The Lattice Structure of L-Contact Relations

Abstract

From the point of view of graded truth approach, we define the notion of a contact relation on the collection of all L -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.

Academic Editor:Rustom M. Mamlook

School of Mathematics, Shandong University of Technology, Zibo, Shandong 255049, China

Received 15 November 2015; Accepted 24 March 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 an [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -valued or [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -contact relation.

In this paper, we want to generalize the theory of contact relations in fuzzy setting. First, Section 2 surveys an overview of contact relations, [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -filter, an [figure omitted; refer to PDF] -relation, and an [figure omitted; refer to PDF] -topology. Section 5 establishes the order preserving correspondence between the set of all [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] and the set of all closed, reflexive, symmetric relations on Ult [figure omitted; refer to PDF] . Section 6 focuses on the algebraic structure of all [figure omitted; refer to PDF] -contact relations.

2. Preliminaries

Let us recall some main notions for each area, that is, contact relations [6-8] and [figure omitted; refer to PDF] -sets [11].

2.1. Contact Relations

We assume familiarity with the notions of Boolean algebra and lattice [16, 17]. Suppose [figure omitted; refer to PDF] is a Boolean algebra, [figure omitted; refer to PDF] is called a binary relation on [figure omitted; refer to PDF] , and all relations are denoted by Rel [figure omitted; refer to PDF] . In [6], first, Düntsch and Winter considered the notion of a contact relation on a Boolean algebra [figure omitted; refer to PDF] .

Definition 1.

Suppose [figure omitted; refer to PDF] , and consider the following properties: for all [figure omitted; refer to PDF] ,

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] , that is, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are not [figure omitted; refer to PDF] -related,

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ,

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] ,

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ,

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is called a contact relation, and [figure omitted; refer to PDF] is called a Boolean contact algebra, if [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] . [figure omitted; refer to PDF] denotes the set of all contact relations on [figure omitted; refer to PDF] .

Second they discussed the set Ult( [figure omitted; refer to PDF] ) which is of all ultrafilters on [figure omitted; refer to PDF] and the set [figure omitted; refer to PDF] of all reflexive and symmetric relations on Ult( [figure omitted; refer to PDF] ) that are closed in the product topology of [figure omitted; refer to PDF] .

Third, they investigated the relation between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and obtained the representation theorem in [6].

Theorem 2.

Suppose that [figure omitted; refer to PDF] is a Boolean algebra. Then, there is a bijective order preserving correspondence between the contact relations on [figure omitted; refer to PDF] and the reflexive and symmetric relations on [figure omitted; refer to PDF] that are closed in the product topology of [figure omitted; refer to PDF] .

Then with the help of Theorem 2, they studied the structure of [figure omitted; refer to PDF] by means of the set [figure omitted; refer to PDF] .

For further information on contact relations and Boolean contact algebras, see [3-8].

2.2. [figure omitted; refer to PDF] -Sets

As a generalization of Zadeh's (classical) notion of a fuzzy set [10], the notion of an [figure omitted; refer to PDF] -set was introduced in [9]. An overview of the theory of [figure omitted; refer to PDF] -sets and [figure omitted; refer to PDF] -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 algebra [figure omitted; refer to PDF] such that

(1) [figure omitted; refer to PDF] is a lattice with the smallest element 0 and the largest element 1,

(2) [figure omitted; refer to PDF] is a commutative monoid; that is, [figure omitted; refer to PDF] is associative, commutative, and holds the identity [figure omitted; refer to PDF] ,

(3) [figure omitted; refer to PDF] form an adjoint pair; that is, [figure omitted; refer to PDF] iff [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] .

Residuated lattice [figure omitted; refer to PDF] is called complete if [figure omitted; refer to PDF] is a complete lattice.

In this paper, we assume that [figure omitted; refer to PDF] is a complete Heyting algebra which is a complete residuated lattice satisfying [figure omitted; refer to PDF] .

For a universe set [figure omitted; refer to PDF] , an [figure omitted; refer to PDF] -set in [figure omitted; refer to PDF] is a mapping [figure omitted; refer to PDF] . [figure omitted; refer to PDF] indicates the truth degree of " [figure omitted; refer to PDF] belongs to [figure omitted; refer to PDF] ." We use the symbol [figure omitted; refer to PDF] to denote the set of all [figure omitted; refer to PDF] -sets in [figure omitted; refer to PDF] . The negation operator is defined: for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] .

