Skip to main content
  • Original research
  • Open access
  • Published:

On L-fuzzy closure operators and L-fuzzy pre-proximities

Abstract

The aim of this paper is to investigate the relations among the L-fuzzy pre-proximities, L-fuzzy closure operators and L-fuzzy co-topologies in complete residuated lattices. We show that there is a Galois correspondence between the category of separated L-fuzzy closure spaces and that of separated L-fuzzy pre-proximity spacesĀ and we give their examples.

Introduction

Closure operators are very useful tool in several areas of mathematical structures with direct applications, both mathematical (e.g, topology, logic) and extra-mathematical (e.g, data mining, knowledge representation). In fuzzy set theory [1, 2], several particular kinds suchĀ asĀ general theory of closure operators which operate with fuzzy sets (so-called fuzzy closure operators) are studied [3,4,5,6].

Ward et al. [7] introduced a complete residuated lattice which is an algebraic structure for many valued logic. BělohlĆ”vek [8] investigated information systems, decision rules and developed the notion of fuzzy contexts using Galois connections with \(R\in L^{X\times Y}\) on a complete residuated lattices. Hƶhle [9] introduced L-fuzzy topologies with algebraic structure L (cqm, quantales, MV-algebra). It has developed in many directions [10,11,12]. Recently, BělohlĆ”vek [13, 14] outlined a general theory of fuzzy closureĀ operators by using the structure of the residuated lattice in place of the usual structure of truth value on [0,Ā 1]. Fang and Yue [15] studied the relationship between L-fuzzy closure systems and L-fuzzy topological spaces from a category viewpoint for a complete residuated lattice L (see also [16]). Ramadan [17] studied the relationship between L-fuzzy interior systems and L-fuzzy topological spaces over complete residuated lattices.

Proximity is an important concept in topology, and it can be considered either as axiomatizations of geometric notions, close to but quite independent of topology, or as convenient tools for an investigation of topological spaces. Hence, proximity has close relations with topology, uniformity and metric. With the development of topology, the theory of proximity makes a massive progress. In the framework of L-topology, many authors generalized the crisp proximity to L-fuzzy setting. Katsaras [18, 19] introduced the concepts of fuzzy topogenous order and fuzzy topogenous structures in completely distributive lattice which are a unified approach to the three spaces: Changā€™s fuzzy topologies [20], Katsarasā€™s fuzzy proximities [21] and Huttonā€™s fuzzy uniformities [22] (see also [23]) . Subsequently, Liu [24], Artico and Moresco [25] extended it into L-fuzzy set theory in view points of Lowenā€™s fuzzy topology [26]. As an extension of Katsarasā€™s definition, El-Dardery [27] introduced L-fuzzy topogenous order in view points of Sostakā€™s fuzzy topology [28], smooth fuzzy topology [29] and Kimā€™s L-fuzzy proximities [30] on strictly two-sided, commutative quantales. L-fuzzy topogenous structures and L-fuzzy proximities [23, 31,32,33,34] have been developed in a slightly different sense.

In this paper, we introduce the notions of L-fuzzy pre-proximities and L-fuzzy closure operators in complete residuated lattices. Moreover, we investigate the relations among the L-fuzzy pre-proximities, L-fuzzy closure operators and L-fuzzy co-topologies. We show that there is a Galois correspondence between the category of separated L-fuzzy closure spaces and that of separated L-fuzzy pre-proximity spaces. In ExampleĀ 19, as an information system as an extension of Pawlakā€™s rough set [35, 36], L-fuzzy pre-proximities, L-fuzzy co-topologies and L-fuzzy closure operators are introduced. By using these concepts, we can apply themĀ to information systems and decision makings [37].

Preliminaries

Definition 1

([8,9,10,11, 38]) An algebra \((L,\wedge ,\vee ,\odot ,\rightarrow ,\bot ,\top )\) is called a complete residuated lattice if it satisfies the following conditions:

  1. (C1)

    \((L,\le ,\vee , \wedge , \bot , \top )\) is a complete lattice with the greatest element \(\top\) and the least element \(\bot\);

  2. (C2)

    \((L, \odot , \top )\) is a commutative monoid;

  3. (C3)

    \(x\odot y \le z\) iff \(x\le y \rightarrow z\) for \(x,y,z\in L\).

In this paper, we assume that \((L, \le , \odot , ^*)\) is a complete residuated lattice with an order reversing involution \(^*\) which is defined by

$$\begin{aligned} x \oplus y = (x^{*} \odot y^{*})^{*}, ~~ x^{*}=x \rightarrow \bot . \end{aligned}$$

For \(\alpha \in L ~and~ f \in L^{X}\), we denote \((\alpha \rightarrow f),(\alpha \odot f),\alpha _{X} \in L^{X}\) as \((\alpha \rightarrow f)(x)=\alpha \rightarrow f(x),\;(\alpha \odot f)(x)=\alpha \odot f(x),\) \(\;\alpha _{X}(x)=\alpha, respectively\)

$$\begin{aligned} \top _{x}(y)=\left\{ \begin{array}{ll} \top ,&{}\;{\text{ if }}\; y=x,\\ \bot , &{} \;{\text{ otherwise }},\\ \end{array} \right. \top ^{*}_{x}(y)=\left\{ \begin{array}{ll} \bot ,&{}\;{\text{ if }}\; y=x,\\ \top , &{} \;{\text{ otherwise }}.\\ \end{array} \right. \end{aligned}$$

Some basic properties of the binary operation \(\odot\) and residuated operation \(\rightarrow\) are collected in the following lemma, and they can be found in many works, for instance [8,9,10,11, 38].

Lemma 2

For each \(x,y,z,x_{i},y_{i},w \in L\), we have the following properties.

  1. (1)

    \(\top \rightarrow x=x\), \(\bot \odot x= \bot ,\)

  2. (2)

    If \(y\le z\), then \(x \odot y \le x \odot z\), \(x \oplus y \le x \oplus z\), \(x \rightarrow y \le x \rightarrow z\) and \(z \rightarrow x \le y \rightarrow x\),

  3. (3)

    \(x \le y\) iff \(x\rightarrow y=\top\).

  4. (4)

    \((\bigwedge _{i} y_{i})^{*}=\bigvee _{i} y^{*}_{i}\), \((\bigvee _{i} y_{i})^{*}=\bigwedge _{i} y^{*}_{i}\),

  5. (5)

    \(x \rightarrow (\bigwedge _{i} y_{i}) = \bigwedge _{i}(x \rightarrow y_{i})\),

  6. (6)

    \((\bigvee _{i} x_{i}) \rightarrow y = \bigwedge _{i}(x_{i} \rightarrow y)\),

  7. (7)

    \(x \odot (\bigvee _{i} y_{i}) = \bigvee _{i} (x \odot y_{i})\),

  8. (8)

    \((\bigwedge _{i} x_{i}) \oplus y = \bigwedge _{i}(x_{i} \oplus y)\),

  9. (9)

    \(( x \odot y) \rightarrow z = x \rightarrow (y \rightarrow z) = y \rightarrow (x \rightarrow z)\),

  10. (10)

    \(x \odot y = (x \rightarrow y^{*})^{*}\) , \(x \oplus y = x^{*} \rightarrow y\) and \(x\rightarrow y=y^{*}\rightarrow x^{*}\),

  11. (11)

    \((x \rightarrow y) \odot ( z \rightarrow w) \le ( x \odot z) \rightarrow ( y \odot w)\),

  12. (12)

    \(x \rightarrow y \le (x \odot z) \rightarrow (y \odot z)\) and \((x \rightarrow y) \odot ( y \rightarrow z) \le x \rightarrow z\),

  13. (13)

    \((x \rightarrow y) \odot ( z \rightarrow w) \le ( x \oplus z) \rightarrow ( y \oplus w)\),

