On Interval-Valued Supra-Fuzzy Syntopogenous Structure

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 S, there is another interval-valued fuzzy syntopogenous structure S called the conjugate of S. 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 suprasyntopogenous space. Throughout this paper, the family of all fuzzy sets on nonempty set X will be denoted by I and the family of all interval-valued fuzzy sets on a nonempty set X will be denoted by π .


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 suprafuzzy topology.

Mathematical Problems in Engineering
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 intervalvalued 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 R on π X there is another interval-valued fuzzy topogenous structure R c in Section 3. In Section 4, we introduce the concept of intervalvalued 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.

Preliminaries
The concept of a supra-fuzzy topological space has been introduced as follows.
Definition 2.1 see 27 . A collection τ * ⊂ I X is a supra-fuzzy topology on X if 0, 1 ∈ τ * and τ * is closed under arbitrary supremum. The pair X, τ * is a supra-fuzzy topological space.

Definition 2.2 see 20 . An interval-valued fuzzy set IVF set for short is a set
The family of all interval-valued fuzzy sets on a given nonempty set X will be denoted by π X .
The IVF set 1 1, 1 is called the universal IVF set and the IVF set 0 0, 0 is called the empty IVF set.

Proposition 2.3 see 20 .
The operations on π X are given by the following: Let A, B ∈ π X :

Mathematical Problems in Engineering
The family η ⊆ π X is called an interval-valued fuzzy topology IVF tpology for short on X if and only if η contains 1, 0, and it is closed under finite intersection and arbitrary union. The pair X, η is called an interval-valued fuzzy topological space IVF top. space for short . Any IVF set A ∈ η is called an open IVF set and the complement of A denoted by A c is called a closed IVF set. The family of all closed interval-valued fuzzy sets is denoted by η c .
Definition 2.5 see 26 . A non empty family η * ⊆ π X is called an interval-valued supra-fuzzy topology IVSF top. for short on X if it contains 0, 1 and it is closed under arbitrary unions.
Definition 2.6 see 26 . A binary relation δ ⊆ π X × π X is called an interval-valued suprafuzzy IVSF proximity on X if it satisfies the following conditions:

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 R be a binary relation on π X . Consider the following axioms: If R satisfies 1 -4 then it is called an interval-valued fuzzy topogenous order IVF topogenous order for short on X.
If R satisfies 1 -3 then it is called an interval-valued supra-fuzzy topogenous order IVSF topogenous order for short on X.

Mathematical Problems in Engineering
Definition 3.2. Let X / φ and S be a nonempty family of IVF topogenous orders on X satisfy the following conditions: Then S is said to be interval-valued fuzzy IVF syntopogenous structure on X. The pair X, S is called an interval-valued fuzzy syntopogenous IVF syntopogenous for short space. If S satisfy the single element, then S is called an IVF topogenous structure on X and the pair X, S is called an IVF topogenous space. Definition 3.3. Let X / φ and S * be a nonempty family of IVSF topogenous orders on X satisfy the conditions s 1 and s 2 . Then S * is called an IVSF syntopogenous space. If S * consists of single element, then it is called an IVSF topogenous structure. If each element of S * is perfect resp., biperfect symmetrical , then it is called a perfect resp., biperfect symmetrical IVSF structure on X.
The following proposition investigates the relation between the family of all IVSF topological spaces and the family of all perfect IVSF topogenous spaces.
Hence ∨C i ∈ τ * and ∨A i ≤ ∨C i ≤ ∨B i . Thus, ∨A i R τ * ∨B i and consequently R τ * is perfect. Thus R τ * is an IVSF topogenous order. Consequently, S τ * {R τ * } is a perfect IVSF topogenous structure.
Conversely, let X, S * be a perfect IVSF topogenous space. Then S * {R * }, where R * is a perfect IVSF order. Define τ R * ⊆ π X by A ∈ τ R * ⇔ AR * A. We show that τ R * is an IVSF topology on X. Since 0R * 0 and 1R * 1, then 0, 1 ∈ τ R * . Also, A i ∈ τ R * for all i ⇒ A i R * A i for all i ⇒ ∨A i R * ∨ A i , because R * is perfect. Hence ∨A i ∈ τ R * and, consequently, τ R * is an IVSF topology on X.
Remark 3.5. One can easily show that R τ R * R * and τ R τ * τ * . Example 3.6. This example is a small form of interval-valued information table of a file containing some patients X {Li, Wang, Zhang, Sun}. We consider the set of symptoms S {Chest-pain, Cough, Stomach-pain, Headache, Temperature}. Each symptom is described by an interval fuzzy sets on X. The symptoms are given in Table 1. Define a binary relation R on π X by A, B ∈ π X ARB if and only if x ≤ y implies A x ≤ B y for all x ∈ X. Therefore, R is biperfect IVSF topogenous order on X. The set of patients are ordered by Li ≤ Wang ≤ Zhang ≤ Sun. So, we have chest-pain R Cough.