Definition 4.

(1) Suppose [figure omitted; refer to PDF] is a system of [figure omitted; refer to PDF] -sets, and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two [figure omitted; refer to PDF] -sets defined as follows, for every [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

(2) Suppose [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -set in [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] [figure omitted; refer to PDF] is a mapping. For every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is called the degree of membership of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a generalization of a system of subsets in the classical case.

Two [figure omitted; refer to PDF] -sets [figure omitted; refer to PDF] in [figure omitted; refer to PDF] are defined: for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Clearly, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are generalizations of the union and the intersection of a system of sets in the classical case (see [11]).

If [figure omitted; refer to PDF] is a system of [figure omitted; refer to PDF] -sets in [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Note 1.

In ordinary set theory, suppose [figure omitted; refer to PDF] is a set and [figure omitted; refer to PDF] is a system of subsets of [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] . Clearly [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the power set of [figure omitted; refer to PDF] .

The classical order [figure omitted; refer to PDF] and equality [figure omitted; refer to PDF] are generalized in fuzzy setting, that is, [figure omitted; refer to PDF] -relation and [figure omitted; refer to PDF] -equality.

[figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -binary relation. The truth degree to which elements [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are related by an [figure omitted; refer to PDF] -relation [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

A binary [figure omitted; refer to PDF] -relation [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -equivalence if it satisfies the following: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] (reflexivity), [figure omitted; refer to PDF] (symmetry), and [figure omitted; refer to PDF] (transitivity). An [figure omitted; refer to PDF] -equivalence is an [figure omitted; refer to PDF] -equality if it satisfies the following: [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] .

An [figure omitted; refer to PDF] -order on [figure omitted; refer to PDF] with an [figure omitted; refer to PDF] -equality relation [figure omitted; refer to PDF] is a binary [figure omitted; refer to PDF] -relation [figure omitted; refer to PDF] which is compatible with respect to [figure omitted; refer to PDF] and satisfies the following: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] (reflexivity), [figure omitted; refer to PDF] (antisymmetry), and [figure omitted; refer to PDF] (transitivity). A set [figure omitted; refer to PDF] equipped with an [figure omitted; refer to PDF] -order [figure omitted; refer to PDF] and an [figure omitted; refer to PDF] -equality [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -ordered set [figure omitted; refer to PDF] .

The subsethood degree [figure omitted; refer to PDF] is defined as follows: for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] We write [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] .

Example 5.

For [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ordered set. In fact, reflexivity and antisymmetry are trivial, and we have to prove transitivity and compatibility. Transitivity is described as follows: [figure omitted; refer to PDF] holds if and only if [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and it is true since [figure omitted; refer to PDF] . In a similar way, we also prove compatibility: [figure omitted; refer to PDF] .

By Example 5, we know that [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -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 all [figure omitted; refer to PDF] -contact relations form an [figure omitted; refer to PDF] -ordered set [figure omitted; refer to PDF] .

Suppose [figure omitted; refer to PDF] is a universe set and [figure omitted; refer to PDF] is the set of all [figure omitted; refer to PDF] -sets in [figure omitted; refer to PDF] . We know that [figure omitted; refer to PDF] might not be a Boolean algebra, where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are defined: [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for every [figure omitted; refer to PDF] , respectively.

A mapping [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -binary relation on [figure omitted; refer to PDF] . The truth degree to which elements [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are related by an [figure omitted; refer to PDF] -relation [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] . The collection of all [figure omitted; refer to PDF] -relations is denoted by [figure omitted; refer to PDF] .

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.

Suppose [figure omitted; refer to PDF] is a universe set, [figure omitted; refer to PDF] is the set of all [figure omitted; refer to PDF] -sets in [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -relation on [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , and consider the following properties:

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -contact relation, and [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -contact algebra, if [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] .

In the paper, we will denote the set of all [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] by CR [figure omitted; refer to PDF] . In [6-8], [figure omitted; refer to PDF] denotes a contact relation. From now on, we write [figure omitted; refer to PDF] instead of [figure omitted; refer to PDF] . Obviously, Definition 6 is a generalization of Definition 1 in fuzzy setting.

