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Categories of Alexandrov L-Fuzzy Pre-uniformities, L-Approximation Operators and Alexandrov L-Fuzzy Topologies

Yong Chan Kim1, A. A. Ramadan2, and Seok Jong Lee3

1Department of Mathematics, Gangneung-Wonju National University, Gangneung, Korea, 2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt, 3Department of Mathematics, Chungbuk National University, Cheongju, Korea
Correspondence to: Seok Jong Lee, (sjl@cbnu.ac.kr)
Received January 18, 2019; Revised March 10, 2019; Accepted March 13, 2019.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

This paper investigates relationships among Alexandrov L-fuzzy pre-uniformities, Alexandrov L-fuzzy topologies, L-lower approximation operators and L-upper approximation operators. It also presents an adjointness between the category of Alexandrov L-fuzzy pre-uniform spaces and reflexive L-fuzzy relations. In addition, it shows that the category of Alexandrov L-fuzzy pre-uniform spaces and the category of L-lower (respectively, upper) approximation spaces are isomorphic.

Keywords : Complete residuated lattice, Alexandrov fuzzy topology, Lower approximation operator, Upper approximation operator, Alexandrov fuzzy pre-uniformity
1. Introduction

Pawlak [1, 2] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. This theory was extended and applied in many directions [315]. An interesting and natural research topic in rough set theory is the study of algebraic structure and topological structure.

Ward and Dilworth [16] introduced a complete residuated lattice which is an important mathematical tool as algebraic structures for many valued logics [38, 10, 1725]. For an extension of classical rough sets, many researchers [3, 4, 6, 21, 24] developed L-lower and L-upper approximation operators in complete residuated lattices. By using this concepts, information systems and decision rules were investigated in complete residuated lattices [6, 7, 17, 18, 20]. Cimoka and Sostak [18] and Ramadan et al. [22, 23]investigated the relationships between L-fuzzy quasi-uniformities and L-fuzzy topologies in complete residuated lattices.

Ma [10] investigated the topological and lattice structures of L-fuzzy rough sets determined by lower and upper sets. Kim and Kim [5] studied the relations between L-fuzzy upper and lower approximation spaces and L-fuzzy quasi-uniform spaces in a strictly two-sided, commutative quantale. Kim and his colleagues [3, 4, 10] investigated the properties of L-lower and L-upper approximate operators and Alexandrov L-fuzzy topologies in complete residuated lattices.

In this paper, we investigate relationships among Alexandrov L-fuzzy pre-uniformities, Alexandrov L-fuzzy topologies, L-lower approximation operators and L-upper approximation operators. We obtain an adjointness between the category of Alexandrov L-fuzzy pre-uniform spaces and reflexive L-fuzzy relations. In addition, we show that the category of Alexandrov L-fuzzy pre-uniform spaces and the category of L-lower (resp. upper) approximation spaces are isomorphic.

### Definition 1.1 ([6, 7, 17])

An algebra (L, ∧, ∨, ⊙, →, ⊥, ⊤) is called a complete residuated lattice if it satisfies the following conditions:

1. (L1) (L, ≤, ∨, ∧, ⊥, ⊤) is a complete lattice with the greatest element ⊤ and the least element ⊥;

2. (L2) (L, ⊙, ⊤) is a commutative monoid;

3. (L3) xyz iff xyz for x, y, zL.

In this paper, we always assume that (L, ≤, ⊙, →, *) is complete residuated lattice with an order reversing involution * defined by

$x*=x→⊥, (x*)*=x.$
2. Preliminaries

For αL, fLX, we denote (αf), (αf), αXLX as (αf)(x) = αf(x), (αf)(x) = αf(x), αX(x) = α, ⊤xLX, ⊤(x,y)LX×X,

$⊤x(y)={⊤,if y=x,⊥,otherwise,⊤(x,y)(z,w)={⊤,if (x,y)=(z,w),⊥,otherwise.$

### Lemma 2.1 ([6, 7, 17])

For each x, y, z, xi, yi, wL, we have the following properties.

1. ⊤ → x = x, ⊥ ⊙ x = ⊥,

2. If yz, then xyxz, xyxz and zxyx,

3. xy iff xy = ⊤,

4. $(⋀iyi)*=⋁iyi*,(⋁iyi)*=⋀iyi*$,

5. x → (∧i yi) = ∧i(xyi),

6. (∨i xi) → y = ∧i(xiy),

7. x ⊙ (∨i yi) = ∨i(xyi),

8. (xy) → z = x → (yz) = y → (xz),

9. xy = (xy*)*, xy = y*x*,

10. (xy) ⊙ (zw) ≤ (xz) → (yw) and (xy) → (xz) ≥ yz,

11. xy ≤ (xz) → (yz) and (xy) ⊙ (yz) ≤ xz,

12. i∈Γxi → ∨i∈Γyi ≥ ∧i∈Γ(xiyi) and ∧i∈Γxi → ∧i∈Γyi ≥ ∧i∈Γ(xiyi).

