# The Lattice Structure of L-Contact Relations

Authors: Xueyou Chen
Journal: Advances in Fuzzy Systems
 (2016)

---

## Abstract

From the point of view of graded truth approach, we define the notion of a contact relation on the collection of allL-sets, discuss the connection to the set of all close, reflexive, and symmetric relations on all L-ultrafilters on X, and investigate the algebraic structure of all L-contact relations.


---

## Body

## 1. Introduction

Contact relations have been studied on two different contexts, the proximity relations [1] and the theory of pointless geometry (topology) [2], since the early 1920s. Recently, it has become a powerful tool in several areas of artificial intelligence, such as qualitative spatial reasoning and ontology building; see [3–8].On the other hand, the notion of anL-set was introduced in [9], as a generalization of Zadeh’s (classical) notion of a fuzzy set [10]. Fuzzy relational modeling processes have been studied, such as fuzzy concept lattices [11].In [12], Winter investigated time-dependent contact structure in Goguen Categories. It turns out that a suitable theory can be defined using an L-valued or L-fuzzy version of a contact relation. In [13–15], we introduced the notion of contact relation in fuzzy setting and discussed some properties of way-below relation, continuous lattice induced by an L-contact relation.In this paper, we want to generalize the theory of contact relations in fuzzy setting. First, Section2 surveys an overview of contact relations, L-sets. Then, Section 3 generalizes the notion of a contact relation in fuzzy setting and presents some examples. Section 4 recalls the notions of an L-filter, an L-relation, and an L-topology. Section 5 establishes the order preserving correspondence between the set of all L-contact relations on L
X and the set of all closed, reflexive, symmetric relations on Ult(
L
X
). Section 6 focuses on the algebraic structure of all L-contact relations.


## 2. Preliminaries

Let us recall some main notions for each area, that is, contact relations [6–8] and L-sets [11].


### 2.1. Contact Relations

We assume familiarity with the notions of Boolean algebra and lattice [16, 17]. Suppose (
B
,
+
,
·
,
∗
,
0,1
) is a Boolean algebra, R
⊆
B
×
B is called a binary relation on B, and all relations are denoted by Rel(
B
). In [6], first, Düntsch and Winter considered the notion of a contact relation on a Boolean algebra B.Definition 1.
SupposeC
∈
R
e
l
(
B
), and consider the following properties: for all x
,
y
,
z
∈
B,(
C
0
)
0
(
-
C
)
x, that is, for x
≠
0, 0 and x are not C-related,
(
C
1
)
x
C
x for x
≠
0,
(
C
2
)
x
C
y
⇒
y
C
x,
(
C
3
)
x
C
y and y
≤
z
⇒
x
C
z,
(
C
4
)
x
C
(
y
+
z
)
⇒
x
C
y or x
C
z.
Cis called a contact relation, and (
B
,
C
) is called a Boolean contact algebra, if C satisfies C
0
–
(
C
4
). C
R
(
B
) denotes the set of all contact relations on B.
Second they discussed the set Ult(B) which is of all ultrafilters on B and the set R
e
l
r
s
c
(
U
l
t
(
B
)
) of all reflexive and symmetric relations on Ult(B) that are closed in the product topology of U
l
t
(
B
)
2.
Third, they investigated the relation betweenC
R
(
B
) and R
e
l
r
s
c
(
U
l
t
(
B
)
) and obtained the representation theorem in [6].Theorem 2.
Suppose thatB is a Boolean algebra. Then, there is a bijective order preserving correspondence between the contact relations on B and the reflexive and symmetric relations on U
l
t
(
B
) that are closed in the product topology of U
l
t
(
B
)
2.Then with the help of Theorem2, they studied the structure of C
R
(
B
) by means of the set R
e
l
r
s
c
(
U
l
t
(
B
)
).For further information on contact relations and Boolean contact algebras, see [3–8].


### 2.2.L-Sets

As a generalization of Zadeh’s (classical) notion of a fuzzy set [10], the notion of an L-set was introduced in [9]. An overview of the theory of L-sets and L-relations (i.e., fuzzy sets and relations in the framework of complete residuated lattices) can be found in [11].Definition 3.
A residuated lattice is an algebraL
=
〈
L
,
⋁
,
⋀
,
⊗
,
→
,
0,1
〉 such that(1)
〈
L
,
⋁
,
⋀
,
0,1
〉 is a lattice with the smallest element 0 and the largest element 1,
(2)
〈
L
,
⊗
,
1
〉 is a commutative monoid; that is, ⊗ is associative, commutative, and holds the identity a
⊗
1
=
a,
(3)
⊗
,
→ form an adjoint pair; that is, a
⊗
b
≤
c iff a
≤
b
→
c holds for all a
,
b
,
c
∈
L.
Residuated latticeL is called complete if 〈
L
,
⋁
,
⋀
〉 is a complete lattice.In this paper, we assume thatL is a complete Heyting algebra which is a complete residuated lattice satisfying a
⊗
b
=
a
∧
b.For a universe setX, an L-set in X is a mapping A
:
X
→
L. A
(
x
) indicates the truth degree of “x belongs to A.” We use the symbol L
X to denote the set of all L-sets in X. The negation operator is defined: for A
∈
L
X, A
∗
(
x
)
=
A
(
x
)
→
0 for every x
∈
X.Definition 4.
(1) Suppose{
A
i
∣
i
∈
I
}
⊆
L
X is a system of L-sets, and ⋁
i
∈
I
A
i and ⋀
i
∈
I
A
i are two L-sets defined as follows, for every x
∈
X:(1)
⋁
i
∈
I
A
i
x
=
⋁
i
∈
I
A
i
x
,
⋀
i
∈
I
A
i
x
=
⋀
i
∈
I
A
i
x
.
(2) SupposeU is an L-set in L
X; that is, U
:
L
X
→
L is a mapping. For every A
∈
L
X, U
(
A
) is called the degree of membership of A in U. Then U is a generalization of a system of subsets in the classical case.
TwoL-sets ⋃
‍
U
,
⋂
‍
U in X are defined: for every x
∈
X,(2)
⋃
U
x
=
⋁
A
∈
L
X
U
A
∧
A
x
,
⋂
U
x
=
⋀
A
∈
L
X
U
A
⟶
A
x
.
Clearly,⋃
‍
U and ⋂
‍
U are generalizations of the union and the intersection of a system of sets in the classical case (see [11]).
IfU is a system of L-sets in L
X, that is, U
=
{
A
i
∣
i
∈
I
}
⊆
L
X, we have ⋃
‍
U
=
⋁
U, and ⋂
‍
U
=
⋀
U.Note 1.
In ordinary set theory, supposeX is a set and {
A
i
∣
i
∈
I
} is a system of subsets of X, if A
i
⊆
X. Clearly {
A
i
∣
i
∈
I
}
⊆
2
X, where 2
X is the power set of X.
The classical order≤ and equality = are generalized in fuzzy setting, that is, L-relation and L-equality.
I
∈
L
X
×
X is called an L-binary relation. The truth degree to which elements x and y are related by an L-relation I is denoted by I
(
x
,
y
) or (
x
I
y
).
A binaryL-relation I on X is an L-equivalence if it satisfies the following: ∀
x
,
y
,
z
∈
X, I
(
x
,
x
)
=
1 (reflexivity), I
(
x
,
y
)
=
I
(
y
,
x
) (symmetry), and I
(
x
,
y
)
⊗
I
(
y
,
z
)
≤
I
(
x
,
z
) (transitivity). An L-equivalence is an L-equality if it satisfies the following: I
(
x
,
y
)
=
1 implies x
=
y.
AnL-order on X with an L-equality relation ≈ is a binary L-relation ⪯ which is compatible with respect to ≈ and satisfies the following: ∀
x
,
y
,
z
∈
X, (
x
⪯
x
)
=
1 (reflexivity), (
x
⪯
y
)
∧
(
y
⪯
x
)
≤
(
x
≈
y
) (antisymmetry), and (
x
⪯
y
)
⊗
(
y
⪯
z
)
≤
(
x
⪯
z
) (transitivity). A set X equipped with an L-order ⪯ and an L-equality ≈ is called an L-ordered set 〈
〈
X
,
≈
〉
,
⪯
〉.
The subsethood degreeS
(
A
,
B
) is defined as follows: for A
,
B
∈
L
X,(3)
S
A
,
B
=
⋀
x
∈
X
A
x
⟶
B
x
,
A
⪯
B
=
S
A
,
B
,
A
≈
B
=
S
A
,
B
∧
S
B
,
A
.We write A
⊆
B, if S
(
A
,
B
)
=
1.Example 5.
For∅
≠
M
⊆
L
X, we obtain that 〈
〈
M
,
≈
〉
,
S
〉 is an L-ordered set. In fact, reflexivity and antisymmetry are trivial, and we have to prove transitivity and compatibility. Transitivity is described as follows: S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
S
(
A
,
C
) holds if and only if S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
A
(
x
)
→
C
(
x
); that is, ∀
x
∈
X, A
(
x
)
⊗
S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
C
(
x
), and it is true since A
(
x
)
⊗
S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
A
(
x
)
⊗
(
A
(
x
)
→
B
(
x
)
)
⊗
(
B
(
x
)
→
C
(
x
)
)
≤
C
(
x
). In a similar way, we also prove compatibility: S
(
A
,
B
)
⊗
(
A
≈
A
′
)
⊗
(
B
≈
B
′
)
≤
S
(
A
′
,
B
′
).By Example5, we know that 〈
〈
L
X
,
≈
〉
,
⪯
〉 is an L-ordered set.


