Definition 9
A mapping \(\delta : L^{X} \times L^{X} \rightarrow L\) is called an L-fuzzy pre-proximity on X if it satisfies the following axioms.
-
(P1)
\(\delta (\top _{X},\bot _{X}) = \delta (\bot _{X}, \top _{X}) = \bot\),
-
(P2)
\(~ \delta (f, g) \ge \bigvee _{x \in X} (f \odot g)(x)\),
-
(P3)
If \(f_{1} \le f_{2}, h_{1} \le h_{2}\), then \(~ \delta (f_{1}, h_{1}) \le \delta (f_{2}, h_{2})\). The pair \((X, \delta )\) is called L-fuzzy pre-proximity space. An L-fuzzy pre-proximity is called an \((L,\odot ,\oplus )\)-fuzzy pre-proximity if
-
(P4)
For every \(f_{1},f_{2}, h_{1}, h_{2}\in L^{X}\) we have
$$\begin{aligned}&\delta (f_{1} \odot f_{2}, h_{1} \oplus h_{2}) \le \delta (f_{1}, h_{1}) \oplus \delta (f_{2}, h_{2}),\\&\delta (f_{1} \oplus f_{2}, h_{1} \odot h_{2}) \le \delta (f_{1}, h_{1}) \oplus \delta (f_{2}, h_{2}). \end{aligned}$$
An L-fuzzy pre-proximity is called an L-fuzzy quasi-proximity on X if it satisfies (P4) and
-
(Q)
\(~ \delta (f, g) \ge \bigwedge _{ h}\{ \delta (f, h) \oplus \delta ( h^{*},g) \}.\) An L-fuzzy quasi-proximity is called an L-fuzzy proximity on X if
-
(P)
\(~ \delta ^{s}= \delta\) where \(~ \delta ^{s} (f, g) = \delta ( g,f)\). An L-fuzzy pre-proximity is called
-
(St)
stratified if \(~ \delta (\alpha \odot f, \alpha \rightarrow g)\le \delta ( f,g)\) and \(~ \delta (\alpha \rightarrow f, \alpha \odot g)\le \delta ( f,g)\),
-
(SE)
separated if \(~ \delta (\top _{x},\top ^{*}_{x})=\delta (\top ^{*}_{x},\top _{x})=\bot\) for each \(x\in X\),
-
(AL)
Alexandrov if \(~ \delta (\bigvee _{i\in \Gamma } f_{i},g) \le \bigvee _{i\in \Gamma } \delta (f_{i},g),~~ \delta (f,\bigvee _{i\in \Gamma } g_{i}) \le \bigvee _{i\in \Gamma } \delta (f,g_{i})\),
-
(GL)
generalized if \(~ \delta ( f,g) \le \bigvee _{x\in X} f(x)\odot \bigvee _{x\in X} g(x).\)
Definition 10
Let \((X,\delta _{X})\) and \((Y,\delta _{Y})\) be L-fuzzy pre-proximity spaces and \(\varphi :(X,\delta _{X}) \rightarrow (Y,\delta _{Y})\) be a mapping. Then, \(D_{\mathcal {\delta }}(\varphi )\) defined by
$$\begin{aligned} D_{\mathcal {\delta }}(\varphi )=\bigwedge _{f,g\in L^{Y}}\Big (\delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }( g))\rightarrow \delta _{Y}(f, g)\Big ) \end{aligned}$$
is called the degree of LF-proximity for \(\varphi\). If \(D_{\mathcal {\delta }}(\varphi )=\top ,\) then \(~ \delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }( g)) \le \delta _{Y}(f, g)\) for each \(f,g\in L^{Y},\) which is exactly the definition of LF-proximity mappings between L-fuzzy pre-proximity spaces.