### Definition 2.2 ([6, 7, 17])

Let X be a set. A map R : X × XL is called an L-fuzzy preorder if it satisfies the following conditions:

1. (E1) reflexive if R(x, x) = ⊤ for all xX,

2. (E2) transitive if R(x, y) ⊙ R(y, z) ≤ R(x, z), for all x, y, zX.

### Lemma 2.3 ([6, 7, 17])

For a given set X, define a binary map S : LX × LXL by

$S(f,g)=⋀x∈X(f(x)→g(x)).$

Then, for each f, g, h, kLX, and αL, the following properties hold.

1. S is an L-fuzzy preorder on LX.

2. fg iff S(f, g) = ⊤.

3. If fg, then S(h, f) ≤ S(h, g) and S(f, h) ≥ S(g, h).

4. S(f, g) ⊙ S(k, h) ≤ S(fk, gh).

5. S(g, h) ≤ S(f, g) → S(f, h).

6. S(f, h) = ∨gLX(S(f, g) ⊙ S(g, h)).

### Definition 2.4 ([3, 4, 20])

A map : LXL is called an Alexandrov L-fuzzy topology on X if it satisfies the following conditions:

1. (AT1) ,

2. (AT2) and , for all {fi}i∈ΓLX.

3. (AT3) and , for all αL, fLX.

The pair (X, ) is called an Alexandrov L-fuzzy topological space.

Let (X, ) and (Y, ) be two Alexandrov L-fuzzy topological spaces. A map ϕ : XY is called L-fuzzy continuous if for each fLY,

$T2(f)≤T1(ϕ←(f)).$

### Definition 2.5 ( [10])

A map : LX×XL is called an Alexandrov L-fuzzy pre-uniformity on X iff the following conditions hold.

1. (AU1) There exists uLX×X such that .

2. (AU2) If vu, then .

3. (AU3) For every uiLX×X, .

4. (AU4) .

5. (AU5) for each αL.

The pair (X, ) is called an Alexandrov L-fuzzy pre-uniform space.

An Alexandrov L-fuzzy pre-uniformity on X is called an Alexandrov L-fuzzy quasi-uniformity if

(AQ) , where

$v∘w(x,z)=⋁y∈Xu(x,y)⊙v(y,z).$

Let (X, ) and (Y, ) be an Alexandrov L-fuzzy quasiuniform spaces, and ϕ : XY ba a map. Then ϕ is said to be L-fuzzy uniformly continuous if , for every vLY×Y.

### Remark 2.6

Let (X, ) be an L-fuzzy pre-uniform space. By (AU1) and (AU2), we have , because u ≤ ⊤X×X for all uLX×X.

### Definition 2.7 ([3])

A map : LXLX is called an L-lower approximation operator on X if

1. (J1) ,

2. (J2) for all fLX,

3. (J3) for all fiLX, and

4. (J4) .

An L-lower approximation operator is said to be topological if (T) for all fLX.

Let (X, ) and (Y, ) be L-lower approximation spaces. is called an L-lower approximation map if, for each gLY,

$ϕ←(JY(g))≤JX(ϕ←(g)).$

### Definition 2.8 ( [3])

A map ℋ : LXLX is called an L-upper approximation operator on X if

1. (H1) ℋ(⊥X) = ⊥X,

2. (H2) ℋ(f) ≥ f for all fLX,

3. (H3) ℋ(∨i∈Γfi) = ∨i∈Γ ℋ(fi) for all fiLX, and

4. (H4) ℋ(αf) = α ⊙ ℋ(f).

An L-upper approximation operator is said to be topological if (T) ℋ(ℋ(f)) = ℋ(f), for all fLX.

Let ℋ and be an L-upper and L-lower approximation on X, respectively. The pair is called a fuzzy rough set for f.

Let (X,ℋX) and (Y,ℋY ) be L-upper approximation spaces. Then ϕ : (X,ℋX) → (Y,ℋY ) is called an L-upper approximation map if, for each gLY,

$ϕ←(HY(g))≥HX(ϕ←(g)).$
3. Categories of Alexandrov L-Fuzzy Preuniformities, L-Approximation Operators and Alexandrov L-Fuzzy Topologies

### Theorem 3.1

Let be an Alexandrov L-fuzzy pre-uniformity on X. For each uLX×X with u−1(x, y) = u(y, x), define a map by

$Us(u)=U(u-1).$

Then the following properties hold.

