On Interval-Valued Supra-Fuzzy Syntopogenous Structure

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

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 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 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 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 𝐴=(𝐴₁,𝐴₂)∈𝐼^𝑋×𝐼^𝑋 such that 𝐴₁≤𝐴₂. 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:(1)𝐴=𝐵⇔𝐴₁=𝐵₁,𝐴₂=𝐵₂,(2)𝐴≤𝐵⇔𝐴₁≤𝐵₁,𝐴₂≤𝐵₂,(3)𝐴∨𝐵=(𝐴₁∨B₁,𝐴₂∨𝐵₂),(4)𝐴∧𝐵=(𝐴₁∧𝐵₁,𝐴₂∧𝐵₂),(5)𝐴^𝑐=(𝐴^𝑐₂,𝐴^𝑐₁).

Definition 2.4 (see [26]). 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 supra-fuzzy (IVSF) proximity on 𝑋 if it satisfies the following conditions: (1)𝐴𝛿𝐵⇒𝐵𝛿𝐴,(2)𝐴𝛿𝐵 or 𝐴𝛿𝐶⇒𝐴𝛿(𝐵∨𝐶),(3)0𝛿^╱1,(4)𝐴𝛿^╱𝐵⇒𝐴≤1−𝐵,(5)𝐴𝛿^╱𝐵⇒∃𝐶∈𝜋^𝑋 s.t. 𝐴𝛿^╱𝐶 and (1−𝐶)𝛿^╱𝐵.

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 𝑅 be a binary relation on 𝜋^𝑋. Consider the following axioms: (1)0𝑅0,(2)𝐴𝑅𝐵⇒𝐴⊆𝐵,(3)𝐴₁≤𝐴𝑅𝐵≤𝐵₁⇒𝐴₁𝑅𝐵₁,(4)𝐴₁𝑅𝐵₁ and 𝐴₂𝑅𝐵₂⇒(𝐴₁∨𝐴₂)𝑅(𝐵₁∨𝐵₂) and (𝐴₁∧𝐴₂)𝑅(𝐵₁∧𝐵₂),(5)𝐴_𝑖𝑅𝐵_𝑖 for all 𝑖⇒(∨𝐴_𝑖)𝑅(∨𝐵_𝑖),(6)𝐴_𝑖𝑅𝐵_𝑖 for all 𝑖⇒(∧𝐴_𝑖)𝑅(∧𝐵_𝑖),(7)𝐴𝑅𝐵⇔(1−𝐵)𝑅(1−𝐴).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.
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 𝑋≠𝜙 and 𝑆 be a nonempty family of IVF topogenous orders on 𝑋 satisfy the following conditions:  (s₁) 𝑅₁,𝑅₂∈𝑆⇒∃𝑅∈𝑆 s.t. 𝑅₁≤𝑅and𝑅₂≤𝑅, (s₂) 𝑅∈𝑆⇒∃𝐿∈𝑆 s.t. 𝑅≤𝐿∘𝐿.
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 (s₁) and (s₂). 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.

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 (𝑋,𝜏^∗) be an IVSF topological space. Define a relation 𝑅_𝜏∗ by 𝐴𝑅_𝜏∗𝐵⇔∃𝐶∈𝜏^∗ s.t. 𝐴≤𝐶≤𝐵. Then 𝑅_𝜏∗ is a perfect IVSF topogenous order on 𝑋. In fact, since 0,1∈𝜏^∗, 0≤0≤0,and1≤1≤1, then 0𝑅_𝜏∗0and 1𝑅_𝜏∗1. Let 𝐴𝑅_𝜏∗𝐵⇒∃𝐶∈𝜏^∗ s.t. 𝐴≤𝐶≤𝐵⇒𝐴≤𝐵. Assume that 𝐴₁≤𝐴𝑅_𝜏∗𝐵≤𝐵₁⇒∃𝐸∈𝜏^∗ s.t. 𝐴₁≤𝐴≤𝐸≤𝐵≤𝐵₁⇒𝐴₁𝑅_𝜏∗𝐵₁. Thus, 𝑅_𝜏∗ is an IVSF topogenous order on 𝑋. We show that 𝑅_𝜏∗ is perfect. Let 𝐴_𝑖𝑅_𝜏∗𝐵_𝑖⇒∃𝐶_𝑖∈𝜏^∗ s.t. 𝐴_𝑖≤𝐶_𝑖≤𝐵_𝑖. 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_𝑖𝑅^∗𝐴_𝑖 for all 𝑖⇒∨𝐴_𝑖𝑅^∗∨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_𝜏𝑅∗=𝑅^∗ and 𝜏_𝑅𝜏∗=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^𝑋𝐴𝑅𝐵 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.