## 2.1. Contact Relations

We assume familiarity with the notions of Boolean algebra and lattice [16, 17]. Suppose (
B
,
+
,
·
,
∗
,
0,1
) is a Boolean algebra, R
⊆
B
×
B is called a binary relation on B, and all relations are denoted by Rel(
B
). In [6], first, Düntsch and Winter considered the notion of a contact relation on a Boolean algebra B.Definition 1.
SupposeC
∈
R
e
l
(
B
), and consider the following properties: for all x
,
y
,
z
∈
B,(
C
0
)
0
(
-
C
)
x, that is, for x
≠
0, 0 and x are not C-related,
(
C
1
)
x
C
x for x
≠
0,
(
C
2
)
x
C
y
⇒
y
C
x,
(
C
3
)
x
C
y and y
≤
z
⇒
x
C
z,
(
C
4
)
x
C
(
y
+
z
)
⇒
x
C
y or x
C
z.
Cis called a contact relation, and (
B
,
C
) is called a Boolean contact algebra, if C satisfies C
0
–
(
C
4
). C
R
(
B
) denotes the set of all contact relations on B.
Second they discussed the set Ult(B) which is of all ultrafilters on B and the set R
e
l
r
s
c
(
U
l
t
(
B
)
) of all reflexive and symmetric relations on Ult(B) that are closed in the product topology of U
l
t
(
B
)
2.
Third, they investigated the relation betweenC
R
(
B
) and R
e
l
r
s
c
(
U
l
t
(
B
)
) and obtained the representation theorem in [6].Theorem 2.
Suppose thatB is a Boolean algebra. Then, there is a bijective order preserving correspondence between the contact relations on B and the reflexive and symmetric relations on U
l
t
(
B
) that are closed in the product topology of U
l
t
(
B
)
2.Then with the help of Theorem2, they studied the structure of C
R
(
B
) by means of the set R
e
l
r
s
c
(
U
l
t
(
B
)
).For further information on contact relations and Boolean contact algebras, see [3–8].