Lemma 11
Let
\((X,\delta )\)
be an
L
-fuzzy pre-proximity space. Then,
$$\begin{aligned} \delta (\alpha \odot f, g)\ge \alpha \odot \delta (f,g) ~ {\text{ iff }} ~ \delta (\alpha \rightarrow f,g)\le \alpha \rightarrow \delta (f,g). \end{aligned}$$
Proof
(1) Let \(~ \delta (\alpha \odot f, g)\ge \alpha \odot \delta ( f,g)\). Then, \(\alpha \odot \delta (\alpha \rightarrow f, g) \le \delta (\alpha \odot (\alpha \rightarrow f), g)\le \delta ( f,g)\). Thus, \(~ \delta (\alpha \rightarrow f, g)\le \alpha \rightarrow \delta ( f,g)\).
Let \(~ \delta (\alpha \rightarrow f, g)\le \alpha \rightarrow \delta ( f,g)\). Then, \(\delta ( f,g)\le \delta (\alpha \rightarrow \alpha \odot f, g) \le \alpha \rightarrow \delta (\alpha \odot f,g)\). Thus, \(~\alpha \odot \delta (f,g) \le \delta (\alpha \odot f,g)\).
From the following theorem, we obtain the L-fuzzy closure operator induced by an L-fuzzy pre-proximity.
Theorem 12
Let \(\delta\) be an L-fuzzy pre-proximity on X. Define \(~ {\mathcal {C}}_{\delta }:L^{X} \rightarrow L^{X}\) as follows:
$$\begin{aligned} {\mathcal {C}}_{\delta }(f)(x) = \bigwedge _{g\in L^{X}}\{g(x) \rightarrow \delta (g,g^{*}) \mid f\le g^{*}\}. \end{aligned}$$
Then,
-
(1)
\(( X, {\mathcal {C}}_{\delta })\) is an L-fuzzy closure space,
-
(2)
If \(\delta\) is stratified, then \({\mathcal {C}}_{\delta }\) is stratified,
-
(3)
If \(\delta\) is separated, then \({\mathcal {C}}_{\delta }\) is separated.
Proof
(1)(C1) Since \(\delta (\top _{X},\bot _{X})=\bot\),
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta }(\bot _{X})(x)&{}= \bigwedge _{g\in L^{X}}\{g(x)\rightarrow \delta (g,g^{*}) \mid \bot _{X}\le g^{*} \}\\ &{} \le (\top _{X}(x) \rightarrow \delta (\top _{X},\bot _{X}))=\bot _{X}(x). \end{array} \end{aligned}$$
(C2) Since \(g \le f^{*}\), then \(~ g \rightarrow \delta (g,g^{*})\ge f^{*}\rightarrow \bot =f.\)
(C3) If \(f\le h\), then
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta }(h)(x) &{}=\bigwedge _{g\in L^{X}}\{( g(x) \rightarrow \delta (g,g^{*}))\mid h\le g^{*}\}\\ &{}\ge \bigwedge _{g\in L^{X}}\{(g(x) \rightarrow \delta (g,g^{*}))\mid f\le g^{*}\}= {\mathcal {C}}_{\delta }(f)(x). \end{array} \end{aligned}$$
(C4) Since
$$\begin{aligned} \begin{array}{clcr} &{}((a\rightarrow b) \oplus (c\rightarrow d))^{*}=(a\rightarrow b)^{*}\odot (c\rightarrow d)^{*} \\ &{} =(a\odot b^{*})\odot (c\odot d^{*})=(a\odot c) \odot (b^{*}\odot d^{*}), \end{array} \end{aligned}$$
then we have \((a\rightarrow b) \oplus (c\rightarrow d)=(a\odot c)\rightarrow (b\oplus d)\). From Lemma 2, we obtain
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta }(f)(x) \oplus {\mathcal {C}}_{\delta }(h)(x) &{} = \bigwedge _{g\in L^{X}}\{(g(x) \rightarrow \delta (g,g^{*}))\mid f\le g^{*}\}\\ &{}\oplus \bigwedge _{k\in L^{X}}\{(k(x) \rightarrow \delta (k,k^{*}))\mid h\le k^{*}\}\\ &{} =\bigwedge _{g,k\in L^{X}}\{( g(x) \odot k(x)) \rightarrow (\delta (g,g^{*})\oplus \delta (k,k^{*}))\mid f\le g^{*}~,h\le k^{*} \}\\ &{}\ge \bigwedge _{g,k\in L^{X}}\{(g\odot k)(x))\rightarrow \delta (g\odot k,g^{*}\oplus k^{*})) \mid f\oplus h\le g^{*}\oplus k^{*} \}\\ &{}\ge {\mathcal {C}}_{\delta }(f \oplus h)(x). \end{array} \end{aligned}$$
Hence, \({\mathcal {C}}_{\delta }\) is an L-fuzzy closure operator on X.