Table 1 

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 𝛿^∗ be an IVSF proximity. Define 𝐴𝑅_𝛿^∗𝐵⇔𝐴𝛿^╱∗(1−𝐵). Then 𝑅_𝛿^∗ is a symmetrical IVSF topogenous order. In fact, since 0𝛿^╱1 and 1𝛿^╱0, then 0𝑅_𝛿^∗0 and 1𝑅_𝛿^∗1. Also 𝐴𝑅_𝛿^∗𝐵⇒𝐴𝛿^╱∗1−𝐵⇒𝐴≤𝐵. Finally, 𝐴₁≤𝐴𝑅_𝛿^∗𝐵≤𝐵₁⇒𝐴₁≤𝐴, 𝐴𝛿^╱∗1−𝐵 and 1−𝐵≥1−𝐵₁. Thus 𝐴𝛿₁^╱∗1−𝐵₁ and therefore 𝐴₁𝑅_𝛿^∗𝐵₁. Consequently (𝑋,𝑆^∗), where 𝑆^∗={𝑅_𝛿^∗} is IVSF topogenous space.
Conversely, let (𝑋,𝑆^∗) a symmetrical IVSF topogenous space. Then 𝑆^∗={𝑅^∗}, where 𝑅^∗ is a symmetrical IVSF topogenous order on 𝑋. Define 𝛿_𝑅^∗ by ^𝐴𝛿_𝑅^{∗𝐵⇔𝐴𝑅}/^∗1−𝐵. We show that 𝛿_𝑅^∗ is an IVSF topogenous order on 𝑋. Assume that 𝐴𝛿_𝑅^∗𝐵, then ^𝐴𝑅/^{∗(1−𝐵)}. Therefore ^𝐵𝑅/^{∗(1−𝐴)} because 𝑅^∗ is symmetric. Hence 𝐵𝛿_𝑅^∗𝐴. Since 0𝑅^∗0 and 1𝑅^∗1, then 0𝛿^╱_𝑅^∗0. Assume that 𝐴𝛿_𝑅^∗𝐵 or 𝐴𝛿_𝑅^∗𝐶. Then ^𝐴𝑅/^{∗(1−𝐵)} or ^𝐴𝑅/^{∗(1−𝐶)}. Suppose that ^𝐴𝛿𝑅/^{∗𝐵∨𝐶}, then ^𝐴𝑅/^{∗(1−(𝐵∨𝐶))}. Thus ^𝐴𝑅/^{∗(1−𝐵)∧(1−𝐶)}. Consequently ^𝐴𝑅/^{∗(1−𝐵)} and ^𝐴𝑅/^{∗(1−𝐶)}. So, 𝐴𝛿^╱_𝑅^∗𝐵 and 𝐴𝛿^╱_𝑅^∗𝐶 a contradiction. Now 𝐴𝛿^╱_𝑅^∗𝐵 implies ^𝐴𝑅/^{∗(1−𝐵)} and hence 𝐴≤(1−𝐵).

4. Interval-Valued Bifuzzy Syntopogenous Space

Proposition 4.1. Let 𝑅 be an IVF topogenous order on 𝑋. Then the relation 𝑅^𝑐 on 𝜋^𝑋 defined by 𝐴𝑅^𝑐𝐵⇔(1−𝐵)𝑅(1−𝐴) is an IVF topogenous order on 𝑋.

Proof. Since 0𝑅0 and 1𝑅1, then 1𝑅^𝑐1 and 0𝑅^𝑐0. Also, 𝐴𝑅^𝑐𝐵⇒(1−𝐵)𝑅(1−𝐴)⇒1−𝐵≤1−𝐴⇒𝐴≤𝐵. Furthermore, 𝐴₁≤𝐴𝑅^𝑐𝐵≤𝐵₁⇒(1−𝐵)𝑅(1−𝐴), 1−𝐵₁≤1−𝐵,1−𝐴≤1−𝐴₁⇒(1−𝐵₁)𝑅(1−𝐴₁)⇒𝐴₁𝑅^𝑐𝐵₁. Finally, 𝐴₁𝑅^𝑐𝐵₁and𝐴₂𝑅^𝑐𝐵₂⇒(1−𝐵₁)𝑅(1−𝐴₁) and (1−𝐵₂)𝑅(1−𝐴₂). Since 𝑅 is a topogenous order, then (1−𝐵₁)∧(1−𝐵₂)𝑅(1−𝐴₁)∧(1−𝐴₂) and hence (1−(𝐵₁∨𝐵₂))𝑅(1−(𝐴₁∨𝐴₂)). Consequently, (𝐴₁∨𝐴₂)𝑅^𝑐(𝐵₁∨𝐵₂). Similarly, we can show that (𝐴₁∧𝐴₂)𝑅^𝑐(𝐵₁∧𝐵₂). Thus 𝑅^𝑐 is an IVF topogenous order on 𝑋.