## 2.2.L-Sets

As a generalization of Zadeh’s (classical) notion of a fuzzy set [10], the notion of an L-set was introduced in [9]. An overview of the theory of L-sets and L-relations (i.e., fuzzy sets and relations in the framework of complete residuated lattices) can be found in [11].Definition 3.
A residuated lattice is an algebraL
=
〈
L
,
⋁
,
⋀
,
⊗
,
→
,
0,1
〉 such that(1)
〈
L
,
⋁
,
⋀
,
0,1
〉 is a lattice with the smallest element 0 and the largest element 1,
(2)
〈
L
,
⊗
,
1
〉 is a commutative monoid; that is, ⊗ is associative, commutative, and holds the identity a
⊗
1
=
a,
(3)
⊗
,
→ form an adjoint pair; that is, a
⊗
b
≤
c iff a
≤
b
→
c holds for all a
,
b
,
c
∈
L.
Residuated latticeL is called complete if 〈
L
,
⋁
,
⋀
〉 is a complete lattice.In this paper, we assume thatL is a complete Heyting algebra which is a complete residuated lattice satisfying a
⊗
b
=
a
∧
b.For a universe setX, an L-set in X is a mapping A
:
X
→
L. A
(
x
) indicates the truth degree of “x belongs to A.” We use the symbol L
X to denote the set of all L-sets in X. The negation operator is defined: for A
∈
L
X, A
∗
(
x
)
=
A
(
x
)
→
0 for every x
∈
X.Definition 4.
(1) Suppose{
A
i
∣
i
∈
I
}
⊆
L
X is a system of L-sets, and ⋁
i
∈
I
A
i and ⋀
i
∈
I
A
i are two L-sets defined as follows, for every x
∈
X:(1)
⋁
i
∈
I
A
i
x
=
⋁
i
∈
I
A
i
x
,
⋀
i
∈
I
A
i
x
=
⋀
i
∈
I
A
i
x
.
(2) SupposeU is an L-set in L
X; that is, U
:
L
X
→
L is a mapping. For every A
∈
L
X, U
(
A
) is called the degree of membership of A in U. Then U is a generalization of a system of subsets in the classical case.
TwoL-sets ⋃
‍
U
,
⋂
‍
U in X are defined: for every x
∈
X,(2)
⋃
U
x
=
⋁
A
∈
L
X
U
A
∧
A
x
,
⋂
U
x
=
⋀
A
∈
L
X
U
A
⟶
A
x
.
Clearly,⋃
‍
U and ⋂
‍
U are generalizations of the union and the intersection of a system of sets in the classical case (see [11]).
IfU is a system of L-sets in L
X, that is, U
=
{
A
i
∣
i
∈
I
}
⊆
L
X, we have ⋃
‍
U
=
⋁
U, and ⋂
‍
U
=
⋀
U.Note 1.
In ordinary set theory, supposeX is a set and {
A
i
∣
i
∈
I
} is a system of subsets of X, if A
i
⊆
X. Clearly {
A
i
∣
i
∈
I
}
⊆
2
X, where 2
X is the power set of X.
The classical order≤ and equality = are generalized in fuzzy setting, that is, L-relation and L-equality.
I
∈
L
X
×
X is called an L-binary relation. The truth degree to which elements x and y are related by an L-relation I is denoted by I
(
x
,
y
) or (
x
I
y
).
A binaryL-relation I on X is an L-equivalence if it satisfies the following: ∀
x
,
y
,
z
∈
X, I
(
x
,
x
)
=
1 (reflexivity), I
(
x
,
y
)
=
I
(
y
,
x
) (symmetry), and I
(
x
,
y
)
⊗
I
(
y
,
z
)
≤
I
(
x
,
z
) (transitivity). An L-equivalence is an L-equality if it satisfies the following: I
(
x
,
y
)
=
1 implies x
=
y.
AnL-order on X with an L-equality relation ≈ is a binary L-relation ⪯ which is compatible with respect to ≈ and satisfies the following: ∀
x
,
y
,
z
∈
X, (
x
⪯
x
)
=
1 (reflexivity), (
x
⪯
y
)
∧
(
y
⪯
x
)
≤
(
x
≈
y
) (antisymmetry), and (
x
⪯
y
)
⊗
(
y
⪯
z
)
≤
(
x
⪯
z
) (transitivity). A set X equipped with an L-order ⪯ and an L-equality ≈ is called an L-ordered set 〈
〈
X
,
≈
〉
,
⪯
〉.
The subsethood degreeS
(
A
,
B
) is defined as follows: for A
,
B
∈
L
X,(3)
S
A
,
B
=
⋀
x
∈
X
A
x
⟶
B
x
,
A
⪯
B
=
S
A
,
B
,
A
≈
B
=
S
A
,
B
∧
S
B
,
A
.We write A
⊆
B, if S
(
A
,
B
)
=
1.Example 5.
For∅
≠
M
⊆
L
X, we obtain that 〈
〈
M
,
≈
〉
,
S
〉 is an L-ordered set. In fact, reflexivity and antisymmetry are trivial, and we have to prove transitivity and compatibility. Transitivity is described as follows: S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
S
(
A
,
C
) holds if and only if S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
A
(
x
)
→
C
(
x
); that is, ∀
x
∈
X, A
(
x
)
⊗
S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
C
(
x
), and it is true since A
(
x
)
⊗
S
(
A
,
B
)
⊗
S
(
B
,
C
)
≤
A
(
x
)
⊗
(
A
(
x
)
→
B
(
x
)
)
⊗
(
B
(
x
)
→
C
(
x
)
)
≤
C
(
x
). In a similar way, we also prove compatibility: S
(
A
,
B
)
⊗
(
A
≈
A
′
)
⊗
(
B
≈
B
′
)
≤
S
(
A
′
,
B
′
).By Example5, we know that 〈
〈
L
X
,
≈
〉
,
⪯
〉 is an L-ordered set.


## 3. L-Contact Relations

In this section, our aim is to define the notion of a contact relation in fuzzy setting, present some examples, and show that allL-contact relations form an L-ordered set 〈
〈
C
R
(
L
X
)
,
≈
〉
,
⪯
〉.SupposeX is a universe set and L
X is the set of all L-sets in X. We know that (
L
X
,
⋃
‍
,
⋂
‍
,
∗
,
0
X
,
1
X
) might not be a Boolean algebra, where 0
X
:
X
→
L and 1
X
:
X
→
L are defined: 0
X
(
x
)
=
0 and 1
X
(
x
)
=
1, for every x
∈
X, respectively.A mappingθ
:
L
X
×
L
X
→
L is called an L-binary relation on L
X. The truth degree to which elements A and B are related by an L-relation θ is denoted by θ
(
A
,
B
). The collection of all L-relations is denoted by R
e
l
(
L
X
).First, from the point of view of graded truth approach, we generalize the notion of a contact relation in fuzzy setting as follows.Definition 6.
SupposeX is a universe set, L
X is the set of all L-sets in X
,
θ
∈
R
e
l
(
L
X
) is an L-relation on L
X, for A
,
B
,
C
∈
L
X, and consider the following properties:(
C
0
)
θ
(
0
X
,
A
)
=
0.
(
C
1
)
θ
(
A
,
A
)
=
1 for A
≠
0
X.
(
C
2
)
θ
(
A
,
B
)
=
θ
(
B
,
A
).
(
C
3
)
B
⊆
C
⇒
θ
(
A
,
B
)
≤
θ
(
A
,
C
).
(
C
4
)
θ
(
A
,
B
∪
C
)
≤
θ
(
A
,
B
)
∨
θ
(
A
,
C
).
θ is called an L-contact relation, and (
L
X
,
θ
) is called an L-contact algebra, if θ satisfies (
C
0
)
–
(
C
4
).In the paper, we will denote the set of allL-contact relations on L
X by CR(
L
X
). In [6–8], C denotes a contact relation. From now on, we write θ instead of C. Obviously, Definition 6 is a generalization of Definition 1 in fuzzy setting.Second, we give some examples ofL-contact relations.Example 7.
ForA
,
B
∈
L
X, we define an L-contact relation:(4)
1
C
R
L
X
A
,
B
=
1
,
i
f
A
≠
0
X
,
B
≠
0
X
,
0
,
o
t
h
e
r
w
i
s
e
.Example 8.
ForA
,
B
∈
L
X, let(5)
O
C
R
L
X
A
,
B
=
1
,
i
f
A
∧
B
≠
0
X
,
0
,
o
t
h
e
r
w
i
s
e
.Then O
C
R
(
L
X
) is a contact relation on L
X.
WhenL
=
2, L
X is the power set of X; that is, L
X
=
2
X; then (
L
X
,
⋃
‍
,
⋂
‍
,
∗
,
0
X
,
1
X
)
=
(
2
X
,
⋃
‍
,
⋂
‍
,
∗
,
∅
,
X
) is a Boolean algebra. Clearly, we have(6)
O
C
R
L
X
A
,
B
=
1
⟺
A
∩
B
≠
∅
,
O
C
R
L
X
A
,
B
=
0
⟺
A
∩
B
=
∅
.Note 2.
In [18], we introduced the relation ρ on L
X. We show that O
C
R
(
L
X
) and ρ are the same in the classical case.Definition 9.
SupposeA
,
B
∈
L
X, and let ρ
(
A
,
B
)
=
⋁
x
∈
X
A
(
x
)
⊗
B
(
x
), which expresses the related degree of A and B with respect to ⊗ [18].
Obviously,ρ is not a contact relation on L
X, since, for every A
∈
L
X, ρ
(
A
,
A
)
=
1 does not hold. But when L
=
2, ρ
(
A
,
B
)
=
1
⇔
A
∩
B
≠
∅; ρ
(
A
,
B
)
=
0
⇔
A
∩
B
=
∅. So the relation ρ is the same with the relation O
C
R
(
L
X
) in the classical case.In the rest of the section, we define⪯ and ≈ on C
R
(
L
X
). For θ
1
,
θ
2
∈
C
R
(
L
X
),(7)
θ
1
⪯
θ
2
=
S
θ
1
,
θ
2
=
⋀
A
,
B
∈
L
X
θ
1
A
,
B
⟶
θ
2
A
,
B
,
θ
1
≈
θ
2
=
S
θ
1
,
θ
2
∧
S
θ
2
,
θ
1
=
⋀
A
,
B
∈
L
X
θ
1
A
,
B
⟷
θ
2
A
,
B
.Thus 〈
〈
C
R
(
L
X
)
,
≈
〉
,
⪯
〉 is an L-ordered set; see [11].In special,θ
1
⊆
θ
2
⇔
S
(
θ
1
,
θ
2
)
=
1
⇔
θ
1
(
A
,
B
)
≤
θ
2
(
A
,
B
) for any A
,
B
∈
L
X.Notice thatO
C
R
(
L
X
) is the smallest element in C
R
(
L
X
). For any A
,
B
∈
L
X and each θ
∈
C
R
(
L
X
),A
∧
B
=
0
X
⇒
O
C
R
(
L
X
)
(
A
,
B
)
=
0
⇒
O
C
R
(
L
X
)
(
A
,
B
)
≤
θ
(
A
,
B
);
A
∧
B
≠
0
X, and let C
=
A
∧
B
≠
0
X;
⇒
1
=
O
C
R
(
L
X
)
(
C
,
C
)
≤
O
C
R
(
L
X
)
(
A
,
B
), and 1
=
θ
(
C
,
C
)
≤
θ
(
A
,
B
);
⇒
O
C
R
(
L
X
)
(
A
,
B
)
=
θ
(
A
,
B
)
=
1.HenceO
C
R
(
L
X
)
⊆
θ.