Second, we give some examples of [figure omitted; refer to PDF] -contact relations.

Example 7.

For [figure omitted; refer to PDF] , we define an [figure omitted; refer to PDF] -contact relation: [figure omitted; refer to PDF]

Example 8.

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] is a contact relation on [figure omitted; refer to PDF] .

When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the power set of [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] [figure omitted; refer to PDF] is a Boolean algebra. Clearly, we have [figure omitted; refer to PDF]

Note 2.

In [18], we introduced the relation [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . We show that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the same in the classical case.

Definition 9.

Suppose [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] , which expresses the related degree of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with respect to [figure omitted; refer to PDF] [18].

Obviously, [figure omitted; refer to PDF] is not a contact relation on [figure omitted; refer to PDF] , since, for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] does not hold. But when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] . So the relation [figure omitted; refer to PDF] is the same with the relation [figure omitted; refer to PDF] in the classical case.

In the rest of the section, we define [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ordered set; see [11].

In special, [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] .

Notice that [figure omitted; refer to PDF] is the smallest element in [figure omitted; refer to PDF] . For any [figure omitted; refer to PDF] and each [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] ;

: [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF] ;

: [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ;

: [figure omitted; refer to PDF] .

Hence [figure omitted; refer to PDF] .

4. L-Filters and L-Topologies

This section is devoted to recall three notions: an [figure omitted; refer to PDF] -filter on [figure omitted; refer to PDF] , an [figure omitted; refer to PDF] -relation, and an [figure omitted; refer to PDF] -topology on the set of all [figure omitted; refer to PDF] -ultrafilters, showing that [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ordered set.

4.1. [figure omitted; refer to PDF] -Filters on [figure omitted; refer to PDF]

We recall the notion of an [figure omitted; refer to PDF] -filter on [figure omitted; refer to PDF] ; see [19, Chapter 3].

Definition 10.

A map [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -filter on [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] fulfills the following axioms:

(F0) [figure omitted; refer to PDF] .

(F1) [figure omitted; refer to PDF] .

(F2) [figure omitted; refer to PDF] .

(F3) [figure omitted; refer to PDF]

Now, we present some examples of [figure omitted; refer to PDF] -filters.

Example 11.

The function [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -filter.

The proof in detail is in [19, Chapter 3].

Example 12.

For every [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , an [figure omitted; refer to PDF] -filter is defined: [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the set of all [figure omitted; refer to PDF] -filters on [figure omitted; refer to PDF] , on which we introduce the partial orderings [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] : [figure omitted; refer to PDF] and thus [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ordered set; see [11].

In special, [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] .

Clearly, [figure omitted; refer to PDF] is the smallest filter on [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] ; and [figure omitted; refer to PDF] .

For instance, when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the power set of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

In [19, Chapter 3], the notion of an [figure omitted; refer to PDF] -filter was investigated. By Theorem [figure omitted; refer to PDF] [19, Chapter 3], we know that the partially ordered set [figure omitted; refer to PDF] has maximal elements. A maximal element in [figure omitted; refer to PDF] is also called an [figure omitted; refer to PDF] -ultrafilter. An [figure omitted; refer to PDF] -filter [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ultrafilter if and only if [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] [19, Theorem [figure omitted; refer to PDF] ].

For every [figure omitted; refer to PDF] , there exists an [figure omitted; refer to PDF] -ultrafilter [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] , that is, Lemma 13.

Lemma 13.

Suppose [figure omitted; refer to PDF] , and then there exists an [figure omitted; refer to PDF] -ultrafilter [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] .

Proof.

For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ,

(1) by Example 12, we obtain an [figure omitted; refer to PDF] -filter [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] ,

(2) by Corollary [figure omitted; refer to PDF] in [19, Chapter 3], there exists an [figure omitted; refer to PDF] -ultrafilter [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Certainly [figure omitted; refer to PDF] .

Lemma 14.

Suppose [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ultrafilter; then we have [figure omitted; refer to PDF] .

Proof.

In [11], the notion of a filter in a residuated lattice [figure omitted; refer to PDF] was introduced. In fact, [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ultrafilter on [figure omitted; refer to PDF] which is a maximal filter in [figure omitted; refer to PDF] . By Lemma [figure omitted; refer to PDF] [11, P55], Lemma 14 holds.

