#### Abstract

A new definition of interval-valued supra-fuzzy syntopogenous (resp., supra-fuzzy proximity) space is given. We show that for any interval-valued fuzzy syntopogenous structure , there is another interval-valued fuzzy syntopogenous structure called the conjugate of . This leads to introducing the concept of interval-valued bifuzzy syntopogenous space. Finally, we show every interval-valued bifuzzy syntopogenous space induces an interval-valued fuzzy supra-syntopogenous space. Throughout this paper, the family of all fuzzy sets on nonempty set will be denoted by and the family of all interval-valued fuzzy sets on a nonempty set will be denoted by .

#### 1. Introduction

In [1], Csaszar introduced the concept of a syntopogenous structure to develop a unified approach to the three main structures of set-theoretic-topology: topologies, uniformities, and proximities. This enables him to evolve a theory including the foundations of the three classical theories of topological spaces, uniform spaces, and proximity spaces. In the case of the fuzzy structures, there are at least three notions of fuzzy syntopogenous structures. The first notion worked out in [2–4] presents a unified approach to the theories of Chang fuzzy topological spaces [5], Hutton fuzzy uniform spaces [6], and Liu fuzzy proximity spaces [7]. The second notion worked out in [8] agrees very well with Lowen fuzzy topological spaces [9], Lowen-Hohle fuzzy uniform spaces [10], and Artico-Moresco fuzzy proximity spaces [11]. The third notion worked out in [12] agrees with the framework of a fuzzifying topology [13]. In [14], Kandil et al. introduced the concepts of a supra-fuzzy topological space and supra-fuzzy proximity space. In [15], Ghanim et al. introduced the concepts of supra-fuzzy syntopogenous space as a generalization of the concepts of supra-fuzzy proximity and supra-fuzzy topology.

Interval-valued fuzzy sets were introduced independently by Zadeh [16], Grattan-Guiness [17], Jahn [18], and Sambuc [19], in the seventies, in the same year. An interval-valued fuzzy set (IVF) is defined by an interval-valued membership function. As a generalization of fuzzy, the concept of intuitionistic fuzzy sets was introduced by Atanassov [20]. In [21], it is shown that the concepts of interval-valued fuzzy sets and intuitionistic fuzzy sets are equivalent and they are extensions of fuzzy sets.

Interval-valued fuzzy sets can be viewed as a generalization of fuzzy sets (Zadeh [22]) that may better model imperfect information which is omnipresent in any conscious decision making [23]. It can be considered as a tool for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge, with an application in supporting medical diagnosis [24, 25]. In [26] new notions of interval-valued supra-fuzzy topology (resp., supra-fuzzy proximity) were introduced by using the notions of interval-valued fuzzy sets.

In this paper, we generalize the concept of supra-fuzzy syntopogenous space by using the notion of interval-valued set. Topology and its generalization proximity and syntopogenous are branches of mathematics which have many real life applications. We believe that the generalized topological structure suggested in this paper will be important base for modification of medical diagnosis, decision making, and knowledge discovery. The other parts of this paper are arranged as follows. In Section 2, We recall and develop some notions and notations concerning supra-fuzzy topological space, interval-valued fuzzy set, interval-valued fuzzy topology, interval-valued supra-fuzzy topology, and interval-valued supra-fuzzy proximity. In Section 3, we introduce the concept of interval-valued supra-fuzzy syntopogenous (resp., supra-fuzzy proximity) space. We show that there is one-to-one correspondence between family of all interval-valued supra-fuzzy topological spaces and the family of all perfect interval-valued supra-fuzzy topogenous spaces. Also, we prove that there is one-to-one correspondence between the family of all interval-valued supra-fuzzy proximity spaces and the family of all symmetrical interval-valued supra-fuzzy topogenous spaces. We show that for any interval-valued fuzzy topogenous order on there is another interval-valued fuzzy topogenous structure in Section 3. In Section 4, we introduce the concept of interval-valued bifuzzy syntopogenous structure by using two interval-valued fuzzy syntopogenous spaces. In the last section we show that with every interval-valued bifuzzy syntopogenous space there is an interval-valued supra-fuzzy syntopogenous associated with this space.

#### 2. Preliminaries

The concept of a supra-fuzzy topological space has been introduced as follows.

*Definition 2.1 (see [27]). *A collection is a supra-fuzzy topology on if 0, 1 and is closed under arbitrary supremum. The pair () is a supra-fuzzy topological space.