  14. (14)

    \(x\odot (x\rightarrow y)\le y\) and \(y \le x\rightarrow (x\odot y),\)

  15. (15)

    \((x\vee y)\odot (z \vee w)\le (x \vee z) \vee (y\odot w)\le (x \oplus z) \vee (y\odot w),\)

  16. (16)

    \(\bigvee _{i\in \Gamma } x_{i} \rightarrow \bigvee _{i\in \Gamma } y_{i}\ge \bigwedge _{i\in \Gamma }(x_{i} \rightarrow y_{i}), ~\) \(\bigwedge _{i\in \Gamma } x_{i} \rightarrow \bigwedge _{i\in \Gamma } y_{i}\ge \bigwedge _{i\in \Gamma }(x_{i} \rightarrow y_{i})\),

  17. (17)

    \((x\odot y)\odot (z \oplus w)\le (x \odot z) \oplus (y\odot w),\)

  18. (18)

    \(z\rightarrow x\le (x\rightarrow y)\rightarrow (z \rightarrow y)\) and \(y\rightarrow z\le (x\rightarrow y)\rightarrow (x \rightarrow z)\).

Definition 3

[14, 16, 39] A map \({\mathcal {C}} :L^{X} \rightarrow L^{X}\) is called an L-fuzzy closure operator on X if \({\mathcal {C}}\) satisfies the following conditions:

  1. (C1)

    \({\mathcal {C}}(\bot _{X}) = \bot _{X}\),

  2. (C2)

    \({\mathcal {C}}(f) \ge f\) for all \(f \in L^{X}\),

  3. (C3)

    If \(f \le g\), then \({\mathcal {C}}(f)\le {\mathcal {C}}(g))\) for all \(f,g\in L^{X}\),

  4. (C4)

    \({\mathcal {C}}(f \oplus g) \le {\mathcal {C}}(f) \oplus {\mathcal {C}}(g)\).

The pair \((X,{\mathcal {C}})\) is called L-fuzzy closure space. An L-fuzzy closure space is called

  1. (T)

    topological if \({\mathcal {C}}({\mathcal {C}}(f)) ={\mathcal {C}}(f) ~~ \forall ~ f \in L^{X}\),

  2. (U)

    stratified if \({\mathcal {C}}(\alpha \rightarrow f) \le \alpha \rightarrow {\mathcal {C}}(f)\) for all \(f\in L^{X}\) and \(\alpha \in L\),

  3. (V)

    co-stratified if \({\mathcal {C}}(\alpha \odot f) \le \alpha \odot {\mathcal {C}}(f)\) for all \(f\in L^{X}\) and \(\alpha \in L\),

  4. (W)

    strong if it is both stratified and co-stratified, i.e, \({\mathcal {C}}(\alpha \odot f)=\alpha \odot {\mathcal {C}}(f)\) for all \(f\in L^{X}\) and \(\alpha \in L\),

  5. (X)

    separated if \({\mathcal {C}}(\top ^{*}_{x})=\top ^{*}_{x}\) for all \(x\in X\),

  6. (Y)

    generalized if \({\mathcal {C}}(f)(x)\ge \bigvee _{x\in X}f(x)\),

  7. (Z)

    Alexandrov if \({\mathcal {C}}(\bigvee _{i\in \Gamma }f_{i})=\bigvee _{i\in \Gamma }{\mathcal {C}}(f_{i})\).

Definition 4

Let \((X, {\mathcal {C}}_{X})\) and \((Y, {\mathcal {C}}_{Y})\) be L-fuzzy closure spaces and \(~\varphi:(X, {\mathcal {C}}_{X})\rightarrow (Y,{\mathcal {C}}_{Y})\) be a mapping. Then, \(D_{C}(\varphi )\) defined by

$$\begin{aligned} D_{C}(\varphi )=\bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big ({\mathcal {C}}_{X}(\varphi ^{\leftarrow }(f))(x)\rightarrow \varphi ^{\leftarrow }({\mathcal {C}}_{Y}(f))(x)\Big ) \end{aligned}$$

is called the degree of LF-closure for \(\varphi\). If \(D_{C}(\varphi )=\top ,\) then \(~~ {\mathcal {C}}_{X}( \varphi ^{\leftarrow }(f)) \le \phi ^{\leftarrow }({\mathcal {C}}_{Y}(f))\) for each \(f\in L^{Y},\) which is exactly the definition of LF-closure mappings between L-fuzzy closure spaces.

Remark 5

An L-fuzzy closure space \((X,{\mathcal {C}})\) is stratified if and only if \({\mathcal {C}}(\alpha \odot f)\ge \alpha \odot {\mathcal {C}}(f)\).

Definition 6

[16, 17, 39] A mapping \({\mathcal {F}} :L^{X} \rightarrow L\) is called L-fuzzy co-topology on X if it satisfies the following conditions:

  1. (T1)

    \(~ {\mathcal {F}}(\bot _{X})= {\mathcal {F}}(\top _{X})= \top\),

  2. (T2)

    \(~ {\mathcal {F}}(f \oplus g) \ge {\mathcal {F}}(f) \odot {\mathcal {F}}(g) ~~ forall ~~ f, g \in L^{X}\),

  3. (T3)

    \(~ {\mathcal {F}}(\bigwedge _{i} f_{i} ) \ge \bigwedge _{i} {\mathcal {F}}(f_{i}) ~~~ forall ~~ \{f_{i}\}_{i \in \Gamma } \subseteq L^{X}\).

The pair \((X,{\mathcal {F}})\) is called L-fuzzy co-topological space. An L-fuzzy co-topological space is said to be

  1. (A)

    stratified if \({\mathcal {F}}(\alpha \odot f)\ge {\mathcal {F}}(f)\),

  2. (B)

    co-stratified if \({\mathcal {F}}(\alpha \rightarrow f)\ge {\mathcal {F}}(f)\),

  3. (C)

    strong if it is both stratified and co-stratified,

  4. (D)

    separated if \({\mathcal {F}}(\top _{x})=\top\) for all \(x\in X\),

  5. (E)

    Alexandrov if \(~ {\mathcal {F}}(\bigvee _{i} f_{i}) \ge \bigwedge _{i} {\mathcal {F}}(f_{i}) ~~ forall ~~ \{f_{i}\}_{i \in \Gamma } \subseteq L^{X}\).

Definition 7

Let \((X,{\mathcal {F}}_{X})\) and \((Y,{\mathcal {F}}_{Y})\) be L-fuzzy co-topological spaces and \(~\varphi :(X, {\mathcal {F}}_{X})\rightarrow (Y,{\mathcal {F}}_{Y})\) be a mapping. Then, \(D_{{\mathcal {F}}}(\phi )\) defined by

$$\begin{aligned} D_{{\mathcal {F}}}(\varphi )=\bigwedge _{f\in L^{Y}}\Big ({\mathcal {F}}_{Y}(f)\rightarrow {\mathcal {F}}_{X}(\varphi ^{\leftarrow }(f))\Big ) \end{aligned}$$

is called the degree of LF-continuous for \(\varphi\). If \(D_{{\mathcal {F}}}(\varphi )=\top ,\) then \(~ {\mathcal {F}}_{Y}( f) \le {\mathcal {F}}_{Y}(\varphi ^{\leftarrow }(f))\) for each \(f\in L^{Y},\) which is exactly the definition of LF-continuous mappings between L-fuzzy co-topological spaces.

Definition 8

[8, 36] Let X be a set. A map \(R :X\times X \rightarrow L\) is called an L-partial order if it satisfies the following conditions

  1. (E1)

    reflexive if \(R(x,x)=\top\) for all \(x\in X\),

  2. (E2)

    transitive if \(R(x,y)\odot R(y,z)\le R(x,z)\) for all \(x,y,z\in X\),

  3. (E3)

    antisymmetric if \(R(x,y)=R(y,x)=\top\), then \(x=y\).