(2)
$$\begin{aligned} \begin{array}{clcr} \alpha \rightarrow {\mathcal {C}}_{\delta }(f)&{}=\alpha \rightarrow \bigwedge _{g\in L^{X}}\{(g(x) \rightarrow \delta (g,g^{*}))\mid f\le g^{*}\}\\ &{} =\bigwedge _{g\in L^{X}}\{((\alpha \odot g(x)) \rightarrow \delta (g,g^{*}))\mid f\le g^{*}\}\\ &{}\ge \bigwedge _{g\in L^{X}} \{(\alpha \odot g(x)) \rightarrow \delta ((\alpha \odot g,\alpha \rightarrow g^{*}))\mid \alpha \rightarrow f\le \alpha \rightarrow g^{*}\}\\ &{}\ge {\mathcal {C}}_{\delta }( \alpha \rightarrow f). \end{array} \end{aligned}$$
(3) By (C2) and
$$\begin{aligned} {\mathcal {C}}_{\delta }(\top _{x}^{*})(x) = \bigwedge _{g\in L^{X}}\{g(x) \rightarrow \delta (g,g^{*}) \mid \top _{x}^{*}\le g^{*}\} \le \top _{x}(x) \rightarrow \delta (\top _{x},\top _{x}^{*})=\top ^{*}_{x}, \end{aligned}$$
we have \({\mathcal {C}}_{\delta }(\top _{x}^{*}) =\top _{x}^{*}.\)
Example 13
Let X be a set and \(R\in L^{X\times X}\) be an L-fuzzy pre-order. Define \(\delta :L^{X}\times L^{X} \rightarrow L\) as
$$\begin{aligned} \delta (f,g) =\bigvee _{x,y\in X} R(x,y)\odot f(x)\odot g(y). \end{aligned}$$
(P1) and (P3) are easily proved.
(P2) For all \(f,g\in L^{X}\),
$$\begin{aligned} \begin{array}{clcr} \delta (f,g) &{}=\bigvee _{x,y\in X} R(x,y)\odot f(x)\odot g(y) \\ &{}\ge \bigvee _{x\in X}R(x,x)\odot f(x)\odot g(x) =\bigvee _{x\in X}f(x)\odot g(x). \end{array} \end{aligned}$$
(P4) For all \(f_{1}, f_{2}, h_{1}, h_{2}\in L^{X}\), by Lemma 2 (17),
$$\begin{aligned} \begin{array}{clcr} \delta (f_{1},h_{1}) \oplus \delta (f_{2},h_{2}) &{} = (\bigvee _{x,y\in X}R(x,y)\odot f_{1}(x)\odot h_{1}(y))\oplus \\ &{}(\bigvee _{z,w\in X}R(z,w)\odot f_{2}(z)\odot h_{2}(w))\\ &{}\ge \bigvee _{x,y,z,w\in X}(R(x,y)\odot R(z,w)\odot f_{1}(x)\odot f_{2}(z))\odot \\ &{}(h_{1}(y)\oplus h_{2}(w))\\ &{}\ge \bigvee _{x,y,w\in X}(R(x,y)\odot R(y,w)\odot f_{1}(x)\odot f_{2}(x))\odot \\ &{}(h_{1}(w)\oplus h_{2}(w))\\ &{}=\bigvee _{x,w\in X}(\bigvee _{y\in X}(R(x,y)\odot R(y,w)) \\ &{} \odot (f_{1}(x)\odot f_{2}(x))\odot (h_{1}(w)\oplus h_{2}(w))\\ &{}=\bigvee _{x,w\in X}(R(x,w)\odot f_{1}(x)\odot f_{2}(x))\odot (\ h_{1}(w)\oplus h_{2}(w)) \\ &{}=\delta (f_{1}\odot f_{2}, h_{1}\oplus h_{2}). \end{array} \end{aligned}$$
Hence, \(\delta\) is an L-fuzzy pre-proximity on X. Since
$$\begin{aligned} \begin{array}{clcr} \delta (\alpha \odot f,\alpha \rightarrow g) &{} =\bigvee _{x,y\in X}(R(x,y)\odot (\alpha \odot f)(x)\odot (\alpha \rightarrow g)(y))\\ &{} \le \bigvee _{x,y\in X}(R(x,y)\odot f(x)\odot g(y))=\delta (f,g), \end{array} \end{aligned}$$