## 4. L-Filters and L-Topologies

This section is devoted to recall three notions: anL-filter on X, an L-relation, and an L-topology on the set of all L-ultrafilters, showing that 〈
〈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
,
≈
〉
,
⪯
〉 is an L-ordered set.


### 4.1.L-Filters on X

We recall the notion of anL-filter on X; see [19, Chapter 3].Definition 10.
A mapν
:
L
X
→
L is called an L-filter on X, if ν fulfills the following axioms:(F0)
ν
(
1
X
)
=
1.
(F1)
A
⊆
B
⇒
ν
(
A
)
≤
ν
(
B
).
(F2)
ν
(
A
)
∧
ν
(
B
)
≤
ν
(
A
∧
B
).
(F3)
ν
(
0
X
)
=
0
.
Now, we present some examples ofL-filters.Example 11.
The functionν defined by ν
(
A
)
=
⋀
x
∈
X
A
(
x
) for every A
∈
L
X is an L-filter.
The proof in detail is in [19, Chapter 3].Example 12.
For everyA
∈
L
X and A
≠
0
X, an L-filter is defined: (8)
ν
A
D
=
1
,
i
f
A
⊆
D
,
0
,
o
t
h
e
r
w
i
s
e
.
LetF
(
L
X
) be the set of all L-filters on X, on which we introduce the partial orderings ⪯ and ≈, for ν
1
,
ν
2
∈
F
(
L
X
):(9)
ν
1
⪯
ν
2
=
S
ν
1
,
ν
2
=
⋀
A
∈
L
X
ν
1
A
⟶
ν
2
A
,
ν
1
≈
ν
2
=
S
ν
1
,
ν
2
∧
S
ν
2
,
ν
1
,and thus 〈
〈
F
(
L
X
)
,
≈
〉
,
⪯
〉 is an L-ordered set; see [11].
In special,ν
1
⊆
ν
2
⇔
S
(
ν
1
,
ν
2
)
=
1; that is, ν
1
(
A
)
≤
ν
2
(
A
) for every A
∈
L
X.
Clearly,0
L
X
~
:
L
X
→
L is the smallest filter on F
(
L
X
), where 0
L
X
~
(
A
)
=
0 for every A
∈
L
X
,
A
≠
1
X; and 0
L
X
~
(
1
X
)
=
1.
For instance, whenL
=
2, L
X
=
2
X is the power set of X, 0
L
X
~
=
{
X
}.
In [19, Chapter 3], the notion of an L-filter was investigated. By Theorem 6.2
.
9 [19, Chapter 3], we know that the partially ordered set (
F
(
L
X
)
,
⊆
) has maximal elements. A maximal element in (
F
(
L
X
)
,
⊆
) is also called an L-ultrafilter. An L-filter ν is an L-ultrafilter if and only if ν
(
A
)
=
(
ν
(
A
∗
)
)
∗ for every A
∈
L
X [19, Theorem 6.2
.
11].
For everyA
∈
L
X, there exists an L-ultrafilter η
A, such that η
A
(
A
)
=
1, that is, Lemma 13.Lemma 13.
SupposeA
∈
L
X
,
A
≠
0
X, and then there exists an L-ultrafilter η
A, such that η
A
(
A
)
=
1.Proof.
ForA
∈
L
X, A
≠
0
X,(1)
by Example12, we obtain an L-filter ν
A satisfying ν
A
(
A
)
=
1,
(2)
by Corollary6.2
.
10 in [19, Chapter 3], there exists an L-ultrafilter η
A, such that ν
A
⊆
η
A. Certainly η
A
(
A
)
=
1.Lemma 14.
Supposeν is an L-ultrafilter; then we have ν
(
B
∪
C
)
≤
ν
(
B
)
∨
ν
(
C
).Proof.
In [11], the notion of a filter in a residuated lattice L was introduced. In fact, ν is an L-ultrafilter on X
⇔
{
ν
(
A
)
∣
A
∈
L
X
} which is a maximal filter in L. By Lemma 2.64 [11, P55], Lemma 14 holds.
Supposeν
1
,
ν
2
∈
U
l
t
L
X, and we define ν
1
∧
ν
2 and ν
1
∨
ν
2 as follows:(10)
ν
1
∧
ν
2
A
=
ν
1
A
∧
ν
2
A
,
ν
1
∨
ν
2
A
=
ν
1
A
∨
ν
2
A
,
f
o
r
a
n
y
A
∈
L
X
.
If{
ν
i
∣
i
∈
I
}
⊆
U
l
t
(
L
X
), ⋁
i
∈
I
ν
i is also defined: (
⋁
i
∈
I
ν
i
)
(
A
)
=
⋁
i
∈
I
ν
i
(
A
) for any A
∈
L
X.


