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On the embedding of convex spaces in stratified Lconvex spaces
SpringerPlus volume 5, Article number: 1610 (2016)
Abstract
Consider L being a continuous lattice, two functors from the category of convex spaces (denoted by CS) to the category of stratified Lconvex spaces (denoted by SLCS) are defined. The first functor enables us to prove that the category CS can be embedded in the category SLCS as a reflective subcategory. The second functor enables us to prove that the category CS can be embedded in the category SLCS 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 Ltopological spaces), we exhibit the difference between (stratified L)topological spaces and (stratified L)convex spaces.
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 latticevalued 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 latticevalued convexities, namely, Lconvexity (Maruyama 2009; Pang and Shi 2016), Mfuzzifying 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 Mfuzzifying convexity; when \(M=2\), an (L, M)convexity is precisely an Lconvexity; and when \(L=M=2\), an (L, M)convexity is precisely a convexity. Similar to (latticevalued) topology, the categorical relationships between convexity and latticedvalued 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 Lconvex spaces as a coreflective subcategory.
In this paper, we shall continue to study the categorical relationships between convex spaces and stratified Lconvex spaces. We shall investigate two embedding functors from the category of convex spaces (denoted by CS) to the category of stratified Lconvex spaces (denoted by SLCS). The first functor enables us to prove that the category CS can be embedded in the category SLCS 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 SLCS 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 latticecontext. 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 Ltopological 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 SLCS ” section, we present the reflective embedding of the category CS in the category SLCS. In “ CS coreflectively embedding in SLCS ” section, we focus on the coreflective embedding of the category CS in the category SLCS. 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
(Gierz et al. 2003) 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 Lsubsets 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 Lsubset 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
(Van De Vel 1993) A subset \({\mathcal {C}}\) of \(2^X\) is called a convex structure on 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 \({\mathcal {C}}\).
The pair \((X,{\mathcal {C}})\) is called a convex space. A mapping \(f:(X,{\mathcal {C}}_X)\longrightarrow (Y,{\mathcal {C}}_Y)\) is called convexitypreserving (CP, in short) provided that \(B \in {\mathcal {C}}_Y\) implies \(f^{1} (B)\in {\mathcal {C}}_X\). The category whose objects are convex spaces and whose morphisms are CP mappings will be denoted by CS.
Definition 2
(Maruyama 2009; Pang and Shi 2016) A subset \({\mathfrak {C}}\) of \(L^X\) is called an Lconvex 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 \({\mathfrak {C}}\).
The pair \((X,{\mathfrak {C}})\) is called an Lconvex 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 Lconvex spaces is called Lconvexitypreserving (LCP, in short) provided that \(\mu \in {\mathfrak {C}}_Y\) implies \(f^\leftarrow (\mu )\in {\mathfrak {C}}_X\). The category whose objects are stratified Lconvex spaces and whose morphisms are LCP mappings will be denoted by SLCS.
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 Bmorphism; 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 Amorphism.
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 SLCS
In this section, we shall present a functor from the category CS to the category SLCS, and then by using it to prove that the category CS can be embedded in the category SLCS as a reflective subcategory.
At first, we fix some notations. For \(a\in L\), \(x\in X\), we denote \(x_a\) as the Lsubset 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
Let \((X,{\mathcal {C}})\) be a convex space and let \({\mathfrak {B}}=\{a\wedge U\ a\in L,U\in {\mathcal {C}}\}.\) Then the set \(\omega _L^1({\mathcal {C}})\) defined below is a stratified Lconvex structure on X,
Proof
(LCS). It is obvious.
(LC2). It is easily seen that \({\mathfrak {B}}\) is closed for the operator \(\wedge\).
For any \(\{\lambda _j\}_{j\in J}\subseteq \omega _L^1({\mathcal {C}})\), let \(\lambda _j={\mathop \bigvee \nolimits _{k\in K(j)}}^\uparrow \mu _{j,k}\). Then
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 \(j\in J\).