\(\delta\) is stratified. Moreover, \(\delta\) is Alexandrov and generalized. By Theorem 12, we obtain a stratified L-fuzzy closure operator \(~ {\mathcal {C}}_{\delta }:L^{X} \rightarrow L^{X}\) as
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta } (f)(x) =\bigwedge _{f\le g^{*}}( g(x) \rightarrow \bigvee _{x,y\in X}(R(x,y)\odot g(x)\odot g^{*}(y))). \end{array} \end{aligned}$$
(1) Let \(R=\top _{X\times X}\) be given. Then, \(~ \delta _{1} (f,g) =\bigvee _{x,y\in X} f(x)\odot g(y)\).
Hence, \(\delta _{1}\) is an L-fuzzy pre-proximity on X. Moreover, \(\delta _{1}\) is stratified, Alexandrov and generalized. Since \(\delta _{1}(\top _{x},\top ^{*}_{x})=\top\), \(\delta _{1}\) is not separated.
By Theorem 12, we obtain a stratified L-fuzzy closure operator \(~ {\mathcal {C}}_{\delta _{1}}:L^{X}\rightarrow L^{X}\) as
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta _{1}} (f) =\bigwedge _{f\le g^{*}}( g(x) \rightarrow (\bigvee _{x,y\in X} g(x)\odot g^{*}(y))). \end{array} \end{aligned}$$
(2) Let \(R=\triangle _{X\times X}\) be given, where
$$\begin{aligned} \triangle _{X\times X}(x,y)=\left\{ \begin{array}{ll} \top ,&{}\;{\text{ if }}\; y=x,\\ \bot , &{} \;{\text{ otherwise }}. \end{array} \right. \end{aligned}$$
Then, \(\delta _{2} (f,g) =\bigvee _{x\in X} f(x)\odot g(x).\) Hence, \(\delta _{2}\) is an L-fuzzy pre-proximity on X. Moreover,
(Q) For all \(f,g \in L^{X}\),
$$\begin{aligned} \begin{array}{clcr} &{}\bigwedge _{h\in L^{X}}(\delta _{2}(f,h) \oplus \delta _{2}(h^{*},g))\\ &{} =\bigwedge _{h\in L^{X}} (\bigvee _{x\in X} (f(x)\odot h(x))\oplus \bigvee _{x\in X} (h^{*}(x)\odot g(x))) ~~~ {\text{(Put } } h=g)\\ &{} \le \bigvee _{x\in X} (f(x)\odot g(x))\oplus \bigvee _{x\in X} (g^{*}(x)\odot g(x)) \\ &{} =\bigvee _{x\in X} (f(x)\odot g(x))\oplus \bot =\delta _{2}(f,g). \end{array} \end{aligned}$$
Hence, \(\delta _{2}\) is an L-fuzzy proximity on X. Since \(\delta _{2}(\top _{x},\top ^{*}_{x})=\bot\), \(\delta _{2}\) is separated. Hence, \(\delta _{2}\) is separated, stratified, Alexandrov and generalized. By Theorem 12, we obtain a strong, separated, generalized and Alexandrov L-fuzzy closure operator \(~ {\mathcal {C}}_{\delta _{2}}:L^{X}\rightarrow L^{X}\) as follows:
$$\begin{aligned} {\mathcal {C}}_{\delta _{2}}(f) =\bigwedge _{f\le g^{*}}( g(x) \rightarrow (\bigvee _{x\in X} g(x)\odot g^{*}(x))) =\bigwedge _{f\le g^{*}}( g(x) \rightarrow \bot )=f. \end{aligned}$$
From the following theorem, we obtain the L-fuzzy pre-proximity induced by an L-fuzzy closure operator.