1. is an Alexandrov L-fuzzy pre-uniformity on X.

2. If is an Alexandrov L-fuzzy quasi-uniformity on X, then is an Alexandrov L-fuzzy quasi-uniformity on X.

### Proof

(1) (AU1), (AU2) and (AU3) can be easily proved.

(AU4) For all uLX×X, we have

$Us(u)=U(u-1)≤⋀x∈Xu-1(x,x)=⋀x∈Xu(x,x)$

(AU5) For all αL and uLX×X,

$Us(α→u)=U(α→u-1)=α→U(u-1)=α→Us(u).$

Hence is an Alexandrov L-fuzzy pre-uniformity on X.

(2) (AQ) For all uLX×X,

$Us(u)=U(u-1)≤⋁{U(v-1)⊙U(w-1)|v-1∘w-1≤u-1}=⋁{Us(v)⊙Us(w)|v∘w≤u}.$

### Theorem 3.2

Let RLX×X be a reflexive L-fuzzy relation. Define a map as

$UR(u)=⋀x,y∈X(R(x,y)→u(x,y)).$

Then the following properties hold.

1. is an Alexandrov L-fuzzy pre-uniformity on X.

2. $URs$ is an Alexandrov L-fuzzy pre-uniformity on X with $URs=UR-1$.

3. If R is an L-fuzzy preorder and there exists xyX such that R(x, y) = ⊥, then is an Alexandrov L-fuzzy quasi-uniformity on X.

### Proof

(AU1) There exists ⊤X×XLX×X such that .

(AU2), (AU3) and (AU5) can be easily proved. (AU4)

$U(u)=⋀x,y∈X(R(x,y)→u(x,y))=⋀x,y∈X(R(x,y)→u(x,y))≤⋀x∈X(R(x,x)→u(x,x))=⋀x∈Xu(x,x).$

Hence is an Alexandrov L-fuzzy pre-uniformity on X.

(2) We obtain $URs:LX×X→L$ as

$URs(u)=U(u-1)=⋀x,y∈X(R(x,y)→u-1(x,y))=⋀x,y∈X(R-1(x,y)→u(x,y))=UR-1(u).$

(3) For v,wLX×X,

$U(v)⊙U(w)=⋀x,y∈X(R(x,y)→v(x,y))⊙⋀z,w∈X(R(z,w)→w(z,w))≤⋀x,y,z∈X(R(x,y)→v(x,y))⊙(R(z,w)→w(z,w))≤⋀x,y,z∈X(R(x,y)⊙R(y,z)→v(x,y)⊙w(y,z))≤⋀x,z∈X(⋁y∈X(R(x,y)⊙R(y,z))→⋁y∈X(v(x,y)⊙w(y,z)))≤⋀x,z∈X(R(x,z)→u(x,y))=U(u).$

Hence .

Since ΔX×X ο u = u and

$UR(ΔX×X)=⋀x,y∈X(R(x,y)→ΔX×X(x,y))=⋁x≠y∈XR*(x,y)=⊤,$

we have .

### Example 3.3

Let ([0, 1], ⊙,→,*, 0, 1) be a complete residuated lattice (ref. [17, 19, 20, 25]) as

$x⊙y=max{0,x+y-1}, x→y=min{1-x+y,1}x*=1-x.$

Let X = {x, y, z} and an L-fuzzy preorder R ∈ [0, 1]X×X as

$R=(10.400.810.50.50.61).$

From Theorem 3.2(3), we obtain Alexandrov L-fuzzy quasiuniformities $UR,URs:LX×X→L$ as

$UR(u)=⋀x,y∈X(R(x,y)→u(x,y)),URs(u)=⋀x,y∈X(R-1(x,y)→u(x,y)).$

### Theorem 3.4

Let (X, ) be an Alexandrov L-fuzzy preuniform space. Define a map by

$RU(x,y)=⋀u∈LX×X(U(u)→u(x,y)).$

Then the following properties hold.

1. is a reflexive L-fuzzy relation such that $RU(x,y)=U*(⊤(x,y)*)$.

2. If (X, ) be an Alexandrov L-fuzzy quasi-uniform space, then is an L-fuzzy preorder.

3. .