Remark 4.2. (1) 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 𝑅₁ and 𝑅₂ be two IVF topogenous orders on 𝑋. If 𝑅₁⊆𝑅₂, then 𝑅₁^𝑐⊆𝑅₂^𝑐 and (𝑅₁∘𝑅₂)^𝑐=𝑅₂^𝑐∘𝑅₁^𝑐.

Proof. Assume that, 𝑅₁⊆𝑅₂. Then 𝐴𝑅₁^𝑐𝐵⇒(1−𝐵)𝑅₁(1−𝐴)⇒(1−𝐵)𝑅₂(1−𝐴)⇒𝐴𝑅₂^𝑐𝐵. Hence, 𝑅₁^𝑐⊆𝑅₂^𝑐. Also, 𝐴(𝑅₁∘𝑅₂)^𝑐𝐵⇒(1−𝐵)(𝑅₁∘𝑅₂)(1−𝐴)⇒∃𝐶∈𝜋^𝑋 s.t. (1−𝐵)𝑅₁𝐶∧𝐶𝑅₂(1−𝐴)⇒∃𝐶∈𝜋^𝑋 s.t. (1−𝐶)𝑅₁^𝑐𝐵∧𝐴𝑅₂^𝑐(1−𝐶)⇒∃1−𝐶∈𝜋^𝑋 s.t. 𝐴𝑅₂^𝑐1−𝐶∧(1−𝐶)𝑅₁^𝑐𝐵⇒𝐴(𝑅₂^𝑐∘𝑅₁^𝑐)𝐵⇒(𝑅₁∘𝑅₂)^𝑐≤𝑅₂^𝑐∘𝑅₁^𝑐. Similarly, we can show that 𝑅₂^𝑐∘𝑅₁^𝑐≤(𝑅₁∘𝑅₂)^𝑐. Consequently, (𝑅₁∘𝑅₂)^𝑐=𝑅₂^𝑐∘𝑅₁^𝑐.

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.

Proof. Since 𝑆≠𝜙, then 𝑆^𝑐≠𝜙. Let 𝑅₁^𝑐,𝑅₂^𝑐∈𝑆^𝑐, then 𝑅₁,𝑅₂∈𝑆. Hence ∃𝑅∈𝑆 s.t. 𝑅₁≤𝑅 and 𝑅₂≤𝑅. Therefore ∃𝑅^𝑐∈𝑆^𝑐 s.t. 𝑅₁^𝑐≤𝑅^𝑐 and 𝑅₂^𝑐≤𝑅^𝑐. Also, 𝑅^𝑐∈𝑆^𝑐⇒𝑅∈𝑆⇒∃𝑅₁∈𝑆 s.t. 𝑅≤(𝑅₁∘𝑅₁)⇒𝑅^𝑐≤(𝑅₁∘𝑅₁)^𝑐=𝑅₁^𝑐∘𝑅₁^𝑐. Thus ∃𝑅^𝑐₁∈𝑆^𝑐 s.t. 𝑅^𝑐≤𝑅₁^𝑐∘𝑅₁^𝑐. 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 𝑆₁,𝑆₂ be two IVF syntopogenous structures on 𝑋. The triple (𝑋,𝑆₁,𝑆₂) is called an interval-valued bifuzzy syntopogenous space. If each 𝑆₁,𝑆₂ is perfect (resp., biperfect symmetrical), then the space (𝑋,𝑆₁,𝑆₂) 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 𝑅₁,𝑅₂ be two interval-valued fuzzy topogenous orders on 𝑋. Then the order 𝑅=𝑅₁∨𝑅₂ defined by 𝐴𝑅𝐵⟺𝐴𝑅₁𝐵or𝐴𝑅₂𝐵 (4.1) is IVSF topogenous order on 𝜋^𝑋.

Proof. Since, 0𝑅_𝑖0 and 1𝑅_𝑖1 for all  𝑖=1,2, then 0𝑅0 and 1𝑅1. Also, 𝐴₁≤𝐴𝑅𝐵≤𝐵₁⇒𝐴₁≤𝐴𝑅_𝑖𝐵≤𝐵₁ for some 𝑖=1,2⇒𝐴₁𝑅_𝑖𝐵₁ for some 𝑖=1,2⇒𝐴₁𝑅𝐵₁. Finally, 𝐴𝑅𝐵⇒𝐴𝑅_𝑖𝐵 for some 𝑖=1,2⇒𝐴≤𝐵. Thus, 𝑅 is an IVSF topogenous order on 𝜋^𝑋.

