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 -sets, discuss the connection to the set of all close, reflexive, and symmetric relations on all -ultrafilters on , and investigate the algebraic structure of all -contact relations.

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 -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 -valued or -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 -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, -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 -filter, an -relation, and an -topology. Section 5 establishes the order preserving correspondence between the set of all -contact relations on and the set of all closed, reflexive, symmetric relations on Ult. Section 6 focuses on the algebraic structure of all -contact relations.

2. Preliminaries

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

2.1. Contact Relations

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

Definition 1. Suppose , and consider the following properties: for all ,, that is, for , and are not -related, for ,, and , or .is called a contact relation, and is called a Boolean contact algebra, if satisfies . denotes the set of all contact relations on .
Second they discussed the set Ult() which is of all ultrafilters on and the set of all reflexive and symmetric relations on Ult() that are closed in the product topology of .
Third, they investigated the relation between and and obtained the representation theorem in [6].

Theorem 2. Suppose that is a Boolean algebra. Then, there is a bijective order preserving correspondence between the contact relations on and the reflexive and symmetric relations on that are closed in the product topology of .

Then with the help of Theorem 2, they studied the structure of by means of the set .

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

2.2. -Sets

As a generalization of Zadeh’s (classical) notion of a fuzzy set [10], the notion of an -set was introduced in [9]. An overview of the theory of -sets and -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 such that(1) is a lattice with the smallest element 0 and the largest element 1,(2) is a commutative monoid; that is, is associative, commutative, and holds the identity ,(3) form an adjoint pair; that is, iff holds for all .Residuated lattice is called complete if is a complete lattice.

In this paper, we assume that is a complete Heyting algebra which is a complete residuated lattice satisfying .

For a universe set , an -set in is a mapping . indicates the truth degree of “ belongs to .” We use the symbol to denote the set of all -sets in . The negation operator is defined: for , for every .

Definition 4. (1) Suppose is a system of -sets, and and are two -sets defined as follows, for every :(2) Suppose is an -set in ; that is, is a mapping. For every , is called the degree of membership of in . Then is a generalization of a system of subsets in the classical case.
Two -sets in are defined: for every ,Clearly, and are generalizations of the union and the intersection of a system of sets in the classical case (see [11]).
If is a system of -sets in , that is, , we have , and .

Note 1. In ordinary set theory, suppose is a set and is a system of subsets of , if . Clearly , where is the power set of .
The classical order and equality are generalized in fuzzy setting, that is, -relation and -equality.
is called an -binary relation. The truth degree to which elements and are related by an -relation is denoted by or .
A binary -relation on is an -equivalence if it satisfies the following: , (reflexivity), (symmetry), and (transitivity). An -equivalence is an -equality if it satisfies the following: implies .
An -order on with an -equality relation is a binary -relation which is compatible with respect to and satisfies the following: , (reflexivity), (antisymmetry), and (transitivity). A set equipped with an -order and an -equality is called an -ordered set .
The subsethood degree is defined as follows: for ,We write , if .

Example 5. For , we obtain that is an -ordered set. In fact, reflexivity and antisymmetry are trivial, and we have to prove transitivity and compatibility. Transitivity is described as follows: holds if and only if ; that is, , , and it is true since . In a similar way, we also prove compatibility: .

By Example 5, we know that is an -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 -contact relations form an -ordered set .

Suppose is a universe set and is the set of all -sets in . We know that might not be a Boolean algebra, where and are defined: and , for every , respectively.

A mapping is called an -binary relation on . The truth degree to which elements and are related by an -relation is denoted by . The collection of all -relations is denoted by .

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 is a universe set, is the set of all -sets in is an -relation on , for , and consider the following properties:. for .... is called an -contact relation, and is called an -contact algebra, if satisfies .

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

Second, we give some examples of -contact relations.

Example 7. For , we define an -contact relation:

Example 8. For , letThen is a contact relation on .
When , is the power set of ; that is, ; then is a Boolean algebra. Clearly, we have

Note 2. In [18], we introduced the relation on . We show that and are the same in the classical case.

Definition 9. Suppose , and let , which expresses the related degree of and with respect to [18].
Obviously, is not a contact relation on , since, for every , does not hold. But when , ; . So the relation is the same with the relation in the classical case.

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

In special, for any .

Notice that is the smallest element in . For any and each , ; , and let ; , and ; .

Hence .

4. L-Filters and L-Topologies

This section is devoted to recall three notions: an -filter on , an -relation, and an -topology on the set of all -ultrafilters, showing that is an -ordered set.

4.1. -Filters on

We recall the notion of an -filter on ; see [19, Chapter 3].

Definition 10. A map is called an -filter on , if fulfills the following axioms:(F0).(F1).(F2).(F3)Now, we present some examples of -filters.

Example 11. The function defined by for every is an -filter.
The proof in detail is in [19, Chapter 3].

Example 12. For every and , an -filter is defined: Let be the set of all -filters on , on which we introduce the partial orderings and , for :and thus is an -ordered set; see [11].
In special, ; that is, for every .
Clearly, is the smallest filter on , where for every ; and .
For instance, when , is the power set of , .
In [19, Chapter 3], the notion of an -filter was investigated. By Theorem [19, Chapter 3], we know that the partially ordered set has maximal elements. A maximal element in is also called an -ultrafilter. An -filter is an -ultrafilter if and only if for every [19, Theorem ].
For every , there exists an -ultrafilter , such that , that is, Lemma 13.