Theorem 14
Let
\((X, {\mathcal {C}})\)
be an
L
-fuzzy closure space. Define a mapping
\(\delta _{{\mathcal {C}}}:L^{X} \times L^{X} \rightarrow L ~\)
by
$$\begin{aligned} \delta _{{\mathcal {C}}}(f,g) = \bigvee _{x\in X} f(x)\odot {\mathcal {C}}(g)(x) ~~~ \forall ~ f,g \in L^{X}. \end{aligned}$$
Then, we have the following properties.
-
(1)
\(\delta _{{\mathcal {C}}}\)
is an
L
-fuzzy pre-proximity,
-
(2)
If \({\mathcal {C}}\) is stratified, then so is \(\delta _{{\mathcal {C}}}\) and \(\delta _{{\mathcal {C}}}( f,\alpha \odot g)\ge \alpha \odot \delta _{{\mathcal {C}}}(f,g)\),
-
(3)
\(\delta _{{\mathcal {C}}}(f,g)\le \bigvee _{h\in L^{X}} (\delta _{{\mathcal {C}}}(f,h)\odot \delta _{{\mathcal {C}}}(h^{*},g))\),the equality holds if \({\mathcal {C}}\) is topological,
-
(4)
If \({\mathcal {C}}\) is topological, then \(\delta _{{\mathcal {C}}}\) is an L-fuzzy quasi-proximity on X,
-
(5)
\({\mathcal {C}}\le {\mathcal {C}}_{\delta _{{\mathcal {C}}}}\), the equality holds if \({\mathcal {C}}\) is topological,
-
(6)
If
\({\mathcal {C}}\)
is separated, then
\(\delta _{{\mathcal {C}}}\)
is separated,
-
(7)
\(\delta _{{\mathcal {C}}_{\delta }} \le \delta\),
-
(8)
If
\({\mathcal {C}}\)
is generalized (resp. Alexandrov), then
\(\delta _{{\mathcal {C}}}\)
is generalized (resp. Alexandrov).
