\), belongs to \({\mathcal {C}}\).

**CS**.

**Definition 2**

*L*-convex structure on

*X*if it satisfies:

- (LC1)
\(0,1\in {\mathfrak {C}}\);

- (LC2)
if \(\{\lambda _j\}_{j\in J}\subseteq {\mathfrak {C}}\), then \(\bigwedge _{j\in J} \lambda _j \in {\mathfrak {C}}\), where \(J\ne \emptyset\);

- (LC3)
if \(\{\lambda _j\}_{j\in J}\subseteq {\mathfrak {C}}\) is directed, then its union, denoted as \(\bigvee _{j\in J}^\uparrow \lambda _j\), belongs to \({\mathfrak {C}}\).

*L*-convex space and it is called stratified if it satisfies moreover (LCS): \(\forall \alpha \in L, \alpha \in {\mathfrak {C}}\).

A mapping \(f:(X,{\mathfrak {C}}_X)\longrightarrow (Y,{\mathfrak {C}}_Y)\) between stratified *L*-convex spaces is called *L*-convexity-preserving (*L*-CP, in short) provided that \(\mu \in {\mathfrak {C}}_Y\) implies \(f^\leftarrow (\mu )\in {\mathfrak {C}}_X\). The category whose objects are stratified *L*-convex spaces and whose morphisms are *L*-CP mappings will be denoted by *SL*-**CS**.

Finally, we recall some categorical notions from Adámek et al. (1990).

**Definition 3**

(Adámek et al. 1990) Suppose that **A** and **B** are concrete categories; \(F: {\mathbf{A}}\longrightarrow {\mathbf{B}}\) and \(G: {\mathbf{B}}\longrightarrow {\mathbf{A}}\) are concrete functors. The pair (*F*, *G*) is called a *Galois correspondence* if \(F\circ G\le {\mathrm{id}}\) in the sense that for each \(Y\in {\mathbf{B}}\), \({\mathrm{id}}_Y: F\circ G(Y)\longrightarrow Y\) is a **B**-morphism; and \({\mathrm{id}}\le G\circ F\) in the sense that for each \(X\in {\mathbf{A}},\) \({\mathrm{id}}_X: X\longrightarrow G\circ F(X)\) 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 \(F\vdash G:{\mathbf{A}}\rightharpoonup {\mathbf{B}}\).

**CS** reflectively embedding in *SL*-**CS**

In this section, we shall present a functor from the category **CS** to the category *SL*-**CS**, and then by using it to prove that the category **CS** can be embedded in the category *SL*-**CS** as a reflective subcategory.

At first, we fix some notations. For \(a\in L\), \(x\in X\), we denote \(x_a\) as the *L*-subset values *a* at *x* and values 0 otherwise. For \(\lambda \in L^X\), let \(pt(\lambda )=\{x_a|\ a\ll \lambda (x)\}\) and let \(Fin(\lambda )\) denote the set of finite subset of \(pt(\lambda )\). Obviously, \(\lambda =\vee pt(\lambda )=\vee \{\vee F|\ F\in Fin(\lambda )\}\).

**Definition 4**

*L*-convex structure on

*X*,

*Proof*

(LCS). It is obvious.

(LC2). It is easily seen that \({\mathfrak {B}}\) is closed for the operator \(\wedge\).

*N*is the set of all choice functions \(h:J\longrightarrow \bigcup _{j\in J}K(j)\) with \(h(j)\in K(j)\) for all \(j\in J\).

- 1.
\(\sigma\) is definable.

- 2.
The set \(\{\sigma (F)|\ F\in Fin(\lambda )\}\) is directed.

- 3.
\(\lambda =\bigvee \{\sigma (F)|\ F\in Fin(\lambda )\}\).

By a combination of (1)–(3), we have proved that \(\lambda \in \omega _L^1({\mathcal {C}})\). \(\square\)

**Lemma 3**

*Let * \((X,{\mathcal {C}})\) *be a convex space. Then * \(\chi _U\in \omega _L^1({\mathcal {C}})\) *iff* \(U\in {\mathcal {C}}\).

