In this section, we assume that L is an order dense chain. Let \({\mathcal {T}}(x_{t},r)=\{g\in L^{X}: x_{t}\in g, {\mathcal {T}}(g)\ge r\}\).
Definition 6.1
Let \((X,{\mathcal {T}})\) be an L-fuzzy topological space and \({\mathcal {G}}\) be an L-fuzzy grill on X. Then, the triplet \((X,{\mathcal {T}},{\mathcal {G}} )\) is called an L-fuzzy grill fuzzy topological space.
Definition 6.2
Let \((X,{\mathcal {T}},{\mathcal {G}} )\) be an L-fuzzy grill fuzzy topological space. The operator \(\Phi _{{\mathcal {G}},{\mathcal {T}}}: L^{X}\times L_{\bot }\rightarrow L^{X}\) which defined by:
$$\begin{aligned} \Phi _{{\mathcal {G}}, {\mathcal {T}}}({f},r)=\bigvee \left\{ x_{t}\in P_{t}(X): {\mathcal {G}}({f}\wedge g)\ge r, \hbox {for each } g\in {\mathcal {T}}(x_{t},r)\right\} \end{aligned}$$
is called the local function associated with L-fuzzy grill \({\mathcal {G}}\) and L-fuzzy topology \({\mathcal {T}},\) simply we denote it by \(\Phi _{{\mathcal {G}}}({f},r)\) .
Theorem 6.3
Let \((X,{\mathcal {T}})\) be an L-fuzzy topological space. Then the following statements hold.
(1) If \({\mathcal {G}}\) is an L-fuzzy grill on X, then \(\Phi _{{\mathcal {G}}}\) is an increasing function; in the sense that \({f}\le g\) implies \(\Phi _{{\mathcal {G}}}({f},r)\le \Phi _{{\mathcal {G}}}({g},r)\).
(2) If \({\mathcal {G}}_{1}\) and \({\mathcal {G}}_{2}\) are two L-fuzzy grills on X with \({\mathcal {G}}_{1}\le {\mathcal {G}}_{2}\), then \(\Phi _{{\mathcal {G}}_{1}}({f},r)\le \Phi _{{\mathcal {G}}_{2}}({f},r)\), \(\forall {f}\in L^{X}, r\in L_{\bot }\).
(3) For any L-fuzzy grill \({\mathcal {G}}\) on X, if \({\mathcal {G}}({f})=\bot\), then \(\Phi _{{\mathcal {G}}}({f},r)=\bot _{X}\), \(\forall r\in L_{\bot }\).
Proof
It is clear. \(\square\)
Theorem 6.4
Let \((X,{\mathcal {T}},{\mathcal {G}} )\) be an L-fuzzy grill fuzzy topological space. Then for all \({f},{g}\in L^{X}\), we have:
(1) \(\Phi _{{\mathcal {G}}}({f}\vee {g},r)\ge \Phi _{{\mathcal {G}}}({f},r)\vee \Phi _{{\mathcal {G}}}({g},r)\), \(r\in L_{\bot }\).
(2) \(\Phi _{{\mathcal {G}}}(\Phi _{{\mathcal {G}}}({f},r),r)\le \Phi _{{\mathcal {G}}}({f},r) =C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r) \le C_{{\mathcal {T}}}({f},r)\), \(r\in L_{\bot }\).
Proof
(1) It is clear.
(2) If \(x_{t}\not \in C_{{\mathcal {T}}}({f},r)\), then there exists \(g\in {\mathcal {T}}(x_{t},r)\) such that \(g\wedge {f}=\bot _{X}\). Then, \({\mathcal {G}}(g\wedge {f},r)={\mathcal {G}}(\bot _{X})=\bot\). Thus, \(x_{t}\not \in\) \(\Phi _{{\mathcal {G}}}({f},r)\). Therefore, \(\Phi _{{\mathcal {G}}}({f},r)\le C_{{\mathcal {T}}}({f},r)\).
Now, we will show that \(C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r)\le \Phi _{{\mathcal {G}}}({f},r)\). Suppose that \(x_{t}\in C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r)\), then for every \(g \in {\mathcal {T}}(x_{t},r)\) we have \(g\wedge \Phi _{{\mathcal {G}}}({f},r)\ne \bot _{X}\). Let \(y_{s}\in g\wedge \Phi _{{\mathcal {G}}}({f},r)\). Then, \(y_{s}\in g\) and \(y_{s}\in \Phi _{{\mathcal {G}}}({f},r)\). Since \(y_{s}\in \Phi _{{\mathcal {G}}}({f},r)\), then for each \(h\in L^{X}\) with \(y_{s}\in h\) and \({\mathcal {T}}(h)\ge r\), we have \({\mathcal {G}}(f\wedge h)\ge r\). Since \(y_{s}\in g\) and \({\mathcal {T}}(g)\ge r\), we have \({\mathcal {G}}(f\wedge g)\ge r\). Therefore, \(x_{t}\in \Phi _{{\mathcal {G}}}({f},r)\). Thus, \(C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r)\le \Phi _{{\mathcal {G}}}({f},r)\), which implies that, \(C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r)= \Phi _{{\mathcal {G}}}({f},r)\). Hence
$$\begin{aligned} \Phi _{{\mathcal {G}}}(\Phi _{{\mathcal {G}}}({f},r),r)\le C_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r) =\Phi _{{\mathcal {G}}}({f},r)\le C_{{\mathcal {T}}}({f},r). \end{aligned}$$
\(\square\)
Remark 6.5
The following example show that the equality in Theorem 6.4(i) does not always hold.