The relationships between L-fuzzy pre-proximities and topological structures

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.

  1. (P1)

    \(\delta (\top _{X},\bot _{X}) = \delta (\bot _{X}, \top _{X}) = \bot\),

  2. (P2)

    \(~ \delta (f, g) \ge \bigvee _{x \in X} (f \odot g)(x)\),

  3. (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

  4. (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

  5. (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

  6. (P)

    \(~ \delta ^{s}= \delta\) where \(~ \delta ^{s} (f, g) = \delta ( g,f)\). An L-fuzzy pre-proximity is called

  7. (St)

    stratified if \(~ \delta (\alpha \odot f, \alpha \rightarrow g)\le \delta ( f,g)\) and \(~ \delta (\alpha \rightarrow f, \alpha \odot g)\le \delta ( f,g)\),

  8. (SE)

    separated if \(~ \delta (\top _{x},\top ^{*}_{x})=\delta (\top ^{*}_{x},\top _{x})=\bot\) for each \(x\in X\),

  9. (AL)

    Alexandrov if \(~ \delta (\bigvee _{i\in \Gamma } f_{i},g) \le \bigvee _{i\in \Gamma } \delta (f_{i},g),~~ \delta (f,\bigvee _{i\in \Gamma } g_{i}) \le \bigvee _{i\in \Gamma } \delta (f,g_{i})\),

  10. (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 \(~ {\mathcal {C}}_{\delta }:L^{X} \rightarrow L^{X}\) asĀ 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. (1)

    \(( X, {\mathcal {C}}_{\delta })\) is an L-fuzzy closure space,

  2. (2)

    If \(\delta\) is stratified, then \({\mathcal {C}}_{\delta }\) is stratified,

  3. (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 LemmaĀ 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 LemmaĀ 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 TheoremĀ 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 TheoremĀ 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 TheoremĀ 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. (1)

    \(\delta _{{\mathcal {C}}}\) is an L -fuzzy pre-proximity,

  2. (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. (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. (4)

    If \({\mathcal {C}}\) is topological, then \(\delta _{{\mathcal {C}}}\) is an L-fuzzy quasi-proximity on X,

  5. (5)

    \({\mathcal {C}}\le {\mathcal {C}}_{\delta _{{\mathcal {C}}}}\), the equality holds if \({\mathcal {C}}\) is topological,

  6. (6)

    If \({\mathcal {C}}\) is separated, then \(\delta _{{\mathcal {C}}}\) is separated,

  7. (7)

    \(\delta _{{\mathcal {C}}_{\delta }} \le \delta\),

  8. (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 LemmaĀ 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. (1)

    \(\delta ^{s}_{{\mathcal {C}}}\) is an L -fuzzy pre-proximity,

  2. (2)

    If \(~{\mathcal {C}}\) is stratified, then \(\delta ^{s}\) is a stratified,

  3. (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. (4)

    If \(~{\mathcal {C}}\) is topological, then \(\delta ^{s}_{{\mathcal {C}}}\) is a L-fuzzy quasi-proximity on X,

  5. (5)

    \({\mathcal {C}}\le {\mathcal {C}}_{\delta ^{s}_{{\mathcal {C}}}}\), the equality holds if \({\mathcal {C}}\) is topological,

  6. (6)

    If \(~{\mathcal {C}}\) is separated, then \(\delta ^{s}_{{\mathcal {C}}}\) is separated,

  7. (7)

    \(\delta ^{s}_{{\mathcal {C}}_{\delta }} \le \delta ^{s}\),

  8. (8)

    If \({\mathcal {C}}\) is generalized (resp. Alexandrov), then \(\delta ^{s}_{{\mathcal {C}}}\) is generalized (resp. Alexandrov).

The relationships between L-fuzzy pre-proximities and L-fuzzy co-topologies

Theorem 16

LetĀ  \(\delta\) be anĀ Alexandrov L-fuzzy pre-proximity on X. Define a mapping \(~ {\mathcal {F}}_{\delta } : L^{X} \rightarrow L ~\) by \(~~ {\mathcal {F}}_{\delta }(f) = \delta ^{*}(f^{*},f)\). Then,

  1. (1)

    \({\mathcal {F}}_{\delta }\) is an L-fuzzy co-topology on X,

  2. (2)

    If \(\delta\) is stratified, then \({\mathcal {F}}_{\delta }\) is strong,

  3. (3)

    If \(\delta\) is separated, then \({\mathcal {F}}_{\delta }\) is separated.

Proof

(1) (T1) \({\mathcal {F}}_{\delta }(\bot _{X})=\delta ^{*}(\bot ^{*}_{X},\bot _{X}) =\top ,~~ {\mathcal {F}}_{\delta }(\top _{X}) =\delta ^{*}(\top ^{*}_{X},\top _{X}) =\top .\)

(T2) \({\mathcal {F}}_{\delta }(f\oplus g)= \delta ^{*}(f^{*}\odot g^{*}, f\oplus g) \ge \delta ^{*}(f^{*},f)\odot \delta ^{*}( g^{*},g)={\mathcal {F}}_{\delta }(f)\odot {\mathcal {F}}_{\delta }(g).\)

(T3) \({\mathcal {F}}_{\delta }(\bigwedge _{i\in \Gamma }f_{i})= \delta ^{*}(\bigvee _{i\in \Gamma }f^{*}_{i},\bigwedge _{i\in \Gamma }f_{i}) \ge \bigwedge _{i\in \Gamma } \delta ^{*}(f^{*}_{i},f_{i})=\bigwedge _{i\in \Gamma }{\mathcal {F}}_{\delta }(f_{i}).\)

(2) \({\mathcal {F}}_{\delta }(\alpha \odot f)= \delta ^{*}(\alpha \rightarrow f^{*},\alpha \odot f) \ge \delta ^{*}(f^{*},f)={\mathcal {F}}_{\delta }(f),\)

$$\begin{aligned} {\mathcal {F}}_{\delta }(\alpha \rightarrow f)= \delta ^{*}(\alpha \odot f^{*}, \alpha \rightarrow f) \ge \delta ^{*}(f^{*},f)={\mathcal {F}}_{\delta }(f). \end{aligned}$$

(3) It is easy.

Theorem 17

Let \((X, {\mathcal {C}})\) be an L -fuzzy closure space. Define the mapping \(~ {\mathcal {F}}_{{\mathcal {C}}_{\delta }} : L^{X} \rightarrow L ~\) by

$$\begin{aligned} {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f) = \bigwedge _{x\in X}({\mathcal {C}}_{\delta }(f)(x)\rightarrow f(x)). \end{aligned}$$

Then,

  1. (1)

    \({\mathcal {F}}_{{\mathcal {C}}_{\delta }}\) is an L-fuzzy co-topology on X with \({\mathcal {F}}_{{\mathcal {C}}_{\delta }} \ge {\mathcal {F}}_{\delta }\),

  2. (2)

    If \({\mathcal {C}}\) is Alexandrov (resp. strong, separated), then \({\mathcal {F}}_{\delta _{{\mathcal {C}}}}\) is Alexandrov (resp. strong, separated).