Interval-Valued Bifuzzy Syntopogenous Space
Thus R c is an IVF topogenous order on X.
The relation R c defined in the previous proposition is called the conjugate of R.
2 If R is perfect or biperfect, then R c is also.

Proposition 4.3.
Let R 1 and R 2 be two IVF topogenous orders on X.
Mathematical Problems in Engineering Proposition 4.4. Let S be an IVF syntopogenous structure on X. Then the family S c {R c : R ∈ S} is also an IVF syntopogenous structure.
Consequently, S c is also IVF syntopogenous structure on X.
Hence, we have the result that on a nonempty set X we have two interval-valued fuzzy syntopogenous structures.
Definition 4.5. Let X a nonempty set. Let S 1 , S 2 be two IVF syntopogenous structures on X. The triple X, S 1 , S 2 is called an interval-valued bifuzzy syntopogenous space. If each S 1 , S 2 is perfect resp., biperfect symmetrical , then the space X, S 1 , S 2 is perfect resp. biperfect symmetrical .
The following proposition shows that two IVF topogenous orders on X can induce an IVSF topogenous order on X. Proposition 4.6. Let R 1 , R 2 be two interval-valued fuzzy topogenous orders on X. Then the order is IVSF topogenous order on π X .
Proof. Since, 0R i 0 and 1R i 1 for all i 1, 2, then 0R0 and 1R1. Also, Thus, R is an IVSF topogenous order on π X .
Proof. Proof. Since 0, 1 ∈ T i for all i 1, 2, then 0, 1 ∈ T * 12 Also, Since, T i i 1, 2 is an IVSF topology, then ∨ i B i 1 ∈ T 1 and ∨ i B i 2 ∈ T 2 . Therefore, ∨ i A i ∈ T * 12 and consequently, T * 12 is an IVSF topology on X.
Proposition 5.4. Let X, S 1 , S 2 be an interval-valued bifuzzy syntopogenous space. Assume that T i is the interval-valued fuzzy topology associated with S i for all i 1, 2 and T * 12 is interval-valued supra-fuzzy topology associated with the space X, T 1 , T 2 . Then T * 12 is the IVSF topology associated with the space X, S * 12 , that is, T * 12 T S * 12 .
Proof. The proof is obvious.
Remark 5.5. In crisp case, let R be a fuzzy topogenous order on I X . Define a binary relation R p as follows: μR p ρ ⇐⇒ ∃ family μ i : i ∈ I s.t. μ ∨ i μ i , μ i Rρ ∀i.

5.1
Then, R p is a perfect fuzzy topogenous order on X finer than R and coarser than any perfect fuzzy semitopogenous order on I X which is finer than R. Let S be a fuzzy syntopogenous structure on X. Define a relation R S on I X by μR S ρ ⇐⇒ μRρ for some R ∈ S.

5.2
The binary relation R S as defined above is a perfect topogenous order on I X . Let R p be the coarsest perfect topogenous order on I X finer than each member of S. X, R p S is the topogenous space. Let R o R p S .

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.