Example 6.6
Let \(X=\{a,b,c,d\}\) and \(L=I\). Define an L-fuzzy topology \({\mathcal {T}}:L^{X}\rightarrow L\) on X by:
$$\begin{aligned} {\mathcal {T}}_{1}({f}) = \left\{ \begin{array}{ll} \top , \,\, &{} \hbox {if }\quad {f} =\bot _{X}, \top _{X} \\ \frac{1}{2}, \,\, &{} \hbox {if }\quad {f}\in \{\chi _{\{a\}}, \chi _{\{a,b\}}\} \\ \bot , \,\, &{} \hbox {otherwise,} \end{array} \right. \end{aligned}$$
Define an L-fuzzy grill \({\mathcal {G}}:L^{X}\rightarrow L\) on X by:
$$\begin{aligned} {\mathcal {G}}_{1}({f}) = \left\{ \begin{array}{ll} \top , \,\, &{} \hbox {if }\quad {f} = \top _{X} \\ \frac{1}{2}, \,\, &{} \hbox {if }\quad {f}\in \{\chi _{\{a,b,c\}}, \chi _{\{a,b,d\}}\} \\ \frac{1}{3}, \,\, &{} \hbox {if }\quad {f}\in \chi _{\{a,b\}} \\ \bot , \,\, &{} \hbox {otherwise,} \end{array} \right. \end{aligned}$$
Then \((X,{\mathcal {T}},{\mathcal {G}})\) is an L-fuzzy grill fuzzy topological space. If \({f}=\chi _{\{a\}}\), \({g}=\chi _{\{b,c\}}\) and \(r=\frac{1}{4}\). Then
\(\Phi _{{\mathcal {G}}}({f},r)\vee \Phi _{{\mathcal {G}}}({g},r)=\bot _{X} < \Phi _{{\mathcal {G}}}({f}\vee {g},r)=\chi _{\{a,b,c\}}\).
Theorem 6.7
Let \((X,{\mathcal {T}},{\mathcal {G}})\) be an L-fuzzy grill fuzzy topological space. Define the operator \(C^{{\mathcal {G}}}_{{\mathcal {T}}}: L^{X}\times L_{\bot }\rightarrow L^{X}\) by:
$$\begin{aligned} C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)={f}\vee \Phi _{{\mathcal {G}}}({f},r). \end{aligned}$$
Then, \(C^{{\mathcal {G}}}_{{\mathcal {T}}}\) satisfies the following properties:
-
(1)
\(C^{{\mathcal {G}}}_{{\mathcal {T}}}(\bot _{X},r)=\bot _{X}\), \(C^{{\mathcal {G}}}_{{\mathcal {T}}}(\top _{X},r)=\top _{X}\), \(\forall r\in L_{\bot }\).
-
(2)
\({f}\le C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)\), \(\forall f\in L^{X}, r\in L_{\bot }\).
-
(3)
\(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)\le C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},s)\) if \(r\le s\).
-
(4)
\(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}\vee {g},r\wedge s)\le C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)\vee C^{{\mathcal {G}}}_{{\mathcal {T}}}({g}, s)\), \(r,s\in L_{\bot }\).
-
(5)
\(C^{{\mathcal {G}}}_{{\mathcal {T}}}(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r),r)=C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)\), \(r\in L_{\bot }\).
Proof
It is straightforward. \(\square\)
Theorem 6.8
Let \((X,{\mathcal {T}},{\mathcal {G}})\) be an L-fuzzy grill fuzzy topological space. Define the map \({\mathcal {T}}_{{\mathcal {G}}}: L^{X}\rightarrow L\) by:
$$\begin{aligned} {\mathcal {T}}_{{\mathcal {G}}}({f})=\bigvee \left\{ r\in L_{\bot }: C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}^{*},r)={f}^{*} \right\} . \end{aligned}$$
Then, \({\mathcal {T}}_{{\mathcal {G}}}\) is an L-fuzzy topology on X.
Proof
(LO1) It is clear.