*Proof*

**Proposition 1**

\(f:(X,{\mathcal {C}}_X)\longrightarrow (Y,{\mathcal {C}}_Y)\) *is CP iff* \(f:(X,\mathcal \omega _L^1({C}_X))\longrightarrow (Y,\omega _L^1({\mathcal {C}}_Y))\) *is* *L*-*CP.*

*Proof*

*L*-CP. \(\square\)

Conversely, let \(f:(X,\mathcal \omega _L^1({C}_X))\longrightarrow (Y,\omega _L^1({\mathcal {C}}_Y))\) be *L*-CP. Then for any \(U\in {\mathcal {C}}_Y\), we have \(\chi _U\in \omega _L^1({\mathcal {C}}_Y)\) and so \({f^{-1}(U)}=f^{\leftarrow }(\chi _ U)\in \omega _L^1({\mathcal {C}}_X)\). It follows by Lemma 3 that \(f^{-1}(U)\in {\mathcal {C}}_X\). Thus \(f:(X,{\mathcal {C}}_X)\longrightarrow (Y,{\mathcal {C}}_Y)\) is CP.

**Proposition 2**

*Let*\((X,{\mathfrak {C}})\)

*be a stratified*

*L*-

*convex space. Then the set*\(\rho _L({\mathfrak {C}})=\{U\in 2^X|\ U\in {\mathfrak {C}}\}\)

*forms a convex structure on*

*X*

*and the correspondence*\((X,{\mathfrak {C}})\mapsto (X,\rho _L({\mathfrak {C}}))\)

*defines a concrete functor*

**Theorem 1**

*The pair * \((\rho _L,\omega _L^1)\) *is a Galois correspondence and* \(\rho _L\) *is a left inverse of* \(\omega _L^1\).

*Proof*

**CS**and \(\omega _L^1\circ \rho _L({\mathfrak {C}})\subseteq {\mathfrak {C}}\) for any \((X,{\mathfrak {C}})\in L\)-

**CS**.

- 1.
\(\rho _L\circ \omega _L^1({\mathcal {C}})={\mathcal {C}}\). It follows immediately by Lemma 3.

- 2.
\(\omega _L^1\circ \rho _L({\mathfrak {C}})\subseteq {\mathfrak {C}}\). Let \(\lambda \in \omega _L^1\circ \rho _L({\mathfrak {C}})\). Then \(\lambda ={\bigvee }^\uparrow _{j\in J}(a_j\wedge {U_j})\) with \(\forall j\in J, a_j\in L, {U_j}\in {\mathfrak {C}}\) and \(\{a_j\wedge {U_j}\}_{j\in J}\) is directed. It follows by the definition of stratified

*L*-convex space that \(\lambda \in {\mathfrak {C}}\).

**Corollary 1**

*The category * **CS** *can be embedded in the category * *SL*-**CS** *as a reflective subcategory.*

**CS** coreflectively embedding in *SL*-**CS**

In this section, we shall give a functor from the category **CS** to the category *SL*-**CS**, and then by using it to prove that the category **CS** can be embedded in the category *SL*-**CS** as a coreflective subcategory. This functor extends Pang and Shi’s functor (2016) from the lattice-context. Precisely, from completely distributive complete lattice to continuous lattice.

Firstly, we fix some notations used in this section.

Let \(\lambda \in L^X\) and \(a\in L\). Then the set \(\lambda _{[a]}:=\{x\in X|\ a\le \lambda (x)\}\) and the set \(\lambda _{(a)}:=\{x\in X|\ a\ll \lambda (x)\}\) are called the *a*-cut and strong *a*-cut of \(\lambda\), respectively.

Let \(a,b\in L\), we say that *a* is wedge below *b* (in symbol, \(a\triangleleft b\)) if for all subsets \(D\subseteq L\), \(y \le \vee D\) always implies that \(x\le d\) for some \(d \in D\). For each \(a\in L\), denote \(\beta (a)=\{b\in L|\ b\triangleleft a\}\).

The following lemma generalizes Huang and Shi’s result from lattice-context. Huang and Shi (2008) defined \(\lambda _{(a)}:=\{x\in X|\ a\triangleleft \lambda (x)\}\) and assumed that *L* being completely distributive complete lattice.