Lemma 13. Suppose , and then there exists an -ultrafilter , such that .

Proof. For , ,(1)by Example 12, we obtain an -filter satisfying ,(2)by Corollary in [19, Chapter 3], there exists an -ultrafilter , such that . Certainly .

Lemma 14. Suppose is an -ultrafilter; then we have .

Proof. In [11], the notion of a filter in a residuated lattice was introduced. In fact, is an -ultrafilter on which is a maximal filter in . By Lemma [11, P55], Lemma 14 holds.
Suppose , and we define and as follows:If , is also defined: for any .

4.2. -Relations on

In the section, we turn to -relations on .

A mapping is also an -relation on . All of them will be denoted by . Suppose , and we define and as follows:

For example, and are defined as follows: and for . Clearly, is the largest element in .

For , it is symmetric if for any . It is reflexive if, for every , .

4.3. -Topologies on

We introduce the notion of an -topology on .

Definition 15. Suppose is called an -topology on , if(1),(2)for every subset of , holds. denotes the closure of in the product topology on . If is a closed set in the product topology , then is called a close relation.
The collection of all close, reflexive, and symmetric relations on is denoted by . Clearly, .
For , we define -order and on :Thus is an -ordered set; see [11].
For , let , and we assume that is a base for the product topology .

5. The Correspondence between and

In [6], Düntsch and Winter proved that there exists a bijective order preserving correspondence between and , that is, Theorem 2.

In Sections 3 and 4, we know that two -ordered sets and are generalizations of and in fuzzy setting, respectively.

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

Step 1. We define a mapping from to and prove that it is one-to-one.
Given , for , put

Then we have the following.

Lemma 16. Suppose is a close, reflexive, and symmetric -relation on ; then is an -contact relation on .

Proof. we have to show that is an -contact relation, that is, to verify satisfies the conditions of Definition 6. Consider the following:.For , by Lemma 13, , and then is obvious., and then By Lemma 14, we haveSo, is an -contact relation on ; that is, .
Furthermore, for every , we define , and thus we obtain a mapping from to . The key is to prove that is one-to-one.

Lemma 17. Suppose is a close, reflexive, and symmetric -relation on , and then and is an injective mapping.

Proof. By the above analysis, the remainder is to prove that is one-to-one, which is equivalent to show for any two close relations ; if , then their images are not equal.
Suppose are two distinct close, reflexive, and symmetric relations, and then there exist , such that . Without loss of generality, we may assume .
(1) Consider , clearly , and there exist , , , such that(2) Since is closed, we have that holds for any with .
If not, there exist , , such thatand we have which leads to a contradiction.
(3) On the other hand, if , thenSince is closed, there exist satisfying a contradiction.
So holds, which implies that is an injective mapping.

Note 3. When , , and there exist , such that , and ; that is, .

Step 2. We define a mapping from to ; that is, for each -contact relation , we consider defining the corresponding .
For an -ultrafilter , an -set in is defined; that is, Obviously, for an -ultrafilter , we have .
Suppose ; in other words, is an -contact relation on , for any -ultrafilters , and let

Therefore we have the following lemma.

Lemma 18. Suppose , then , for , and is an injective mapping.

Proof. The closeness, symmetry, and reflexivity of follow directly from the above definition.
By the above proof and Lemma 17, we know that is also an injective mapping. This completes the proof.

Note 4. When , , and , holds.

Step 3. By Lemmas 16 and 17, we obtain two mappings , in the step, and we wish to prove that preserve the order .
First, we prove that preserves the order .

Lemma 19. preserves the order .

Proof. For each , we defineWe have to prove that preserves the order ; that is, . ConsiderSecond, we also prove that preserves the order .

Lemma 20. preserves the order .

Proof. For , letWe have to prove that preserves the order ; that is to say, . Consider

Note 5. Suppose is a Boolean algebra, and by Lemmas 18, 19, and 20, we obtain the corresponding results: for , if , then holds. Similarly, for , if , then also holds.

So, there exists a bijective order preserving correspondence between the set of -contact relations on and the set of all close, reflexive, symmetric relations on , that is, the following proposition, which is a fuzzification of Theorem 2.

Proposition 21. Suppose is the set of all -sets on , and then there exists a bijective order preserving correspondence between the set of all -contact relations on and the set of all close, reflexive, symmetric relations on .

6. The Structure of Contact Relations

In this section, we consider two problems; the first is discussing the structure of , and the second is obtaining the corresponding results about the .

First, we focus on the structure of .

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

Furthermore, is also closed with the operator defined as follows:

On , the largest element is ; the smallest element is , where, for , , if ; , otherwise.

Second, with the help of Proposition 21, we obtain the corresponding results on .

Let be the set of all -contact relations on . It is closed with the two operators: and . That is, suppose :

Thus we obtain the following.

Proposition 22. Suppose is the collection of all -contact relations on , and then it forms a complete lattice with respect to and , on which the largest element is ; the smallest element is .

Note 6. Consider(1).For any ,(i)  , and   ;(ii)  .If not, ; by (i), we obtain , a contradiction. Thus .(2)Consider .For any ,(i)  ,  ≥ ;(ii)   .

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 form an -ordered complete lattice with the two operators and , and investigated the correspondence between the contact relations on and the close, reflexive, symmetric relations on .

Competing Interests

The author declares that they have no competing interests.