New Developments in Mathematical Control and Information for Fuzzy Systems
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F. M. Sleim, Heba I. Mustafa, "On IntervalValued SupraFuzzy Syntopogenous Structure", Mathematical Problems in Engineering, vol. 2012, Article ID 854942, 8 pages, 2012. https://doi.org/10.1155/2012/854942
On IntervalValued SupraFuzzy Syntopogenous Structure
Abstract
A new definition of intervalvalued suprafuzzy syntopogenous (resp., suprafuzzy proximity) space is given. We show that for any intervalvalued fuzzy syntopogenous structure , there is another intervalvalued fuzzy syntopogenous structure called the conjugate of . This leads to introducing the concept of intervalvalued bifuzzy syntopogenous space. Finally, we show every intervalvalued bifuzzy syntopogenous space induces an intervalvalued fuzzy suprasyntopogenous space. Throughout this paper, the family of all fuzzy sets on nonempty set will be denoted by and the family of all intervalvalued 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 settheoretictopology: 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], LowenHohle fuzzy uniform spaces [10], and ArticoMoresco 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 suprafuzzy topological space and suprafuzzy proximity space. In [15], Ghanim et al. introduced the concepts of suprafuzzy syntopogenous space as a generalization of the concepts of suprafuzzy proximity and suprafuzzy topology.
Intervalvalued fuzzy sets were introduced independently by Zadeh [16], GrattanGuiness [17], Jahn [18], and Sambuc [19], in the seventies, in the same year. An intervalvalued fuzzy set (IVF) is defined by an intervalvalued 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 intervalvalued fuzzy sets and intuitionistic fuzzy sets are equivalent and they are extensions of fuzzy sets.
Intervalvalued 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 intervalvalued suprafuzzy topology (resp., suprafuzzy proximity) were introduced by using the notions of intervalvalued fuzzy sets.
In this paper, we generalize the concept of suprafuzzy syntopogenous space by using the notion of intervalvalued 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 suprafuzzy topological space, intervalvalued fuzzy set, intervalvalued fuzzy topology, intervalvalued suprafuzzy topology, and intervalvalued suprafuzzy proximity. In Section 3, we introduce the concept of intervalvalued suprafuzzy syntopogenous (resp., suprafuzzy proximity) space. We show that there is onetoone correspondence between family of all intervalvalued suprafuzzy topological spaces and the family of all perfect intervalvalued suprafuzzy topogenous spaces. Also, we prove that there is onetoone correspondence between the family of all intervalvalued suprafuzzy proximity spaces and the family of all symmetrical intervalvalued suprafuzzy topogenous spaces. We show that for any intervalvalued fuzzy topogenous order on there is another intervalvalued fuzzy topogenous structure in Section 3. In Section 4, we introduce the concept of intervalvalued bifuzzy syntopogenous structure by using two intervalvalued fuzzy syntopogenous spaces. In the last section we show that with every intervalvalued bifuzzy syntopogenous space there is an intervalvalued suprafuzzy syntopogenous associated with this space.
2. Preliminaries
The concept of a suprafuzzy topological space has been introduced as follows.
Definition 2.1 (see [27]). A collection is a suprafuzzy topology on if 0, 1 and is closed under arbitrary supremum. The pair () is a suprafuzzy topological space.
Definition 2.2 (see [20]). An intervalvalued fuzzy set (IVF set for short) is a set such that . The family of all intervalvalued 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 intervalvalued 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 intervalvalued 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 intervalvalued fuzzy sets is denoted by .
Definition 2.5 (see [26]). A non empty family is called an intervalvalued suprafuzzy 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 intervalvalued suprafuzzy (IVSF) proximity on if it satisfies the following conditions: (1),(2) or ,(3),(4),(5) s.t. and .
3. IntervalValued Fuzzy Topogenous Order
In this article, we define a new concept of topogenous structure by using the concept of intervalvalued 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 intervalvalued fuzzy topogenous order (IVF topogenous order for short) on .
If satisfies (1)–(3) then it is called an intervalvalued suprafuzzy topogenous order (IVSF topogenous order for short) on .
The intervalvalued 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 intervalvalued fuzzy (IVF) syntopogenous structure on . The pair () is called an intervalvalued 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 onetoone 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 intervalvalued information table of a file containing some patients = {Li, Wang, Zhang, Sun}. We consider the set of symptoms = {Chestpain, Cough, Stomachpain, 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 chestpain 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 onetoone 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. IntervalValued 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 intervalvalued fuzzy syntopogenous structures.
Definition 4.5. Let a nonempty set. Let be two IVF syntopogenous structures on . The triple is called an intervalvalued 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 intervalvalued 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. IntervalValued SupraFuzzy Syntopogenous Space Associated with IntervalValued Bifuzzy Syntopogenous Spaces
Proposition 5.1. Let be intervalvalued 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 intervalvalued suprafuzzy 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 intervalvalued bifuzzy syntopogenous space. Assume that is the intervalvalued fuzzy topology associated with for all and is intervalvalued suprafuzzy 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 intervalvalued 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.
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Copyright © 2012 F. M. Sleim and Heba I. Mustafa. 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.