Classical Scott topology on complete lattices and dcpos is studied in [2]. After that, many works have been done to generalize that theory on posets [1, 6]. Recently, fuzzy Scott topology has been investigated on fuzzy ordered sets with the necessary condition that fuzzy joins of all directed fuzzy set exist (i.e. dcpos) [22]. In the absence of any sort of join, the previous result is invalid, so an additional consideration for fuzzy Scott topology on *L*-posets is needed. This is our motivation for this section.

#### Definition 4.1

*[34*, *36*, *37] L*-*filter on X is a map* 𝓕 : *L*^{X} → *L satisfying that*:

(LF1) 𝓕(0) = 0 *and* 𝓕(1) = 1;

(LF2) ∀ *A*, *B* ∈ *L*^{X}, 𝓕(*A* ∧ *B*) = 𝓕(*A*) ∧ 𝓕(*B*).

An *L*-filter is called stratified if it satisfies the following condition:
$$\begin{array}{}(SF)\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\forall}a\in L,\mathcal{F}(a)\ge a\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}or\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathcal{F}(a\wedge B)\ge a\wedge \mathcal{F}(B).\end{array}$$

The set of all stratified *L*-filters on *X* will be denoted by
$\begin{array}{}{\mathbb{F}}_{L}^{s}(X)\end{array}$.

#### Example 4.3

*][34*, *35]]*

*For any x* ∈ *X*, *define a map* [*x*] : *L*^{X} → *L as* [*x*](*A*) = *A*(*x*) *for every A* ∈ *L*^{X}. *Then* [*x*] *is a stratified L*-*filter*, *called the principal L*-*filter of x*.

*Let* (*X*, *τ*) *be an L*-*fuzzy topological space and x* ∈ *X*. *Define*
$\begin{array}{}{\mathcal{U}}_{\tau}^{x}\end{array}$ : *L*^{X} → *L by*
$$\begin{array}{}\mathrm{\forall}A\in {L}^{X},{\mathcal{U}}_{\tau}^{x}(A)={\displaystyle \underset{B\le A}{\bigvee}B(x)\wedge \tau (B).}\end{array}$$

*Then*
$\begin{array}{}{\mathcal{U}}_{\tau}^{x}\end{array}$ *is an L*-*filter*, *and it is stratified if τ is enriched*.

*Let* (*X*, *δ*) *be an L*-*topological space and x* ∈ *X*. *Define*
$\begin{array}{}{\mathcal{U}}_{\delta}^{x}\end{array}$ : *L*^{X} → *L by* ∀ *A* ∈ *L*^{X}, $\begin{array}{}{\mathcal{U}}_{\tau}^{x}\end{array}$(*A*) = *A*^{∘}(*x*), *where* ∘ *is the L*-*interior operator of* (*X*, *δ*). *Then*
$\begin{array}{}{\mathcal{U}}_{\delta}^{x}\end{array}$ *is an L*-*filter*, *and if δ is stratified then so is* $\begin{array}{}{\mathcal{U}}_{\delta}^{x}\end{array}$.

#### Definition 4.4

([34, 35, 38]). *A stratified L*-*generalized convergence structure on X is a map R* :
$\begin{array}{}{\mathbb{F}}_{L}^{s}(X)\end{array}$ × *X* → *L satisfying that*

∀ *x* ∈ *X*, *R*([*x*], *x*) = 1;

∀ *x* ∈ *X*, ∀ 𝓕, 𝓖 ∈
$\begin{array}{}{\mathbb{F}}_{L}^{s}(X)\end{array}$, 𝓕 ≤ 𝓖 ⇒ *R*(𝓕, *x*) ≤ *R*(𝓖, *x*);

If *R* is a stratified *L*-generalized convergence structure on *X*, *x* ∈ *X*. Define
$\begin{array}{}{\mathcal{U}}_{R}^{x}\end{array}$ : *L*^{X} → *L* by
$$\begin{array}{}\mathrm{\forall}A\in {L}^{X},{\mathcal{U}}_{R}^{x}(A)={\displaystyle \underset{\mathcal{F}\in {\mathbb{F}}_{L}^{s}(X)}{\bigwedge}(R(\mathcal{F},x)\to \mathcal{F}(A)).}\end{array}$$

Then we have the following theorem.

#### Theorem 4.5

([34]). *Each stratified L*-*generalized convergence structure R on X induces an enriched L*-*fuzzy topology τ*_{R} on X given by
$$\begin{array}{}\mathrm{\forall}A\in {L}^{X},{\tau}_{R}(A)={\displaystyle \underset{x\in X}{\bigwedge}\left(A(x)\to \underset{\mathcal{F}\in {\mathbb{F}}_{L}^{s}(X),\mathcal{F}\ge {\mathcal{U}}_{R}^{x}}{\bigwedge}\mathcal{F}(A)\right).}\end{array}$$ *and a stratified L*-*topology δ*_{R} = {*A* ∈ *L*^{X} : *τ*_{R}(*A*) = 1}.