**Lemma 4**

*Let * \(\lambda \in L^X\) *and* \(a\in L\). *Then* (1) \(\lambda _{[a]}=\bigcap _{b\ll a}\lambda _{(b)}\); (2) \(\lambda _{(a)}=\bigcup _{a\ll b}\lambda _{[b]}\).

*Proof*

- 1.
For any \(b\ll a\), it follows by Lemma 1 (2) that \(\lambda _{[a]}\subseteq \lambda _{(b)}\). Thus \(\lambda _{[a]}\subseteq \bigcap _{b\ll a}\lambda _{(b)}\). Conversely, let \(x\in \bigcap _{b\ll a}\lambda _{(b)}\). Then by Lemma 1 (1) we get \(\forall b\ll a\), \(\lambda (x)\ge b\), it follows that \(\lambda (x)\ge \vee \Downarrow a=a\), i.e., \(x\in \lambda _{[a]}\). Thus \(\lambda _{[a]}\supseteq \bigcap _{b\ll a}\lambda _{(b)}\).

- 2.
For any \(a\ll b\), it follows by Lemma 1 (2) that \(\lambda _{[b]}\subseteq \lambda _{(a)}\). Thus \(\lambda _{(a)}\supseteq \bigcup _{a\ll b}\lambda _{[b]}\). Conversely, let \(x\in \lambda _{(a)}\). Then by Lemma 1 (3) we have \(b\in L\) such that \(a\ll b\ll \lambda (x)\), it follows by Lemma 1 (1) that \(\lambda (x)\ge b\), i.e., \(x\in \lambda _{[b]}\). Thus \(\lambda _{(a)}\subseteq \bigcup _{a\ll b}\lambda _{[b]}\).

**Lemma 5**

*Let * \(\{\lambda _j\}_{j\in J}\subseteq L^X\) *be directed. Then* \((\bigvee ^{\uparrow }_{j\in J}\lambda _j)_{(a)}=\bigcup ^{\uparrow }_{j\in J}(\lambda _j)_{(a)}\) *for any* \(a\in L\).

*Proof*

Let \(x\in (\bigvee ^{\uparrow }_{j\in J}\lambda _j)_{(a)}\), i.e., \(a\ll \bigvee ^{\uparrow }_{j\in J}\lambda _j(x)\). Then it follows immediately from Lemma 1 (4) that \(a\ll \lambda _j(x)\), i.e., \(x\in (\lambda _j)_{(a)}\) for some \(j\in J\). This means \((\bigvee ^{\uparrow }_{j\in J}\lambda _j)_{(a)}\subseteq \bigcup ^{\uparrow }_{j\in J}(\lambda _j)_{(a)}\). The reverse inclusion holds obviously.

The way below relation \(\ll\) on *L* is called multiplicative (Gierz et al. 2003) if \(a\ll b, c\) implies \(a\ll b\wedge c\).

**Lemma 6**

*Assume that the way below relation * \(\ll\) *on* *L* *is multiplicative. Then for any* \(\lambda \in L^X\) *and any* \(a\in L\), *the set* \(\{\lambda _{(b)}|\ a\ll b, b\in L\}\) *is directed if it is nonempty.*

*Proof*

For any \(\lambda _{(b)},\lambda _{(c)}\) with \(a\ll b, c\), it follows by Lemma 1 (2) that \(\lambda _{(b)},\lambda _{(c)}\subseteq \lambda _{(b\wedge c)}\). In addition, by the multiplicative condition we have \(a\ll b\wedge c\). This proves that \(\{\lambda _{(b)}|\ a\ll b, b\in L\}\) is directed.

Pang and Shi (2016) proved a similar result when *L* being a completely distributive complete lattice with the condition \(\beta (a\wedge b)=\beta (a)\cap \beta (b)\) for any \(a,b\in L\). It is easily seen that this condition is equivalent to that the wedge below relation on *L* is multiplicative. \(\square\)

**Definition 5**

*L*be multiplicative. Then the set \(\omega _L^2({\mathcal {C}})\) defined below is a stratified

*L*-convex structure on

*X*,

*Proof*

The proofs of (LCS) and (LC2) are obvious. We only check (LC3) below.