4. If RLX×X is reflexive, then .

### Proof

(1) For any xX, we have from (AU2). Then

$RU(x,x)=⋀u∈LX(U(u)→u(x,x))=⊤.$

Thus is an reflexive L-fuzzy relation. For each $u=⋀x,y∈X(u*(x,y)→⊤(x,y)*)$, we have

$RU(x,y)=⋀u∈LX×X(U(u)→u(x,y))=⋀u∈LX×X(U(⋀x,y∈X(u*(x,y)→⊤(x,y)*))→u(x,y))=⋀u∈LX×X(⋀x,y∈X(u*(x,y)→U(⊤(x,y)*))→u(x,y))=⋀u∈LX×X(⋀x,y∈X(U*(⊤(x,y)*)→u(x,y))→u(x,y))≥U*(⊤(x,y)*).$

Also, we have

$RU(x,y)=⋀u∈LX×X(U(u)→u(x,y))≤U(⊤(x,y)*)→⊤(x,y)*(x,y)=U*(⊤(x,y)*).$

Hence $RU(x,y)=U*(⊤(x,y)*)$.

(2) For each x, y, zX and v ο wu,

$RU(x,y)⊙RU(y,z)=⋀v∈LX×X(U(v)→v(x,y))⊙⋀w∈LX×X(U(w)→w(y,z))=⋀v,w∈LX×X(U(v)⊙U(w)→v(x,y)⊙w(y,z))=⋀u∈LX×X(U(u)→u(x,z))=RU(x,z)(by AQ).$

(3) For each uLX×X,

$URU(u)=⋀x,y∈X(RU(x,y)→u(x,y))=⋀x,y∈X(⋀v∈LX×X(U(v)→v(x,y))→u(x,y))≥⋀x,y∈X((U(u)→u(x,y))→u(x,y))≥U(u).$

(4) For each x, yX, we have

$RUR(x,y)=⋀u∈LX×X(UR(u)→u(x,y))=⋀u∈LX×X(⋀x,y∈X(R(x,y)→u(x,y))→u(x,y))≥⋀u∈LX×X((R(x,y)→u(x,y))→u(x,y))≥R(x,y).$

Also, we have

$RUR(x,y)≤(UR(⊤(x,y)*)→⊤(x,y)*(x,y))=(R(x,y)→⊤(x,y)*(x,y))→⊤(x,y)*(x,y)=R(x,y).$

Hence, .

### Theorem 3.5

Let (X, ) be an Alexandrov L-pre-uniform space. Define two maps by

$JU(f)(x)=⋀y∈X(U*(⊤(x,y)*)→f(y)),HU(f)(x)=⋁y∈X(U*(⊤(x,y)*)⊙f(y)).$

Then the following properties hold.

1. is an L-lower approximation operator such that $JU(⊤y*)(x)=U(⊤(x,y)*)$.

2. is an L-upper approximation operator such that $HU(⊤y)(x)=U*(⊤(x,y)*)$.

3. If (X, ) be an Alexandrov L-fuzzy quasi-uniform space, then is a topological L-lower approximation operator.

4. If (X, ) be an Alexandrov L-fuzzy quasi-uniform space, then is a topological L-upper approximation operator.

### Proof

(1) Since $U(⊤(x,x)*)≤⊤(x,x)*(x,x)=⊥$ from (AU4), we have (J2)

$JU(f)(x)≤U*(⊤(x,x)*)→f(x)=f(x).$

(J1), (J3), (J4) and (2) are easily proved.

(3) By Theorem 3.4(2), $⋁y∈X(U*(⊤(x,y)*)⊙U*(⊤(y,z)*))=⋁y∈X(RU(x,y)⊙RU(y,z))=RU(x,z)=U*(⊤(x,z)*)$. For each xX, fLX,

$JU(JU(f))(x)=⋀y∈X(U*(⊤(x,y)*)→JU(f)(y))=⋀y∈X(U*(⊤(x,y)*)→⋀z∈X(U*(⊤(y,z)*)→f(z)))=⋀z∈X(⋁y∈X(U*(⊤(x,y)*)⊙U*(⊤(y,z)*))→f(z)))=⋀z∈X(U*(⊤(x,z)*)→f(z)))=JU(f)(x).$

(4) For each xX, fLX,

$HU(HU(f))(x)=⋁y∈X(U*(⊤(x,y)*)⊙HU(f)(y))=⋁y∈X(U*(⊤(x,y)*)⊙⋁z∈X(U*(⊤(x,y)*)⊙f(z)))=⋁z∈X(⋁y∈XU*(⊤(x,y)*)⊙U*(⊤(y,z)*))⊙f(z)))=⋁z∈X(U*(⊤(x,z)*)⊙f(z)).$

### Theorem 3.6

Let (X, ) and (X,ℋ) be L-lower and L-upper approximation spaces. Define two maps as

$UJ(u)=⋀x,y∈X(J*(⊤y*)(x)→u(x,y)),UH(u)=⋀y∈X(H(⊤y)(x)→u(x,y)).$

Then the following properties hold.