Proof
(1) (P1) Since \({\mathcal {C}}(\bot _{X})=\bot _{X} ~\) and \(~ {\mathcal {C}}(\top _{X})=\top _{X}\), we have
$$\begin{aligned} \begin{array}{clcr} &{}\delta _{{\mathcal {C}}}(\top _{X},\bot _{X}) = \bigvee _{x\in X}(\top _{X}(x)\odot {\mathcal {C}}(\bot _{X})(x))=\bot,\\ &{} \delta _{{\mathcal {C}}}(\bot _{X},\top _{X}) = \bigvee _{x\in X}(\bot _{X}(x)\odot {\mathcal {C}}(\top _{X})(x))=\bot . \end{array} \end{aligned}$$
(P2) Since \({\mathcal {C}}(f)\ge f\), we have
$$\begin{aligned} \delta _{{\mathcal {C}}}(f,g) = \bigvee _{x\in X} f(x)\odot {\mathcal {C}}(g)(x) \ge \bigvee _{x\in X} f(x)\odot g(x). \end{aligned}$$
(P3) If \(f \le f_{1}\) and \(g \le g_{1}\), then \({\mathcal {C}}(g)\le {\mathcal {C}}(g_{1})\). Thus,
$$\begin{aligned} \delta _{{\mathcal {C}}}(f,g) = \bigvee _{x\in X} f(x)\odot {\mathcal {C}}(g)(x) \le \bigvee _{x\in X} f_{1}(x) \odot {\mathcal {C}}(g_{1})(x) = \delta _{{\mathcal {C}}}(f_{1},g_{1}). \end{aligned}$$
(P4)
$$\begin{aligned} \begin{array}{clcr} \delta _{{\mathcal {C}}}(f_{1},g_{1}) \oplus \delta _{{\mathcal {C}}}(f_{2},g_{2}) &{} = \bigvee _{x\in X}(f_{1}(x) \odot {\mathcal {C}}(g_{1})(x))\oplus (\bigvee _{x\in X} f_{2}(x) \odot {\mathcal {C}}(g_{2})(x))\\ &{}\ge \bigvee _{x\in X}(f_{1}(x) \odot {\mathcal {C}}(g_{1})(x))\oplus (f_{2}(x) \odot {\mathcal {C}}(g_{2})(x)) \\ &{}~~ {\text{(by Lemma 2(13)) }} \\ &{}\ge \bigvee _{x\in X}(f_{1}(x) \odot f_{2}(x)) \odot ({\mathcal {C}}(g_{1})(x)\oplus {\mathcal {C}}(g_{2})(x)) \\ &{}\ge \bigvee _{x\in X}(f_{1}(x) \odot f_{2}(x)) \odot {\mathcal {C}}(g_{1}\oplus g_{2})(x) =\delta _{{\mathcal {C}}}(f_{1}\oplus f_{2},g_{1}\oplus g_{2}). \end{array} \end{aligned}$$
Hence, \(\delta _{{\mathcal {C}}}\) is an L-fuzzy pre-proximity on X.
(2) If \({\mathcal {C}}\) is a stratified, we have
$$\begin{aligned} \begin{array}{clcr} \delta _{{\mathcal {C}}}(\alpha \odot f,\alpha \rightarrow g)&{}=\bigvee _{x\in X}(\alpha \odot f)(x)\odot {\mathcal {C}}(\alpha \rightarrow g)(x)\\ &{}\le \bigvee _{x\in X} \alpha \odot f(x)\odot (\alpha \rightarrow {\mathcal {C}}(g)(x))\\ &{}\le \bigvee _{x\in X} f(x)\odot {\mathcal {C}}(g)(x)=\delta _{{\mathcal {C}}}(f,g), \\ &{} ~ \\
\delta _{{\mathcal {C}}}(f,\alpha \odot g)&{}=\bigvee _{x\in X}f(x)\odot {\mathcal {C}}(\alpha \odot g)(x)\\ &{}\ge \bigvee _{x\in X}f(x)\odot \alpha \odot {\mathcal {C}}(g)(x)\\ &{}=\alpha \odot (\bigvee _{x\in X}f(x)\odot {\mathcal {C}}(g)(x))=\alpha \odot \delta _{{\mathcal {C}}}(f,g). \end{array} \end{aligned}$$
(3)
$$\begin{aligned} \begin{array}{clcr} &{}\delta ^{*}_{{\mathcal {C}}}(f,h) \odot \delta ^{*}_{{\mathcal {C}}}(h^{*},g)\\ &{} =\Big (\bigvee _{x\in X}f(x)\odot {\mathcal {C}}(h)(x)\Big )^{*}\odot \Big (\bigvee _{x\in X}h^{*}(x)\odot {\mathcal {C}}(g)(x)\Big )^{*} \\ &{}= \bigwedge _{x\in X}(f(x)\rightarrow {\mathcal {C}}^{*}(h)(x))\odot \bigwedge _{x\in X}(h^{*}(x)\rightarrow {\mathcal {C}}^{*}(g)(x)) \\ &{}~~~~~~ {\text{(Since } } {\mathcal {C}}^{*}(h)\le h^{*})\\ &{} \le \bigwedge _{x\in X}(f(x)\rightarrow h^{*}(x))\odot \bigwedge _{x\in X}(h^{*}(x)\rightarrow {\mathcal {C}}^{*}(g)(x)) \\ &{} \le \bigwedge _{x\in X}(f(x)\rightarrow {\mathcal {C}}^{*}(g)(x)) =\delta ^{*}_{{\mathcal {C}}}(f,g). \end{array} \end{aligned}$$
Hence, \(\delta _{{\mathcal {C}}}(f,g)\le \bigwedge _{h\in L^{X}} (\delta _{{\mathcal {C}}}(f,h)\oplus \delta _{{\mathcal {C}}}(h^{*},g))\).
