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Categories of Alexandrov L-Fuzzy Pre-uniformities, L-Approximation Operators and Alexandrov L-Fuzzy Topologies
International Journal of Fuzzy Logic and Intelligent Systems 2019;19(1):28-39
Published online March 25, 2019
© 2019 Korean Institute of Intelligent Systems.

Yong Chan Kim, A. A. Ramadan, and Seok Jong Lee

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)=xX(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

vw(x,z)=yXu(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)xXu-1(x,x)=xXu(x,x)

(AU5) For all αL and uLX×X,

Us(αu)=U(αu-1)=αU(u-1)=αUs(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-1w-1u-1}={Us(v)Us(w)|vwu}.

Theorem 3.2

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

UR(u)=x,yX(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,yX(R(x,y)u(x,y))=x,yX(R(x,y)u(x,y))xX(R(x,x)u(x,x))=xXu(x,x).

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

(2) We obtain URs:LX×XL as

URs(u)=U(u-1)=x,yX(R(x,y)u-1(x,y))=x,yX(R-1(x,y)u(x,y))=UR-1(u).

(3) For v,wLX×X,

U(v)U(w)=x,yX(R(x,y)v(x,y))z,wX(R(z,w)w(z,w))x,y,zX(R(x,y)v(x,y))(R(z,w)w(z,w))x,y,zX(R(x,y)R(y,z)v(x,y)w(y,z))x,zX(yX(R(x,y)R(y,z))yX(v(x,y)w(y,z)))x,zX(R(x,z)u(x,y))=U(u).

Hence .

Since ΔX×X ο u = u and

UR(ΔX×X)=x,yX(R(x,y)ΔX×X(x,y))=xyXR*(x,y)=,

we have .

Example 3.3

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

xy=max{0,x+y-1},xy=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×XL as

UR(u)=x,yX(R(x,y)u(x,y)),URs(u)=x,yX(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)=uLX×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)=uLX(U(u)u(x,x))=.

Thus is an reflexive L-fuzzy relation. For each u=x,yX(u*(x,y)(x,y)*), we have

RU(x,y)=uLX×X(U(u)u(x,y))=uLX×X(U(x,yX(u*(x,y)(x,y)*))u(x,y))=uLX×X(x,yX(u*(x,y)U((x,y)*))u(x,y))=uLX×X(x,yX(U*((x,y)*)u(x,y))u(x,y))U*((x,y)*).

Also, we have

RU(x,y)=uLX×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)=vLX×X(U(v)v(x,y))wLX×X(U(w)w(y,z))=v,wLX×X(U(v)U(w)v(x,y)w(y,z))=uLX×X(U(u)u(x,z))=RU(x,z)(by AQ).

(3) For each uLX×X,

URU(u)=x,yX(RU(x,y)u(x,y))=x,yX(vLX×X(U(v)v(x,y))u(x,y))x,yX((U(u)u(x,y))u(x,y))U(u).

(4) For each x, yX, we have

RUR(x,y)=uLX×X(UR(u)u(x,y))=uLX×X(x,yX(R(x,y)u(x,y))u(x,y))uLX×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)=yX(U*((x,y)*)f(y)),HU(f)(x)=yX(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), yX(U*((x,y)*)U*((y,z)*))=yX(RU(x,y)RU(y,z))=RU(x,z)=U*((x,z)*). For each xX, fLX,

JU(JU(f))(x)=yX(U*((x,y)*)JU(f)(y))=yX(U*((x,y)*)zX(U*((y,z)*)f(z)))=zX(yX(U*((x,y)*)U*((y,z)*))f(z)))=zX(U*((x,z)*)f(z)))=JU(f)(x).

(4) For each xX, fLX,

HU(HU(f))(x)=yX(U*((x,y)*)HU(f)(y))=yX(U*((x,y)*)zX(U*((x,y)*)f(z)))=zX(yXU*((x,y)*)U*((y,z)*))f(z)))=zX(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,yX(J*(y*)(x)u(x,y)),UH(u)=yX(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,

yX(J*(z*)(y)J(y*)(x))=J(yX(J*(z*)(y)y*))(x)=J(J(z*))(x)=J(z*)(x).

Then, yX(J*(x*)(y)J*(y*)(x))=J*(z*)(x).

UJ(v)UJ(w)=x,yX(J*(y*)(x)v(x,y))z,wX(J*(w*)(z)w(z,w))x,y,zX(J*(y*)(x)v(x,y))(J*(w*)(z)w(z,w))x,y,zX(J*(y*)(x)J*(z*)(y)v(x,y)w(y,z))x,zX(yX(J*(y*)(x)J*(z*)(y))yX(v(x,y)w(y,z))x,zX(J*(z*)(x)u(x,z))=U(u).