1. is an Alexandrov L-fuzzy preuniformity such that $UJ(⊤(x,y)*)=H(⊤y)(x)$.

2. is an Alexandrov L-fuzzy preuniformity such that $UJ(⊤(x,y)*)=J(⊤y*)(x)$.

3. If is topological and there exists x, yX with $J(⊤y*)(x)=⊤$, then is an Alexandrov L-fuzzy quasiuniformity.

4. If ℋ is topological and there exists x, yX with ℋ(⊤y)(x) = ⊥, then is an Alexandrov L-fuzzy quasiuniformity.

5. and .

6. and .

### Proof

(1) and (2). Since $J*(⊤x*)(x)≥⊤x(x)=⊤$ and ℋ(⊤x)(x) ≥ ⊤x(x) = ⊤, it is similarly proved as Theorem 3.2.

(3) For each x, zX,

$⋀y∈X(J*(⊤z*)(y)→J(⊤y*)(x))=J(⋀y∈X(J*(⊤z*)(y)→⊤y*))(x)=J(J(⊤z*))(x)=J(⊤z*)(x).$

Then, $⋁y∈X(J*(⊤x*)(y)⊙J*(⊤y*)(x))=J*(⊤z*)(x)$.

$UJ(v)⊙UJ(w)=⋀x,y∈X(J*(⊤y*)(x)→v(x,y))⊙⋀z,w∈X(J*(⊤w*)(z)→w(z,w))≤⋀x,y,z∈X(J*(⊤y*)(x)→v(x,y))⊙(J*(⊤w*)(z)→w(z,w))≤⋀x,y,z∈X(J*(⊤y*)(x)⊙J*(⊤z*)(y)→v(x,y)⊙w(y,z))≤⋀x,z∈X(⋁y∈X(J*(⊤y*)(x)⊙J*(⊤z*)(y))→⋁y∈X(v(x,y)⊙w(y,z))≤⋀x,z∈X(J*(⊤z*)(x)→u(x,z))=U(u).$

Hence .

Since ΔX×X ο u = u and

$UJ(ΔX×X)=⋀x,y∈X(J*(⊤y*))(x)→ΔX×X(x,y))=⋁x≠y∈XU(⊤y*)(x)=⊤,$

we have .

(4) It is similarly proved as (3).

(5) Since $JU(⊤y*)(x)=⋀x,y∈X(U*(⊤(x,y)*)→⊤y*(y))=(U*(⊤(x,y)*)$, we have

$UJU(u)=⋀x,y∈X(JU*(⊤y*)(x)→u(x,y))=⋀x,y∈X(U*(⊤(x,y)*)→u(x,y))=⋀x,y∈X(u*(x,y)→U(⊤(x,y)*))=U(⋀x,y∈X(u*(x,y)→(⊤(x,y)*))=U(n).$

Since $UJ(⊤(x,y)*)=⋀x,y∈X(J*(⊤y*)(x)→⊤(x,y)*(x,y))=J(⊤y*)(x)$, we have

$JUJ(f)(x)=⋀y∈X(UJ*(⊤(x,y)*)→f(y))=⋀y∈X(J*(⊤y*)(x)→f(y))=⋀y∈X(f*(y)→J(⊤y*)(x))=J(⋀y∈X(f*(y)→(⊤y*))(x)=J(f)(x).$

(6) Since $UU(⊤y)(x)=⋁y∈X(U*(⊤(x,y)*)⊙⊤y(y))=U(⊤(x,y)*)$, we have

$UHU(u)=⋀x,y∈X(HU(⊤y)(x)→u(x,y))=⋀x,y∈X(U*(⊤(x,y)*)→u(x,y))=⋀x,y∈X(u*(x,y)→U(⊤(x,y)*))=U(⋀x,y∈X(u*(x,y)→⊤(x,y)*))=U(u).$

Since $UH(⊤(x,y)*)=⋀x,y∈X(H(⊤y)(x)→⊤(x,y)*(x,y))=H*(⊤y)(x)$, we have

$HUH(f)(x)=⋁y∈X(UH*(⊤(x,y)*)⊙f(y))=⋁y∈X(H(⊤y)(x)⊙f(y))=H(⋁y∈X(f(y)⊙⊤y))(x)=H(f)(x).$

### Theorem 3.7

Let (X, ) and (Y, ) be Alexandrov L-fuzzy preuniform spaces. Let be L-fuzzy uniformly continuous. Then the following properties hold.