*C*being closed for intersection and directed union. \(\square\)

*L*-convex space. Then Pang and Shi (2016) defined \(\iota _L({\mathfrak {C}})\) as the finest convex structure on

*X*which contains all \(\lambda _{[a]}\) for all \(\lambda \in {\mathfrak {C}}, a\in L\). They proved that the correspondence \((X,{\mathfrak {C}})\mapsto (X,\iota _L({\mathfrak {C}}))\) defined a concrete functor

*L*being multiplicative, we get the following results. \(\square\)

**Theorem 2**

*The pair * \((\omega _L^2, \iota _L)\) *is a Galois correspondence and* \(\iota _L\) *is a left inverse of * \(\omega _L^2\).

**Corollary 2**

*The category * **CS** *can be embedded in the category* *SL*-**CS** *as a coreflective subcategory.*

*Remark 1*

Let us replace the convex space \((X,{\mathcal {C}})\) in Definition 4 and Definition 5 with a topological space \((X,{\mathcal {T}})\). Then \(\omega _L^2\) defines an embedding functor from the category of topological spaces to the category of stratified *L*-topological spaces. This functor was first proposed by Lowen (1976) for \(L=[0,1]\) and then extended by many researchers (Höle and Kubiak 2007; Lai and Zhang 2005; Liu and Luo 1997; Wang 1988; Warner 1990). If we further remove the directed condition in \(\omega _L^1\) then we also get an embedding functor from the category of topological spaces to the category of stratified *L*-topological spaces. By the definition of stratified *L*-topology, it is easily seen that \(\omega _L^1({\mathcal {T}})\subseteq \omega _L^2({\mathcal {T}})\). Conversely, if \(\lambda \in \omega _L^2({\mathcal {T}})\) then \(\lambda =\bigvee _{a\in L}(a\wedge \lambda _{[a]})\in \omega _L^1({\mathcal {T}})\). Thus \(\omega _L^1=\omega _L^2\) and it follows the following well known result. That is, the category of topological spaces can be embedded in the category of stratified *L*-topological spaces as a both reflective and coreflective subcategory.

*Remark 2*

Does **CS** can be embedded in *L*-**CS** as a both reflective and coreflective subcategory? Now, we can not answer it. For a convex space \((X,{\mathcal {C}})\), the inclusion \(\omega _L^1({\mathcal {C}})\subseteq \omega _L^2({\mathcal {C}})\) holds obviously. But the reverse inclusion seems do not hold. The reason is that for an *L*-subset \(\lambda \in L^X\), the set \(\{a\wedge \lambda _{[a]}|\ a\in L\}\) is generally not directed.

At last, we give two interesting examples to distinguish (*L*-)convex space from (*L*-) topological spaces.

*Example 1*

An upper set *U* on *L* is called Scott open if for each directed set \(D \subseteq L\), \(\vee ^\uparrow D\in U\) implies that \(d\in U\) for some \(d\in D\). It is known that the Scott open sets on *L* form a topology *L*, called the Scott topology (Gierz et al. 2003). It is not difficult to check that the Scott open sets on *L* do not form a convex structure on *L* since they are not closed for intersection.

*Example 2*

An *L*-filter (Höhle and Rodabaugh 1999) on a set *X* is a function \({\mathcal {F}}:L^X\longrightarrow L\) such that for all \(\lambda ,\mu \in L^X\), (F1) \({\mathcal {F}}(0)=0\), \({\mathcal {F}}(1)=1\); (F2) \(\lambda \le \mu \Rightarrow {\mathcal {F}}(\lambda )\le {\mathcal {F}}(\mu )\); (F3) \({\mathcal {F}}(\lambda )\wedge {\mathcal {F}}(\mu )\le {\mathcal {F}}(\lambda \wedge \mu )\).