### 4.2.L-Relations on Ult
(
L
X
)

In the section, we turn toL-relations on U
l
t
(
L
X
).A mappingR
:
U
l
t
(
L
X
)
×
U
l
t
(
L
X
)
→
L is also an L-relation on U
l
t
(
L
X
). All of them will be denoted by R
e
l
(
U
l
t
(
L
X
)
). Suppose R
1
,
R
2
∈
R
e
l
(
U
l
t
(
L
X
)
), and we define R
1
∧
R
2 and R
1
∨
R
2 as follows: (11)
R
1
∧
R
2
ν
1
,
ν
2
=
R
1
ν
1
,
ν
2
∧
R
2
ν
1
,
ν
2
,
R
1
∨
R
2
ν
1
,
ν
2
=
R
1
ν
1
,
ν
2
∨
R
2
ν
1
,
ν
2
,
f
o
r
a
n
y
ν
1
,
ν
2
∈
U
l
t
L
X
.For example,1
U
l
t
(
L
X
) and 0
U
l
t
(
L
X
) are defined as follows: 1
U
l
t
(
L
X
)
(
ν
1
,
ν
2
)
=
1 and 0
U
l
t
(
L
X
)
(
ν
1
,
ν
2
)
=
0 for ν
1
,
ν
2
∈
U
l
t
(
L
X
). Clearly, 1
U
l
t
(
L
X
) is the largest element in R
e
l
(
U
l
t
(
L
X
)
).ForR
∈
R
e
l
(
U
l
t
(
L
X
)
), it is symmetric if R
(
ν
1
,
ν
2
)
=
R
(
ν
2
,
ν
1
) for any ν
1
,
ν
2
∈
U
l
t
(
L
X
). It is reflexive if, for every ν
∈
U
l
t
(
L
X
), R
(
ν
,
ν
)
=
1.


### 4.3.L-Topologies on Ult
(
L
X
)

We introduce the notion of anL-topology on U
l
t
(
L
X
).Definition 15.
SupposeT
⊆
U
l
t
(
L
X
) is called an L-topology on U
l
t
(
L
X
), if(1)
ν
1
,
ν
2
∈
T
⇒
ν
1
∧
ν
2
∈
T,
(2)
for every subset{
ν
i
∣
i
∈
I
} of T, ⋁
i
∈
I
ν
i
∈
T holds.
C
L
(
R
) denotes the closure of R in the product topology T
×
T on U
l
t
(
L
X
)
×
U
l
t
(
L
X
). If R is a closed set in the product topology T
×
T, then R is called a close relation.
The collection of all close, reflexive, and symmetric relations onU
l
t
(
L
X
) is denoted by R
e
l
r
s
c
(
U
l
t
(
L
X
)
). Clearly, 1
U
l
t
(
L
X
)
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
).
ForR
1
,
R
2
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), we define L-order ⪯ and ≈ on R
e
l
r
s
c
(
U
l
t
(
L
X
)
):(12)
S
R
1
,
R
2
=
R
1
⪯
R
2
=
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
⟶
R
2
ν
1
,
ν
2
,
R
1
≈
R
2
=
S
R
1
,
R
2
∧
S
R
2
,
R
1
.
Thus〈
〈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
,
≈
〉
,
⪯
〉 is an L-ordered set; see [11].
ForA
∈
L
X, let h
(
A
)
=
{
ν
∣
ν
(
A
)
≠
0
}, and we assume that {
h
(
A
)
×
h
(
B
)
∣
A
,
B
∈
L
X
} is a base for the product topology T
×
T.


## 4.1.L-Filters on X

We recall the notion of anL-filter on X; see [19, Chapter 3].Definition 10.
A mapν
:
L
X
→
L is called an L-filter on X, if ν fulfills the following axioms:(F0)
ν
(
1
X
)
=
1.
(F1)
A
⊆
B
⇒
ν
(
A
)
≤
ν
(
B
).
(F2)
ν
(
A
)
∧
ν
(
B
)
≤
ν
(
A
∧
B
).
(F3)
ν
(
0
X
)
=
0
.
Now, we present some examples ofL-filters.Example 11.
The functionν defined by ν
(
A
)
=
⋀
x
∈
X
A
(
x
) for every A
∈
L
X is an L-filter.
The proof in detail is in [19, Chapter 3].Example 12.
For everyA
∈
L
X and A
≠
0
X, an L-filter is defined: (8)
ν
A
D
=
1
,
i
f
A
⊆
D
,
0
,
o
t
h
e
r
w
i
s
e
.
LetF
(
L
X
) be the set of all L-filters on X, on which we introduce the partial orderings ⪯ and ≈, for ν
1
,
ν
2
∈
F
(
L
X
):(9)
ν
1
⪯
ν
2
=
S
ν
1
,
ν
2
=
⋀
A
∈
L
X
ν
1
A
⟶
ν
2
A
,
ν
1
≈
ν
2
=
S
ν
1
,
ν
2
∧
S
ν
2
,
ν
1
,and thus 〈
〈
F
(
L
X
)
,
≈
〉
,
⪯
〉 is an L-ordered set; see [11].
In special,ν
1
⊆
ν
2
⇔
S
(
ν
1
,
ν
2
)
=
1; that is, ν
1
(
A
)
≤
ν
2
(
A
) for every A
∈
L
X.
Clearly,0
L
X
~
:
L
X
→
L is the smallest filter on F
(
L
X
), where 0
L
X
~
(
A
)
=
0 for every A
∈
L
X
,
A
≠
1
X; and 0
L
X
~
(
1
X
)
=
1.
For instance, whenL
=
2, L
X
=
2
X is the power set of X, 0
L
X
~
=
{
X
}.
In [19, Chapter 3], the notion of an L-filter was investigated. By Theorem 6.2
.
9 [19, Chapter 3], we know that the partially ordered set (
F
(
L
X
)
,
⊆
) has maximal elements. A maximal element in (
F
(
L
X
)
,
⊆
) is also called an L-ultrafilter. An L-filter ν is an L-ultrafilter if and only if ν
(
A
)
=
(
ν
(
A
∗
)
)
∗ for every A
∈
L
X [19, Theorem 6.2
.
11].
For everyA
∈
L
X, there exists an L-ultrafilter η
A, such that η
A
(
A
)
=
1, that is, Lemma 13.Lemma 13.
SupposeA
∈
L
X
,
A
≠
0
X, and then there exists an L-ultrafilter η
A, such that η
A
(
A
)
=
1.Proof.
ForA
∈
L
X, A
≠
0
X,(1)
by Example12, we obtain an L-filter ν
A satisfying ν
A
(
A
)
=
1,
(2)
by Corollary6.2
.
10 in [19, Chapter 3], there exists an L-ultrafilter η
A, such that ν
A
⊆
η
A. Certainly η
A
(
A
)
=
1.Lemma 14.
Supposeν is an L-ultrafilter; then we have ν
(
B
∪
C
)
≤
ν
(
B
)
∨
ν
(
C
).Proof.
In [11], the notion of a filter in a residuated lattice L was introduced. In fact, ν is an L-ultrafilter on X
⇔
{
ν
(
A
)
∣
A
∈
L
X
} which is a maximal filter in L. By Lemma 2.64 [11, P55], Lemma 14 holds.
Supposeν
1
,
ν
2
∈
U
l
t
L
X, and we define ν
1
∧
ν
2 and ν
1
∨
ν
2 as follows:(10)
ν
1
∧
ν
2
A
=
ν
1
A
∧
ν
2
A
,
ν
1
∨
ν
2
A
=
ν
1
A
∨
ν
2
A
,
f
o
r
a
n
y
A
∈
L
X
.
If{
ν
i
∣
i
∈
I
}
⊆
U
l
t
(
L
X
), ⋁
i
∈
I
ν
i is also defined: (
⋁
i
∈
I
ν
i
)
(
A
)
=
⋁
i
∈
I
ν
i
(
A
) for any A
∈
L
X.