If \({\mathcal {C}}\) is topological, then
$$\begin{aligned} \begin{array}{clcr} &{}\bigvee _{h\in L^{X}}(\delta^{*}_{{\mathcal {C}}}(f,h) \odot \delta^{*}_{{\mathcal {C}}}(h^{*},g))\\ &{}=\bigvee _{h\in L^{X}}(\bigwedge _{x\in X} (f(x)\rightarrow {\mathcal {C}}^{*}(h)(x)))\odot (\bigwedge _{x\in X}(h^{*}(x)\rightarrow {\mathcal {C}}^{*}(g)(x))) \\ &{} ~~~ {\text{(put }}{\mathcal {C}}(g)=h)\\ &{}\ge \bigwedge _{x\in X}(f(x)\rightarrow {\mathcal {C}}^{*}({\mathcal {C}}(g))(x))\odot (\bigwedge _{x\in X}({\mathcal {C}}^{*}(g)(x)\rightarrow {\mathcal {C}}(g^{*})(x)) \\ &{} =\bigwedge _{x\in X}(f(x)\rightarrow {\mathcal {C}}^{*}(g)(x))=\delta ^{*}_{{\mathcal {C}}}(f,g). \end{array} \end{aligned}$$
(4) By (3), it is trivial.
(5) From Lemma 2, we have,
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta _{{\mathcal {C}}}}(f)(x) &{}=\bigwedge _{g\in L^{X}}\{ \bigwedge _{x\in X}(g(x)\rightarrow \delta _{{\mathcal {C}}}(g,g^{*}))\mid f\le g^{*}\}\\ &{}= \bigwedge _{g\in L^{X}}\{\bigwedge _{x\in X}(g(x)\rightarrow (\bigvee _{x\in X}g(x)\odot {\mathcal {C}}(g^{*})(x)))\mid f\le g^{*}\}\\ &{}=\{\big (\bigvee _{g\in L^{X}}g(x)\odot \bigwedge _{x\in X}({\mathcal {C}}(g^{*})(x)\rightarrow g^{*}(x))\big )^{*}\mid f\le g^{*}\}\\ &{}\ge \{\big (\bigvee _{g\in L^{X}}(\bigwedge _{x\in X}({\mathcal {C}}(f)(x)\rightarrow g^{*}(x))\odot g(x))\big )^{*}\mid {\mathcal {C}}(f)\le {\mathcal {C}}(g^{*})\}\\ &{}=\Big (\bigvee _{g\in L^{X}}(\bigwedge _{x\in X}(g(x)\rightarrow {\mathcal {C}}^{*}(f)(x))\odot g(x))\Big )^{*}\ge {\mathcal {C}}(f)(x). \end{array} \end{aligned}$$
If \({\mathcal {C}}\) is topological, then
$$\begin{aligned} \begin{array}{clcr} {\mathcal {C}}_{\delta _{{\mathcal {C}}}}(f) (x) &{}=\bigwedge _{g\in L^{X}}\{ g(x)\rightarrow \delta _{{\mathcal {C}}}(g,g^{*})\mid f\le g^{*} \}\\ &{} =\{\Big (\bigvee _{g\in L^{X}}g(x)\odot \bigwedge _{x\in X}({\mathcal {C}}(g^{*})(x)\rightarrow g^{*}(x))\Big )^{*}\mid f\le g^{*}\} \\ &{} ~~{\text{(Put } } g^{*}={\mathcal {C}}(f))\\ &{} \le \Big ({\mathcal {C}}^{*}(f)(x)\odot \bigwedge _{x\in X}({\mathcal {C}}({\mathcal {C}}(f)(x))\rightarrow {\mathcal {C}}(f)(x))\Big )^{*}= {\mathcal {C}}(f)(x). \end{array} \end{aligned}$$
(6) \(\delta ^{*}_{{\mathcal {C}}_{\delta }}(\top _{x},\top ^{*}_{x})=\bigwedge _{x\in X}({\mathcal {C}}_{\delta }(\top ^{*}_{x})(x)\rightarrow \top ^{*}_{x}(x))=\top\).
