The Lattice Structure of L-Contact Relations

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 L-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, Section 2 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 Lcontact relations on L and the set of all closed, reflexive, symmetric relations on Ult(L). Section 6 focuses on the algebraic structure of all L-contact relations. 2. Preliminaries


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][4][5][6][7][8].
On the other hand, the notion of an L-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][14][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, Section 2 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 Lcontact relations on   and the set of all closed, reflexive, symmetric relations on Ult(  ).Section 6 focuses on the algebraic structure of all L-contact relations.
2.1.Contact Relations.We assume familiarity with the notions of Boolean algebra and lattice [16,17].Suppose (, +, ⋅, * , 0, 1) 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  ∈ Rel(), and consider the following properties: for all , ,  ∈ , ( 0 ) 0(−), that is, for  ̸ = 0, 0 and  are not -related, ( Second they discussed the set Ult() which is of all ultrafilters on  and the set Rel rsc (Ult()) of all reflexive and symmetric relations on Ult() that are closed in the product topology of Ult() 2 .

Advances in Fuzzy Systems
Third, they investigated the relation between CR() and Rel rsc (Ult()) 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 () 2 .
Then with the help of Theorem 2, they studied the structure of CR() by means of the set Rel rsc (Ult()).
Residuated lattice L is called complete if ⟨, ⋁, ⋀⟩ is a complete lattice.
In this paper, we assume that L is a complete Heyting algebra which is a complete residuated lattice satisfying  ⊗  =  ∧ .
For a universe set , an L-set in  is a mapping  :  → .() indicates the truth degree of " belongs to ."We use the symbol   to denote the set of all L-sets in .The negation operator is defined: for  ∈   ,  * () = () → 0 for every  ∈ .Definition 4. (1) Suppose {  |  ∈ } ⊆   is a system of Lsets, and ⋁ ∈   and ⋀ ∈   are two L-sets defined as follows, for every  ∈ : (2) Suppose U is an L-set in   ; that is, U :   →  is a mapping.For every  ∈   , U() is called the degree of membership of  in U. Then U is a generalization of a system of subsets in the classical case.
Two L-sets ⋃ U, ⋂ U in  are defined: for every  ∈ , Clearly, ⋃ U and ⋂ U are generalizations of the union and the intersection of a system of sets in the classical case (see [11]).
If U is a system of L-sets in   , that is, U = {  |  ∈ } ⊆   , we have ⋃ U = ⋁U, and ⋂ U = ⋀U.Note 1.In ordinary set theory, suppose  is a set and , where 2  is the power set of .
The classical order ≤ and equality = are generalized in fuzzy setting, that is, L-relation and L-equality.
∈  × is called an L-binary relation.The truth degree to which elements  and  are related by an L-relation  is denoted by (, ) or ().

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 L-contact relations form an L-ordered set ⟨⟨CR(  ), ≈⟩, ⪯⟩.
A mapping  :   ×   →  is called an L-binary relation on   .The truth degree to which elements  and  are related by an L-relation  is denoted by (, ).The collection of all L-relations is denoted by Rel(  ).
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 Lsets in ,  ∈ Rel(  ) is an L-relation on   , for , ,  ∈   , and consider the following properties: is called an L-contact relation, and (  , ) is called an L-contact algebra, if  satisfies ( 0 )-( 4 ).
In the paper, we will denote the set of all L-contact relations on   by CR(  ).In [6][7][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 L-contact relations.
In the rest of the section, we define ⪯ and ≈ on CR(  ).

L-Filters and L-Topologies
This section is devoted to recall three notions: an L-filter on , an L-relation, and an L-topology on the set of all L-ultrafilters, showing that ⟨⟨Rel rsc (Ult(  )), ≈⟩, ⪯⟩ is an Lordered set.
In this section, we continue to investigate the relation between two L-ordered sets ⟨⟨Rel rsc (Ult(  )), ≈⟩, ⪯⟩ and ⟨⟨CR(  ), ≈⟩, ⪯⟩, 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 Rel rsc (Ult(  )) to CR(  ) and prove that it is one-to-one.
Lemma 16.Suppose  is a close, reflexive, and symmetric Lrelation on (  ); then   is an L-contact relation on   .
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  1 ,  2 ∈ Rel rsc (Ult(  )); if  1 ̸ =  2 , then their images   1 ,   2 are not equal.
Suppose  1 ,  2 are two distinct close, reflexive, and symmetric relations, and then there exist 2 ).Without loss of generality, we may assume , and there exist ,  ∈   , ] 0 1 () ̸ = 0, ] 0 2 () ̸ = 0, such that (2) Since  2 is closed, we have that Advances in Fuzzy Systems and we have which leads to a contradiction.
For an L-ultrafilter ], an L-set in  is defined; that is, Obviously, for an L-ultrafilter ], we have ⋂ ] ̸ = 0  .Suppose  ∈ CR(  ); in other words,  is an L-contact relation on   , for any L-ultrafilters ] 1 , ] 2 ∈ Ult(  ), and let Therefore we have the following lemma.
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.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 ⪯.

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 Lordered complete lattice with the two operators ∑ and ∏, and investigated the correspondence between the contact relations on   and the close, reflexive, symmetric relations on Ult(  ).
2) for every subset {]  |  ∈ } of , ⋁ ∈ ]  ∈  holds.CL() denotes the closure of  in the product topology  ×  on Ult(  ) × Ult(  ).If  is a closed set in the product topology  × , then  is called a close relation.