## 4.2.L-Relations on Ult
(
L
X
)

In the section, we turn toL-relations on U
l
t
(
L
X
).A mappingR
:
U
l
t
(
L
X
)
×
U
l
t
(
L
X
)
→
L is also an L-relation on U
l
t
(
L
X
). All of them will be denoted by R
e
l
(
U
l
t
(
L
X
)
). Suppose R
1
,
R
2
∈
R
e
l
(
U
l
t
(
L
X
)
), and we define R
1
∧
R
2 and R
1
∨
R
2 as follows: (11)
R
1
∧
R
2
ν
1
,
ν
2
=
R
1
ν
1
,
ν
2
∧
R
2
ν
1
,
ν
2
,
R
1
∨
R
2
ν
1
,
ν
2
=
R
1
ν
1
,
ν
2
∨
R
2
ν
1
,
ν
2
,
f
o
r
a
n
y
ν
1
,
ν
2
∈
U
l
t
L
X
.For example,1
U
l
t
(
L
X
) and 0
U
l
t
(
L
X
) are defined as follows: 1
U
l
t
(
L
X
)
(
ν
1
,
ν
2
)
=
1 and 0
U
l
t
(
L
X
)
(
ν
1
,
ν
2
)
=
0 for ν
1
,
ν
2
∈
U
l
t
(
L
X
). Clearly, 1
U
l
t
(
L
X
) is the largest element in R
e
l
(
U
l
t
(
L
X
)
).ForR
∈
R
e
l
(
U
l
t
(
L
X
)
), it is symmetric if R
(
ν
1
,
ν
2
)
=
R
(
ν
2
,
ν
1
) for any ν
1
,
ν
2
∈
U
l
t
(
L
X
). It is reflexive if, for every ν
∈
U
l
t
(
L
X
), R
(
ν
,
ν
)
=
1.


## 4.3.L-Topologies on Ult
(
L
X
)

We introduce the notion of anL-topology on U
l
t
(
L
X
).Definition 15.
SupposeT
⊆
U
l
t
(
L
X
) is called an L-topology on U
l
t
(
L
X
), if(1)
ν
1
,
ν
2
∈
T
⇒
ν
1
∧
ν
2
∈
T,
(2)
for every subset{
ν
i
∣
i
∈
I
} of T, ⋁
i
∈
I
ν
i
∈
T holds.
C
L
(
R
) denotes the closure of R in the product topology T
×
T on U
l
t
(
L
X
)
×
U
l
t
(
L
X
). If R is a closed set in the product topology T
×
T, then R is called a close relation.
The collection of all close, reflexive, and symmetric relations onU
l
t
(
L
X
) is denoted by R
e
l
r
s
c
(
U
l
t
(
L
X
)
). Clearly, 1
U
l
t
(
L
X
)
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
).
ForR
1
,
R
2
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), we define L-order ⪯ and ≈ on R
e
l
r
s
c
(
U
l
t
(
L
X
)
):(12)
S
R
1
,
R
2
=
R
1
⪯
R
2
=
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
⟶
R
2
ν
1
,
ν
2
,
R
1
≈
R
2
=
S
R
1
,
R
2
∧
S
R
2
,
R
1
.
Thus〈
〈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
,
≈
〉
,
⪯
〉 is an L-ordered set; see [11].
ForA
∈
L
X, let h
(
A
)
=
{
ν
∣
ν
(
A
)
≠
0
}, and we assume that {
h
(
A
)
×
h
(
B
)
∣
A
,
B
∈
L
X
} is a base for the product topology T
×
T.