*Definition 2.2 (see [20]). *An interval-valued fuzzy set (IVF set for short) is a set such that . The family of all interval-valued fuzzy sets on a given nonempty set will be denoted by .

The IVF set = (1, 1) is called the universal IVF set and the IVF set = (0, 0) is called the empty IVF set.

Proposition 2.3 (see [20]). *The operations on are given by the following: Let , :*(1),,(2),,(3),(4),(5).

*Definition 2.4 (see [26]). *The family is called an interval-valued fuzzy topology (IVF tpology for short) on if and only if contains , , and it is closed under finite intersection and arbitrary union. The pair () is called an interval-valued fuzzy topological space (IVF top. space for short). Any IVF set is called an open IVF set and the complement of denoted by is called a closed IVF set. The family of all closed interval-valued fuzzy sets is denoted by .

*Definition 2.5 (see [26]). *A non empty family is called an interval-valued supra-fuzzy topology (IVSF top. for short) on if it contains , and it is closed under arbitrary unions.

*Definition 2.6 (see [26]). *A binary relation is called an interval-valued supra-fuzzy (IVSF) proximity on if it satisfies the following conditions: (1),(2) or ,(3),(4),(5) s.t. and .

#### 3. Interval-Valued Fuzzy Topogenous Order

In this article, we define a new concept of topogenous structure by using the concept of interval-valued fuzzy sets.

*Definition 3.1. *Let be a binary relation on . Consider the following axioms: (1),(2),(3),(4) and and ,(5) for all ,(6) for all ,(7).If satisfies (1)–(4) then it is called an interval-valued fuzzy topogenous order (IVF topogenous order for short) on .

If satisfies (1)–(3) then it is called an interval-valued supra-fuzzy topogenous order (IVSF topogenous order for short) on .

The interval-valued fuzzy topogenous order is called perfect (resp., biperfect, symmetrical) if it satisfies the condition (5) (resp., (6), (7)).

*Definition 3.2. *Let and be a nonempty family of IVF topogenous orders on satisfy the following conditions: (s_{1}) s.t. and, (s_{2}) s.t. .

Then is said to be interval-valued fuzzy (IVF) syntopogenous structure on . The pair () is called an interval-valued fuzzy syntopogenous (IVF syntopogenous for short) space. If satisfy the single element, then is called an IVF topogenous structure on and the pair () is called an IVF topogenous space.

*Definition 3.3. *Let and be a nonempty family of IVSF topogenous orders on satisfy the conditions (s_{1}) and (s_{2}). Then is called an IVSF syntopogenous space. If consists of single element, then it is called an IVSF topogenous structure. If each element of is perfect (resp., biperfect symmetrical), then it is called a perfect (resp., biperfect symmetrical) IVSF structure on .

The following proposition investigates the relation between the family of all IVSF topological spaces and the family of all perfect IVSF topogenous spaces.

Proposition 3.4. *There is one-to-one correspondence between the family of all IVSF topological spaces and the family of all perfect IVSF topogenous spaces.*

*Proof. *Let be an IVSF topological space. Define a relation by s.t. . Then is a perfect IVSF topogenous order on . In fact, since , and, then and . Let s.t. . Assume that s.t. . Thus, is an IVSF topogenous order on . We show that is perfect. Let s.t. . Hence and . Thus, and consequently is perfect. Thus is an IVSF topogenous order. Consequently, is a perfect IVSF topogenous structure.

Conversely, let () be a perfect IVSF topogenous space. Then , where is a perfect IVSF order. Define by . We show that is an IVSF topology on . Since and , then . Also, for all for all , because is perfect. Hence and, consequently, is an IVSF topology on .

*Remark 3.5. *One can easily show that and .

*Example 3.6. *This example is a small form of interval-valued information table of a file containing some patients = {*Li*, *Wang*, *Zhang*, *Sun*}. We consider the set of symptoms = {*Chest-pain*, *Cough, Stomach-pain*, *Headache*, *Temperature*}. Each symptom is described by an interval fuzzy sets on . The symptoms are given in Table 1. Define a binary relation on by if and only if implies for all . Therefore, is biperfect IVSF topogenous order on . The set of patients are ordered by *Li* ≤ *Wang* ≤ *Zhang *≤ *Sun*. So, we have chest-pain Cough.

The following proposition shows the relation between the family of all IVSF proximity spaces and the family of all symmetrical IVSF topogenous spaces.