Elicited by the well-known results, we aim to study topologies on an *L*-poset, then the consideration of a kind of convergence structures on it will be effective. To reach that goal we begin with the discussion of the lower bound of a stratified *L*-filter.

Let (*X*, *e*) be an *L*-poset and 𝓕 ∈
$\begin{array}{}{\mathbb{F}}_{L}^{s}(X)\end{array}$. Define 𝓕^{l} ∈ *L*^{X} by
$$\begin{array}{}{\displaystyle \mathrm{\forall}x\in X,{\mathcal{F}}^{l}(x)=\underset{A\in {L}^{X}}{\bigvee}\mathcal{F}(A)\wedge {A}^{l}(x).}\end{array}$$

#### Proposition 4.6

([22]). *Let* (*X*, *e*) *be an L*-*poset and* 𝓕, 𝓖 ∈ 𝔽_{L}(*X*). *Then*

𝓕^{l}(*x*) = 𝓕(↑ *x*);

𝓕 ≤ 𝓖 ⇒ 𝓕^{l} ≤ 𝓖^{l};

∀ *x* ∈ *X*, [*x*]^{l} = ↓ *x*;

For a fuzzy ideal on (*X*, *e*), define 𝓕_{I} : *L*^{X} → *L* by
$$\begin{array}{}{\displaystyle \mathrm{\forall}A\in {L}^{X},{\mathcal{F}}_{I}(A)=\underset{x\in X}{\bigvee}(I(x)\wedge sub(\uparrow x,A)).}\end{array}$$

Then 𝓕_{I} is a stratified *L*-filter on *X*, and
$\begin{array}{}{\mathcal{F}}_{I}^{l}\end{array}$ = *I* (refer to [22] for detail).

Let (*X*, *e*) be an *L*-poset. Define a map *S* : 𝔽_{L}(*X*) × *X* → *L* by
$$\begin{array}{}{\displaystyle \mathrm{\forall}(\mathcal{F},x)\in {\mathbb{F}}_{L}(X)\times X,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}S(\mathcal{F},x)=\underset{I\in {\mathcal{I}}_{L}(X)}{\bigvee}(sub(I,{\mathcal{F}}^{l})\wedge {I}^{ul}(x)).}\end{array}$$

It is easily seen that *S* is a stratified *L*-generalized convergence structure on *X*, and *S*(𝓕, *x*) can be interpreted as the degree of 𝓕 Scott converges to *x*. Moreover, we define
$\begin{array}{}{\mathcal{U}}_{S}^{x}\end{array}$ : *L*^{X} → *L* by
$$\begin{array}{}{\displaystyle \mathrm{\forall}A\in {L}^{X},{\mathcal{U}}_{S}^{x}(A)=\underset{\mathcal{F}\in {\mathbb{F}}_{L}^{s}(X)}{\bigwedge}(S(\mathcal{F},x)\to \mathcal{F}(A)).}\end{array}$$

Then
$\begin{array}{}{\mathcal{U}}_{S}^{x}\end{array}$ is a stratified *L*-filter.

By Theorem 4.5, there is an enriched *L*-fuzzy topology associated with *S*. We denote it as *σ*_{LF}(*X*, *e*)(*σ*_{LF}(*X*) for short), that is,
$$\begin{array}{}{\displaystyle \mathrm{\forall}A\in {L}^{X},{\sigma}_{LF}(X,e)(A)=\underset{x\in X}{\bigwedge}(A(x)\to {\mathcal{U}}_{S}^{x}(A)).}\end{array}$$

Furthermore, let
$$\begin{array}{}{\sigma}_{L}(X,e)=\{A\in {L}^{X}:{\sigma}_{LF}(X,e)(A)=1\}.\end{array}$$

Then *σ*_{L}(*X*, *e*) (*σ*_{L}(*X*) for short) is a stratified *L*-topology called fuzzy Scott topology on (*X*, *e*). We say an *L*-subset *A* is fuzzy Scott open if *A* ∈ *σ*_{L}(*X*).

#### Proposition 4.7

*For A* ∈ *L*^{X}, *the following are equivalent*:

*A is fuzzy Scott open;*

∀ *x* ∈ *X*, *A*(*x*) ≤ ⋀_{I ∈ 𝓘L(X)} (*I*^{ul}(*x*) → 𝓕_{I}(*A*));

*A is an upper L*-*set*, *and A*(*x*) ≤ ⋀_{I ∈ 𝓘L(X)} (*I*^{ul}(*x*) → ⋁_{y ∈ X}(*I*(*y*) ∧ *A*(*y*))) *for all x* ∈ *X*.