Suppose [figure omitted; refer to PDF] , and we define [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is also defined: [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] .

4.2. [figure omitted; refer to PDF] -Relations on [figure omitted; refer to PDF]

In the section, we turn to [figure omitted; refer to PDF] -relations on [figure omitted; refer to PDF] .

A mapping [figure omitted; refer to PDF] is also an [figure omitted; refer to PDF] -relation on [figure omitted; refer to PDF] . All of them will be denoted by [figure omitted; refer to PDF] . Suppose [figure omitted; refer to PDF] , and we define [figure omitted; refer to PDF] and [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF]

For example, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are defined as follows: [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Clearly, [figure omitted; refer to PDF] is the largest element in [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] , it is symmetric if [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . It is reflexive if, for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

4.3. [figure omitted; refer to PDF] -Topologies on [figure omitted; refer to PDF]

We introduce the notion of an [figure omitted; refer to PDF] -topology on [figure omitted; refer to PDF] .

Definition 15.

Suppose [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -topology on [figure omitted; refer to PDF] , if

(1) [figure omitted; refer to PDF] ,

(2) for every subset [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] holds.

[figure omitted; refer to PDF] denotes the closure of [figure omitted; refer to PDF] in the product topology [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is a closed set in the product topology [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is called a close relation.

The collection of all close, reflexive, and symmetric relations on [figure omitted; refer to PDF] is denoted by [figure omitted; refer to PDF] . Clearly, [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] -order [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Thus [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -ordered set; see [11].

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] , and we assume that [figure omitted; refer to PDF] is a base for the product topology [figure omitted; refer to PDF] .

5. The Correspondence between [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

In [6], Düntsch and Winter proved that there exists a bijective order preserving correspondence between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, Theorem 2.

In Sections 3 and 4, we know that two [figure omitted; refer to PDF] -ordered sets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are generalizations of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in fuzzy setting, respectively.

In this section, we continue to investigate the relation between two [figure omitted; refer to PDF] -ordered sets [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , obtaining a fuzzication of Theorem 2 [6]. For this, we define two mappings: [figure omitted; refer to PDF] and we prove that they are the order preserving correspondences. We divide the work into three steps.

Step 1.

We define a mapping [figure omitted; refer to PDF] from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and prove that it is one-to-one.

Given [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , put [figure omitted; refer to PDF]

Then we have the following.

Lemma 16.

Suppose [figure omitted; refer to PDF] is a close, reflexive, and symmetric [figure omitted; refer to PDF] -relation on [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -contact relation on [figure omitted; refer to PDF] .

Proof.

we have to show that [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -contact relation, that is, to verify [figure omitted; refer to PDF] satisfies the conditions of Definition 6. Consider the following:

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

[figure omitted; refer to PDF] : For [figure omitted; refer to PDF] , by Lemma 13, [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF]

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] is obvious.

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF]

[figure omitted; refer to PDF] : By Lemma 14, we have [figure omitted; refer to PDF]

So, [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -contact relation on [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] .

Furthermore, for every [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] , and thus we obtain a mapping [figure omitted; refer to PDF] from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] . The key is to prove that [figure omitted; refer to PDF] is one-to-one.

Lemma 17.

Suppose [figure omitted; refer to PDF] is a close, reflexive, and symmetric [figure omitted; refer to PDF] -relation on [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an injective mapping.

Proof.

By the above analysis, the remainder is to prove that [figure omitted; refer to PDF] is one-to-one, which is equivalent to show for any two close relations [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , then their images [figure omitted; refer to PDF] are not equal.

Suppose [figure omitted; refer to PDF] are two distinct close, reflexive, and symmetric relations, and then there exist [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Without loss of generality, we may assume [figure omitted; refer to PDF] .

(1) Consider [figure omitted; refer to PDF] , clearly [figure omitted; refer to PDF] , and there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF]

(2) Since [figure omitted; refer to PDF] is closed, we have that [figure omitted; refer to PDF] holds for any [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

If not, there exist [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] and we have [figure omitted; refer to PDF] which leads to a contradiction.

(3) On the other hand, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] is closed, there exist [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] a contradiction.