Proposition 3.7. *There is one-to-one correspondence between the family of all IVSF proximity spaces and the family of all symmetrical IVSF topogenous spaces.*

*Proof. *Let be an IVSF proximity. Define . Then is a symmetrical IVSF topogenous order. In fact, since and , then and . Also . Finally, , and . Thus and therefore . Consequently (), where is IVSF topogenous space.

Conversely, let a symmetrical IVSF topogenous space. Then , where is a symmetrical IVSF topogenous order on . Define by . We show that is an IVSF topogenous order on . Assume that , then . Therefore because is symmetric. Hence . Since and , then . Assume that or . Then or . Suppose that , then . Thus . Consequently and . So, and a contradiction. Now implies and hence .

#### 4. Interval-Valued Bifuzzy Syntopogenous Space

Proposition 4.1. *Let be an IVF topogenous order on . Then the relation on defined by is an IVF topogenous order on .*

* Proof. *Since and , then and . Also, . Furthermore, , . Finally, and and . Since is a topogenous order, then and hence . Consequently, . Similarly, we can show that . Thus is an IVF topogenous order on .

*Remark 4.2. *(1) The relation defined in the previous proposition is called the conjugate of .

(2) If is perfect or biperfect, then is also.

Proposition 4.3. *Let and be two IVF topogenous orders on . If , then and .*

*Proof. *Assume that, . Then . Hence, . Also, s.t. s.t. s.t. . Similarly, we can show that . Consequently, .

Proposition 4.4. *Let be an IVF syntopogenous structure on . Then the family is also an IVF syntopogenous structure.*

*Proof. *Since , then . Let , then . Hence s.t. and . Therefore s.t. and . Also, s.t. . Thus s.t. . Consequently, is also IVF syntopogenous structure on .

Hence, we have the result that on a nonempty set we have two interval-valued fuzzy syntopogenous structures.

*Definition 4.5. *Let a nonempty set. Let be two IVF syntopogenous structures on . The triple is called an interval-valued bifuzzy syntopogenous space. If each is perfect (resp., biperfect symmetrical), then the space is perfect (resp. biperfect symmetrical).

The following proposition shows that two IVF topogenous orders on can induce an IVSF topogenous order on .

Proposition 4.6. *Let be two interval-valued fuzzy topogenous orders on . Then the order defined by
**
is IVSF topogenous order on .*

*Proof. *Since, and for all , then and . Also, for some for some . Finally, for some . Thus, is an IVSF topogenous order on .

#### 5. Interval-Valued Supra-Fuzzy Syntopogenous Space Associated with Interval-Valued Bifuzzy Syntopogenous Spaces

Proposition 5.1. *Let be interval-valued bifuzzy syntopogenous space. Then the family is an IVSF syntopogenous structure on .*

*Proof. *(s_{1}) Let and and for some s.t. and s.t. . So, s.t. .

(s_{2}) , s.t. s.t. and s.t. . Since , then and . Therefore and . So and consequently is an IVSF syntopogenous structure on .

*Remark 5.2. *The structure is called the interval-valued supra-fuzzy syntopogenous associated with the space .

Proposition 5.3. *Let be an IVF topological spaces. Then the family is an IVSF topology on .*

*Proof. *Since for all , then Also, , . Since, is an IVSF topology, then and . Therefore, and consequently, is an IVSF topology on .

Proposition 5.4. *Let be an interval-valued bifuzzy syntopogenous space. Assume that is the interval-valued fuzzy topology associated with for all and is interval-valued supra-fuzzy topology associated with the space . Then is the IVSF topology associated with the space , that is, .*

*Proof. *The proof is obvious.

*Remark 5.5. *In crisp case, let be a fuzzy topogenous order on . Define a binary relation as follows:
Then, is a perfect fuzzy topogenous order on finer than and coarser than any perfect fuzzy semitopogenous order on which is finer than .

Let S be a fuzzy syntopogenous structure on . Define a relation on by
The binary relation as defined above is a perfect topogenous order on . Let be the coarsest perfect topogenous order on finer than each member of . is the topogenous space. Let .

#### 6. Conclusion

Computer scientists used the relation concepts in many areas of life such as artificial intelligence, knowledge discovery, decision making and medical diagnosis [28]. One of the important theories depend on relations is the rough set theory [29]. Many authors studied the topological properties of rough set [30, 31]. In this paper we generalized the topological concepts by introducing the notions of IVSF syntopogenous structures. We can use the relations obtained from an interval-valued information systems to generate an IVSF syntopogenous structure and use the mathematical properties of these structures to discover the knowledge in the information modelings.

#### Acknowledgment

The authors are thankful to both the referees for their valuable suggestions for correcting this paper.