(LC3) Let \(\{\lambda _j\}_{j\in J}\subseteq \omega _L^1({\mathcal {C}})\) be directed and \(\lambda _j={\bigvee }^\uparrow _{k\in K(j)}\mu _{j,k}\). We denote
Let \(\sigma :Fin(\lambda )\longrightarrow {\mathfrak {B}}\) be a function defined by

1.
\(\sigma\) is definable.
Let \(F\in Fin(\lambda )\). Then For each \(x_a\in F\), we have \(a\ll \lambda (x)={\mathop \bigvee \nolimits _{j\in J}}^\uparrow \lambda _j(x)\). It follows by Lemma 1 (4) that \(a\ll \lambda _{j_{x_a}}(x)\) for some \(j_{x_\alpha }\in J\). Since \(\{\lambda _j\}_{j\in J}\) is directed then there exists a \(j\in J\), denote as \(j_F\), such that \(\lambda _{j_{x_a}}\le \lambda _{j_F}\) for all \(j_{{x_a}}\). By Lemma 1 (2) we get \(a\ll \lambda _{j_F}(x)\). This shows that \(F\in Fin(\lambda _{j_F})\). By a similar discussion on \(\lambda _{j_F}\) we have that \(F\in Fin(\mu _{j_F,k_F})\) for some \(k_{F}\in K(j_{F})\). It follows that \(\vee F\le \mu _{j_F,k_F}\).
We have proved that for any \(F\in Fin(\lambda )\), there exists a \(\mu _{j,k}\) such that \(\vee F\le \mu _{j,k}\). Because \({\mathfrak {B}}\) is closed for \(\wedge\), we get that \(\sigma\) is definable.

2.
The set \(\{\sigma (F)\ F\in Fin(\lambda )\}\) is directed.
It is easily seen that \(\sigma\) is orderpreserving. That is, if \(F_1, F_2\in Fin(\lambda )\) and \(F_1\subseteq F_2\) then \(\sigma (F_1)\le \sigma (F_2)\). Thus for any \(F_1,F_2\in Fin(\lambda )\), it follows that \(F_1\cup F_2\in Fin(\lambda )\) and \(\sigma (F_1), \sigma (F_2)\le \sigma (F_1\cup F_2)\). This shows that \(\{\sigma (F)\ F\in Fin(\lambda )\}\) is directed.

3.
\(\lambda =\bigvee \{\sigma (F)\ F\in Fin(\lambda )\}\).
For any \(F\in Fin(\lambda )\), it is easily observed that \(\vee F\le \sigma (F)\le \lambda .\) Then it follows that
This means that \(\lambda =\bigvee \{\sigma (F)\ F\in Fin(\lambda )\}\).
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
The sufficiency is obvious. We check the necessity. Let \(\chi _U\in \omega _L^1({\mathcal {C}})\). Then
with \(\forall j\in J, a_j\in L, U_j\in {\mathcal {C}}\) and \(\{a_j\wedge {U_j}\}_{j\in J}\) is directed. Without loss of generality, we assume that \(a_j\ne 0\) for all \(j\in J\). It is easily seen that \(\{U_j\}_{j\in J}\) is directed. In the following we check that
On one hand, it is obvious that \(U\supseteq U_j\) for any \(j\in J\) and so \(U\supseteq {\bigcup }^\uparrow _{j\in J}U_j\). On the other hand, for any \(x\in U\) we have
which means \(x\in U_j\) for some \(j\in J\). Thus \(U\subseteq {\bigcup }^\uparrow _{j\in J}U_j\) as desired.
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 LCP.
Proof
Let \(f:(X,{\mathcal {C}}_X)\longrightarrow (Y,{\mathcal {C}}_Y)\) be CP. Then for any \(\lambda ={\bigvee }^\uparrow _{j\in J}(a_j\wedge {U_j})\in \omega _L^1({\mathcal {C}}_Y)\), we have
It follows that \(f:(X,\mathcal \omega _L^1({C}_X))\longrightarrow (Y,\omega _L^1({\mathcal {C}}_Y))\) is LCP. \(\square\)
Conversely, let \(f:(X,\mathcal \omega _L^1({C}_X))\longrightarrow (Y,\omega _L^1({\mathcal {C}}_Y))\) be LCP. 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.