So [figure omitted; refer to PDF] holds, which implies that [figure omitted; refer to PDF] is an injective mapping.

Note 3.

When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] there exist [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] .

Step 2.

We define a mapping [figure omitted; refer to PDF] from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] ; that is, for each [figure omitted; refer to PDF] -contact relation [figure omitted; refer to PDF] , we consider defining the corresponding [figure omitted; refer to PDF] .

For an [figure omitted; refer to PDF] -ultrafilter [figure omitted; refer to PDF] , an [figure omitted; refer to PDF] -set in [figure omitted; refer to PDF] is defined; that is, [figure omitted; refer to PDF]

Obviously, for an [figure omitted; refer to PDF] -ultrafilter [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] .

Suppose [figure omitted; refer to PDF] ; in other words, [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -contact relation on [figure omitted; refer to PDF] , for any [figure omitted; refer to PDF] -ultrafilters [figure omitted; refer to PDF] , and let [figure omitted; refer to PDF]

Therefore we have the following lemma.

Lemma 18.

Suppose [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is an injective mapping.

Proof.

The closeness, symmetry, and reflexivity of [figure omitted; refer to PDF] follow directly from the above definition.

By the above proof and Lemma 17, we know that [figure omitted; refer to PDF] is also an injective mapping. This completes the proof.

Note 4.

When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] holds.

Step 3.

By Lemmas 16 and 17, we obtain two mappings [figure omitted; refer to PDF] , in the step, and we wish to prove that [figure omitted; refer to PDF] preserve the order [figure omitted; refer to PDF] .

First, we prove that [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] .

Lemma 19.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] .

Proof.

For each [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF]

We have to prove that [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

Second, we also prove that [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] .

Lemma 20.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] .

Proof.

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF]

We have to prove that [figure omitted; refer to PDF] preserves the order [figure omitted; refer to PDF] ; that is to say, [figure omitted; refer to PDF] . Consider [figure omitted; refer to PDF]

Note 5.

Suppose [figure omitted; refer to PDF] is a Boolean algebra, and by Lemmas 18, 19, and 20, we obtain the corresponding results: for [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] holds. Similarly, for [figure omitted; refer to PDF] [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] also holds.

So, there exists a bijective order preserving correspondence between the set of [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] and the set of all close, reflexive, symmetric relations on [figure omitted; refer to PDF] , that is, the following proposition, which is a fuzzification of Theorem 2.

Proposition 21.

Suppose [figure omitted; refer to PDF] is the set of all [figure omitted; refer to PDF] -sets on [figure omitted; refer to PDF] , and then there exists a bijective order preserving correspondence between the set of all [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] and the set of all close, reflexive, symmetric relations on [figure omitted; refer to PDF] .

6. The Structure of Contact Relations

In this section, we consider two problems; the first is discussing the structure of [figure omitted; refer to PDF] , and the second is obtaining the corresponding results about the [figure omitted; refer to PDF] .

First, we focus on the structure of [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the set of all close, reflexive, and symmetric relations on [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is a close set in the product topology [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . From the topological point of view, we know that [figure omitted; refer to PDF] is closed with finite join and infinity meet. In other words, suppose [figure omitted; refer to PDF] , and we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Furthermore, [figure omitted; refer to PDF] is also closed with the operator defined as follows: [figure omitted; refer to PDF]

On [figure omitted; refer to PDF] , the largest element is [figure omitted; refer to PDF] ; the smallest element is [figure omitted; refer to PDF] , where, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] , otherwise.

Second, with the help of Proposition 21, we obtain the corresponding results on [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be the set of all [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] . It is closed with the two operators: [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . That is, suppose [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

Thus we obtain the following.

Proposition 22.

Suppose [figure omitted; refer to PDF] is the collection of all [figure omitted; refer to PDF] -contact relations on [figure omitted; refer to PDF] , and then it forms a complete lattice with respect to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , on which the largest element is [figure omitted; refer to PDF] ; the smallest element is [figure omitted; refer to PDF] .

Note 6.

Consider

(1) [figure omitted; refer to PDF] .

For any [figure omitted; refer to PDF] ,

(i) [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] , and [figure omitted; refer to PDF]

: [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF]

: [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] .