1. is an order preserving map.

2. is an L-lower approximation map.

3. is an L-upper approximation map.

### Proof

(1) For any x, zX,

$RU(x,z)=⋀u∈LX×X(U(u)→u(x,y))≤⋀(ϕ×ϕ)←(v)∈LX×X(U((ϕ×ϕ)←(v))→(ϕ×ϕ)←(v)(x,y))≤⋀v∈LY×Y(V(v)→v(ϕ(x),ϕ(z)))=RV(ϕ(x),ϕ(z)).$

(2) Since $V(⊤(ϕ(x),ϕ(z))*)≤U((ϕ×ϕ)←(⊤(ϕ(x),ϕ(z))*))≤U(⊤(x,z)*)$, we have

$JU(ϕ←(f))(x)=⋀z∈X(U*(⊤(x,y)*)→ϕ←(f)(z))≥⋀z∈X(V*(⊤(ϕ(x),ϕ(z))*)→f(ϕ(z)))≥⋀y∈Y(V*(⊤(ϕ(x),y)*)→f(y))=ϕ←(JV(f))(x).$

(3) Since $V(⊤(ϕ(x),ϕ(z))*)≤U((ϕ×ϕ)←(⊤(ϕ(x),ϕ(z))*))≤U(⊤(x,z)*)$, we have

$HU(ϕ←(f))(x)=⋁z∈X(U*(⊤(x,z)*)⊙ϕ←(f)(z))≤⋁z∈X(V*(⊤(ϕ(x),ϕ(z))*)⊙f(ϕ(z)))≤⋁y∈Y(V*(⊤(ϕ(x),y)*)⊙f(y))=ϕ←(HV(f))(x).$

### Theorem 3.8

1. Let ϕ : (X,RX) → (Y,RY) be an order preserving map. Then ϕ : (X, URX) → (Y, URY) is L-fuzzy uniformly continuous.

2. Let be an L-lower approximation map. Then is L-fuzzy uniformly continuous.

3. Let be an L-upper approximation map. Then is L-fuzzy uniformly continuous.

### Proof

(1) For any x, zX,

$URX((ϕ×ϕ)←(v))=⋀x,y∈X(RX(x,y)→(ϕ×ϕ)←(v)(x,y))≥⋀x,y∈X(RY(ϕ(x),ϕ(y))→v(ϕ(x),ϕ(y)))≥⋀z,w∈Y(RY(z,w)→v(z,w))=URY(v).$

(2) Since is an L-lower approximation map, we have

$ϕ←(JY(⊤ϕ(y)*)(x)≤JX(ϕ←(⊤ϕ(y)*))(x)≤JX(⊤y*)(x),UJX((ϕ×ϕ)←(v))=⋀x,y∈X(JX*(⊤y*)(x)→(ϕ×ϕ)←(v)(x,y))≥⋀x,y∈X(JY*(⊤ϕ(y)*(ϕ(x))→v(ϕ(x),ϕ(y)))≥⋀z,w∈Y(JY*(⊤z*)(w)→v(z,w))=UJY(v).$

(3) Since is an L-upper approximation map, we have ℋY(⊤ϕ(y))(ϕ(x)) = ϕ(ℋY(⊤ϕ(y))(x) ≥ ℋX(ϕ( ⊤ϕ(y))(x) ≥ ℋX(⊤y)(x).

$UHX((ϕ×ϕ)←(v))=⋀x,y∈X(HX(⊤y)(x)→(ϕ×ϕ)←(v)(x,y))≥⋀x,y∈X(HY(⊤ϕ(x)(ϕ(y))→v(ϕ(x),ϕ(y)))≥⋀z,w∈Y(HY(⊤z)(w)→v(z,w))=UHY(v).$

### Definition 3.9 ( [17, 20])

Suppose that are concrete functors.

1. and are said to be isomorphic if and .

2. The pair (F,G) is called a Galois correspondence between and if for each , idY : F ο G(Y) → Y is a -morphism, and for each , idX : XG ο F(X) is a -morphism.

If (F,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.

Let AUS be denote the category of Alexandrov L-fuzzy preuniform spaces and uniform continuous maps for morphisms. Let FRS be denote the category of L-fuzzy reflexive relations and order preserving maps for morphisms. Let LAP (resp. UAP) be denote the category of L-lower (resp. L-upper) approximation spaces and L-lower (resp. L-upper) approximation maps for morphisms.