The set of *L*-filters on *X* is denoted by \({\mathcal {F}}_L(X)\). Since \({\mathcal {F}}_L(X)\) is a subset of \(L^{(L^X)}\), hence, there is a natural partial order on \({\mathcal {F}}_L(X)\) inherited from \(L^{(L^X)}\). Precisely, for \({\mathcal {F}}, {\mathcal {G}}\in {\mathcal {F}}_L(X)\), \({\mathcal {F}}\le {\mathcal {G}}\Leftrightarrow \forall \lambda \in L^X,{\mathcal {F}}(\lambda )\le {\mathcal {G}}(\lambda ).\)

It is known that \({\mathcal {F}}_L(X)\) is closed for intersection, but is not closed for union (Fang 2010; Jäger 2001). In the following, we check that \({\mathcal {F}}_L(X)\) is closed for directed union.

- 1.
Let \(Y=L^X\) and \({\mathfrak {C}}=\{0,1\}\cup {\mathcal {F}}_L(X)\). Then it is easily seen that \({\mathfrak {C}}\) is an

*L*-convex structure on*Y*but not an*L*-topology on*Y*. - 2.
If we call a function \({\mathcal {F}}:L^X\longrightarrow L\) satisfying (F2) and (F3) as a nearly

*L*-filter on*X*. Let \({\mathcal {F}}_L^N(X)\) denote the set of nearly*L*-filters on*X*. Then it is easily seen that \({\mathcal {F}}_L^N(X)\) is a stratified*L*-convex structure on*Y*but not a stratified*L*-topology on*Y*.

*M*, similar to (1)–(2), we can define (stratified)

*L*-convex structure on

*M*.

## Conclusions

When *L* being a continuous lattice, an embedding functor from the category **CS** to *SL*-**CS** is introduced, then it is used to prove that the category **CS** can be embedded in the category *SL*-**CS** as a reflective subcategory. When *L* being a continuous lattice with a multiplicative condition, Pang and Shi’s functor (2016) is generalized from the lattice context, then it is used to prove that the category **CS** can be embedded in the category *SL*-**CS** as a coreflective subcategory. It is well known that the category of topological spaces can be embedded in the category of stratified *L*-topological spaces as a both reflective and coreflective subcategory. But, we find that the category of convex spaces seem not be embedded in the category of stratified *L*-convex spaces as a both reflective and coreflective subcategory. This shows the difference between (stratified *L*-)topological spaces and (stratified *L*-) convex spaces from categorical sense.

## Notes

### Authors' contributions

All of the authors have significant contributions to this paper and the final form of this paper is approved by all of them. Both authors read and approved the final manuscript.

### Acknowledgements

The authors thank the reviewers and the 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) and the Ke Yan Foundation of Liaocheng University (318011505).

### Competing interests

The authors declare that they have no competing interests.