It is easily seen that the correspondence \((X,{\mathcal {C}})\mapsto (X,\omega _L^1({\mathcal {C}}))\) defines an embedding functor
Proposition 2
(Pang and Shi 2016) Let \((X,{\mathfrak {C}})\) be a stratified Lconvex 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
It is sufficient to show that \(\rho _L\circ \omega _L^1({\mathcal {C}})={\mathcal {C}}\) for any \((X,{\mathcal {C}})\in\) 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 Lconvex space that \(\lambda \in {\mathfrak {C}}\).
\(\square\)
Corollary 1
The category CS can be embedded in the category SLCS as a reflective subcategory.
CS coreflectively embedding in SLCS
In this section, we shall give a functor from the category CS to the category SLCS, and then by using it to prove that the category CS can be embedded in the category SLCS as a coreflective subcategory. This functor extends Pang and Shi’s functor (2016) from the latticecontext. 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 acut and strong acut 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 latticecontext. 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
Let \((X,{\mathcal {C}})\) be a convex space and the way below relation \(\ll\) on L be multiplicative. Then the set \(\omega _L^2({\mathcal {C}})\) defined below is a stratified Lconvex structure on X,
Proof
The proofs of (LCS) and (LC2) are obvious. We only check (LC3) below.
Let \(\{\lambda _j\}_{j\in J}\subseteq \omega _L^2({\mathcal {C}})\) be directed and \(a\in L\). Then
It follows immediately that \({\bigvee _{j\in J}}^\uparrow \lambda _j\in \omega _L^2({\mathcal {C}})\) by \((\lambda _j)_{[c]}\in {\mathcal {C}}\) for any \(j\in J, c\in L\) and C being closed for intersection and directed union. \(\square\)
Similar to Pang and Shi (2016), we can prove that the correspondence \((X,{\mathcal {C}})\mapsto (X,\omega _L^2({\mathcal {C}}))\) defines an embedding functor
Let \((X,{\mathfrak {C}})\) be a stratified Lconvex 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
Similar to Pang and Shi (2016), when the way below relation \(\ll\) on 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 SLCS 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 Ltopological 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 Ltopological spaces. By the definition of stratified Ltopology, 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 Ltopological spaces as a both reflective and coreflective subcategory.
Remark 2
Does CS can be embedded in LCS 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 Lsubset \(\lambda \in L^X\), 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 Lfilter (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 Lfilters 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.
Let \(\{{\mathcal {F}}_j\}_{j\in J}\subseteq {\mathcal {F}}_L(X)\) be directed. Then it is readily seen that \(\bigvee ^\uparrow _{j\in J}{\mathcal {F}}_j\) satisfies the conditions (F1) and (F2). Taking \(\lambda ,\mu \in L^X\), then
Thus \(\bigvee ^\uparrow _{j\in J}{\mathcal {F}}_j\) satisfies the condition (F3). We have proved 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 Lconvex structure on Y but not an Ltopology on Y.

2.
If we call a function \({\mathcal {F}}:L^X\longrightarrow L\) satisfying (F2) and (F3) as a nearly Lfilter on X. Let \({\mathcal {F}}_L^N(X)\) denote the set of nearly Lfilters on X. Then it is easily seen that \({\mathcal {F}}_L^N(X)\) is a stratified Lconvex structure on Y but not a stratified Ltopology on Y.
Note that \(L^X\) forms a continuous lattice. If replacing \(L^X\) with a continuous lattice M, similar to (1)–(2), we can define (stratified) Lconvex structure on M.
Conclusions
When L being a continuous lattice, an embedding functor from the category CS to SLCS is introduced, then it is used to prove that the category CS can be embedded in the category SLCS 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 SLCS as a coreflective subcategory. It is well known that the category of topological spaces can be embedded in the category of stratified Ltopological 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 Lconvex 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.
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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).
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Jin, Q., Li, L. On the embedding of convex spaces in stratified Lconvex spaces. SpringerPlus 5, 1610 (2016). https://doi.org/10.1186/s4006401632555
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DOI: https://doi.org/10.1186/s4006401632555