### Theorem 3.10

1. F : AUSFRS defined as is a functor.

2. G : FRSAUS defined as is a functor.

3. The pair (F,G) is a Galois correspondence between AUS and FRS.

4. Two categories AUS and LAP are isomorphic.

5. Two categories AUS and UAP are isomorphic.

### Proof

(1) and (2) are follows from Theorems 3.2, 3.4, 3.7 and 3.8.

(3) By Theorem 3.4(4), if (X,RX) is an L-fuzzy reflexive relation, then F(G(X,RX) = (X,RURX = RX). Hence, the identity map idX : (X,RX) → (X, RURX) = F(G(X,RX) is an order preserving map. Moreover, if (X, ) is an Alexandrov L-fuzy pre-uniform space, then by Theorem 3.4(3), we have . Hence the identity map is uniform continuous. Therefore (F,G) is a Galois correspondence.

(4) By Theorems 3.5 and 3.7, H : AUSLAP defined as is a functor. By Theorems 3.6 and 3.8, K : LAPAUS defined as is a functor. By Theorem 3.5(6),

$K(H(X,UX))=K(X,JUX)=(X,UJUX=UX),H(K(X,JX))=H(X,UJX)=(X,JUJX=JX).$

Hence, two categories AUS and LAP are isomorphic.

(5) By Theorems 3.5 and 3.7, P : AUSUAP defined as is a functor. By Theorems 3.6 and 3.8, Q : UAPAUS defined as is a functor. By Theorem 3.5(5),

$Q(P(X,UX))=Q(X,HUX)=(X,UHUX=UX),P(Q(X,HX))=P(X,UHX)=(X,HUHX=HX).$

Hence, two categories AUS and UAP are isomorphic.

### Theorem 3.11

1. Let be an L-lower approximation operator. Define as . Then is an L-upper approximation operator.

2. Let be an L-upper approximation operator. Define as . Then is an L-lower approximation operator.

3. Let be an Alexandrov L-fuzzy topology. Define as . Then is an Alexandrov L-fuzzy topology.

### Proof

(1) For fLX and αL, we have and . The remaining can be proved by similar way.

### Lemma 3.12

For every fLX, we define uf, $uf-1:X×X→L$ by:

$uf(x,y)=f(x)→f(y),uf-1(x,y)=uf(y,x).$

Then we have the following statements

1. X×X = uX = uX.

2. For every ufLX×X, we have uf ο ufuf.

3. ufugufg, for each f, gLX.

4. $uf-1=uf*$.

5. u(∨i∈Γfi) ≥ ∧i∈Γufi and u(∧i∈Γfi) ≥ ∧i∈Γufi.

6. uαfuf and uαfuf.

7. uf ≥ ∧zX uz and $uf≥⋀z∈Xu⊤z-1$.

### Proof

(1) ⊤X×X(x, y) = ⊤ = uX(x, y) = ⊥X(x) → ⊥X(y) = ⊤X(x)→ ⊤X(y) = uX (x, y).

(2) For each x, zX,

$(uf∘uf)(x,z)=⋁y∈Xuf(x,y)⊙uf(y,z)=⋁y∈X((f(x)→f(y))⊙(f(y)→f(z)))≤f(x)→f(z)=uf(x,z).$

(3) For each x, yX,

$(uf⊙ug)(x,y)=uf(x,y)⊙ug(x,y)=(f(x)→f(y))⊙(g(x)→g(y))≤f(x)⊙g(x)→f(y)⊙g(y)=uf⊙g(x,y).$

(4)

$uf-1(x,y)=uf(y,x)=f(y)→f(x)=f*(x)→f*(y)=uf*(x,y).$

(5) From Lemma 2.2(12), we have

$u(⋁i∈Γfi)(x,y)=⋁i∈Γfi(x)→⋁i∈Γfi(y)≥⋀i∈Γ(fi(x)→fi(y))=⋀i∈Γufi,u(⋀i∈Γfi)=⋀i∈Γfi(x)→⋀i∈Γfi(y)≥⋀i∈Γ(fi(x)→fi(y))=⋀i∈Γufi.$

(6) From Lemma 2.2(10) and (11), we have

$uα→f(x,y)=(α→f(x))→(α→f(y))≥f(x)→f(y)=uf(x,y)uα⊙f(x,y)=(α⊙f(x))→(α⊙f(y))≥f(x)→f(y)=uf(x,y).$

(7) Since f = ∨zX(f(z) ⊙ ⊤z) and $f=⋀z∈X(f*(z)→⊤z*)$, by (6) we have uf ≥ ∧zX uz and $uf≥⋀z∈Xu⊤z-1$.