Proof

(1) (T1) \({\mathcal {F}}_{{\mathcal {C}}_{\delta }}(\top _{X})= \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(\top _{X})(x) \rightarrow \top _{X}(x)) = \top ,\)

$$\begin{aligned} {\mathcal {F}}_{{\mathcal {C}}_{\delta }} (\bot _{X}) = \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(\bot _{X})(x) \rightarrow \bot _{X}(x)) =\top . \end{aligned}$$

(T2)

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{{\mathcal {C}}_{\delta }} (f \oplus g) &{}=\bigwedge _{x \in X}({\mathcal {C}}_{\delta }(f \oplus g)(x) \rightarrow (f \oplus g)(x)) \\ &{} \ge \bigwedge _{x \in X}(({\mathcal {C}}_{\delta }(f)(x) \oplus {\mathcal {C}}_{\delta }(g)(x)) \rightarrow (f(x) \oplus g(x))) ~~~{\text{(by Lemma 2(13)) }} \\ &{} \ge \bigwedge _{x \in X}( {\mathcal {C}}_{\delta }(f)(x)\rightarrow f(x) )\odot \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(g)(x) \rightarrow g(x)) \\ &{} = {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f) \odot {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(g). \end{array} \end{aligned}$$

(T3) By LemmaĀ 2(16), we have

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(\bigwedge _{i\in \Gamma }f_{i}) &{}= \bigwedge _{x\in X}({\mathcal {C}}_{\delta }(\bigwedge _{i\in \Gamma }f_{i})(x)\rightarrow (\bigwedge _{i\in \Gamma }f_{i})(x))\\ &{}\ge \bigwedge _{x\in X}(\bigwedge _{i\in \Gamma }{\mathcal {C}}_{\delta }(f_{i})(x)\rightarrow \bigwedge _{i\in \Gamma }f_{i}(x))\\ &{}\ge \bigwedge _{i\in \Gamma }(\bigwedge _{x\in X}({\mathcal {C}}_{\delta }(f_{i})(x)\rightarrow f_{i}(x)))=\bigwedge _{i\in \Gamma }{\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f_{i}). \end{array} \end{aligned}$$

Hence, \({\mathcal {F}}_{{\mathcal {C}}_{\delta }}\) is an L-fuzzy co-topology on X.

Moreover,

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f)&{}=\bigwedge _{x\in X}({\mathcal {C}}_{\delta }(f)(x)\rightarrow f(x) )\\ &{}=\big(\bigvee _{x\in X}{\mathcal {C}}_{\delta }(f)(x)\odot f^{*}(x)\big)^{*}\\ &{}\ge \big(\bigvee _{x\in X}f(x)\odot f^{*}(x)\big)^{*}\ge \delta ^{*}(f^{*},f)={\mathcal {F}}_{\delta }(f). \end{array} \end{aligned}$$

(2)

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(\bigvee _{i\in \Gamma }f_{i}) &{}= \bigwedge _{x\in X}({\mathcal {C}}_{\delta }(\bigvee _{i\in \Gamma } f_{i})(x)\rightarrow \bigvee _{i\in \Gamma } f_{i}(x))\\ &{}= \bigwedge _{x\in X}(\bigvee _{i\in \Gamma }{\mathcal {C}}_{\delta }(f_{i})(x)\rightarrow \bigvee _{i\in \Gamma } f_{i}(x)))\\ &{}\ge \bigwedge _{i\in \Gamma }(\bigwedge _{x\in X}({\mathcal {C}}_{\delta }(f_{i})(x)\rightarrow f_{i}(x)))=\bigwedge _{i\in \Gamma }{\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f_{i}). \end{array} \end{aligned}$$

Hence, \({\mathcal {F}}_{{\mathcal {C}}_{\delta }}\) is anĀ Alexandrov L-fuzzy co-topology on X. By LemmaĀ 2(14)(18), we have

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{{\mathcal {C}}_{\delta }} (\alpha \odot f) &{}= \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(\alpha \odot f)(x) \rightarrow (\alpha \odot f(x)))\\ &{} \ge \bigwedge _{x \in X}((\alpha \odot {\mathcal {C}}_{\delta }(f)(x)) \rightarrow (\alpha \odot f(x))) \\ &{}\ge \bigwedge _{x \in X}(f(x) \rightarrow {\mathcal {C}}_{\delta }(f)(x)) = {\mathcal {F}}_{{\mathcal {C}}_{\delta }}(f), \\ &{}~ \\ {\mathcal {F}}_{{\mathcal {C}}_{\delta }} (\alpha \rightarrow f) &{}= \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(\alpha \rightarrow f)(x) \rightarrow (\alpha \rightarrow f(x)))\\ &{}\ge \bigwedge _{x \in X}((\alpha \rightarrow {\mathcal {C}}_{\delta }(f)(x)) \rightarrow (\alpha \rightarrow f(x))) \\ &{} \ge \bigwedge _{x \in X}({\mathcal {C}}_{\delta }(f)(x) \rightarrow f(x)) = {\mathcal {F}}_{{\mathcal {C}}_{\delta }} (f). \end{array} \end{aligned}$$

Other cases are easily proved.

Theorem 18

Let \((X, \delta )\) be an L-fuzzy pre-proximity space. Then, the mapping \(~ {\mathcal {F}}^{(1)}_{\delta } : L^{X} \rightarrow L ~\) defined by \(~ {\mathcal {F}}^{(1)}_{\delta }(f) = \bigwedge _{x\in X}(\delta (f,\top _{x})\rightarrow f(x))\) is an L-fuzzy co-topology on X. Moreover, if \(\delta\) is Alexandrov and \(~ \delta (\alpha \odot f, g)\ge \alpha \odot \delta (f,g)\), then \({\mathcal {F}}_{\delta }^{(1)}(f^{*})\ge {\mathcal {F}}_{\delta }(f)\).

Proof

(1) (T1) \({\mathcal {F}}^{(1)}_{\delta }(\bot _{X})=\bigwedge _{x\in X}(\delta (\bot _{X}, \top _{x})\rightarrow \bot _{X}(x))=\top ,\)

$$\begin{aligned} {\mathcal {F}}^{(1)}_{\delta }(\top _{x})=\bigwedge _{x\in X}(\delta (\top _{X}, \top _{x})\rightarrow \top _{X}(x))=\top . \end{aligned}$$

(T2)

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}^{(1)}_{\delta }(f\oplus g)&{}= \bigwedge _{x\in X}(\delta (f\oplus g,\top _{x})\rightarrow (f\oplus g)(x) )\\ &{}\ge \bigwedge _{x\in X}((\delta (f,\top _{x})\oplus \delta(g,\top _{x}))\rightarrow (f(x)\oplus g(x)))\\ &{}\ge \bigwedge _{x\in X}(\delta (f,\top _{x})\rightarrow f(x)) \odot \bigwedge _{x\in X}(\delta (g,\top _{x})\rightarrow g(x))\\ &{}\ge {\mathcal {F}}^{(1)}_{\delta }(f)\odot {\mathcal {F}}^{(1)}_{\delta }(g). \end{array} \end{aligned}$$

(T3)

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}^{(1)}_{\delta }(\bigwedge _{i\in \Gamma }f_{i})&{}= \bigwedge _{x\in X}(\delta (\bigwedge _{i\in \Gamma }f_{i}, \top _{x})\rightarrow \bigwedge _{i\in \Gamma }f_{i}(x))\\ &{}=\bigwedge _{i\in \Gamma } \bigwedge _{x\in X}(\delta (\bigwedge _{i\in \Gamma }f_{i}, \top _{x})\rightarrow f_{i}(x))\\ &{}\ge \bigwedge _{i\in \Gamma }\bigwedge _{x\in X} (\delta (f_{i}, \top _{x})\rightarrow f_{i}(x))=\bigwedge _{i\in \Gamma }{\mathcal {F}}^{(1)}_{\delta }(f_{i}). \end{array} \end{aligned}$$

Moreover, if \(\delta\) is Alexandrov, then

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}^{(1)}_{\delta }(\bigvee _{i\in \Gamma }f_{i}) &{}= \bigwedge _{x\in X}(\delta (\bigvee _{i\in \Gamma }f_{i}, \top _{x})\rightarrow \bigvee _{i\in \Gamma }f_{i}(x))\\ &{}= \bigwedge _{x\in X}(\bigvee _{i\in \Gamma }\delta (f_{i}, \top _{x})\rightarrow \bigvee _{i\in \Gamma }f_{i}(x))\\ &{}\ge \bigwedge _{i\in \Gamma }\bigwedge _{x\in X} (\delta (f_{i}, \top _{x})\rightarrow f_{i}(x)) =\bigwedge _{i\in \Gamma }{\mathcal {F}}^{(1)}_{\delta }(f_{i}). \end{array} \end{aligned}$$

Hence, \({\mathcal {F}}^{(1)}_{\delta }\) is Alexandrov L-fuzzy co-topology on X.