#### Proof

(1) ⇒(2). Since *A* is fuzzy Scott open, so for all *x* ∈ *X*, we have
$$\begin{array}{}{\displaystyle A(x)\le \underset{\mathcal{F}\in {\mathbb{F}}_{L}^{s}(X)}{\bigwedge}(S(\mathcal{F},x)\to \mathcal{F}(A))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}(S({\mathcal{F}}_{I},x)\to {\mathcal{F}}_{I}(A))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}\left({I}^{ul}(x)\to {\mathcal{F}}_{I}(A)\right).}\end{array}$$

(2) ⇒(3). Above all, *A*(*x*) ≤ ⋀_{I ∈ 𝓘L(X)} (*I*^{ul}(*x*) → 𝓕_{I}(*A*)) follows immediately from (2) and the fact that *sub* (↑ *x*, *A*) ≤ *A*(*x*). Next, for ∀ *x*, *y* ∈ *X*,
$$\begin{array}{}{\displaystyle A(x)\wedge e(x,y)\le e(x,y)\wedge \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}\left({I}^{ul}(x)\to {\mathcal{F}}_{I}(A)\right)}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le e(x,y)\wedge \left((\downarrow x{)}^{ul}(x)\to {\mathcal{F}}_{\downarrow x}(A)\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=e(x,y)\wedge {\mathcal{F}}_{\downarrow x}(A)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le e(x,y)\wedge sub(\uparrow x,A)\le A(y).\end{array}$$

It implies that *A* is an upper *L*-set.

(3) ⇒(1). For any *x* ∈ *X* and 𝓕 ∈
$\begin{array}{}{\mathbb{F}}_{L}^{s}(X)\end{array}$,
$$\begin{array}{}{\displaystyle A(x)\wedge S(\mathcal{F},x)=A(x)\wedge \underset{I\in {\mathcal{I}}_{L}(X)}{\bigvee}\left(sub(I,{\mathcal{F}}^{l})\wedge {I}^{ul}(x)\right)}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\displaystyle \underset{I\in {\mathcal{I}}_{L}(X)}{\bigvee}\left(A(x)\wedge {I}^{ul}(x)\wedge sub(I,{\mathcal{F}}^{l})\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{I\in {\mathcal{I}}_{L}(x)}{\bigvee}\underset{y\in x}{\bigvee}\left(A(y)\wedge I(y)\wedge sub(I,{\mathcal{F}}^{l})\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{y\in X}{\bigvee}\left(A(y)\wedge {\mathcal{F}}^{l}(y)\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\underset{y\in X}{\bigvee}(A(y)\wedge \mathcal{F}(\uparrow y))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{y\in X}{\bigvee}\mathcal{F}(A(y)\wedge \uparrow y)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \mathcal{F}(A).}\end{array}$$

So *A* is fuzzy Scott open. □

#### Theorem 4.8

*If* (*X*, *e*) *is a continuous L*-*poset*, *and x* ∈ *X*, *then* ⇑ *x is fuzzy Scott open where* ⇑ *x*(*y*) = ⇓ *y*(*x*) *for every y* ∈ *X*.

#### Proof

First, ⇑ *x* is an upper *L*-set obviously for ∀ *x* ∈ *X*.

For ∀ *y* ∈ *X*, ∀ *I* ∈ 𝓘_{L}(*X*), we have ⇑ *x*(*y*) = ⇓ *y*(*x*) = ⋁_{z ∈ X}(⇓ *z*(*x*) ∧ ⇓ *y*(*z*)). Since ⇓ *y*(*z*) = ⋀_{J ∈ 𝓘L(X)}(*J*^{ul}(*y*) → *J*(*z*)) ≤ *I*^{ul}(*y*) → *I*(*z*), so
$$\begin{array}{}\Uparrow x(y)\le \underset{z\in X}{\bigvee}\left(\Uparrow x(z)\wedge \left({I}^{ul}(y)\to I(z)\right)\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\le \underset{z\in X}{\bigvee}\left({I}^{ul}(y)\to (\Uparrow x(z)\wedge I(z))\right)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\le {I}^{ul}(y)\to \underset{z\in X}{\bigvee}(\Uparrow x(z)\wedge I(z)).\end{array}$$

As follows by Proposition 4.7(3), ⇑ *x* is fuzzy Scott open. □

#### Theorem 4.9

*If* (*X*, *e*) *is a continuous L*-*poset*, *and A* ∈ *L*^{X}, *then A is fuzzy Scott open iff A is an upper L*-*set*, *and A*(*x*) = ⋁_{y ∈ X}(*A*(*y*) ∧ ⇓ *x*(*y*)) *for all x* ∈ *X*.