### Theorem 3.13

Let be an Alexandrov L-fuzzy pre-uniformity on X. Define a map . Then the following properties hold.

1. is an Alexandrov L-fuzzy topology with $TUs=TUs$.

2. There exists such that

$TU(f)=⋀x,y∈X(RU(x,y)→(f(x)→f(y))).$

3. , where .

4. , where .

### Proof

(1) (AT1) and .

(AT2) By Lemma 3.12(5), we have and .

(AT3) By Lemma 3.12(6), we have and . Hence, is an Alexandrov L-fuzzy topology on X. Moreover, $TUs(f)=TU(f*)=U(uf*)=Us(uf)=TUs(f)$.

(2) For $uf=⋀x,y∈X(uf*(x,y)→⊤(x,y)*)$, there exists $RU(x,y)=U*(⊤(x,y)*)$ such that

$TU(f)=U(uf)=U(⋀x,y∈X(uf*(x,y)→⊤(x,y)*)=⋀x,y∈X(uf*(x,y)→U(⊤(x,y)*))=⋀x,y∈X(U*(⊤(x,y)*)→uf(x,y))=⋀x,y∈X(RU(x,y)→(f(x)→f(y))).$

(3)

$TJU(f)=S(f,JU(f))=⋀x∈X(f(x)→JU(f)(x))=⋀x∈X(f(x)→⋀z∈X(RU(x,z)→f(z)))=⋀x,z∈X(f(x)→(RU(x,z)→f(z)))=⋀x,z∈X(RU(x,z)→(f(x)→f(z)))=TU(f).$

(4) From the fact $RUs(y,x)=Us*(⊤(y,x)*)=U*(⊤(y,x)-1*)=U*(⊤(x,y)*)=R(x,y)$,

$THU(f)=S(HU(f),f)=⋀x∈X(HU(f)(x)→f(x)))=⋀x∈X((⋁y∈X(RU(x,y)⊙(f(y))→f(x))))=⋀x,y∈X(RU(x,y)→(f(y)→f(x)))=⋀x,z∈X(RUs(y,x)→(f(y)→f(x)))=TUs(f).$

### Example 3.14

Consider R defined in Example 3.3. By Theorem 3.13, we obtain two Alexandrov L-fuzzy topologies $TUR,TURs:LX→L$, where

$TUR(f)=⋀x,y∈X(R(x,y)→(f(x)→f(y))),TURs(f)=URs(uf)=UR(uf*)=⋀x,y∈X(R(x,y)→(f*(x)→f*(y)))=⋀x,y∈X(R(x,y)→(f(y)→f(x))).$

### Theorem 3.15

Let (X, ) and (Y, ) be two L-fuzzy preuniform spaces. Let be L-fuzzy uniformly continuous. Then is L-fuzzy continuous.

### Proof

For x, yX,

$(ϕ×ϕ)←(uf)(x,y)=uf(ϕ(x),ϕ(y))=f(ϕ(x))→f(ϕ(y))=ϕ←(f)(x)→ϕ←(f)(y)=uϕ←(f)(x,y).$

Thus,

$TU(ϕ←(f))=U(uϕ←(f))=U((ϕ×ϕ)←(uf))≥V(uf)=TV(f).$
4. Conclusion

We obtained an adjointness between the category of Alexandrov L-fuzzy pre-uniform spaces and reflexive L-fuzzy relations. In addition, we showed that the category of Alexandrov L-fuzzy pre-uniform spaces and the category of L-lower (resp. upper) approximation spaces are isomorphic.

Conflict of Interest

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Biographies

Yong Chan Kim received his B.S., M.S., and Ph.D. degrees in Mathematics from Yonsei University, Seoul, Korea, in 1982, 1984, and 1991, respectively. He is currently a professor of Gangneung-Wonju National University. His research interests include fuzzy topology and fuzzy logic.

E-mail: yck@gwnu.ac.kr

A. A. Ramadan received his B.S., M.S., and Ph.D. degrees from Assuit University. He is currently a professor of Beni-Suef University. His research interest is a fuzzy logic.

Seok Jong Lee received his M.S. and Ph.D. degrees from Yonsei University in 1986 and 1990, respectively. He is a professor at the Department of Mathematics, Chungbuk National University since 1989. His research interests include general topology and fuzzy topology. He is a member of KIIS and KMS.

E-mail: sjl@cbnu.ac.kr

March 2020, 20 (1)
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