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

Proposition 5.1. Let (𝑋,𝑆₁,𝑆₂) be interval-valued bifuzzy syntopogenous space. Then the family 𝑆^∗₁₂={𝑅=𝑅₁∨𝑅₂∶𝑅₁∈𝑆₁,𝑅₂∈𝑆₂} is an IVSF syntopogenous structure on 𝑋.

Proof. (s₁) Let 𝑅,𝐿∈𝑆^∗₁₂⇒𝑅=𝑅₁∨𝑅₂ and 𝐿=𝐿₁∨𝐿₂𝑅_𝑖and𝐿_𝑖∈𝑆_𝑖 for some 𝑖=1,2⇒∃𝐾₁∈𝑆₁ s.t. 𝑅₁∨𝐿₁=𝐾₁ and ∃𝐾₂∈𝑆₁ s.t. 𝑅₂∨𝐿₂=𝐾₂. So, ∃𝐾=𝐾₁∨𝐾₂∈𝑆^∗₁₂ s.t. 𝑅∨𝐿≤𝐾₁∨𝐾₂=𝐾.
(s₂) 𝑅∈𝑆^∗₁₂⇒∃𝑅₁∈𝑆₁, 𝑅₂∈𝑆₂ s.t. 𝑅=𝑅₁∨𝑅₂⇒∃𝐿₁∈𝑆₁ s.t. 𝑅₁≤𝐿₁∘𝐿₁ and ∃𝐿₂∈𝑆₂ s.t. 𝑅₂≤𝐿₂∘𝐿₂. Since 𝐿₁,𝐿₂≤𝐿₁∨𝐿₂, then 𝐿₁∘𝐿₁≤𝐿₁∨𝐿₂∘𝐿₁∨𝐿₂ and 𝐿₂∘𝐿₂≤𝐿₁∨𝐿₂∘𝐿₁∨𝐿₂. Therefore 𝑅₁≤𝐿₁∘𝐿₁≤𝐿₁∨𝐿₂∘𝐿₁∨𝐿₂ and 𝑅₂≤𝐿₂∘𝐿₂≤𝐿₁∨𝐿₂∘𝐿₁∨𝐿₂. So 𝑅₁∨𝑅₂≤(𝐿₁∨𝐿₂)∘(𝐿₁∨𝐿₂) and consequently 𝑆^∗₁₂ is an IVSF syntopogenous structure on 𝑋.

Remark 5.2. The structure 𝑆^∗₁₂ is called the interval-valued supra-fuzzy syntopogenous associated with the space (𝑋,𝑆₁,𝑆₂).

Proposition 5.3. Let (𝑋,𝑇_𝑖)𝑖=1,2 be an IVF topological spaces. Then the family 𝑇^∗₁₂={𝐴=𝐴₁∨𝐴₂∶𝐴_𝑖∈𝑇_𝑖𝑖=1,2} is an IVSF topology on 𝑋.

Proof. Since 0,1∈𝑇_𝑖 for all 𝑖=1,2, then 0,1∈𝑇^∗₁₂ Also, 𝐴_𝑖∈𝑇^∗₁₂⇒𝐴_𝑖=𝐵^𝑖₁∨𝐵^𝑖₂, 𝐵^𝑖₁∈𝑇₁,𝐵^𝑖₂∈𝑇₂⇒∨_𝑖𝐴_𝑖=∨_𝑖(𝐵^𝑖₁∨𝐵^𝑖₂)=(∨_𝑖𝐵^𝑖₁)∨(∨_𝑖𝐵^𝑖₂). Since, 𝑇_𝑖𝑖=1,2 is an IVSF topology, then (∨_𝑖𝐵^𝑖₁)∈𝑇₁ and (∨_𝑖𝐵^𝑖₂)∈𝑇₂. Therefore, ∨_𝑖𝐴_𝑖∈𝑇^∗₁₂ and consequently, 𝑇^∗₁₂ is an IVSF topology on 𝑋.

Proposition 5.4. Let (𝑋,𝑆₁,𝑆₂) 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,𝑇₁,𝑇₂). 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^𝑝𝜌⟺∃family𝜇_𝑖∶𝑖∈𝐼s.t.𝜇=∨_𝑖𝜇_𝑖,𝜇_𝑖𝑅𝜌∀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^𝑆𝜌⟺𝜇𝑅𝜌forsome𝑅∈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.

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.