If \(~ \delta (\alpha \odot f, g)\ge \alpha \odot \delta ( f,g)\), then

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{\delta }(f)=\delta ^{*}(f^{*},f)&{}=\delta ^{*}(f^{*},\bigvee _{x\in X}f(x)\odot \top _{x})\\ &{}\le \bigwedge _{x\in X}(f(x)\rightarrow \delta ^{*}(f^{*},\top _{x}))\\ &{}=\bigwedge _{x\in X}(\delta (f^{*},\top _{x})\rightarrow f^{*}(x)) ={\mathcal {F}}^{(1)}_{\delta }(f^{*}). \end{array} \end{aligned}$$

Example 19

Let \(X=\{h_{i} \mid i=\{1,...,3\}\}\) with \(h_{i}\)Ā =Ā house and \(Y=\{e,b,w,c,i\}\) with eĀ =Ā expensive, bĀ =Ā beautiful, w=wooden, cĀ =Ā creative, iĀ =Ā in the green surroundings. Let \(([0,1],\odot ,\rightarrow ,^{*}, 0,1)\) be a complete residuated lattice as

$$\begin{aligned} \begin{array}{clcr} x\odot y=\max \{0, x+y-1\},~ x\rightarrow y=\min \{1-x+y,1\}, ~ x^{*}=1-x. \end{array} \end{aligned}$$

Let \(I \in [0,1]^{X\times Y}\) be a fuzzy information as follows:

$$\begin{aligned} \begin{array}{ccccccc} I &{} e&{} b &{} w&{} c &{} i \\ h_{1} &{} 0.7 &{} 0.6 &{} 0.5 &{} 0.9 &{} 0.2 \\ h_{2} &{} 0.6 &{} 0.8 &{} 0.4 &{} 0.3 &{} 0.5 \\ h_{3} &{} 0.4 &{} 0.9 &{} 0.8 &{} 0.6 &{} 0.6 \end{array} \end{aligned}$$

Define [0,Ā 1]-fuzzy pre-orders \(R_{X}^{Y}, ~ R_{X}^{\{b,w \}}\in [0,1]^{X\times X}\) by

$$\begin{aligned} \begin{array}{clcr} R_{X}^{Y}(h_{i},h_{j}) &{}=\bigwedge _{y\in Y}(I(h_{i},y)\rightarrow I(h_{j},y)),\\ R_{X}^{\{b,w \}}(h_{i},h_{j}) &{}=\bigwedge _{y\in \{b,w \}}(R(h_{i},y)\rightarrow R(h_{j},y)), \end{array}\\ R_{X}^{Y}=\left( \begin{array}{ccc} 1 &{} 0.4 &{} 0.7 \\ 0.7 &{} 1 &{} 0.8 \\ 0.6 &{} 0.6 &{} 1 \\ \end{array} \right) , R_{X}^{\{b,w \}}=\left( \begin{array}{ccc} 1 &{} 0.9 &{} 1 \\ 0.8 &{} 1 &{} 1 \\ 0.7 &{} 0.6 &{} 1 \end{array} \right) . \end{aligned}$$

(1) For each \(R\in \{R_{X}^{Y}, R_{X}^{\{b,w \}}\}\), by ExampleĀ 13, we obtain a stratified, Alexandrov and generalized [0,Ā 1]-fuzzy pre-proximity \(\delta _{R}:[0,1]^{X}\times [0,1]^{X}\rightarrow [0,1]\) as

$$\begin{aligned} \begin{array}{clcr} \delta _{R} (f,g) =\bigvee _{h_{i},h_{j}\in X} R_{X}^{Y}(h_{i},h_{j})\odot f(h_{i})\odot g(h_{j}). \end{array} \end{aligned}$$

By TheoremĀ 12, we obtain a stratified [0,Ā 1]-fuzzy closure operator \(~ C_{\delta _{R}}:[0,1]^{X}\rightarrow [0,1]^{X}\) as

$$\begin{aligned} \begin{array}{clcr} C_{\delta _{R}} (f)(h_{i}) &{}= \bigwedge _{g\in L^{X}}((S(f,g^{*}) \odot g(h_{i})) \rightarrow \delta _{R}(g,g^{*}))\\ &{} = \bigwedge _{g\in L^{X}}\Big ( (S(f,g^{*})\odot g(h_{i}))\\ &{} \rightarrow \big ( \bigvee _{h_{j},h_{k}\in X} R_{X}^{Y}(h_{j},h_{k}) \odot g(h_{j}) \odot g^{*}(h_{k})\big )\Big ). \end{array} \end{aligned}$$

By TheoremĀ 16, we obtain a strong [0,Ā 1]-fuzzy co-topology \({\mathcal {F}}_{\delta _{R}}:[0,1]^{X}\rightarrow [0,1]\) as

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{\delta _{R}}(f) =\delta _{R}^{*}(f^{*},f) &{}=(\bigvee _{h_{i},h_{j}\in X} R_{X}^{Y}(h_{i},h_{j})\odot f^{*}(h_{i})\odot f(h_{j}))^{*}\\ &{}=\bigwedge _{h_{i},h_{j}\in X}(R_{X}^{Y}(h_{i},h_{j})\odot f(h_{j})\rightarrow f(h_{i})). \end{array} \end{aligned}$$

Since

$$\begin{aligned} \begin{array}{clcr} \delta _{R} (f,\top _{h_{j}}) =\bigvee _{h_{i},h_{j}\in X} R_{X}^{Y}(h_{i},h_{j}) \odot f(h_{i})\odot \top _{h_{j}}(h_{j}) = \bigvee _{h_{i}\in X} R(h_{i},h_{j})\odot f(h_{i}), \end{array} \end{aligned}$$

by TheoremĀ 18, we obtain [0,Ā 1]-fuzzy co-topology \({\mathcal {F}}^{(1)}_{\delta _{R}}:[0,1]^{X}\rightarrow [0,1]\) as

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}^{(1)}_{\delta _{R}}(f) &{} =\bigwedge _{h_{j}\in X}(\delta _{R} (f,\top _{h_{j}}) \rightarrow f(h_{j})) \\ &{}=\bigwedge _{h_{j}\in X}\big ((\bigvee _{h_{i}\in X} R(h_{i},h_{j}) \odot f(h_{i})) \rightarrow f(h_{j})\big )\\ &{}=\bigwedge _{h_{i},h_{j}\in X}((R(h_{i},h_{j})\odot f(h_{i}))\rightarrow f(h_{j})). \end{array} \end{aligned}$$

(2) For each \(R\in \{R_{X}^{Y}, R_{X}^{\{b,w \}}\}\), we obtain a strong, generalized, topological and Alexandrov [0,Ā 1]-fuzzy closure operator \(~C_{R}:[0,1]^{X}\rightarrow [0,1]^{X}\) as

$$\begin{aligned} \begin{array}{clcr} C_{R} (f)(h_{j}) =\bigvee _{h_{i}\in X} R(h_{i},h_{j})\odot f(h_{i}). \end{array} \end{aligned}$$