#### Proof

Necessity: Suppose *A* is fuzzy Scott open, then by Proposition 4.7, *A* is an upper *L*-set and for all *x* ∈ *X*, we have
$$\begin{array}{}A(x)\le (\Downarrow x{)}^{ul}(x)\to {\mathcal{F}}_{\Downarrow x}(A)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\mathcal{F}}_{\Downarrow x}(A)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle =\underset{y\in X}{\bigvee}(\Downarrow x(y)\wedge sub(\uparrow y,A))}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \le \underset{y\in X}{\bigvee}(\Downarrow x(y)\wedge A(y)).}\end{array}$$

Clearly, *A*(*x*) = ⋁_{y ∈ X}(*A*(*y*) ∧↓ *x*(*y*)) ≥ ⋁_{y ∈ X}(*A*(*y*) ∧ ⇓ *x*(*y*)). Therefore, *A*(*x*) = ⋁_{y ∈ X}(*A*(*y*) ∧ ⇓ *x*(*y*)).

Sufficiency: By Proposition 4.7, it suffices to show that for all *x* ∈ *X*, *I* ∈ 𝓘_{L}(*X*), *A*(*x*) ∧ *I*^{ul}(*x*) ≤ ⋁_{y ∈ X}(*I*(*y*) ∧ *A*(*y*)). In fact, by the conditions,
$$\begin{array}{}{\displaystyle A(x)\wedge {I}^{ul}(x)=\underset{y\in X}{\bigvee}\left(A(y)\wedge \Downarrow x(y)\wedge {I}^{ul}(x)\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\le \underset{y\in X}{\bigvee}\left(A(y)\wedge {I}^{ul}(x)\wedge ({I}^{ul}(x)\to I(y))\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\le \underset{y\in X}{\bigvee}(A(y)\wedge I(y)).}\end{array}$$ □

#### Definition 4.10

*Let* (*X*, *e*) *be a L*-*poset*, *for any λ* ∈ *L and x* ∈ *X*, *define* ⇑_{λ}*X* = *λ* ∧ ⇑ *x*.

It is worth noting that ⇑_{λ}*X* ∈*σ L*(*X*) in a continuous *L*-poset (*X*, *e*) since *σ*_{L}(*X*) is stratified.

#### Theorem 4.11

*Let* (*X*, *e*) *be a continuous L*-*poset*, *then* {⇑_{λ}*X* : *λ* ∈ *L*, *x* ∈ *X*} *is a basis of σ*_{L}(*X*).

#### Proof

If *A* ∈ *σ*_{L}(*X*), then by the above theorem, for any *x* ∈ *X*, *A*(*x*) = ⋁_{y ∈ X}(*A*(*y*) ∧ ⇓ *x*(*y*)) = ⋁_{y ∈ X}(*A*(*y*) ∧ ⇑ *y*(*x*)) = ⋁_{y ∈ X} ⇑_{A(y)}*y*(*x*). This implies that *A* = ⋁_{y ∈ X} ⇑_{A(y)}*y*. □

#### Theorem 4.12

*Let* (*X*, *e*) *be an L*-*poset*, *then for any x* ∈ *X*, (↑ *x*)^{∘} ≤ ⇑ *x and* (↑ *x*)^{∘} = ⇑ *x when* (*X*, *e*) *is continuous*, *where* ∘ *is the L*-*interior operator with respect to σ*_{L}(*X*).

#### Proof

Suppose *A* ∈ *σ*_{L}(*X*) with *A* ≤ ↑ *x*. Then by Proposition 4.7, for all *y* ∈ *X*, we have
$$\begin{array}{}{\displaystyle A(y)\le \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}\left({I}^{ul}(y)\to \underset{z\in Z}{\bigvee}(I(z)\wedge A(z))\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}\left({I}^{ul}(y)\to \underset{z\in Z}{\bigvee}(I(z)\wedge e(x,z))\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\le \underset{I\in {\mathcal{I}}_{L}(X)}{\bigwedge}\left({I}^{ul}(y)\to I(x)\right)}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\Uparrow x(y).\end{array}$$

So (↑ *x*)^{∘} = ⋁{*A* ∈ *σ*_{L}(*X*) : *A* ≤ ↑ *x*} ≤ ⇑ *x*. If (*X*, *e*) is continuous, then ⇑ *x* ∈ *σ*_{L}(*X*) and ⇑ *x* ≤ ↑ *x*. It implies ⇑ *x* ≤ (↑ *x*)^{∘}, and so (↑ *x*)^{∘} = ⇑ *x*. □

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