## 5. The Correspondence betweenRel
rsc
(
Ult
(
L
X
)
) and CR
(
L
X
)

In [6], Düntsch and Winter proved that there exists a bijective order preserving correspondence between R
e
l
r
s
c
(
U
l
t
(
B
)
) and C
R
(
B
), that is, Theorem 2.In Sections3 and 4, we know that two L-ordered sets 〈
〈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
,
≈
〉
,
⪯
〉 and 〈
〈
C
R
(
L
X
)
,
≈
〉
,
⪯
〉 are generalizations of R
e
l
r
s
c
(
U
l
t
(
B
)
) and C
R
(
B
) in fuzzy setting, respectively.In this section, we continue to investigate the relation between twoL-ordered sets 〈
〈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
,
≈
〉
,
⪯
〉 and 〈
〈
C
R
(
L
X
)
,
≈
〉
,
⪯
〉, obtaining a fuzzication of Theorem 2 [6]. For this, we define two mappings:(13)
q
:
R
e
l
r
s
c
U
l
t
L
X
⟶
C
R
L
X
,
p
:
C
R
L
X
⟶
R
e
l
r
s
c
U
l
t
L
X
,and we prove that they are the order preserving correspondences. We divide the work into three steps.Step 1.
We define a mappingq from R
e
l
r
s
c
(
U
l
t
(
L
X
)
) to C
R
(
L
X
) and prove that it is one-to-one.
GivenR
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), for A
,
B
∈
L
X, put(14)
θ
R
A
,
B
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
.Then we have the following.Lemma 16.
SupposeR is a close, reflexive, and symmetric L-relation on U
l
t
(
L
X
); then θ
R is an L-contact relation on L
X.Proof.
we have to show thatθ
R is an L-contact relation, that is, to verify θ
R satisfies the conditions of Definition 6. Consider the following:(
C
0
)
θ
R
(
0
X
,
A
)
=
0.
(
C
1
)
ForA
≠
0
X, by Lemma 13, η
A
(
A
)
=
1, and then (15)
θ
R
A
,
A
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
A
≥
R
η
A
,
η
A
∧
η
A
A
∧
η
A
A
=
1
.
(
C
2
)
θ
R
(
A
,
B
)
=
θ
R
(
B
,
A
) is obvious.
(
C
3
)
B
⊆
C
⇒
ν
(
B
)
≤
ν
(
C
), and then (16)
θ
R
A
,
B
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
≤
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
C
=
θ
R
A
,
C
.
(
C
4
)
By Lemma14, we have(17)
θ
R
A
,
B
∪
C
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
∪
C
≤
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
∨
ν
2
C
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
∨
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
C
L
i
s
a
c
o
m
p
l
e
t
e
H
e
y
t
i
n
g
a
l
g
e
b
r
a
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
∨
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
C
=
θ
R
A
,
B
∨
θ
R
A
,
C
.
So,θ
R is an L-contact relation on L
X; that is, θ
R
∈
C
R
(
L
X
).
Furthermore, for everyR
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), we define q
(
R
)
=
θ
R
∈
C
R
(
L
X
), and thus we obtain a mapping q from R
e
l
r
s
c
(
U
l
t
(
L
X
)
) to C
R
(
L
X
). The key is to prove that q is one-to-one.Lemma 17.
SupposeR is a close, reflexive, and symmetric L-relation on U
l
t
(
L
X
), and then q
:
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
→
C
R
(
L
X
) and q
(
R
)
=
θ
R
∈
C
R
(
L
X
) is an injective mapping.Proof.
By the above analysis, the remainder is to prove thatq is one-to-one, which is equivalent to show for any two close relations R
1
,
R
2
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
); if R
1
≠
R
2, then their images θ
R
1
,
θ
R
2 are not equal.
SupposeR
1
,
R
2 are two distinct close, reflexive, and symmetric relations, and then there exist ν
1
0
,
ν
2
0
∈
U
l
t
(
L
X
), such that R
1
(
ν
1
0
,
ν
2
0
)
≠
R
2
(
ν
1
0
,
ν
2
0
). Without loss of generality, we may assume R
1
(
ν
1
0
,
ν
2
0
)
≰
R
2
(
ν
1
0
,
ν
2
0
).
(1) ConsiderR
1
(
ν
1
0
,
ν
2
0
)
≰
R
2
(
ν
1
0
,
ν
2
0
), clearly ν
1
0
≠
ν
2
0, and there exist A
,
B
∈
L
X, ν
1
0
(
A
)
≠
0, ν
2
0
(
B
)
≠
0, such that(18)
R
1
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
≰
R
2
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
.
(2) SinceR
2 is closed, we have that R
1
(
ν
1
0
,
ν
2
0
)
∧
ν
1
0
(
A
)
∧
ν
2
0
(
B
)
≰
R
2
(
ν
1
,
ν
2
)
∧
ν
1
(
A
)
∧
ν
2
(
B
) holds for any ν
1
,
ν
2
∈
U
l
t
(
L
X
) with ν
1
(
A
)
≠
0
,
ν
2
(
B
)
≠
0.
If not, there existω
1
,
ω
2
∈
U
l
t
(
L
X
), ω
1
(
A
)
≠
0
,
ω
2
(
B
)
≠
0, such that(19)
R
1
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
≤
R
2
ω
1
,
ω
2
∧
ω
1
A
∧
ω
2
B
,and we have (20)
R
1
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
≤
C
L
R
2
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
=
R
2
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
,which leads to a contradiction.
(3) On the other hand, ifθ
R
1
=
θ
R
2, then(21)
R
1
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
≤
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
=
θ
R
1
A
,
B
=
θ
R
2
A
,
B
=
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
2
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
.
SinceR
2 is closed, there exist η
1
,
η
2
∈
U
l
t
(
L
X
) satisfying (22)
R
1
ν
1
0
,
ν
2
0
∧
ν
1
0
A
∧
ν
2
0
B
≤
R
2
η
1
,
η
2
∧
η
1
A
∧
η
2
B
,a contradiction.
Soθ
R
1
≰
θ
R
2 holds, which implies that q is an injective mapping.Note 3.
WhenL
=
2, q
(
R
)
=
{
ν
1
×
ν
2
∣
R
(
ν
1
,
ν
2
)
=
1
}, and θ
R
(
A
,
B
)
=
1
⇔ there exist ν
1
,
ν
2
∈
U
l
t
(
L
X
), such that A
∈
ν
1
,
B
∈
ν
2, and R
(
ν
1
,
ν
2
)
=
1; that is, ν
1
×
ν
2
⊆
q
(
R
).Step 2.
We define a mappingp from C
R
(
L
X
) to R
e
l
r
s
c
(
U
l
t
(
L
X
)
); that is, for each L-contact relation θ
∈
C
R
(
L
X
), we consider defining the corresponding R
θ
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
).
For anL-ultrafilter ν, an L-set in X is defined; that is, (23)
⋂
ν
:
X
⟶
L
,
⋂
ν
x
=
⋀
A
∈
L
X
ν
A
⟶
A
x
,
f
o
r
e
v
e
r
y
x
∈
X
.
Obviously, for anL-ultrafilter ν, we have ⋂
‍
ν
≠
0
X.
Supposeθ
∈
C
R
(
L
X
); in other words, θ is an L-contact relation on L
X, for any L-ultrafilters ν
1
,
ν
2
∈
U
l
t
(
L
X
), and let(24)
R
θ
ν
1
,
ν
2
=
θ
⋂
ν
1
,
⋂
ν
2
.Therefore we have the following lemma.Lemma 18.
Supposeθ
∈
C
R
(
L
X
), then p
:
C
R
(
L
X
)
→
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), for θ
∈
C
R
(
L
X
), and p
(
θ
)
=
C
L
(
R
θ
)
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
) is an injective mapping.Proof.
The closeness, symmetry, and reflexivity ofR
θ follow directly from the above definition.
By the above proof and Lemma17, we know that p is also an injective mapping. This completes the proof.Note 4.
WhenL
=
2, p
(
θ
)
=
{
A
×
B
∣
θ
(
A
,
B
)
=
1
}, and R
(
ν
1
,
ν
2
)
=
1
⇔
∀
A
∈
ν
1
,
B
∈
ν
2, θ
(
A
,
B
)
=
1 holds.Step 3.
By Lemmas16 and 17, we obtain two mappings p
,
q, in the step, and we wish to prove that q
,
p preserve the order ⪯.
First, we prove thatq preserves the order ⪯.Lemma 19.
q
:
R
e
l
r
s
c
(
U
l
t
(
L
X
)
)
→
C
R
(
L
X
) preserves the order ⪯.Proof.
For eachR
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), we define(25)
q
:
R
e
l
r
s
c
U
l
t
L
X
⟶
C
R
L
X
,
q
R
=
θ
R
∈
C
R
L
X
.
We have to prove thatq preserves the order ⪯; that is, (
R
1
⪯
R
2
)
≤
(
q
(
R
1
)
⪯
q
(
R
2
)
). Consider(26)
q
R
1
⪯
q
R
2
=
θ
R
1
⪯
θ
R
2
=
⋀
A
,
B
∈
L
X
θ
R
1
A
,
B
⟶
θ
R
2
A
,
B
=
⋀
A
,
B
∈
L
X
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
⟶
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
2
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
=
⋀
A
,
B
∈
L
X
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
⟶
⋁
ν
1
,
ν
2
∈
U
l
t
L
X
R
2
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
≥
⋀
A
,
B
∈
L
X
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
⟶
R
2
ν
1
,
ν
2
∧
ν
1
A
∧
ν
2
B
≥
⋀
A
,
B
∈
L
X
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
1
ν
1
,
ν
2
⟶
R
2
ν
1
,
ν
2
≥
R
1
⪯
R
2
.
Second, we also prove thatp preserves the order ⪯.Lemma 20.
p
:
C
R
(
L
X
)
→
R
e
l
r
s
c
(
U
l
t
(
L
X
)
) preserves the order ⪯.Proof.
Forθ
∈
C
R
(
L
X
), let(27)
p
:
C
R
L
X
⟶
R
e
l
r
s
c
U
l
t
L
X
,
p
θ
=
C
L
R
θ
∈
R
e
l
r
s
c
U
l
t
L
X
.
We have to prove thatp preserves the order ⪯; that is to say, (
θ
1
⪯
θ
2
)
≤
(
p
(
θ
1
)
⪯
p
(
θ
2
)
). Consider(28)
p
θ
1
⪯
p
θ
2
=
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
p
θ
1
ν
1
,
ν
2
⟶
p
θ
2
ν
1
,
ν
2
=
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
C
L
R
θ
1
ν
1
,
ν
2
⟶
C
L
R
θ
2
ν
1
,
ν
2
≥
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
R
θ
1
ν
1
,
ν
2
⟶
R
θ
2
ν
1
,
ν
2
=
⋀
ν
1
,
ν
2
∈
U
l
t
L
X
θ
1
⋂
ν
1
,
⋂
ν
2
⟶
θ
2
⋂
ν
1
,
⋂
ν
2
≥
⋀
A
,
B
∈
L
X
θ
1
A
,
B
⟶
θ
2
A
,
B
=
θ
1
⪯
θ
2
.Note 5.
SupposeB is a Boolean algebra, and by Lemmas 18, 19, and 20, we obtain the corresponding results: for θ
1
,
θ
2
∈
C
R
(
B
), if θ
1
⊆
θ
2, then p
(
θ
1
)
⊆
p
(
θ
2
) holds. Similarly, for R
1
,
R
2
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), if R
1
⊆
R
2, then q
(
R
1
)
⊆
q
(
R
2
) also holds.So, there exists a bijective order preserving correspondence between the set ofL-contact relations on L
X and the set of all close, reflexive, symmetric relations on U
l
t
(
L
X
), that is, the following proposition, which is a fuzzification of Theorem 2.Proposition 21.
SupposeL
X is the set of all L-sets on X, and then there exists a bijective order preserving correspondence between the set of all L-contact relations on L
X and the set of all close, reflexive, symmetric relations on U
l
t
(
L
X
).