Hence .

Since ΔX×X ο u = u and

UJ(ΔX×X)=x,yX(J*(y*))(x)ΔX×X(x,y))=xyXU(y*)(x)=,

we have .

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

(5) Since JU(y*)(x)=x,yX(U*((x,y)*)y*(y))=(U*((x,y)*), we have

UJU(u)=x,yX(JU*(y*)(x)u(x,y))=x,yX(U*((x,y)*)u(x,y))=x,yX(u*(x,y)U((x,y)*))=U(x,yX(u*(x,y)((x,y)*))=U(n).

Since UJ((x,y)*)=x,yX(J*(y*)(x)(x,y)*(x,y))=J(y*)(x), we have

JUJ(f)(x)=yX(UJ*((x,y)*)f(y))=yX(J*(y*)(x)f(y))=yX(f*(y)J(y*)(x))=J(yX(f*(y)(y*))(x)=J(f)(x).

(6) Since UU(y)(x)=yX(U*((x,y)*)y(y))=U((x,y)*), we have

UHU(u)=x,yX(HU(y)(x)u(x,y))=x,yX(U*((x,y)*)u(x,y))=x,yX(u*(x,y)U((x,y)*))=U(x,yX(u*(x,y)(x,y)*))=U(u).

Since UH((x,y)*)=x,yX(H(y)(x)(x,y)*(x,y))=H*(y)(x), we have

HUH(f)(x)=yX(UH*((x,y)*)f(y))=yX(H(y)(x)f(y))=H(yX(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)=uLX×X(U(u)u(x,y))(ϕ×ϕ)(v)LX×X(U((ϕ×ϕ)(v))(ϕ×ϕ)(v)(x,y))vLY×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)=zX(U*((x,y)*)ϕ(f)(z))zX(V*((ϕ(x),ϕ(z))*)f(ϕ(z)))yY(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)=zX(U*((x,z)*)ϕ(f)(z))zX(V*((ϕ(x),ϕ(z))*)f(ϕ(z)))yY(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,yX(RX(x,y)(ϕ×ϕ)(v)(x,y))x,yX(RY(ϕ(x),ϕ(y))v(ϕ(x),ϕ(y)))z,wY(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,yX(JX*(y*)(x)(ϕ×ϕ)(v)(x,y))x,yX(JY*(ϕ(y)*(ϕ(x))v(ϕ(x),ϕ(y)))z,wY(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,yX(HX(y)(x)(ϕ×ϕ)(v)(x,y))x,yX(HY(ϕ(x)(ϕ(y))v(ϕ(x),ϕ(y)))z,wY(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×XL 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 ufzXuz-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,

(ufuf)(x,z)=yXuf(x,y)uf(y,z)=yX((f(x)f(y))(f(y)f(z)))f(x)f(z)=uf(x,z).

(3) For each x, yX,

(ufug)(x,y)=uf(x,y)ug(x,y)=(f(x)f(y))(g(x)g(y))f(x)g(x)f(y)g(y)=ufg(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(iΓfi)(x,y)=iΓfi(x)iΓfi(y)iΓ(fi(x)fi(y))=iΓufi,u(iΓfi)=iΓfi(x)iΓfi(y)iΓ(fi(x)fi(y))=iΓufi.

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

uαf(x,y)=(αf(x))(αf(y))f(x)f(y)=uf(x,y)uαf(x,y)=(αf(x))(αf(y))f(x)f(y)=uf(x,y).

(7) Since f = ∨zX(f(z) ⊙ ⊤z) and f=zX(f*(z)z*), by (6) we have uf ≥ ∧zX uz and ufzXuz-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,yX(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,yX(uf*(x,y)(x,y)*), there exists RU(x,y)=U*((x,y)*) such that

TU(f)=U(uf)=U(x,yX(uf*(x,y)(x,y)*)=x,yX(uf*(x,y)U((x,y)*))=x,yX(U*((x,y)*)uf(x,y))=x,yX(RU(x,y)(f(x)f(y))).

(3)

TJU(f)=S(f,JU(f))=xX(f(x)JU(f)(x))=xX(f(x)zX(RU(x,z)f(z)))=x,zX(f(x)(RU(x,z)f(z)))=x,zX(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)=xX(HU(f)(x)f(x)))=xX((yX(RU(x,y)(f(y))f(x))))=x,yX(RU(x,y)(f(y)f(x)))=x,zX(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:LXL, where

TUR(f)=x,yX(R(x,y)(f(x)f(y))),TURs(f)=URs(uf)=UR(uf*)=x,yX(R(x,y)(f*(x)f*(y)))=x,yX(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

No potential conflict of interest relevant to this article was reported.


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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.

E-mail: aramadan58@hotmail.com


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