(LO2) Suppose that there exist \({f}_{1},{f}_{2}\in L^{X}\) such that
$$\begin{aligned} {\mathcal {T}}_{{\mathcal {G}}}({f}_{1}\wedge {f}_{2})\not \ge {\mathcal {T}}_{{\mathcal {G}}}({f}_{1})\wedge {\mathcal {T}}_{{\mathcal {G}}}({f}_{2}). \end{aligned}$$
By the definitions of \({\mathcal {T}}_{{\mathcal {G}}}({f}_{1})\) and \({\mathcal {T}}_{{\mathcal {G}}}({f}_{2})\), there exist \(r_{1}, r_{2}\in L_{\bot }\) with \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{1}^{*},r)={f}_{1}^{*}\) and \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{2}^{*},r)={f}_{2}^{*}\) such that \({\mathcal {T}}_{{\mathcal {G}}}({f}_{1}\wedge {f}_{2})\not \ge r_{1}\wedge r_{2}\). From Theorem 6.7(4),
$$\begin{aligned} C^{{\mathcal {G}}}_{{\mathcal {T}}}(({f}_{1}\wedge {f}_{2})^{*},r_{1}\wedge r_{2})= \,& {} C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{1}^{*}\vee {f}_{2}^{*},r_{1}\wedge r_{2}) \\\le \,& {} C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{1}^{*},r_{1})\vee C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{2}^{*}, r_{2}) \\\le \,& {} {f}_{1}^{*}\vee {f}_{2}^{*} \\= \,& {} ({f}_{1}\wedge {f}_{2})^{*} . \end{aligned}$$
By Theorem 6.7(2), \(C^{{\mathcal {G}}}_{{\mathcal {T}}}(({f}_{1}\wedge {f}_{2})^{*},r_{1}\wedge r_{2})=({f}_{1}\wedge {f}_{2})^{*}\). Then, \({\mathcal {T}}_{{\mathcal {G}}}({f}_{1}\wedge {f}_{2})\ge r_{1}\wedge r_{2}\). It is contradiction. Hence, \({\mathcal {T}}_{{\mathcal {G}}}({f}_{1}\wedge {f}_{2})\ge {\mathcal {T}}_{{\mathcal {G}}}({f}_{1})\wedge {\mathcal {T}}_{{\mathcal {G}}}({f}_{2}), \forall {f}_{1},{f}_{2}\in L^{X}.\)
(LO3) Suppose that there exist \(\{{f}_{i}:i\in \Gamma \}\subseteq L^{X}\) such that:
$$\begin{aligned} {\mathcal {T}}_{{\mathcal {G}}}(\bigvee _{i\in \Gamma }{f}_{i})\not \ge \bigwedge _{i\in \Gamma }{\mathcal {T}}_{{\mathcal {G}}}({f}_{i}). \end{aligned}$$
Since L is an order dense chain, there exists \(r_{0 }\in L_{\bot }\) such that
$$\begin{aligned} {\mathcal {T}}_{{\mathcal {G}}}(\bigvee _{i\in \Gamma }{f}_{i})< r_{0 }\le \bigwedge _{i\in \Gamma }{\mathcal {T}}_{{\mathcal {G}}}({f}_{i}). \end{aligned}$$
Since \(\bigwedge _{i\in \Gamma }{\mathcal {T}}_{{\mathcal {G}}}({f}_{i})\ge r_{0 }\). Then \({\mathcal {T}}_{{\mathcal {G}}}({f}_{i})\ge r_{0 }\), \(\forall i\in \Gamma\). This implies that: \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{i}^{*},r_{0 })={f}_{i}^{*}\), \(\forall i\in \Gamma\). Let \({f}=\bigvee _{i\in \Gamma }{f}_{i}\). Then, \({f}_{i}\le {f}\), \(\forall i\in \Gamma\). Therefore, \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}^{*},r_{0 })\le C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{i}^{*},r_{0 })\), \(\forall i\in \Gamma\). Then
$$\begin{aligned} C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}^{*},r_{0 })\le \,& {} {\bigwedge }_{i\in \Gamma }C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}_{i}^{*},r_{0 })\\= \,& {} {\bigwedge }_{i\in \Gamma }{f}_{i}^{*}\\= \,& {} \left( {\bigvee }_{i\in \Gamma }{f}_{i}\right) ^{*}\\= \,& {} {f}^{*} . \end{aligned}$$
Thus, \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f}^{*},r_{0 })= {f}^{*}\). Then, \({\mathcal {T}}_{{\mathcal {G}}}(\bigvee _{i\in \Gamma }{f}_{i})={\mathcal {T}}_{{\mathcal {G}}}({f})\ge r_{0 }\), a contradiction. Thus, \({\mathcal {T}}_{{\mathcal {G}}}(\bigvee _{i\in \Gamma }{f}_{i})\ge \bigwedge _{i\in \Gamma }{\mathcal {T}}_{{\mathcal {G}}}({f}_{i})\), for each \(\{{f}_{i}:i\in \Gamma \}\subseteq L^{X}\). \(\square\)
Theorem 6.9
Let \((X,{\mathcal {T}})\) be an L-fuzzy topological space. Then the following statements hold.