By TheoremĀ 14, we obtain a generalized, topological and Alexandrov [0,Ā 1]-fuzzy quasi-proximity \(\delta _{C_{R}}\) as

$$\begin{aligned} \begin{array}{clcr} \delta _{C_{R}}(f,g) &{}= \bigvee _{h_{i}\in X} f(h_{i}) \odot C_{R}(g)(h_{i}) \\ &{}= \bigvee _{h_{i}\in X} f(h_{i}) \odot (\bigvee _{h_{j}\in X} R(h_{j},h_{i}) \odot g(h_{j})) \\ &{}= \bigvee _{h_{i},h_{j}\in X} R(h_{j},h_{i}) \odot f(h_{i}) \odot g(h_{j}). \end{array} \end{aligned}$$

By TheoremĀ 16, we obtain [0,Ā 1]-fuzzy co-topologies \({\mathcal {F}}_{\delta _{C_{R}}}\) and \({\mathcal {F}}^{(1)}_{\delta _{C_{R}}}\) as follows:

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}_{\delta _{C_{R}}}(f) =\delta _{C_{R}}^{*}(f^{*},f) &{}=(\bigvee _{h_{i},h_{j}\in X} R(h_{i},h_{j}) \odot f^{*}(h_{i})\odot f(h_{j}))^{*}\\ &{}=\bigwedge _{h_{i},h_{j}\in X}(R_{X}^{Y}(h_{i},h_{j})\odot f(h_{j})\rightarrow f(h_{i})). \end{array} \end{aligned}$$

Also we have

$$\begin{aligned} \begin{array}{clcr} {\mathcal {F}}^{(1)}_{\delta _{C_{R}}}(f) &{}=\bigwedge _{h_{j}\in X}(\delta _{R} (f,\top _{h_{j}})\rightarrow f(h_{j}))\\ &{}=\bigwedge _{h_{j}\in X}((\bigvee _{h_{i}\in X} R(h_{i},h_{j}) \odot f(h_{i})) \rightarrow f(h_{j}))\\ &{}=\bigwedge _{h_{i},h_{j}\in X}((R(h_{i},h_{j})\odot f(h_{i}))\rightarrow f(h_{j})). \end{array} \end{aligned}$$

Galois correspondences

Theorem 20

Let \(\varphi :X\rightarrow Y\) be a mapping. Then

  1. (1)

    \(D_{\delta }(\varphi )\le D_{{\mathcal {C}}_{\delta }}(\varphi )\),

  2. (2)

    \(D_{\delta }(\varphi )= D_{{\mathcal {F}}_{\delta }}(\varphi )\),

  3. (3)

    \(D_{\delta }(\varphi )\le D_{\mathcal {F_{\delta }}^{(1)}}(\varphi )\).

Proof

(1) By LemmaĀ 2(18), we have

$$\begin{aligned} \begin{array}{clcr} D_{{\mathcal {C}}_{\delta }}(\varphi )&{}=\bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big ({\mathcal {C}}_{\delta _{X}}(\varphi ^\leftarrow (f))(x)\rightarrow \varphi ^{\leftarrow }({\mathcal {C}}_{\delta _{Y}}(f))(x)\Big )\\ &{} =\bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big (\bigwedge _{h\in L^X}\{ h(x)\rightarrow \delta _{X}(h,h^{*})~\mid ~\varphi ^\leftarrow (f)\le h^{*}\}\\ &{} \rightarrow \bigwedge _{g\in L^{Y}}\{ g(y)\rightarrow \delta _{Y}(g,g^{*})~\mid ~f\le g^{*}\}\Big )\\ &{}\ge \bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big (\bigwedge _{h\in L^X}\{ h(x)\rightarrow \delta _{X}(h,h^{*})~\mid ~\varphi ^\leftarrow (f)\le h^{*}\}\\ &{} \rightarrow \bigwedge _{g\in L^{Y}}\{ g(\varphi (x))\rightarrow \delta _{Y}(g,g^{*})~\mid ~\varphi ^{\leftarrow }(f)\le \varphi ^{\leftarrow }(g^{*})\}\Big) \\ &{}\ge \bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big ((\varphi ^{\leftarrow }(g) (x)\rightarrow \delta _{X}(\varphi ^{\leftarrow}(g),\varphi ^{\leftarrow }(g^{*})))\rightarrow (\varphi ^{\leftarrow }(g) (x)\rightarrow \delta _{Y}(g,g^{*}))\Big)\\ &{} \ge \bigwedge _{f,g\in L^{Y}}\Big (\delta _{X}(\varphi ^{\leftarrow }(g),\varphi ^{\leftarrow }(g^{*})) \rightarrow \delta _{Y}(g,g^{*})\Big )= D_{\delta }(\varphi ). \end{array} \end{aligned}$$

(2)

$$\begin{aligned} \begin{array}{clcr} D_{{\mathcal {F}}_{\delta }}(\varphi )&{}=\bigwedge _{f\in L^{Y}}\Big ( {\mathcal {F}}_{\delta _{Y}}(f)\rightarrow {\mathcal {F}}_{\delta _{X}}(\varphi ^\leftarrow (f))\Big )\\ &{}=\bigwedge _{f\in L^{Y}}\Big (\delta ^{*}_{Y}(f^{*},f)\rightarrow \delta ^{*}_{X}(\varphi ^\leftarrow (f^{*}),\varphi ^\leftarrow (f))\Big )\\ &{}=\bigwedge _{f\in L^{Y}}\Big (\delta _{X}(\varphi ^\leftarrow (f^{*}),\varphi ^\leftarrow (f))\rightarrow \delta _{Y}(f^{*},f))\Big )= D_{\delta }(\varphi ). \end{array} \end{aligned}$$

(3)

$$\begin{aligned} \begin{array}{clcr} D_{{\mathcal {F}}_{\delta }^{(1)}}(\varphi) &{}=\bigwedge _{f\in L^{Y}}\Big ( {\mathcal {F}}_{\delta _{Y}}^{(1)}(f)\rightarrow {\mathcal {F}}_{\delta _{X}}^{(1)}(\varphi ^\leftarrow (f))\Big)\\ &{}=\bigwedge _{f\in L^{Y}}\Big (\bigwedge _{y\in Y}(\delta _{Y}(f,\top _{y})\rightarrow f(y)) \\&{}\rightarrow \bigwedge _{x\in X}(\delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }(\top _{\varphi (x)})\rightarrow \varphi ^{\leftarrow }(f)(x))\Big)\\ &{}\ge \bigwedge _{f\in L^{Y}}\Big (\bigwedge _{x\in X}(\delta _{Y}(f,\top _{\varphi (x)})\rightarrow f(\varphi (x)))\\&{} \rightarrow \bigwedge _{x\in X}(\delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }(\top _{\varphi (x)})\rightarrow f(\varphi (x))\Big)\\ &{}\ge \bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big (\delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }(\top _{\varphi (x)})\rightarrow \delta _{Y}(f,\top _{\varphi (x)})\Big)\\ &{}=\bigwedge _{f\in L^{Y}}\Big (\delta _{X}(\varphi ^{\leftarrow }(f),\varphi ^{\leftarrow }(\top _{\varphi (x)})\rightarrow \delta _{Y}(f,\top _{\varphi (x)})\Big)=D_{\delta }(\varphi). \end{array} \end{aligned}$$

Theorem 21

Let \(\varphi :X\rightarrow Y\) be a mapping. Then,

  1. (1)

    \(D_{{\mathcal {C}}}(\varphi )\le D_{\delta _{{\mathcal {C}}}}(\varphi )\),

  2. (2)

    \(D_{{\mathcal {C}}_{\delta }}(\varphi )\le D_{{\mathcal {F}}_{\delta }}(\varphi ).\)