### References

- Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories. Wiley, New YorkMATHGoogle Scholar
- Chepoi V (1994) Separation of two convex sets in convexity structures. J Geom 50:30–51MathSciNetCrossRefMATHGoogle Scholar
- Eckhoff J (1968) Radon’s theorem in convex product structures I. Monatsh Math 72:303–314MathSciNetCrossRefMATHGoogle Scholar
- Fang JM (2010) Stratified
*L*-ordered convergence structures. Fuzzy Sets Syst 161:2130–2149MathSciNetCrossRefMATHGoogle Scholar - Goguen JA (1967)
*L*-fuzzy sets. J Math Anal Appl 18:145–174MathSciNetCrossRefMATHGoogle Scholar - Gierz G, Hofmann KH, Keimel K, Lawson JD, Mislove MW, Scott DS (2003) Continuous lattices and domains. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- Höhle U, Rodabaugh SE (1999) Mathematics of fuzzy sets: logic, topology and measure theory, The handbooks of fuzzy sets series, vol 3. Kluwer Academic Publishers, BostonMATHGoogle Scholar
- Höle U, Kubiak T (2007) Many valued topologies and lower semicontinuity. Semigroup Forum 75:1–17MathSciNetCrossRefMATHGoogle Scholar
- Huang HL, Shi FG (2008)
*L*-fuzzy numbers and their properties. Inf Sci 178:1141–1151MathSciNetCrossRefMATHGoogle Scholar - Jäger G (2001) A category of
*L*-fuzzy convergence spaces. Quaest Math 24:501–517MathSciNetCrossRefMATHGoogle Scholar - Lassak M (1977) On metric B-convexity for which diameters of any set and its hull are equal. Bull Acad Polon Sci 25:969–975MathSciNetMATHGoogle Scholar
- Lowen R (1976) Fuzzy topological spaces and fuzzy compactness. J Math Anal Appl 56:623–633MathSciNetCrossRefMATHGoogle Scholar
- Lai HL, Zhang DX (2005) On the embedding of top in the category of stratified
*L*-topological spaces. Chin Ann Math Ser B 26:219–228MathSciNetCrossRefMATHGoogle Scholar - Li LQ, Jin Q (2011) On adjunctions between Lim, SL-Top, and SL-Lim. Fuzzy Sets Syst 182:66–78MathSciNetCrossRefMATHGoogle Scholar
- Li LQ, Li QG (2015) On enriched
*L*-topologies: base and subbase. J Intell Fuzzy Syst 28(6):2423–2432MathSciNetCrossRefGoogle Scholar - Liu YM, Luo MK (1997) Fuzzy Topol. World Scientific Publishing, SingaporeGoogle Scholar
- Maruyama Y (2009) Lattice-valued fuzzy convex geometry. RIMS Kokyuroku 164:22–37Google Scholar
- Menger K (1928) Untersuchungen über allgemeine Metrik. Math Ann 100:75–163MathSciNetCrossRefMATHGoogle Scholar
- Pang B, Shi FG (2016) Subcategories of the category of
*L*-convex spaces. Fuzzy Sets Syst. doi: 10.1016/j.fss.2016.02.014 Google Scholar - Shi F-G, Xiu Z-Y (2014) A new approach to the fuzzification of convex structures. J Appl Math 2014:249183. doi: 10.1155/2014/249183
- Shi FG, Li EQ (2016) The restricted hull operator of M-fuzzifying convex structures. J Intell Fuzzy Syst 30(1):409–421CrossRefGoogle Scholar
- Sierkama G (1975) Carathéodory and Helly-numbers of convex-product-structures. Pac J Math 61:272–282Google Scholar
- Philip S (2010) Studies on fuzzy matroids and related topics, PhD thesis, Cochin University of Science and TechnologyGoogle Scholar
- Rosa MV (1994a) On fuzzy topology fuzzy convexity spaces and fuzzy local convexity. Fuzzy Sets Syst 62:97–100MathSciNetCrossRefMATHGoogle Scholar
- Rosa MV (1994b) A study of fuzzy convexity with special reference to separation properties, PhD thesis, Cochin University of Science and TechnologyGoogle Scholar
- Soltan VP (1983) D-convexity in graphs. Sov Math Dokl 28:419–421MathSciNetMATHGoogle Scholar
- Š̆ostak AP (1985) On a fuzzy topological structure. Rend Ciecolo Mat Palermo (Suppl. Ser. II) 11:89–103Google Scholar
- Van De Vel M (1984) Binary convexities and distributive lattices. Proc Lond Math Soc 48:1–33MathSciNetCrossRefMATHGoogle Scholar
- Van De Vel M (1993) Theory of convex structures. Elsevier, AmsterdamMATHGoogle Scholar
- Van Mill J (1977) Supercompactness and Wallman spaces. Mathematical Centre Tracts 85, Mathematisch Centrum, AmsterdamGoogle Scholar
- Varlet JC (1975) Remarks on distributive lattices. Bull Acad Polon Sci 23:1143–1147MathSciNetMATHGoogle Scholar
- Wang GJ (1988) Theory of
*L*-fuzzy topological spaces. Shanxi normal University press, Xi’anGoogle Scholar - Warner MW (1990) Fuzzy topology with respect to continuous lattices. Fuzzy Sets Syst 35:85–91MathSciNetCrossRefMATHGoogle Scholar
- Xiu ZY (2015) Studies on \((L,M)\)-fuzzy convexity spaces and their related theory, PhD thesis, Beijing Institute of TechnologyGoogle Scholar
- Ying MS (1991) A new approach to fuzzy topology (I). Fuzzy Sets Syst 39:303–321MathSciNetCrossRefMATHGoogle Scholar
- Yue YL, Fang JM (2005) Generated
*L*-fuzzy topological spaces. Fuzzy Sets Syst 154:103–117MathSciNetCrossRefMATHGoogle Scholar

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