If not, [figure omitted; refer to PDF] ; by (i), we obtain [figure omitted; refer to PDF] , a contradiction. Thus [figure omitted; refer to PDF] .

(2) Consider [figure omitted; refer to PDF] .

For any [figure omitted; refer to PDF] ,

(i) [figure omitted; refer to PDF]

: [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] ≥ [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF]

: [figure omitted; refer to PDF]

: [figure omitted; refer to PDF]

: [figure omitted; refer to PDF] .

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 on [figure omitted; refer to PDF] form an [figure omitted; refer to PDF] -ordered complete lattice with the two operators [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and investigated the correspondence between the contact relations on [figure omitted; refer to PDF] and the close, reflexive, symmetric relations on [figure omitted; refer to PDF] .

References

[1] S. A. Naimpally, B. D. Warrack Proximity Spaces , Cambridge University Press, Cambridge, UK, 1970.

[2] T. de Laguna, "Point, line, and surface, as sets of solids," The Journal of Philosophy , vol. 19, no. 17, pp. 449-461, 1922.

[3] B. Bennett, I. Düntsch, M. Aiello, J. van Benthem, I. Pratt-Hartmann, "Algebras, axioms, and topology," Handbook of Spatial Logics , pp. 99-159, Kluwer Academic, Dordrecht, Netherlands, 2007.

[4] G. Dimov, D. Vakarelov, "Contact algebras and region-based theory of space: a proximity approach-I," Fundamenta Informaticae , vol. 74, no. 2-3, pp. 209-249, 2006.

[5] G. Dimov, D. Vakarelov, "Contact algebras and region-based theory of space: a proximity approach-II," Fundamenta Informaticae , vol. 74, no. 2-3, pp. 251-282, 2006.

[6] I. Düntsch, M. Winter, R. Berghammer, B. Möller, G. Struth, "The lattice of contact relations on a Boolean algebra," Relations and Kleene Algebra in Computer Science , vol. 4988, of Lecture Notes in Computer Science, pp. 99-109, 2008.

[7] I. Düntsch, D. Vakarelov, "Region-based theory of discrete spaces: a proximity approach," Annals of Mathematics and Artificial Intelligence , vol. 49, no. 1-4, pp. 5-14, 2007.

[8] I. Düntsch, M. Winter, "A representation theorem for Boolean contact algebras," Theoretical Computer Science , vol. 347, no. 3, pp. 498-512, 2005.

[9] J. A. Goguen, " L -fuzzy sets," Journal of Mathematical Analysis and Applications , vol. 18, pp. 145-174, 1967.

[10] L. A. Zadeh, "Fuzzy sets," Information and Computation , vol. 8, pp. 338-353, 1965.

[11] R. Belohlavek Fuzzy Relational Systems: Foundations and Principles , Kluwer Academic, New York, NY, USA, 2002.

[12] M. Winter, "Time-dependent contact structures in Goguen Categories," Relational Methods in Computer Science , vol. 3929, of Lecture Notes in Computer Science, pp. 249-262, Springer, Berlin, Germany, 2006.

[13] X. Y. Chen, "Contact relation and way-below relation in fuzzy setting," Communications in Computer and Information Science , vol. 308, no. 2, pp. 379-386, 2012.

[14] X. Y. Chen, "Contact relation and continuous lattice in fuzzy setting," International Journal of Advancements in Computing Technology , vol. 4, no. 17, pp. 599-606, 2012.

[15] X. Y. Chen, "Contact relation on de Morgan algebra," In press Annals of Fuzzy Mathematics and Informatics

[16] B. Davey, H. Priestley Introduction to Lattices and Order , Cambridge University Press, New York, NY, USA, 2002., 2nd.

[17] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott A Compendium of Continuous Lattices , Springer, Berlin, Germany, 1980.

[18] X. Y. Chen, Q. G. Li, "Construction of rough approximations in fuzzy setting," Fuzzy Sets and Systems , vol. 158, no. 23, pp. 2641-2653, 2007.

[19] U. Höhle, A. P. Sostak, U. Höhle, S. E. Rodabaugh, "Axiomatic foundations of fixed-basis fuzzy topology," Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory , pp. 123-272, Kluwer Academic, New York, NY, USA, 1999.