Proof

(1) From LemmaĀ 2(18), we have

$$\begin{aligned} \begin{array}{clcr} D_{\delta _{{\mathcal {C}}}}(\varphi )&{}= \bigwedge _{f,g\in L^{Y}}\Big (\delta _{{\mathcal {C}}_{X}}(\varphi ^{\leftarrow }(f),\phi ^{\leftarrow }(g))\rightarrow \delta _{{\mathcal {C}}_{Y}}(f,g)\Big )\\ &{}=\bigwedge _{f,g\in L^{Y}}\Big (\bigvee _{x \in X} \varphi ^{\leftarrow }(f)(x)\odot {\mathcal {C}}_{X}(\varphi ^{\leftarrow }(g))(x)\rightarrow \bigvee _{y \in Y} f(y)\odot {\mathcal {C}}_{Y}(g)(y)\Big ) \\ &{}\ge \bigwedge _{f,g\in L^{Y}}\Big (\bigvee _{x \in X} f(\varphi (x))\odot {\mathcal {C}}_{X}(\varphi ^{\leftarrow }(g))(x)\rightarrow \bigvee _{x \in X} f(\varphi (x))\odot {\mathcal {C}}_{Y}(g)(\varphi (x))\Big )\\ . &{}\ge \bigwedge _{g\in L^{Y}}\bigwedge _{x \in X}\Big ( {\mathcal {C}}_{X}(\varphi ^{\leftarrow }(g))(x)\rightarrow \varphi ^{\leftarrow }({\mathcal {C}}_{Y}(g))(x) \Big )=D_{{\mathcal {C}}}(\varphi ). \\ \end{array} \end{aligned}$$

(2) From LemmaĀ 2(18), we have

$$\begin{aligned} \begin{array}{clcr} D_{{\mathcal {F}}_{{\mathcal {C}}_{\delta}}}(\varphi) &=\bigwedge _{f\in L^{Y}}\Big ( {\mathcal {F}}_{{\mathcal {C}}_{\delta _{Y}}}(f) \rightarrow{\mathcal {F}}_{{\mathcal {C}}_{\delta _{X}}}(\varphi ^{\leftarrow}(f)) \Big) \\ &=\bigwedge _{f\in L^{Y}}\Big(\bigwedge _{y\in Y}( {\mathcal {C}}_{\delta _{Y}}(f)(y)\rightarrow f(y)) \\&{} \rightarrow \bigwedge _{x\in X} ({\mathcal {C}}_{\delta _{X}}(\varphi ^{\leftarrow }(f))(x)\rightarrow \varphi ^{\leftarrow }(f)(x))\Big)\\ &\ge \bigwedge_{f\in L^{Y}}\Big(\bigwedge _{x\in X}( {\mathcal {C}}_{\delta _{Y}}(f)(\varphi (x))\rightarrow f(\varphi (x))) \\&{} \rightarrow \bigwedge _{x\in X} ({\mathcal {C}}_{\delta _{X}}(\varphi ^{\leftarrow }(f))(x)\rightarrow \varphi ^{\leftarrow }(f)(x))\Big)\\ &= \bigwedge _{f\in L^{Y}}\Big (\bigwedge _{x\in X}( \varphi ^{\leftarrow }({\mathcal {C}}_{\delta _{Y}}(f))(x)\rightarrow \varphi ^{\leftarrow }(f)(x)) \\&{} \rightarrow \bigwedge _{x\in X} ({\mathcal {C}}_{\delta _{X}}(\varphi^{\leftarrow }(f))(x)\rightarrow \varphi^{\leftarrow }(f)(x)) \Big)\\ &\ge \bigwedge _{f\in L^{Y}}\bigwedge _{x\in X}\Big({\mathcal {C}}_{\delta _{X}}(\varphi ^{\leftarrow }(f))(x) \rightarrow \varphi ^{\leftarrow }({\mathcal {C}}_{\delta _{Y}}(f))(x)\Big)=D_{{\mathcal {C}}_{\delta }}(\varphi). \end{array} \end{aligned}$$

Definition 22

[40] Suppose that \(F:{\mathcal {D}} \rightarrow {\mathcal {C}},\) \(G:{\mathcal {C}}\rightarrow {\mathcal {D}}\) are concrete functors. The pair (F,Ā G) is called a Galois correspondence between \({\mathcal {C}}\) and \({\mathcal {D}}\) if for each \(Y\in {\mathcal {C}},\) \(id_{Y}:F\circ G(Y)\rightarrow Y\) is a \({\mathcal {C}}\)-morphism, and for each \(X\in {\mathcal {D}}\), \(id_{X}:X\rightarrow G \circ F(X)\) is a \({\mathcal {D}}\)-morphism.

If (F,Ā G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G, or equivalently that G is a right adjoint of F.

The category of separated L-fuzzy pre-proximity spaces with LF-proximity mappings as morphisms is denoted by SPROX.

The category of separated LF-fuzzy closure spaces with LF-closure mappings as morphisms is denoted by SFC.

From TheoremsĀ 12 andĀ 20, we obtain a concrete functor \(\Theta : \mathbf{SPROX} \rightarrow \mathbf{SFC}\) defined as

$$\begin{aligned} \Theta (X,\delta )=(X,{\mathcal {C}}_{\delta }), \Theta (\phi )=\phi . \end{aligned}$$

From TheoremsĀ 14 andĀ 21, we obtain a concrete functor \(\Gamma : \mathbf{SFC} \rightarrow \mathbf{SPROX}\) defined as

$$\begin{aligned} \Gamma (X,{\mathcal {C}})=(X,\delta _{{\mathcal {C}}}), \Gamma (\phi )=\phi . \end{aligned}$$

Theorem 23

\(\Gamma : \mathbf{SFC} \rightarrow \mathbf{SPROX}\) is a left adjoint of \(\Theta : \mathbf{SPROX} \rightarrow \mathbf{SFC}\), i.e, \((\Theta ,\Gamma )\) is a Galois correspondence.

Proof

By TheoremĀ 14(5), if \({\mathcal {C}}_{X}\) is a separated L-fuzzy closure operator on a set X, then \(\Theta (\Gamma (C_{X}))={\mathcal {C}}_{\delta _{{\mathcal {C}}_{X}}}\ge {\mathcal {C}}_{X}\). Hence, the identity map \(id_{X}: (X,{\mathcal {C}}_{X})\rightarrow (X,{\mathcal {C}}_{\delta _{{\mathcal {C}}_{X}}})=(X,\Theta (\Gamma (C_{X})))\) is an LF-closure map. Moreover, if \(\delta _{Y}\) is a separated L-fuzzy pre-proximity on a set Y, by TheoremĀ 14(7),Ā we haveĀ  \(\Gamma (\Theta (\delta _{Y}))=\delta _{{\mathcal {C}}_{\delta _{Y}}}\le \delta _{Y}\). Hence, the identity map \(id_{Y}: (Y,\Gamma (\Theta (\delta _{Y})))\rightarrow (Y,\delta _{Y})\) is an LF-proximity map. Therefore, \((\Theta ,\Gamma )\) is a Galois correspondence.

Conclusion.

In this paper, L-fuzzy pre-proximities and L-fuzzy closure operators in complete residuated lattice are investigated. From a given L-fuzzy pre-proximity \(\delta\), we can obtain an L-fuzzy closure operator \({\mathcal {C}}_{\delta }\) (see TheoremĀ 12). Conversely, for given L-fuzzy closure space \({\mathcal {C}}\), we obtain L-fuzzy pre-proximity \(\delta _{{\mathcal {C}}}\) (see TheoremĀ 14) and L-fuzzy co-topologies \({\mathcal {F}}_{\delta }\) and \({\mathcal {F}}_{{\mathcal {C}}_{\delta }}\) (TheoremsĀ 16,Ā 17,Ā 18). It is also shown that there is a Galois correspondence between the category of (separated) L-fuzzy closureĀ spaces and that of (separated) L-fuzzy pre-proximity spaces (TheoremĀ 21). We give ExampleĀ 19 as a viewpoint of the topological structure for fuzzy information and fuzzy rough sets in a complete residuated lattice.