(7)
$$\begin{aligned} \begin{array}{clcr} \delta _{{\mathcal {C}}_{\delta }}(f,g) &{}=\bigvee _{x\in X} f(x)\odot {\mathcal {C}}_{\delta }(g)(x)\\ &{}=\bigvee _{x\in X} f(x)\odot \Big (\bigvee _{h\le g^{*}}\delta ^{*}(h,h^{*})\odot h(x)\Big )^{*}\\ &{}\le \bigvee _{x\in X}f(x)\odot \Big (\bigvee _{h\le g^{*}}(\bigwedge _{x\in X}(h(x)\rightarrow h(x))\odot h(x))\Big )^{*}\\ &{}\le \bigvee _{x\in X}f(x)\odot g(x)\le \delta (f,g). \end{array} \end{aligned}$$
(8) It is easily proved from definitions.
Corollary 15
Let
\((X, {\mathcal {C}})\)
be an
L
-fuzzy closure space. Define a mapping
\(~\delta ^{s}_{{\mathcal {C}}}:L^{X} \times L^{X} \rightarrow L ~\)
by
$$\begin{aligned} \delta ^{s}_{{\mathcal {C}}}(f,g) = \bigvee _{x\in X}g(x)\odot {\mathcal {C}}(f)(x) ~~~ \forall ~ f,g \in L^{X}. \end{aligned}$$
Then, we have the following properties.
-
(1)
\(\delta ^{s}_{{\mathcal {C}}}\)
is an
L
-fuzzy pre-proximity,
-
(2)
If
\(~{\mathcal {C}}\)
is stratified, then
\(\delta ^{s}\)
is a stratified,
-
(3)
\(\delta ^{s}_{{\mathcal {C}}}(f,g)\le \bigvee _{h\in L^{X}} (\delta ^{s}_{{\mathcal {C}}}(f,h)\odot \delta ^{s}_{{\mathcal {C}}}(h^{*},g))\), the equality holds if \({\mathcal {C}}\) is topological,
-
(4)
If \(~{\mathcal {C}}\) is topological, then \(\delta ^{s}_{{\mathcal {C}}}\) is a L-fuzzy quasi-proximity on X,
-
(5)
\({\mathcal {C}}\le {\mathcal {C}}_{\delta ^{s}_{{\mathcal {C}}}}\), the equality holds if \({\mathcal {C}}\) is topological,
-
(6)
If
\(~{\mathcal {C}}\)
is separated, then
\(\delta ^{s}_{{\mathcal {C}}}\)
is separated,
-
(7)
\(\delta ^{s}_{{\mathcal {C}}_{\delta }} \le \delta ^{s}\),
-
(8)
If
\({\mathcal {C}}\)
is generalized (resp. Alexandrov), then
\(\delta ^{s}_{{\mathcal {C}}}\)
is generalized (resp. Alexandrov).