-
(1)
If \({\mathcal {G}}_{1}\) and \({\mathcal {G}}_{2}\) are L-fuzzy grills on X with \({\mathcal {G}}_{1}\le {\mathcal {G}}_{2}\), then \({\mathcal {T}}_{{\mathcal {G}}_{1}}\le {\mathcal {T}}_{{\mathcal {G}}_{2}}\).
-
(2)
If \({\mathcal {G}}\) is an L-fuzzy grill on X and \({f}\in L^{X}\) with \({\mathcal {G}}({f})=\bot\), then there exists \(r\in L_{\bot }\) such that \({\mathcal {T}}_{{\mathcal {G}}}({f}^{*})\ge r\).
-
(3)
For any \({f}\in L^{X}\), \(r\in L_{\bot }\) and for any L-fuzzy grill \({\mathcal {G}}\) on X, \({\mathcal {T}}_{{\mathcal {G}}}((\Phi _{{\mathcal {G}}}({f},r))^{*})\ge r\).
-
(4)
If \({f}\in L^{X}\), \(r\in L_{\bot }\) with \({\mathcal {T}}_{{\mathcal {G}}}({f}^{*})\ge r\), then \(\Phi _{{\mathcal {G}}}({f},r)\le {f}\).
Proof
(1) Let \(r\in L_{\bot }\) such that \({\mathcal {T}}_{{\mathcal {G}}_{2}}({f})\ge r\). Then \(C^{{\mathcal {G}}_{2}}_{{\mathcal {T}}}({f}^{*},r)={f}^{*}\). Thus, \({f}^{*}\vee \Phi _{{\mathcal {G}}_{2}}({f}^{*},r)={f}^{*}\). This implies that \(\Phi _{{\mathcal {G}}_{2}}({f}^{*},r)\le {f}^{*}\). By Theorem 6.3(2), we have \(\Phi _{{\mathcal {G}}_{1}}({f}^{*},r)\le {f}^{*}\). This implies that \({f}^{*}\vee \Phi _{{\mathcal {G}}_{1}}({f}^{*},r)={f}^{*}\). Thus, \(C^{{\mathcal {G}}_{1}}_{{\mathcal {T}}}({f}^{*},r)={f}^{*}\), which implies that \({\mathcal {T}}_{{\mathcal {G}}_{1}}({f})\ge r\). Thus, \({\mathcal {T}}_{{\mathcal {G}}_{2}}\le {\mathcal {T}}_{{\mathcal {G}}_{1}}\).
(2) Let \({\mathcal {G}}\) be an L-fuzzy grill, \(r\in L_{\bot }\) and \({f}\in L^{X}\) with \({\mathcal {G}}({f})=\bot\). Then by Theorem 6.3(3), \(\Phi _{{\mathcal {G}}}({f},r)=\bot _{X}\). Thus \(C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)={f}\vee \Phi _{{\mathcal {G}}}({f},r)={f}\). This implies that \({\mathcal {T}}_{{\mathcal {G}}}({f}^{*})\ge r\).
(3) Let \({f}\in L^{X}\) and \(r\in L_{\bot }\). For any L-fuzzy grill \({\mathcal {G}}\) on X , we have
$$\begin{aligned} C^{{\mathcal {G}}}_{{\mathcal {T}}}(\Phi _{{\mathcal {G}}}({f},r),r)=\Phi _{{\mathcal {G}}}({f},r)\vee \Phi _{{\mathcal {G}}}(\Phi _{{\mathcal {G}}}({f},r),r)=\Phi _{{\mathcal {G}}}({f},r).\,\,\,\,\,\,\, (\hbox {by Theorem } 6.4(2)) \end{aligned}$$
Thus, \({\mathcal {T}}_{{\mathcal {G}}}((\Phi _{{\mathcal {G}}}({f},r))^{*})\ge r\).
(4) Let \({f}\in L^{X}\) and \(r\in L_{\bot }\) with \({\mathcal {T}}_{{\mathcal {G}}}({f}^{*})\ge r\). Suppose that \(x_{t}\not \in {f}= C^{{\mathcal {G}}}_{{\mathcal {T}}}({f},r)={f}\vee \Phi _{{\mathcal {G}}}({f},r)\), which implies that \(x_{t}\not \in \Phi _{{\mathcal {G}}}({f},r)\). Thus, \(\Phi _{{\mathcal {G}}}({f},r)\le {f}\). \(\square\)