## 6. The Structure of Contact Relations

In this section, we consider two problems; the first is discussing the structure ofR
e
l
r
s
c
(
U
l
t
(
L
X
)
), and the second is obtaining the corresponding results about the C
R
(
L
X
).First, we focus on the structure ofR
e
l
r
s
c
(
U
l
t
(
L
X
)
).LetR
e
l
r
s
c
(
U
l
t
(
L
X
)
) be the set of all close, reflexive, and symmetric relations on U
l
t
(
L
X
). For each R
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), R is a close set in the product topology T
×
T on U
l
t
(
L
X
)
×
U
l
t
(
L
X
). From the topological point of view, we know that R
e
l
r
s
c
(
U
l
t
(
L
X
)
) is closed with finite join and infinity meet. In other words, suppose {
R
i
∣
i
∈
I
}
⊆
R
e
l
r
s
c
(
U
l
t
(
L
X
)
), and we have ⋁
j
=
1
m
R
j
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
) and ⋀
i
∈
I
R
i
∈
R
e
l
r
s
c
(
U
l
t
(
L
X
)
).Furthermore,R
e
l
r
s
c
(
U
l
t
(
L
X
)
) is also closed with the operator defined as follows: (29)
∑
i
∈
I
R
i
=
C
L
⋁
i
∈
I
R
i
,
that  is  to  say
,
∑
i
∈
I
R
i
∈
R
e
l
r
s
c
U
l
t
L
X
.OnR
e
l
r
s
c
(
U
l
t
(
L
X
)
), the largest element is 1
U
l
t
(
L
X
); the smallest element is 1
U
l
t
(
L
X
)
r
s
c, where, for ν
1
,
ν
2
∈
U
l
t
(
L
X
), 1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
=
1, if ν
1
=
ν
2; 1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
=
0, otherwise.Second, with the help of Proposition21, we obtain the corresponding results on C
R
(
L
X
).LetC
R
(
L
X
) be the set of all L-contact relations on L
X. It is closed with the two operators: ∑
‍ and ∏. That is, suppose {
θ
i
∣
i
∈
I
}
⊆
C
R
(
L
X
):(30)
∑
i
∈
I
θ
i
=
q
C
L
⋁
i
∈
I
p
θ
i
,
∏
i
∈
I
θ
i
=
q
⋀
i
∈
I
p
θ
i
.Thus we obtain the following.Proposition 22.
SupposeC
R
(
L
X
) is the collection of all L-contact relations on L
X, and then it forms a complete lattice with respect to ∑
‍ and ∏, on which the largest element is 1
C
R
(
L
X
); the smallest element is O
C
R
(
L
X
).Note 6.
Consider(1)
q
(
1
U
l
t
(
L
X
)
r
s
c
)
=
O
C
R
(
L
X
).
For anyA
,
B
∈
L
X,(i)
O
C
R
(
L
X
)
(
A
,
B
)
=
1
⇒
A
∧
B
≠
0
X
⇒
1
U
l
t
(
L
X
)
r
s
c
(
η
A
∧
B
,
η
A
∧
B
)
=
1, and η
A
∧
B
(
A
)
=
1
,
η
A
∧
B
(
B
)
=
1
⇒
q
(
1
U
l
t
(
L
X
)
r
s
c
)
(
A
,
B
)
=
⋁
ν
1
,
ν
2
∈
U
l
t
(
L
X
)
1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
∧
ν
1
(
A
)
∧
ν
2
(
B
)
≥
1
U
l
t
(
L
X
)
r
s
c
(
η
A
∧
B
,
η
A
∧
B
)
∧
η
A
∧
B
(
A
)
∧
η
A
∧
B
(
B
)
=
1;
(ii)
q
(
1
U
l
t
(
L
X
)
r
s
c
)
(
A
,
B
)
=
0
⇒
⋁
ν
1
,
ν
2
∈
U
l
t
(
L
X
)
1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
∧
ν
1
(
A
)
∧
ν
2
(
B
)
=
0
⇒
A
∧
B
=
0
X.
If not,A
∧
B
≠
0
X; by (i), we obtain 1
U
l
t
(
L
X
)
r
s
c
(
η
A
∧
B
,
η
A
∧
B
)
∧
η
A
∧
B
(
A
)
∧
η
A
∧
B
(
B
)
=
1, a contradiction. Thus ⇒
O
C
R
(
L
X
)
(
A
,
B
)
=
0.(2)
Considerp
(
O
C
R
(
L
X
)
)
=
1
U
l
t
(
L
X
)
r
s
c.
For anyν
1
,
ν
2
∈
U
l
t
(
L
X
),(i)
1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
=
1
⇒
ν
1
=
ν
2
⇒
⋂
‍
ν
1
=
⋂
‍
ν
2,
⇒
p
(
O
C
R
(
L
X
)
)
(
ν
1
,
ν
2
)
=
C
L
(
O
C
R
(
L
X
)
)
(
ν
1
,
ν
2
) ≥ O
C
R
(
L
X
)
(
⋂
‍
ν
1
,
⋂
‍
ν
2
)
=
1;
(ii)
p
(
O
C
R
(
L
X
)
)
(
ν
1
,
ν
2
)
=
0
⇒
⋂
‍
ν
1
≠
⋂
‍
ν
2
⇒
ν
1
≠
ν
2
⇒
1
U
l
t
(
L
X
)
r
s
c
(
ν
1
,
ν
2
)
=
0.


## 7. Conclusion

In the paper, from the point of view of graded truth approach, we introduced the notion of a contact relation in fuzzy setting, proved all contact relations onL
X form an L-ordered complete lattice with the two operators ∑
‍ and ∏, and investigated the correspondence between the contact relations on L
X and the close, reflexive, symmetric relations on U
l
t
(
L
X
).



---

*Source: 3942416-2016-04-18.xml*