In the future, the concepts of L-fuzzy pre-proximity spaces, information systems and decision rules with a view point of applications to multi-attribute decision-making will beĀ investigated in residuated lattices.

Availability of data and materials

Not applicable.

References

  1. Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145ā€“174 (1967)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  2. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338ā€“353 (1965)

    ArticleĀ  Google ScholarĀ 

  3. Bandler, W., Kohout, L.: Special properties, closures and interiors of crisp and fuzzy relations. Fuzzy Sets Syst. 26(3), 317ā€“331 (1988)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  4. Biacino, L., Gerla, G.: Closure systems and L-subalgebras. Inf. Sci. 33, 181ā€“195 (1984)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  5. Biacino, L., Gerla, G.: An extension principle for closure operators. J. Math. Anal. Appl. 198, 1ā€“24 (1996)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  6. Bodenhofer, U., De Cock, M., Kerre, E.E.: Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems. Int. J. Gen. Syst. 32(4), 343ā€“360 (2003)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  7. Ward, M., Dilworth, R.P.: Residuated lattices. Trans. Am. Math. Soc. 45, 335ā€“354 (1939)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  8. BělohlĆ”vek, R.: Fuzzy Relational Systems. Kluwer Academic Publishers, New York (2002)

    BookĀ  Google ScholarĀ 

  9. Hƶhle, U., Rodabaugh, S.E.: Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory. The Handbooks of Fuzzy Sets Series 3. Kluwer Academic Publishers, Boston (1999)

    BookĀ  Google ScholarĀ 

  10. Rodabaugh, S.E., Klement, E.P.: Topological and Algebraic Structures in Fuzzy Sets. The Handbook of Recent Developments in the Mathematics of Fuzzy Sets. Kluwer Academic Publishers, Boston (2003)

    BookĀ  Google ScholarĀ 

  11. Turunen, E.: Mathematics Behind Fuzzy Logic. Springer, Heidelberg (1999)

    MATHĀ  Google ScholarĀ 

  12. Zhang, D.: An enriched category approach to many valued topology. Fuzzy Sets Syst. 158, 349ā€“366 (2007)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  13. BělohlĆ”vek, R.: Fuzzy closure operators. J. Math. Anal. Appl. 262, 473ā€“489 (2001)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  14. BělohlĆ”vek, R.: Fuzzy closure operators II. Soft Comput.Ā 7(1), 53ā€“64 (2002)

    ArticleĀ  Google ScholarĀ 

  15. Fang, J., Yue, Y.: L-fuzzy closure systems. Fuzzy Sets Syst. 161, 1242ā€“1252 (2010)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  16. Shi, F.G., Pang, B.: Categories isomorphic to the category of L-fuzzy closure system spaces. Iran. J. Fuzzy Syst. 10, 127ā€“146 (2013)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  17. Ramadan, A.A.: L-fuzzy interior systems. Comput. Math. Appl. 62, 4301ā€“4307 (2011)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  18. Katsaras, A.K., Petalas, C.G.: A unified theory of fuzzy topologies, fuzzy proximities and fuzzy uniformities. Rev. Roum. Math. Pures Appl. 28, 845ā€“896 (1983)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  19. Katsaras, A.K.: Fuzzy syntopogenous structures compatible with Lowen fuzzy uniformities and Artico-Moresco fuzzy proximities. Fuzzy Sets Syst. 36, 375ā€“393 (1990)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  20. Chang, C.L.: Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182ā€“190 (1968)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  21. Katsaras, A.K.: Fuzzy proximity spaces. J. Math. Anal. Appl. 68, 100ā€“110 (1979)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  22. Hutton, B.: Uniformities in fuzzy topological spaces. J. Math. Anal. Appl. 58, 74ā€“79 (1977)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  23. Bayoumi, F.: On L-proximities of the internal type. Fuzzy Sets Syst. 157, 1941ā€“1955 (2006)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  24. Liu, Wang-jin: Fuzzy proximity spaces redefined. Fuzzy Sets Syst. 15, 241ā€“248 (1985)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  25. Artico, G., Moresco, R.: Fuzzy proximities and totally bounded fuzzy uniformities. J. Math. Anal. Appl. 99, 320ā€“337 (1984)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  26. Lowen, R.: Fuzzy topological spaces and fuzzy compactness. J. Math. Anal. Appl. 56, 621ā€“633 (1976)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  27. El-Dardery, M., Ramadan, A.A., Kim, Y.C.: L-fuzzy topogenous orders and L-fuzzy topologies. J. Intell. Fuzzy Syst. 24, 685ā€“691 (2013)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  28. Å ostak, A.: On a fuzzy topological structure. Suppl. Rend. Circ. Mat. Palermo Ser. II(11), 89ā€“103 (1985)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  29. Ramadan, A.A.: Smooth topological spaces. Fuzzy Sets Syst. 48, 371ā€“375 (1992)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  30. Kim, Y.C., Min, K.C.: L-fuzzy proximities and L-fuzzy topologies. Inf. Sci. 173, 93ā€“113 (2005)

    ArticleĀ  Google ScholarĀ 

  31. Čimoka, D., Å ostak, A.P.: L-fuzzy syntopogenous structures, part I: fundamentals and application to L-fuzzy topologies, L-fuzzy proximities and L-fuzzy uniformities. Fuzzy Sets Syst. 232, 74ā€“97 (2013)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  32. Kim, Y.C., Oh, J.-M.: Alexandrov L-fuzzy pre-proximities. Mathematics 7(1), 1ā€“15 (2019)

    Google ScholarĀ 

  33. Ramadan, A.A., Elkordy, E.H., Kim, Y.C.: Perfect L-fuzzy topogenous spaces, L-fuzzy quasi-proximities and L-fuzzy quasi-uniform spaces. J. Int. Fuzzy Syst. 28, 2591ā€“2604 (2015)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  34. Ramadan, A.A., Usama, M.A., Reham, M.A.: On L-fuzzy pre-proximities and L-fuzzy interior operators. Ann. Fuzzy Math. Inf. 17(2), 191ā€“204 (2019)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  35. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341ā€“356 (1982)

    ArticleĀ  Google ScholarĀ 

  36. Wang, C.Y.: Topological characterizations of generalized fuzzy rough sets. Fuzzy Sets Syst. 312, 109ā€“125 (2017)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  37. Zhang, Li., Zhan, J.: Covering-based generalized IF rough sets with applications to multi-attribute decision-making. Inf. Sci. 478, 275ā€“302 (2019)

    ArticleĀ  MathSciNetĀ  Google ScholarĀ 

  38. HƔjek, P.: Metamathematices of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)

    BookĀ  Google ScholarĀ 

  39. Oh, J.-M., Kim, Y.C.: L-Fuzzy closure operators and L-fuzzy cotopologies. J. Math. Comput. Sci. 9(2), 131ā€“145 (2019)

    Google ScholarĀ 

  40. AdƔmek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990)

    MATHĀ  Google ScholarĀ 

Download references

Acknowledgements

Not applicable.

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Contributions

All authors jointly worked on the results, and they read and approved the final manuscript.

Corresponding author

Correspondence to A. A. Ramadan.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ramadan, A.A., Elkordy, E.H. & Usama, M.A. On L-fuzzy closure operators and L-fuzzy pre-proximities. J Egypt Math Soc 29, 9 (2021). https://doi.org/10.1186/s42787-021-00117-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s42787-021-00117-8

Keywords

Mathematics Subject Classification